Pdwhpäwlfd sdud Lqjhqlhuðd Vdud Dudqfleld Fduydmdo Mdlph Phqd Orufd
5
Frqwhqwv L
Lqwurgxflöq do Fäofxor
4 Orv Qýphurv Uhdohv 414 Lqwurgxfflöq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 415 D{lrpdv gh Fxhusr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41514 Hvwuxfwxud gh Juxsr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41515 D{lrpdv gh U frpr Fxhusr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41516 Srwhqfldv Hqwhudv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41517 Frqmxqwr uhvwulfflöq | frqmxqwr vroxflöq gh xqd hfxdflöq 41518 Uhvroxflöq gh sureohpdv gh sodqwhr 1 1 1 1 1 1 1 1 1 1 1 1 416 D{lrpdv gh Rughq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41614 Udð} q0ìvlpd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41615 s Vlvwhpdv gh Hfxdflrqhv qr Olqhdohv 1 1 1 1 1 1 1 1 1 1 1 1 1 5 hv luudflrqdo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41616 41617 Lqhfxdflrqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41618 Od hfxdflöq gh vhjxqgr judgr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41619 Ydoru devroxwr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 417 D{lrpd gho Vxsuhpr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 418 Hmhuflflrv Sursxhvwrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41814 Hvwuxfwxudv gh Juxsr | Fxhusr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41815 Sureohpdv gh Sodqwhr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41816 Hmhuflflrv gh Vlvwhpdv gh Hfxdflrqhv 1 1 1 1 1 1 1 1 1 1 1 1 41817 Hmhuflflrv Dfhufd gh orv D{lrpdv gh Rughq 1 1 1 1 1 1 1 1 5 Ixqflrqhv 514 Lqwurgxfflöq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 515 Grplqlr | Uhfruulgr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 516 Uhsuhvhqwdflöq Juäfd 1 1 1 1 1 1 1 1 1 1 1 1 1 51614 Udðfhv phgldqwh juäfdv 1 1 1 1 1 1 1 1 517 Dsolfdflrqhv gh Ixqflrqhv/ Prghodflöq 1 1 1 1 51714 Surjudpdflöq Olqhdo 1 1 1 1 1 1 1 1 1 1 518 Dojheud gh Ixqflrqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 51814 Frpsrvlflöq gh ixqflrqhv 1 1 1 1 1 1 1 51815 Ixqflrqhv Fuhflhqwhv | Ghfuhflhqwhv 1 519 Odv ixqflrqhv H{srqhqfldohv | Orjduðwplfdv 1 51: Hmhuflflrv Sursxhvwrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 51:14 Hmhuflflrv gh Grplqlr | Uhfruulgr 1 1 51:15 Hmhuflflrv gh Juäfd 1 1 1 1 1 1 1 1 1 1 51:16 Hmhuflflrv gh Prghodflöq Pdwhpäwlfd 51:17 Hmhuflflrv gh Frpsxhvwd gh ixqflrqhv 51:18 Hmhuflflrv gh Orjdulwprv 1 1 1 1 1 1 1 1 6
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
< < < 44 47 54 55 57 67 6< 78 84 8: 96 :4 ;5 ;< ;< 4> 5> 6> = = =j @ Q ^ Q ^ i3j
Hq ] frq od vxpd +., vh yhulfd=
l, Fodxvxud=
Q @ iq = q 5 Qj=
Sdud wrgr d> e 5 ] / vh wlhqh txh d . e 5 ]
ll, Dvrfldwlylgdg=
Sdud wrgr d> e> f 5 ]
lll, H{lvwhqfld gh qhxwur=
+d . e, . f @ d . +e . f,
H{lvwh h @ 3 5 ] / wdo txh sdud wrgr d 5 ]
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ly, H{lvwhqfld gh lqyhuvr=
Sdud wrgr d 5 ] / h{lvwh +d, 5 ] wdo txh
d . +d, @ +d, . d @ 3
y, Frqpxwdwlylgdg=
Sdud wrgr d> e 5 ]
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4151 D[LRPDV GH FXHUSR
44
Hq fdpelr/ vl frqvlghudprv ] edmr od rshudflöq surgxfwr +, vh yhulfdq +l,/ +ll,/ +lll, +hq hvwh fdvr 4 kdfh ho sdsho gh qhxwur, | +y,1 Vlq hpedujr qr vh yhulfd od surslhgdg gh lqyhuvr/ +ly,/ sxhvwr txh= Gdgr d 5 ] qr vlhpsuh h{lvwh d3 5 ] wdo txh dd3 @ h @ 4 Gluhprv txh ] edmr od rshudflöq gh vxpd +., hv xq juxsr | dqrwduhprv +]> ., hv juxsr> hq fdpelr/ +]> , qr hv juxsr1
Ghqlflöq 414 Gluhprv txh hv xqd rshudflöq+elqduld%, hq J vl sdud wrgr {> | hq J h{lvwh xq ýqlfr { | hq J1
Qrwd= Ho khfkr txh { | hvwä hq J | qr hq rwur frqmxqwr vh dfrvwxpeud d ghflu txh fxpsoh od surslhgdg gh fodxvxud1
41514
Hvwuxfwxud gh Juxsr
Ghqlflöq 415 Vhd J xq frqmxqwr qr ydfðr | xqd rshudflöq hq J1 Gluhprv txh J hv xq juxsr edmr od rshudflöq vl odv vljxlhqwhv wuhv dupdflrqhv vrq flhuwdv l, Dvrfldwlylgdg=
Sdud wrgr {> |> } hq J/ vh fxpsoh
Hq vðperorv ll, H{lvwhqfld gh hohphqwr qhxwur=
+{ |, } @ { +| },
+;{> |> } 5 J,++{ |, } @ { +| },,
H{lvwh h/ hohphqwr qhxwur hq J/ wdo txh sdud wrgr { hq J/ vh wlhqh {h@h{@{
Hq vðperorv
+ , hv juxsr1 Vl dghpäv vh yhulfd>
ly, Frqpxwdwlylgdg=
Sdud wrgr {> | hq J/ vh fxpsoh Hq vðperorv
{| @|{
+;{> | 5 J,+{ | @ | {,
hqwrqfhv +J> , hv xq juxsr Deholdqr r vlpsohphqwh J hv xq juxsr Deholdqr/ vxehqwhqglhqgr txh kd| xqd rshudflöq +,1 4 Qrwhprv
od lpsruwdqfld gho rughq gh orv fxdqwlfdgruhv / ! +frpsduh frq ll ,1
FKDSWHU 41 ORV QÝPHURV UHDOHV
45 Hmhpsor 4 Vhd
P5 +], @
d e f g
= d> e> f> g 5 ]
ho frqmxqwr gh pdwulfhv fxdgudgdv gh rughq 5 frq frhflhqwhv hq ] | +., od rshudflöq ghqlgd sru 3 d . d3 e . e3 d e d e3 @ > . f g f3 g3 f . f3 g . g3 hqwrqfhv P5 +], hv xq juxsr1
Vroxflöq= Yhultxhprv txh P5 +], fxpsoh frq odv surslhgdghv gh juxsr
l, Dvrfldwlylgdg= 3 d e d Vhdq > f g f3 @ @ @ @ @ @
e3 g3
33 d > f33
e33 g33
shuwhqhflhqwhv d P5 +],1 Vh wlhqh vxfhvlydphqwh=
33 d3 e3 d e33 . . f3 g3 f33 g33 d . d3 e . e3 d33 e33 . / sru ghqlflöq gh . 3 3 f.f g.g f33 g33 +d . d3 , . d33 +e . e3 , . e33 +f . f3 , . f33 +g . g3 , . g33 d . +d3 . d33 e . +e3 . e33 / dvrfldwlylgdg hq ] f . +f3 . f33 g . +g3 . g33 3 d e d . d33 e3 . e33 . f3 . f33 g3 . g33 f g 33 3 e33 d d e d e3 . . f3 g3 f33 g33 f g d e f g
ll, H{lvwhqfld gho qhxwur= 3 3 d e H{lvwh 5 P5 +], wdo txh sdud wrgr 5 P5 +], vh wlhqh 3 3 f g 3 3 d e d e 3 3 d e . @ . @ 3 3 f g f g 3 3 f g
lll, H{lvwhqfld gh hohphqwrv lqyhuvrv= d e d e Sdud wrgr 5 P5 +], h{lvwh 5 P5 +], wdo txh f g f g d e d e d e d e 3 3 . @ . @ f g f g f g f g 3 3
Revhuydflöq= Pxfkdv yhfhv vh suhwhqgh ghprvwudu txh xq frqmxqwr hv juxsr prvwudqgr txh 5 ö 6 hohphqwrv sduwlfxoduhv/ wrpdgrv do d}du/ vdwlvidfhq odv surslhgdghv gh fodxvxud/ dvrfldwlylgdg/ h{lvwhqfld gh hohphqwr qhxwur | h{lvwhqfld gh hohphqwr lqyhuvr1 Wdo dfflöq qr ghpxhvwud txh xq frqmxqwr hv juxsr> wdpsrfr vhuä xqd ghprvwudflöq yäolgd vl vh wrpd xq prqwöq gh hohphqwrv gho frqmxqwr1 Od ýqlfd irupd gh ghprvwudu hvwr hv wrpäqgrorv wrgrv/ yhulfdqgr txh wrgrv hq frqmxqwr fxpsohq odv surslhgdghv gh fodxvxud/ dvrfldwlylgdg/ h{lvwhqfld gh hohphqwr qhxwur | h{lvwhqfld gh hohphqwrv lqyhuvrv +Qrwh orv fxdqwlfdgruhv xvdgrv hq odv surslhgdghv,1 Hq ho fdvr txh vh txlhud suredu txh xq frqmxqwr qr hv juxsr/ edvwd ghprvwudu txh h{lvwh do phqrv dojxqd surslhgdg txh qr vh fxpsod1 Vhjýq od uhjod lqglfd od qhjdflöq gh od sursrvlflöq=
4151 D[LRPDV GH FXHUSR
hv=
46
Wrgrv orv hohphqwrv gh J fxpsohq wrgdv odv surslhgdghv H{lvwh do phqrv xq hohphqwr gh J txh qr fxpsoh dojxqd surslhgdg
Sru hvr hv txh fxdqgr vh txlhuh suredu txh xq frqmxqwr J hv xq juxsr vh wrpdq xqrv hohphqwrv gh J jhqìulfrv txh uhsuhvhqwdq d fxdohvtxlhud gh orv hohphqwrv gh J1
Hmhpsor 5 Vhd
E@
d 4 e 3
= d> e 5 ]
frq od vxpd +., ghqlgd hq ho hmhpsor 41 Hqwrqfhv +E> ., qr hv xq juxsr/ sxhvwr txh qr vh yhulfd od surslhgdg gh fodxvxud1 Hq hihfwr> 4 4 5 4 5E | 5E 4 3 3 3 shur
4 4 4 3
.
5 4 3 3
@
6 5 4 3
95 E
|d txh od hqwudgd vxshulru ghuhfkd gh od pdwul} uhvxowdqwh qr hv 4/ frqglflöq qhfhvduld sdud hvwdu hq ho frqmxqwr E1 Hmhpsor 6 Vhd F @ i+{> |, = { 5 ]> | 5 Qj | vhd od rshudflöq ghqlgd sru +{4 > |4 , +{5 > |5 , @ +{4 {5 > |4 . |5 ,
Hqwrqfhv +F> , qr hv juxsr1 Hq hihfwr> +4> 5,> +6> 8,> +4> 7, 5 F Sru rwud sduwh
++4> 5, +6> 8,, +4> 7, @ +4 6> 5 . 8, +4> 7, @ +4> 44,
+4> 5, ++6> 8, +4> 7,, @ +4> 5, +6 +4,> 8 . 7, @ +6> 44,
Frpr +4> 44, 9@ +6> 44,/ hqwrqfhv qr vh fxpsoh od surslhgdg dvrfldwlyd1 Hmhpsor 7 Vhd
T @
qd
r = d 5 ]> e 5 ] i3j >
e oodpdgr ho frqmxqwr gh orv Qýphurv Udflrqdohv1 Frqvlghuh odv rshudflrqhv vxpd | surgxfwr ghqlgdv sru= d . e d e
f g f g
@ @
dg . ef eg df eg
.ef 5 T | df Qrwh txh dgeg eg 5 T/ sxhvwr txh/ dg . ef 5 ]/ df 5 ] | eg 5 ] i3j1 +T> ., hv juxsr | +T i3j> , hv juxsr1 Vh ghmd frpr hmhuflflr od yhulfdflöq gh odv uhvshfwlydv surslhgdghv1
Hmhpsor 8 = Vhdq J @ i+{> |, = { 5 T > | 5 ]j | od rshudflöq ghqlgd sru= +{> |, +{3 > | 3 , @ +{{3 > | . | 3 ,
Ghwhuplqhprv vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1
FKDSWHU 41 ORV QÝPHURV UHDOHV
47 d, Ho hohphqwr qhxwur gh J hv +3> 3,1
e, H{lvwh xq hohphqwr qhxwur sdud J1 f, Ho lqyhuvr gh +4> 7, hv +3> 7,1 g, +J> , hv xq juxsr1
h, J fxpsoh od frqpxwdwlylgdg1 Vroxflöq=
d, Idovr/ h{lvwh +5> 6, 5 J/ wdo txh
+5> 6, +3> 3, @ +3> 6, 9@ +5> 6,
e, Yhugdghur/ srutxh h{lvwh ho hohphqwr +4> 3, 5 J wdo txh sdud wrgr +{> |, 5 J> +{> |, +4> 3, @ +4> 3, +{> |, @ +{> |,
f, Idovr>
+4> 7, +3> 7, @ +3> 3, 9@ +4> 3,
g, Idovr/ srutxh h{lvwh +3> 3, 5 J wdo txh sdud wrgr +{> |, 5 J> hv ghflu/ +3/ 3, qr wlhqh lqyhuvr1
+3> 3, +{> |, @ +3> |, 9@ +4> 3,
h, Yhugdghur/ sxhv sdud wrgr +{> |,> +{3 > | 3 , 5 J>
oxhjr
41515
+{> |, +{3 > |3 , @ +{{3 > | . | 3 , @ +{3 {> | 3 . |, @ +{3 > | 3 , +{> |, +{> |, +{3 > |3 , @ +{3 > | 3 , +{> |,
D{lrpdv gh U frpr Fxhusr
Dgplwluhprv od h{lvwhqfld gh xq frqmxqwr U fx|rv hohphqwrv vh oodpduäq qýphurv uhdohv1 Hq ho frqmxqwr U vh wlhqhq odv rshudflrqhv ixqgdphqwdohv= Od vxpd r dglflöq= d . e Ho surgxfwr r pxowlsolfdflöq= d e Frqvlgìuhqvh orv vljxlhqwhv d{lrpdv l, 3 5 U / 4 5 U / 3 9@ 4
ll, +U > ., hv juxsr deholdqr
lll, ++U i3j,> , hv juxsr deholdqr ly, Glvwulexwlylgdg=
;d> e> f 5 U = d+e . f, @ de . df
Vh glfh txh ho frqmxqwr U surylvwr gh wrgrv orv d{lrpdv gdgrv dqwhulruphqwh hv xq Fxhusr | vh hvfuleh> +U> .> , hv xq fxhusr1 Orv hohphqwrv qhxwurv gh U vrq ho 3 | ho 4 +sdud od dglflöq | pxowlsolfdflöq uhvshfwlydphqwh, shur qxqfd khprv sxhvwr hq gxgd txh hvwrv vrq orv ýqlfrv frq hvwd surslhgdg1 Hq hihfwr/ sru hmhpsor 4 hv ho qhxwur pxowlsolfdwlyr/ vhjýq hvwdeohfh od sursrvlflöq vljxlhqwh1
4151 D[LRPDV GH FXHUSR
48
Sursrvlflöq 416 Hq orv qýphurv uhdohv whqhprv l, Ho qhxwur dglwlyr hv ýqlfr
ll, Ho qhxwur pxowlsolfdwlyr hv ýqlfr
lll, Ho lqyhuvr dglwlyr gh xq qýphur uhdo hv ýqlfr
ly, Ho lqyhuvr pxowlsolfdwlyr gh xq qýphur uhdo qr qxor hv ýqlfr
Vlq gxgd/ sdud od pd|ruðd gh qrvrwurv hvwd vhuä od sulphud ghprvwudflöq gh xqlflgdg | vlhqgr dvðhv exhqr kdfhu qrwdu txh pxfkdv yhfhv ghehuhprv suredu txh dojýq hqwh pdwhpäwlfr hv ýqlfr1 Or xvxdo hq hvwrv fdvrv hv xvdu ho pìwrgr gho devxugr txh frqvlvwh hq vxsrqhu +hq hvwrv sureohpdv, txh kd| grv hohphqwrv +hv ghflu/ txh qr hv ýqlfr, txh fxpsohq od surslhgdg txh orv ghqh +sru hmhpsor vhu qhxwur, | edmr hvwh vxsxhvwr oohjdu d xqd iudvh pdwhpäwlfd txh vhd idovd1 Ghprvwudflöq= +l, Vxsrqjdprv txh 3> 33 5 U vrq grv qhxwurv dglwlyrv glvwlqwrv +3 9@ 33 ,/ hv ghflu/ 3 | 3* fxpsohq ;d 5 U d @ 3 . d @ d . 3 ;e 5 U e . 33 @ 33 . e @ e
+414, +415,
33 @ 3 . 33 @ 33 . 3
+416,
3 . 33 @ 33 . 3 @ 3
+417,
Frpr 33 5 U/ hq sduwlfxodu vh sxhgh wrpdu d @ 33 hq 414/ dvðwhqhprv txh Dqäorjdphqwh frqvlghudqgr e @ 3 hq 415 vh wlhqh Frqhfwdqgr odv fdghqdv 416 | 417 vh wlhqh txh
33 @ 3 . 33 @ 33 . 3 @ 3
hv ghflu/
33 @ 3
or txh kdfh xqd frqwudglfflöq frq 3 9@ 33 Vh ghmdq frpr hmhuflflr sdud ho ohfwru odv ghprvwudflrqhv gh odv sduwhv +ll,/ +lll, | +ly, gh od sursrvlflöq1 Revhuydflöq= Hv lpsruwdqwh dfodudu txh ho pìwrgr gho devxugr hv px| ýwlo sdud ghprvwudu sursrvlflrqhv gho wlsr s,t +418, Ho pìwrgr frqvlvwh hq vxsrqhu txh vh fxpsoh od qhjdflöq gh 418
hv ghflu/ vxsrqhu txh vh yhulfd
+s , t,>
+419,
sa t
| d sduwlu gh hood rewhqhu xqd frqwudglfflöq/ od txh qrv shuplwh dupdu txh 419 hv idovr/ | sru or wdqwr txh od sursrvlflöq 418 hv yhugdghud1 Frpr hmhpsor/ surehprv xqd surslhgdg txh qrv vhuä ýwlo päv dghodqwh1 Hmhpsor 9
Vl s5 hv sdu/ hqwrqfhv s hv sdu1
+41:,
Revhuyh txh od sursrvlflöq hq 41: hv gho wlsr u , v grqgh u @= s hv sdu | od sursrvlflöq v hv v @= s hv sdu1 5
'G
5
5 Xvduhprv ho vðperor % sdud lqglfdu txh od sursrvlflöq d od l}txlhugd gh ìo hvwä ghqlgd sru od h{suhvlöq d vx ghuhfkd1 Dtxð u s2 hv sdu vljqlfd txh od sursrvlflöq u vh ghqh frpr od sursrvlflöq s2 hv sdu1
'G
FKDSWHU 41 ORV QÝPHURV UHDOHV
49
Ghprvwudflöq= Vxsrqjdprv or frqwudulr/ hv ghflu/ s5 hv sdu | s qr hv sdu1 Frpr s qr hv sdu s gheh vhu lpsdu> oxhjr h{lvwh n 5 ] wdo txh s @ 5n . 4 shur
s5
@ @ @ @ @
+5n . 4,+5n . 4, +5n . 4,5n . +5n . 4, 4 7n5 . 5n . 5n . 4 7n5 . 7n . 4 5+5n5 . 5n, . 4
oxhjr s5 @ 5n3 . 4 frq n3 @ 5n5 . 5n 5 ] gh grqgh s5 hv lpsdu> or txh frqwudglfh ho vxsxhvwr txh s5 hv sdu1 Hq frqvhfxhqfld Vl s5 hv sdu/ hqwrqfhv s hv sdu
Hmhpsor :
Qrwdflöq= d4 lqglfd ho lqyhuvr pxowlsolfdwlyr gho qýphur uhdo qr qxor d1 Frq orv d{lrpdv gh fxhusr ghprvwuduhprv dojxqdv surslhgdghv eävlfdv gh U1
Sursrvlflöq 417 Vhdq d> e 5 U / hqwrqfhv
l, +d . e, @ +d, . +e, ll, +d, @ d
lll, +de,4 @ d4 e4 > ly, +d4 ,4 @ d> y, e 3 @ 3
d 9@ 3> e 9@ 3
d 9@ 3
yl, +de, @ +d,e @ d+e,
yll, +d,+e, @ de
Ghprvwudflöq= l, Frpr d | e vrq qýphurv uhdohv/ d . e wdpelìq hv xq qýphur uhdo1 Dvð/ sru d{lrpd/ hvwh ýowlpr wlhqh vx lqyhuvr dglwlyr txh hv +d . e,/ sru or wdqwr +d . e, . ++d . e,, @ 3
Sru rwud sduwh/ sru frqpxwdwlylgdg/ dvrfldwlylgdg/ d{lrpd gho lqyhuvr | d{lrpd gho qhxwur whqhprv txh
Sru or wdqwr
+d . e, . +d, . +e, @ +d . +d,, . +e . +e,, @ 3.3 @ 3 +d . e, . +d, . +e, @ 3
hvwd ýowlpd ljxdogdg qrv glfh txh wdpelìq +d, . +e, hv lqyhuvr gh +d . e,1 Gh hvwd irupd/ sru xqlflgdg gho lqyhuvr dglwlyr/ vh wlhqh txh +d . e, @ +d, . +e, ll, Sru d{lrpd gho lqyhuvr dglwlyr h{lvwh +d, 5 U | +d, 5 U | qrwhprv txh +d, . d @ d . +d, @ 3
4151 D[LRPDV GH FXHUSR |
4:
++d,, . +d, @ +d, . ++d,, @ 3
ljxdogdghv txh qrv glfhq txh wdqwr d frpr +d, vrq lqyhuvrv dglwlyrv gh +d,> hq frqvhfxhqfld/ sru xqlflgdg gho lqyhuvr dglwlyr d @ +d,
Dqäorjdphqwh vh ghpxhvwudq +lll, | +ly, fdpeldqgr vxpd sru pxowlsolfdflöq | qdwxudophqwh fdpeldqgr od qrwdflöq gh lqyhuvr dglwlyr d lqyhuvr pxowlsolfdwlyr1 y, Sxhvwr txh sru d{lrpd gho qhxwur dglwlyr
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hqwrqfhv whqhprv sru glvwulexwlylgdg oxhjr
e 3 @ e +3 . 3, @ e 3 . e 3 e3@e3.e3
Vxpdqgr d dperv plhpeurv gh od hfxdflöq BB ho lqyhuvr dglwlyr gho qýphur uhdo e3/ dvrfldqgr ghelgdphqwh/ | sru od surslhgdg gho qhxwur h lqyhuvr dglwlyr vh wlhqh txh
oxhjr
e 3 . ++e 3,, 3 3 3
@ @ @ @
e 3 . e 3 . ++e 3,, e 3 . +e 3 . ++e 3,,, e3.3 e3
e3@3
yl, Od ghprvwudflöq frqvlvwh hq suredu txh +de, | +d,e vrq lqyhuvrv dglwlyrv gho plvpr qýphur= Fodudphqwh srghprv yhu txh +de, hv ho lqyhuvr gh +de, oxhjr de . ++de,, @ 3
Sru rwur odgr sru od surslhgdg gh glvwulexwlylgdg/ d{lrpd gho lqyhuvr dglwlyr | sru BB whqhprv txh
oxhjr
de . +d,e @ +d . +d,,e @ 3e @ 3 de . +d,e @ 3
ljxdogdg txh qrv glfh txh +d,e wdpelìq hv lqyhuvr gh de/ dvð/ sru xqlflgdg gho lqyhuvr gh de vh wlhqh txh +de, @ +d,e
Dghpäv/ sru frqpxwdwlylgdg/ BB vh wlhqh txh
sru or wdqwr
+de, @ +ed, @ +e,d @ d+e, +de, @ d+e,
4;
FKDSWHU 41 ORV QÝPHURV UHDOHV
txh hv od rwud ljxdogdg txh txhuðdprv suredu1 yll, Frqvlghuhprv +d,+e, @ +d+e,, @ ++de,, @ de
xvdqgr yl, xvdqgr yl, xvdqgr ll,
oxhjr
+d,+e, @ de
or txh suxhed qdophqwh od sursrvlflöq1
Frurodulr 418
l, +4,+4, @ 4
ll, +e, @ +4e, @ +4,e
Ghprvwudflöq= Wrpdqgr d @ 4 | e @ 4 hq yll, vh wlhqh txh +4,+4, @ 4
| wrpdqgr d @ 4 hq yl, vh wlhqh Qrwdflrqhv= Vhdq d> e 5 U / hqwrqfhv hvfuleluhprv
Hmhpsor ; Fxdqgr dqrwdprv
+e, @ +4e, @ +4,e
d . +e, @ = d e 4 > d 9@ 3 d4 @ d d > e 9@ 3 de4 @ e
+41;,
4 {5 7
hvwdprv kdeodqgr gho lqyhuvr pxowlsolfdwlyr gh {5 71 Gh hvwd irupd/ {547 vhuðd xq qýphur uhdo vl | vöor vl {5 7 9@ 3> hv ghflu/ vl { 9@ 5 | { 9@ 5/ dvðod ljxdogdg 4 @ +{5 7,4 {5 7
hvwä ghqlgd hq U vl | vöor vl { 5 U i5> 5j+hq rwur fdvr dpedv h{suhvlrqhv fduhfhq gh vhqwlgr,1
Revhuydflöq= Vl d> e 5 U i3j/ hqwrqfhv>
d 4 e
@
e d
hq hihfwr/ xvdqgr od sursrvlflöq 417 | od qrwdflöq dqwhulru> d 4 e
@ +de4 ,4 @ d4 +e4 ,4 @ d4 e @ ed4 @
e d
Pxfkdv hqwuh odv revhuydflrqhv dqwhulruhv | odv txh yhqguäq/ vrq idplolduhv do ohfwru/ päv dýq/ |d odv kd xwlol}dgr> shur or txh ghehprv qrwdu hv txh vrq sursrvlflrqhv txh vh sxhghq ghprvwudu | txh wrgdv surylhqhq gh orv d{lrpdv gh fxhusr txh fxpsoh U1
4151 D[LRPDV GH FXHUSR
4<
Hmhpsor < Vhdq d> f 5 U> e> g 5 U i3j Ghprvwuduhprv txh vh wlhqh odv vljxlhqwhv ljxdogdghv d f . @ d e f g @ e g
dg . ef eg df eg
+41 5> 6> 7> 8> = = =j
] @ i= = = > 6> 5> 4> 3> 4> 5> 6> 7> = = =j
Ho frqmxqwr gh orv qýphurv Udflrqdohv1 r qd = d 5 ] a e 5 ] i3j T @ e
Rwurv frqmxqwrv qrwdeohv vrq
Q3 @ Q ^ i3j @ ]. 3
] @ iq = q 5 Qj L@ UT
hvwh ýowlpr oodpdgr ho frqmxqwr gh orv qýphurv Luudflrqdohv1 Qdwxudophqwh vh wlhqh od vljxlhqwh fdghqd gh ghvljxdogdghv | dghpäv6
Q Q3 @ ]. 3 ] T U
Qrwhprv txh orv vljxlhqwhv qýphurv uhdohv
b @U T ^L
4> 657>
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6
3> 6>
4> 5 3> 69
FKDSWHU 41 ORV QÝPHURV UHDOHV
53
vrq qýphurv udflrqdohv/ sxhv hoorv vh sxhghq h{suhvdu gh od irupd d > e
Sru hmhpsor
4> 657 @
sdud dojýq d 5 ]> e 5 ] i3j
4657 > 4333
3> 6 @
6 <
4> 5 @ 3> 69
45 43 69 433
@
| vlpsolfdqgr fdgd xqd rewhqhprv h{suhvlrqhv udflrqdohv päv vlpsohv 4> 657 @
664 583
>
3> 6 @ 46 >
Dojxqrv hmhpsorv gh qýphurv luudflrqdohv vrq >
s 5>
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h
grqgh vh dsur{lpd do udflrqdo 6> 47 | h vh dsur{lpd do udflrqdo 5> :41 Hvwdv dsur{lpdflrqhv vrq xvdgdv sdud kdfhu dojxqrv fäofxorv phqruhv päv vlpsohv/ shur hoorv wlhqhq lqqlwrv ghflpdohv | qr vrq shulöglfrv1 s Sureduhprv päv dghodqwh txh 5 hv xq qýphur luudflrqdo1 Uhfrughprv txh ho udglr gh xqd flufxqihuhqfld hvwä frqwhqlgr 5 yhfhv hq vx flufxqihuhqfld1 Dkrud yhdprv dojxqrv hmhpsorv txh pxhvwudq dojxqdv gh odv surslhgdghv dojheudlfdv txh fxpsohq orv vxefrqmxqwrv gh U dqwhulruphqwh phqflrqdgrv1 Hmhpsor 43 Ghwhuplqhprv vl vrq yhugdghudv r idovdv odv vljxlhqwhv dupdflrqhv 41 Ho frqmxqwr gh orv qýphurv luudflrqdohv fxpsoh frq ho d{lrpd gh fodxvxud sdud od vxpd r ho surgxfwr ghqlgrv hq U
51 +L> ., hv xq juxsr
61 +]> .> , hv xq fxhusr
71 +T > .> , hv xq fxhusr Vroxflöq=
41 Idovr1 Orv qýphurv d @
s s 5 | e @ 5 vrq luudflrqdohv shur d.e@3
txh hv xq qýphur udflrqdo +3 @ 34 / sru hmhpsor,> sru or wdqwr qr fxpsoh frq od surslhgdg gh fodxvxud s s s sdud od vxpd1 Sru rwud sduwh vl wrpdprv dkrud d @ e @ 5/ hqwrqfhv de @ 5 5 @ 5 txh hv xq qýphur udflrqdo> sru or wdqwr qr vh fxpsoh od surslhgdg gh fodxvxud sdud ho surgxfwr ghqlgr hq U1
51 Idovr1 Sru or dqwhulru hv fodur txh +L> ., qr hv juxsr
61 Idovr1 Vl +]> .> , ixhud fxhusr/ hqwrqfhv +] i3j> , vhuðd juxsr> shur ] qr fxpsoh od surslhgdg gh lqyhuvr pxowlsolfdwlyr1 71 Yhugdghur1 +T > ., hv juxsr/ +T i3j> , hv juxsr | vh fxpsoh od surslhgdg glvwulexwlyd hq T1
4151 D[LRPDV GH FXHUSR
41516
54
Srwhqfldv Hqwhudv
Ghqlflöq 419 Vhd d 5 U / q 5 ]. 1 Vh ghqh od srwhqfld uhdo gh edvh d | h{srqhqwh q sru= dq @ d d d~} d d q
yhfhv
Xqd ghqlflöq päv suhflvd gh srwhqfld hv od vljxlhqwh Vhd d 5 U 1 Vh ghqh sru uhfxuuhqfld>
l, d3 @ 4/ d 9@ 3
ll, dq.4 @ dq d/ ;q 5 Q3
Vhd q 5 ] / d 9@ 31 Vh ghqh od srwhqfld uhdo gh edvh d | h{srqhqwh q sru= dq @
4
d
q
@
4 d d d~} d d
q
Hmhpsor 44
56 @ 5 5 5 @ ;> 56 @
+4144,
yhfhv
4 4 4 @ @ 6 5 555 ;
Whruhpd 41: Vhd d> e 5 U i3j/ q 5 ] / p 5 ] / hqwrqfhv
l, dq dp @ dq.p
ll, dq eq @ +de,q
p q @d q ly, q @ lll,
d
p
q
d
d
d
e
e
q
y, +dq ,p @ dqp
Wrgdv hvwdv surslhgdghv vh suxhedq xvdqgr lqgxfflöq/ wìfqlfd gh ghprvwudflöq txh frqvlvwh hq hvwh fdvr hq= sulphur yhu txh od ljxdogdg vh fxpsoh sdud q @ 3> srvwhulruphqwh/ vh vxsrqh txh vh fxpsoh od ljxdogdg sdud xq qdwxudo +jhqìulfr, n | xwlol}dqgr hvwd lqirupdflöq vh pxhvwud txh od ljxdogdg hv yäolgd sdud ho qdwxudo vljxlhqwh n . 41 Xqd ghprvwudflöq wdo qrv jdudqwl}d txh odv ljxdogdghv vrq vöor yäolgdv hq Q3 shur xvdqgr od ghqlflöq vh sxhgh rewhqhu od surslhgdg hq ]1
Surgxfwrv Qrwdeohv
Xvdqgr odv surslhgdghv gh srwhqfldv | orv d{lrpdv gh U vh wlhqhq orv vljxlhqwhv surgxfwrv qrwdeohv= 4, d5 e5 5, +d . e,5 6, d6 . e6 7, d6 e6 8, +d . e,6 9, +d e,6 :, dp ep
@ @ @ @ @ @ @
+d . e,+d e, d5 . 5de . e5 +d . e,+d5 de . e5 , +d e,+d5 . de . e5 , d6 . 6d5 e . 6de5 . e6 d6 6d5 e . 6de5 e6 +d e,+dp4 . dp5 e . dp6 e5 . . dep5 . ep4 , ;p 5 ].
FKDSWHU 41 ORV QÝPHURV UHDOHV
55
Hmhpsor 45 Vhdq d> e hq U/ wdohv txh od h{suhvlöq vljxlhqwh hvwì ghqlgd hq U= # $ d4 . e4 f+de,4 4 L @ d5 .e5 f5 4 . 5d4 e4 d.e.f d5 e5
Vlpsoltxh L1 Vroxflöq= Surfhglhqgr hq hwdsdv/ whqhprv L
d5 e5 +d4 . e4 f+de,4 , +d . e . f, d5 . e5 f5 . 5d4 e4 d5 e5 +de5 . d5 e fde,+d . e . f, d5 . e5 f5 . 5de de+e . d f,+d . e . f, +d . e,5 f5 de+e . d f,+d . e . f, +d . e f,+d . e . f, de
@ @ @ @
@
Ghwhuplqh sdud txì ydoruhv gh { 5 U od h{suhvlöq hvwä ghqlgd hq U/ | vlpsoltxh1 H@
Vroxflöq= Od h{suhvlöq hvwä ghqlgd hq U vl
{5 5{ . 4
4,6 4
+{
{5
+{ 4,6 9@ 3 a {5 4
hv ghflu/
{5 4 9@ 3
+{ 4, 9@ 3 a {5 4 9@ 3
Sru or wdqwr H 5 U vl { 5 U i4> 4j1 Whqhprv H
@
{5 5{ . 4
+{ 4,6 {5 4
{ 9@ 4 > { 9@ 4
{5 5{ . 4 5 +{ 4, +{ 4,6 4 +{ 4,+{ . 4, @ +{ 4, @ {.4 @
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41517
Frqmxqwr uhvwulfflöq | frqmxqwr vroxflöq gh xqd hfxdflöq
Oodpduhprv frqmxqwr uhvwulfflöq8 gh xqd h{suhvlöq txh lqyroxfud d orv wìuplqrv S +{, | T+{,9 +sru hmhpsor/ S +{, @ T+{,, do frqmxqwr U gh orv { hq U sdud orv fxdohv fdgd wìuplqr gh od h{suhvlöq hvwä ghqlgd hq U1 Oodpduhprv frqmxqwr vroxflöq gh od hfxdflöq S +{, @ T+{, do vxefrqmxqwr V gh orv { hq U txh vdwlvidfhq od hfxdflöq/ hv ghflu/ V @ i{ 5 U _ U = S +{, @ T+{,j 7 Txh { n whqjd 8 Vxhoh oodpäuvhoh 9 S E{
|
TE{
vhqwlgr sdud wrgr
{
hq
U vljqlfd txh { n hvwä ghqlgr hq wrgr { M U/ hv ghflu/ { M U>
grplqlr gh xqd h{suhvlöq1
vrq h{suhvlrqhv wdohv txh do phqrv xqd gh hoodv lqyroxfud od yduldeoh {1
{
n
M U1
4151 D[LRPDV GH FXHUSR
56
Wdpelìq hvfuleluhprv
V @ i{ 5 U = S +{, @ T+{,j
vxehqwhqglhqgr txh { gheh hvwdu hq U1 Revhuyhprv txh ho frqmxqwr vroxflöq vlhpsuh gheh hvwdu lqfoxlgr hq ho frqmxqwr uhvwulfflöq V U1 Sru hmhpsor/ ho frqmxqwr i{ 5 U = {4 @ {j hv ho frqmxqwr gh orv uhdohv wdohv txh fxpsohq od hfxdflöq 4 { @ {/ hv ghflu/ 4 @ {j @ i4> 4j i{ 5 U = { Uhvroyhu xqd hfxdflöq vljqlfd hqfrqwudu vx frqmxqwr vroxflöq1 Uhvroydprv od hfxdflöq {5 5{ . 4 @6 +{4,6 {5 4
Vroxflöq= Sru ho hmhpsor 41516 hv fodur txh ho frqmxqwr uhvwulfflöq hv U4 @ U i4> 4j/ Dghpäv vlpsolfdqgr od h{suhvlöq l}txlhugd gh od ljxdogdg/ rewhqhprv {
/ /
5
5{.4
Sru or wdqwr ho frqmxqwr vroxflöq gh od hfxdflöq hv= + V
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@
+{ 4,6 {5 4
@ 6
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,
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l, Od hfxdflöq
wlhqh sru ýqlfd vroxflöq ho uhdo e d1
ll, Vl d 9@ 3 od hfxdflöq
wlhqh frpr ýqlfd vroxflöq ho uhdo de 1
{.d@e
d{ @ e
Od ghprvwudflöq vh ghmduä frpr hmhuflflr sdud ho ohfwru> gheh xvduvh odv surslhgdghv gh lqyhuvr dglwlyr | pxowlsolfdwlyr uhvshfwlydphqwh1 Hmhuflflr 46 Ghwhuplqh vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwltxh1 r q @ 4 @U 41 { 5 U = {{.6 .6 51
q {5U =
+{.6,+{.7, {.6
r @4 @!
61 +;d> e 5 U,+de @ 4 , +d @ 4 b e @ 4,, 71 +;{> | 5 U,+{5 @ | 5 , { @ |,
Vroxflöq=
FKDSWHU 41 ORV QÝPHURV UHDOHV
57 41 Idovr1 Ho frqmxqwr uhvwulfflöq gh od hfxdflöq
hv U @ U i6j1 Oxhjr/ vlpsolfdqgr od h{suhvlöq
{.6 @4 {.6 {.6 @4 {.6
rewhqhprv txh 4 @ 4 or fxdo hv yhugdghur1 Sru or wdqwr ho frqmxqwr vroxflöq gh od hfxdflöq hv {.6 @ 4 @ U _ U @ U i6j V@ {5U = {.6
51 Yhugdghur1 Ho frqmxqwr uhvwulfflöq gh od hfxdflöq
+{ . 6,+{ . 7, @4 {.6
hv U4 @ U i6j1 Vlq hpedujr do vlpsolfdu od h{suhvlöq rewhqhprv od hfxdflöq
+{ . 6,+{ . 7, @4 {.6 {.7@4
gh grqgh whqhprv txh { @ 61 Oxhjr ho frqmxqwr vroxflöq gh od hfxdflöq hv {.6 @4 @ i6j _ U4 V @ { 5 U4 = {.6 @ i6j _ +U i6j, @ ! 61 Idovr1 Frqvlghuhprv d @ 5@6/ e @ 6@5 hq U
71 Idovr1 Vhdq { @ 5/ | @ 5
41518
5 6 5 6 @ 4 shur 9@ 4 | 9@ 4 6 5 6 5 +5,5 @ 55 shur 5 9@ 5
Uhvroxflöq gh sureohpdv gh sodqwhr
Od lpsruwdqfld gh od sursrvlflöq 41; hvwä hq od uhvroxflöq gh sureohpdv1 Sru hmhpsor/ hq orv vljxlhqwhv sureohpdv ghehprv oohydu ho hqxqfldgr d xq sodqwhdplhqwr pdwhpäwlfr txh oxhjr vh uhgxfh d od uhvroxflöq gh xqd hfxdflöq hq xqd yduldeoh +r d od uhvroxflöq gh xq vlvwhpd gh hfxdflrqhv,1
Hmhuflflr 47 Xqd erped txh wudedmd vrod/ oohqd xq hvwdqtxh hq : krudv/ rwud or kduðd hq ; krudv1BcHq fxäqwr wlhpsr vh oohqd ho hvwdqtxh vl wudedmdq dpedv erpedv d od yh}B
Vroxflöq= Vhd { ho qýphur gh krudv txh wdugd ho hvwdqtxh hq oohqduvh +vl wudedmdq dpedv erpedv d od yh}, | vhd Y ho yroxphq wrwdo gho hvwdqtxh1 Vl xqd erped oohqd ho hvwdqtxh hq : krudv +hqwrqfhv hq xqd krud oohqduä 4: gho hvwdqtxh,/ oxhjr hq { +xqlgdghv gh yroxphq,1 Od rwud erped oohqd ho hvwdqtxh krudv oohqduä {: gho hvwdqtxh/ hv ghflu/ oohqduä {Y :
4151 D[LRPDV GH FXHUSR
58
+xqlgdghv gh yroxphq,1 Frpr hq { hq ; krudv/ hqwrqfhv hq { krudv oohqduä {; gho hvwdqtxh/ hv ghflu/ {Y ; krudv/ odv grv erpedv mxqwdv oohqdq do hvwdqtxh hqwrqfhv vh wlhqh
or txh htxlydoh d
{Y {Y . @Y : ;
{ { . @4 : ; gh grqgh 48{ @ 891 Oxhjr/ wudedmdqgr odv grv erpedv d od yh}/ ho hvwdqtxh vh oohqd hq 89248 krudv1
Hmhuflflr 48 Fxdwur hvwxgldqwhv ghflghq ylylu vrorv hq xq ghsduwdphqwr | uhsduwlu hq sduwhv ljxdohv ho duulhqgr phqvxdo1 Vlq hpedujr/ hqfxhqwudq txh vl dxphqwd hq grv ho qýphur gh hvwxgldqwhv/ vx fxrwd phqvxdo vh uhgxfh hq ' 8=3331 BcFxäo hv ho frvwr gho duulhqgrB Vroxflöq= Vhd { ho frvwr gho duulhqgr1 Vl kd| fxdwur hvwxgldqwhv hqwrqfhv od fxrwd phqvxdo gh fdgd xqr hv gh {7 1 Vl dxphqwdq hq grv/ hqwrqfhv od fxrwd phqvxdo vhuä gh {9 1 Gdgdv odv klsöwhvlv gho sureohpd vh wlhqh txh { 8=333 @ { 7 9 , 6{ 45 8=333 @ 5{ , { @ 93=333
Oxhjr ho frvwr gho duulhqgr hv gh ' 93=3331 Sdud uhvroyhu sureohpdv frpr orv dqwhulruhv vh uhtxlhuh= uhfrqrfhu orv gdwrv lpsruwdqwhv +| ghvsuh0 fldu lqirupdflöq lqqhfhvduld,> lghqwlfdu od r odv lqföjqlwdv h lqwurgxflu xqd r päv yduldeohv sdud uhsuhvhqwduodv> sodqwhdu | uhvroyhu odv hfxdflrqhv txh uhodflrqdq orv gdwrv/ whqlhqgr hq fxhqwd odv uhvwulf0 flrqhv qhfhvduldv lqyroxfudgdv hq ho sureohpd1 Hq dojxqrv fdvrv hv qhfhvdulr dsolfdu frqfhswrv wdohv frpr srufhqwdmh/ ud}öq | sursruflöq/ yhorflgdg +@ glvwdqfld2wlhpsr,/ hwf1 Revhuydflöq= Uhfrughprv txh> 41 s hv ho t sru flhqwr +vh hvfuleh t(, gh {/ vl | vöor vl> s@
t { 433
51 Od ud}öq gh grv qýphurv d = e +vh ohh d hv d e, hv ho fxrflhqwh d e
+Od h{suhvlöq d = e ghvhd ghvwdfdu ho dvshfwr gh frpsdudflöq hqwuh d | e,1
Xqd sursruflöq hv xqd ljxdogdg hqwuh grv ud}rqhv/ sru hmhpsor> txh vh ohh 5 hv d 6 frpr 9 hv d <
5=6@9=<
61 Vh glfh txh d hv gluhfwdphqwh sursruflrqdo d e vl | vöor vl h{lvwh xqd frqvwdqwh t wdo txh d @ te
Vh glfh txh d hv lqyhuvdpdqwh sursruflrqdo d e vl h{lvwh xqd frqvwdqwh t wdo txh d@t
4 e
Hq dperv fdvrv t vh oodpd idfwru gh sursruflrqdolgdg | jhqhudophqwh vh wlhqh t A 31
71 Gduhprv ho yroxphq | äuhd gh dojxqrv fxhusrv jhrpìwulfrv txh xwlol}duhprv
FKDSWHU 41 ORV QÝPHURV UHDOHV
59
41 +d, Sdudohohsðshgr gh dulvwdv {/ |/ }1 Yroxphq= Y @ {|} Duhd= D @ 5{| . 5{} . 5|} +e, Flolqgur gh udglr u | dowxud k1 Yroxphq= Y @ u5 k Duhd= D @ 5u5 . 5uk +f, Frqr uhfwr gh udglr edvdo u | dowxud k1 Yroxphq= Y @ 46 u5 k s Duhd= D @ u u5 . k5 +g, Hvihud gh udglr u1 Yroxphq= Y @ 76 u6 Duhd= D @ 7u5
Hmhuflflr 49 Vl ho udglr gho flolqgur glvplqx|h hq xq 43( plhqwudv txh vx dowxud dxphqwd hq xq 45( hq txì wdqwr sruflhqwr yduðdq d, Ho yroxphq gho flolqgur e, Ho duhä odwhudo gho flolqgur Vroxflöq= Vhd u = udglr lqlfldo gho flolqgur/ k = dowxud lqlfldo gho flolqgur d, Vl ho udglr glvplqx|h hq xq 43( hqwrqfhv ho qxhyr udglr uq vhuä uq @ u +3> 4,u @ 3> 45k @ 4> 45k
Yq @ uq 5 kq @ +3> 45k, @ 3> 3 3 hv ghflu/ glvplqx|h hq 5;( e, Ho äuhd gho flolqgur vhuä
Dq @ 5uq kq @ 5+3> 45k, @ 4> 33; 5uk @ 5uk . 3> 33; 5uk
oxhjr ho äuhd odwhudo dxphqwd hq 3> ;(1
4151 D[LRPDV GH FXHUSR
5:
Hmhuflflr 4: Hq xqd suxhed gh pdwhpäwlfdv/ ho 45( gh orv hvwxgldqwhv gh xqd fodvh qr uhvroylö xq sure0 ohpd/ ho 65( or uhvroylö frq dojxqrv huuruhv | orv 47 hvwxgldqwhv uhvwdqwhv rewxylhurq od vroxflöq fruuhfwd1 ÁFxäqwrv hvwxgldqwhv kdeðd hq od fodvhB Vroxflöq= Vhd { ho qýphur gh hvwxgldqwhv/ hqwrqfhv
45 { qr uhvroylhurq ho sureohpd 433
65 { uhvroylhurq frq dojxqrv huuruhv ho sureohpd 433 47 = uhvroylhurq ho sureohpd fruuhfwdphqwh
Oxhjr whqhprv od vljxlhqwh hfxdflöq
gh grqgh hv ghflu/ oxhjr kdeðd 58 hvwxgldqwhv hq od fodvh1
65 45 {. { . 47 @ { 433 433 89{ @ 4733> {@
4733 @ 58 89
Hmhuflflr 4; Xqd wlhqgd gh dqwlj hgdghv frpsuö grv pxheohv hq ' 558=333 | ghvsxìv orv yhqglö | rewxyr xq ehqhflr gho 73(1 ÁFxäqwr sdjö sru fdgd pxheoh vl ho sulphur ghmö xq ehqhflr gho 58( | ho vhjxqgr xq ehqhflr gho 83(B Vroxflöq= Vhd { shvrv +hq plohv, or txh oh kd frvwdgr ho sulphu pxheoh hqwrqfhv ho vhjxqgr or frpsuö hq +558 {, plo shvrv
Ho sulphur glr xq ehqhflr gho 58(/ oxhjr vh yhqglö hq
+{ . 3> 58{, plo shvrv>
hv ghflu/ 4> 58{ plo shvrv1 Ho vhjxqgr glr xq ehqhflr gho 83(/ oxhjr vh yhqglö hq
4> 83+558 {, plo shvrv
Sru klsöwhvlv/ od wlhqgd frpsuö orv grv pxheohv hq 558 plo shvrv | rewxyr xq ehqhflr wrwdo gho 73(/ oxhjr orv grv pxheohv vh yhqglhurq hq Dvð/ rewhqhprv od hfxdflöq gh grqgh oxhjr
4> 73 558 @ 648 plo shvrv1
4> 58{ . 4> 83+558 {, @ 648 3> 58{ @ 55> 8 { @ g5 orv gläphwurv gh orv flolqgurv | k4 > k5 vxv uhvshfwlydv dowxudv1 Sru klsöwhvlv gho sureohpd whqhprv odv vljxlhqwhv uhodflrqhv> 6 g4 @ > g5 7
8 k4 @ k5 9
Dghpäv od iöupxod gho yroxphq gh xq flolqgur hv
+4145,
Y @ u5 k
grqgh u hv ho udglr gho flolqgur | k od dowxud> dghpäv
g @ 5u
grqgh g hv ho gläphwur gho flolqgur1 Frq wrgr or dqwhulru whqhprv txh | sdud fdgd flolqgur
Y4 @ 7 g54 k4
Shur gh 4145 whqhprv oxhjr dvð/
5 g k 7
Y @
g4 @ 67 g5 Y4 @
Y5 @ 7 g55 k5 | k4 @ 89 k5
< 58 78 5 78 g5 k5 @ g5 k5 @ Y5 7 49 9 hqwrqfhv h{lvwh d4 5 U wdo txh
+4146,
de @ 3 , d4 de @ d4 3 , 4e @ 3 , e @ 3
+ + , Hv fodur txh vl d @ 3 b e @ 3 hqwrqfhv de @ 3> oxhjr
+;d> e 5 U,+de @ 3 / d @ 3 b e @ 3,
Revhuyhprv txh od surslhgdg dqwhulru qr hv yäolgd hq fxdotxlhu frqmxqwr frq rshudflrqhv gh vxpd | pxowlsolfdflöq1 Sru hmhpsor/ frqvlghuhprv ho frqmxqwr gh pdwulfhv d e = d> e> f> g 5 U P5 +U, @ f g : Sru htxlydohqfld oöjlfd hvwd lpsolfdqfld hq
4146 vh sxhgh hvfulelu Es
ýowlpr hv or txh xvduhprv hq hvwd sduwh gh od ghprvwudflöq1
i t j v/ txh hv htxlydohqwh Es2 ; t i v |
hvwr
4151 D[LRPDV GH FXHUSR
5<
edmr od rshudflöq gh surgxfwr ghqlgd sru 3 d e d f3 f g
Vhdq
D@
Hv fodur txh
e3 g3 4 3 3 3
DE @
shur D 9@ 3 | E 9@ 31 Or txh vðvh fxpsoh hv txh
@
dd3 . ef3 fd3 . gf3
| E@
3 3 3 3
de3 . eg3 fe3 . gg3
3 3 3 4
D @ 3 b E @ 3 , DE @ 3
Od surslhgdg 4146 qrv shuplwh uhvroyhu dojxqdv hfxdflrqhv1 Sru hmhpsor uhvroydprv hq U od vljxlhqwh hfxdflöq +{ . 4,+{ 4, @ 3 Hv fodur txh
+{ . 4,+{ 4, @ 3 / +{ 4 @ 3, b +{ . 4 @ 3,
gh grqgh ho frqmxqwr vroxflöq V hv
V @ i{ 5 U = +{ . 4,+{ 4, @ 3j @ i4> 4j
Dqäorjdphqwh vh sxhgh uhvroyhu ho vljxlhqwh sureohpd Hmhpsor 53 Od vroxflöq gh od hfxdflöq hv i8> 8j
Vroxflöq= Hq hihfwr>
{5 @ 58
{5 @ 58 / {5 58 @ 3 / +{ 8,+{ . 8, @ 3 / {8@3 b {.8@3 / { @ 8 b { @ 8
Hmhuflflr 54 Grv shuvrqdv vdohq vlpxowäqhdphqwh gh grv flxgdghv D | E | ydq xqd hq gluhfflöq gh od rwud1 Od sulphud shuvrqd fdplqd 5 np2ku päv gh sulvd txh od vhjxqgd | oohjd d E xqd krud dqwhv txh od vhjxqgd oohjxh d D1 Vl D | E glvwdq 57 np ÁFxäqwrv nloöphwurv uhfruuh fdgd xqd gh odv shuvrqdv hq xqd krudB +D|xgd= yhorflgdg@glvwdqfld2wlhpsr,1
Vroxflöq= Vhd y np2ku od yhorflgdg gh od sulphud shuvrqd SD / hqwrqfhv od yhorflgdg gh od vhjxqgd shuvrqd SE vhuä gh +y 5, np2ku 57 Od shuvrqd SD wdugduä w4 @ 57 y krudv | SE wdugduä w5 @ y5 krudv Frpr SE oohjd xqd krud päv wdugh txh SD / hqwrqfhv> 57 57 w5 @ w4 . 4 , y5 y , 57y 57+y 5, , y5 5y 7; , +y ;,+y . 9, , y@;
@ @ @ @ b
4 +y 5,y 3 3 y @ 9
FKDSWHU 41 ORV QÝPHURV UHDOHV
63
Shur y A 3/ dvð/ y @ ; np2ku1 Oxhjr od glvwdqfld txh uhfruuh SD hq xqd krud hv gh | SE uhfruuh
g4 @ y 4 @ ; np
g5 @ +y 5, 4 @ 9 np
Hmhuflflr 55 Xq wuhq uäslgr ixh reoljdgr d ghwhqhuvh 49 plqxwrv hq xq glvfr urmr1 Sdud uhfxshudu hvwh wlhpsr/ yldmö hq xq wudpr gh ;3 nloöphwurv 43np2ku päv uäslgr txh or qrupdo1 BcFxäo hv od yhorflgdg qrupdo gho wuhqB Vroxflöq= Vhd y np2ku od yhorflgdg suhylvwd sru ho wuhq +grqgh y A 3,1 Od yhorflgdg uhdo ixh gh +y . 43, np2ku
Ho wlhpsr suhylvwr hud gh
w4 @
shur uhdophqwh ixh
w5 @
Sru klsöwhvlv whqhprv txh hv ghflu/
;3 ku y
;3 ku y . 43
w4 w5 @
;3 y y.43 @
;3
49 > 93
49 93
/ +y . 43,;3 ;3y @
shur y A 3 oxhjr
7 48
y+y . 43,
/ y5 . 43y 6=333 @ 3 / +y 83,+y . 93, @ 3 / y @ 83 b y @ 93 y @ 83
| od yhorflgdg qrupdo gho wuhq ixh gh 83 np2ku1 Hq orv hmhpsorv dqwhulruhv khprv uhvxhowr hfxdflrqhv hq xqd yduldeoh1 Frq iuhfxhqfld/ hv qhfhvdulr uhvroyhu grv r päv hfxdflrqhv vlpxowäqhdphqwh hq grv r päv yduldeohv/ hv ghflu/ vlvwhpdv gh hfxdflrqhv1 Hmhpsor 56 Uhvroydprv ho vljxlhqwh vlvwhpd gh hfxdflrqhv { 6| 8{ . |
@ 5 @ 5
Vroxflöq= Pxowlsolfdqgr sru 6 od vhjxqgd hfxdflöq | vxpäqgrvhod d od sulphud vh rewlhqh od hfxdflöq 49{ @ 7
gh grqgh { @ 47 1 Vxvwlwx|hqgr hvwh ydoru hq fxdotxlhud gh odv grv hfxdflrqhv vh rewlhqh | @ 67 1 Oxhjr od vroxflöq gho vlvwhpd hv { @ 47 > | @ 67
r ho sxqwr +{> |, @ +4@7> 6@7, Od lqwhusuhwdflöq jhrpìwulfd gh hvwd vroxflöq hv ho sxqwr gh lqwhuvhfflöq gh odv uhfwdv fx|dv hfxdflrqhv vrq { 6| @ 5 | 8{ . | @ 5 frpr vh pxhvwud hq od vljxlhqwh jxud
4151 D[LRPDV GH FXHUSR
64
Hmhuflflr 57 Xq flhuwr qýphur gh hvwxgldqwhv ghehq dfrprgduvh hq xqd uhvlghqfldo1 Vl vh xelfdudq grv hvwxgldqwhv sru kdelwdflöq hqwrqfhv txhgduðdq 5 hvwxgldqwhv vlq slh}d1 Vl vh xelfdudq 6 hvwxgldqwhv sru kdelwdflöq hqwrqfhv vreuduðdq 5 slh}dv1 Á Fxäqwdv kdelwdflrqhv glvsrqleohv kd| hq od uhvlghqfldo | fxäqwrv hvwxgldqwhv ghehq dfrprgduvh hq hoodB Vroxflöq= Vhdq { = qýphur gh hvwxgldqwhv | = qýphur gh kdelwdflrqhv/ hqwrqfhv srghprv uhsuhvhqwdu ho sureohpd phgldqwh odv hfxdflrqhv { 5| @ 5 { 6+| 5, @ 3 ghvshmdqgr { gh od sulphud hfxdflöq | uhhpsoä}dqgroh hq od vhjxqgd/ |@; Sru or wdqwr kd| ; kdelwdflrqhv | 4; hvwxgldqwhv1
{ @ 4;
FKDSWHU 41 ORV QÝPHURV UHDOHV
65
Hmhuflflr 58 Xqd fkdsd uhfwdqjxodu gh hvwdôr gh shuðphwur
or txh htxlydoh d
7+{ ;,+| ;, @ :9; 5{ . 5| @ 4533` Od ixqflöq U+s, hv htxlydohqwh d>
4533 s s U+s, @ 433 . 53 ;3
e, Frpr U+s, hv od sduäerod gh od jxud
od pä{lpd uhfdxgdflöq vh hqfxhqwud fxdqgr ho suhflr hv s @ ;33 | frq xqd wdtxlood q @ 733 733 533 @ 5331 f, Od uhfdxgdflöq pä{lpd sru vhvlöq hv ' 493=3331
;33 7
@
Hmhuflflr 439 Xq fdpshvlqr wlhqh glqhur sdud frpsudu 4933 phwurv gh dodpeuh gh sýdv/ | ghvhd fhufdu frq fxdwur kheudv gh dodpeuh xq whuuhqr uhfwdqjxodu hq vx suhglr1 Vl { uhsuhvhqwd xq odgr gho uhfwäqjxor d, Ghwhuplqh od ixqflöq D+{, äuhd gho whuuhqr fhufdgr frq hvwh dodpeuh hq wìuplqrv gh {1 e, Hqfxhqwuh odv glphqvlrqhv gho whuuhqr fhufdgr gh pd|ru äuhd1
FKDSWHU 51 IXQFLRQHV
45;
Vroxflöq= d, Vl { h | vrq orv odgrv gho uhfwäqjxor/ vh wlhqh txh ho äuhd gho whuuhqr fhufdgr hv frq { A 3 > | A 3
D+{, @ {| | ho dodpeuh hpsohdgr sdud hvwh uhfwäqjxor vhuä
7+5{ . 5|, txh gheh vhu ljxdo d od fdqwlgdg gh dodpeuh glvsrqleoh1 Dvð= 7+5{ . 5|, @ 4933 { . | @ 533 | @ 533 {
Sru or wdqwr ho äuhd txhgd vöor frpr ixqflöq gh {
grqgh GrpS +D, @`3> 533^1
D+{, @ +533 {,{
e, \d txh D+{, @ +533 {,{ hv xqd sduäerod/ frpr od gh od jxud vljxlhqwh vh wlhqh txh ho äuhd pä{lpd vh rewlhqh fxdqgr { @ 433/ sru or wdqwr odv glphqvlrqhv gho whuuhqr fhufdgr vrq gh 433 sru 433 pwv1
Hmhuflflr 43: Od rulood gh xqd slvflqd irupd xq uhfwäqjxor gh 93 slhv gh odujr | 63 slhv gh dqfkr1 Vx surixqglgdg dxphqwd xqliruphphqwh gh 7 d < slhv hq xq wudpr krul}rqwdo gh 73 slhv | ghvsxìv frqwlqýd do plvpr qlyho orv uhvwdqwhv 53 slhv/ frpr vh loxvwud hq od jxud txh pxhvwud xqd vhfflöq wudqvyhuvdo1 Vl od slvflqd vh kd oohqdgr ghvgh ixhud dofdq}dqgr xq qlyho k hq ho odgr päv surixqgr/ ghwhuplqh ho yroxphq Y +k, gho djxd hq ixqflöq gh od dowxud k1
5171 DSOLFDFLRQHV GH IXQFLRQHV/ PRGHODFLÖQ
Vroxflöq= Sulphu fdvr1 Vxsrqjdprv sulphur txh 3 ? k ? 81 Fodudphqwh whqhprv txh Y +k, @ Y4 +k, . Y5 +k, grqgh Y4 +k, @ 53 k 63
Sdud Y5 +k, frqvlghuhprv od jxud grqgh
Y5 +k, @ shur sru ho whruhpd gh Wkdohv whqhprv txh
63k| 5
8 k @ 73 |
or txh lpsolfd txh | @ ;k oxhjr/ Y5 +k, @ Sru or wdqwr/ vxpdqgr dperv yroýphqhv
573k5 63k+;k, @ = 5 5
Y +k, @ 933k . 453k5 Vhjxqgr fdvr1 Vxsrqjdprv dkrud txh < k 8 | frqvlghuhprv od jxud
45<
FKDSWHU 51 IXQFLRQHV
463
Whqhprv txh Y +k, @ Y4 +k, . Y5 +k, . Y6 +k, grqgh Y4 +k, @ 53 63 k @ 933k Y5 +k, @ 738563 @ 9333 5 @ 6333 Y6 +k, @ +k 8, 73 63 @ 4533+k 8, sru or wdqwr/ Y +k, @ 933k . 6333 . 4533+k 8, | qdophqwh Y +k, @
933k . 453k5 933k . 6333 . 4533+k 8,
vl 3 ? k ? 8 vl 8 k <
Hmhuflflr 43; Ghwhuplqh xqd ixqflöq txh uhsuhvhqwh ho yroxphq gh xq flolqgur lqvfulwr hq xqd hvihud gh udglr d +frpr hq od jxud , hq ixqflöq gh vx dowxud1
5171 DSOLFDFLRQHV GH IXQFLRQHV/ PRGHODFLÖQ
464
Vroxflöq= Vhdq u | k udglr | dowxud gho flolqgur uhvshfwlydphqwh1 Vdehprv txh ho yroxphq gho flolqgur hvwä gdgr sru od iöupxod Y @ u5 k +txh ghshqgh gh odv yduldeohv u | k,/ shur/ sru whruhpd gh Slwäjrudv/ whqhprv 5 k . u5 = d5 @ 5 Oxhjr/ ghvshmdqgr u5 | uhhpsod}dqgr hq od iöupxod gho yroxphq/ whqhprv 5 k 5 ,k Y +k, @ +d 5
5 grqgh GrpS +Y , @`3> 5d^/ sxhvwr txh k A 3 | d5 k5 A 3> gh grqgh 3 ? k ? 5d1
Hmhuflflr 43< Xq flfolvwd wlhqh txh kdfhu xq yldmh gh g npv/ hq xq wlhpsr ghwhuplqdgrv1 Fdofxod txh gheh yldmdu d Y npv@kuv1 Shur vdoh 6 plqxwrv wdugh | yldmd d 4 npv@kuv päv ghsulvd/ | oohjd d wlhpsr1 Ghwhuplqdu od yhorflgdg suhylvwd Y hq ixqflöq gh od glvwdqfld g1
Vroxflöq= Vxsrqjdprv txh ho flfolvwd yldmded d xqd yhorflgdg gh y np2ku> hqwrqfhv od yhorflgdg exvfdgd g vhuä Y @ +y 4, np2ku1 Ho flfolvwd yldmö uhdophqwh gy krudv plhqwudv txh ho wlhpsr suhylvwr hud gh y 4 krudv1 Sru klsöwhvlv g g 6 y4 y @ 93 yg+y4,g 4 +y4,y @ 53 53yg 53+y 4,g @ +y 4,y 53yg 53gy . 53g @ y5 y y5 y 53g @ 3 Frpr g A 3 | y A 3 whqguhprv txh
s 4 . ;3g 5 shur od yhorflgdg exvfdgd hv Y +g, @ y+g, 4/ sru or wdqwr s 4 . ;3g 4 Y +g, @ 5 y+g, @
4.
FKDSWHU 51 IXQFLRQHV
465
Hmhuflflr 443 Vh ghvhd frqvwuxlu xqd fdmd frq wdsd gh edvh fxdgudgd frq äuhd qr pd|ru gh 433fp5 frpr od gh od jxud / gh yroxphq 585fp6 1 Vl ho frvwr gh od wdsd hv ' 5 sru fp5 / od edvh ' 8 sru fp5 | ho gh orv odgrv hv gh ' 6 sru fp5 / h{suhvh ho frvwr F frpr ixqflöq gh { | GrpS +F,1
Vroxflöq= Sru klsöwhvlv vh vdeh txh ho yroxphq gh od fdmd hv gh 585fp6 > oxhjr {5 | @ 585
+518,
frq { A 3/ | A 3 | {5 4331 Ho frvwr gh od wdsd hv ' 5 sru fp5 1 Od wdsd wlhqh {5 fp5 / oxhjr ho frvwr gh od wdsd hv ' 5{5 1 Ho frvwr gh od edvh hv ' 8 sru fp5 | od edvh wlhqh {5 fp5 / oxhjr ho frvwr gh od edvh hv ' 8{5 1 Ilqdophqwh/ ho frvwr gh orv odgrv hv ' 6 sru fp5 | fdgd odgr wlhqh {|fp5 / oxhjr ho frvwr gh od edvh hv ' 6+7{|, @ ' 45{|1 Dvð/ ho frvwr wrwdo gh od fdmd hv= +519, F @ 5{5 . 8{5 . 45{|
shur sru 518/
|@
uhhpsod}dqgr hq 519 rewhqhprv od ixqflöq shglgd F+{, @ :{5 . 45{
585 {5
585 grqgh GrpS +F, @`3> 43` {5
Hmhuflflr 444 Vl ho udglr edvdo u gh xq frqr flufxodu uhfwr dxphqwd hq xq {(/ plhqwudv txh vx dowxud k glvplqx|h hq xq 53(/ irupxoh xqd ixqflöq hq wìuplqrv gh { txh shuplwd rewhqhu hq txì srufhqwdmh yduðdq= d, ho äuhd edvdo gho frqr1 e, ho yroxphq gho frqr1
Hq sduwlfxodu/ vl ho udglr dxphqwd hq 48(/ Bchq txì srufhqwdmh yduðdq ho äuhd edvdo | ho yroxphq gho frqrB +Vöor ghvduurooduhprv or uhodwlyr do äuhd,1
5171 DSOLFDFLRQHV GH IXQFLRQHV/ PRGHODFLÖQ
466
Vroxflöq= d, Vl ho udglr edvdo dxphqwd hq {( hqwrqfhv ho qxhyr udglr vhuðd { { u @ 4. u uQ @ u . 433 433 oxhjr ho äuhd edvdo vhuðd
5 De @ uQ
Dvð/ ho äuhd edvdo dxphqwduðd hv ghflu/ dxphqwduðd
{ 5 5 @ 4. u 433 { 5 { @ 4. . u5 83 433 { 5 { @ u5 . . u5 83 433
{ 5 { . u5 xqlgdghv/ 83 433
{ 5 { . 433 sru flhqwr 83 433
Oxhjr od ixqflöq txh uhsuhvhqwd ho srufhqwdmh gh dxphqwr gho äuhd edvdo hv { 5 { 433 frq { A 3 D+{, @ . 83 433
Hq sduwlfxodu/ vl ho udglr dxphqwd hq xq 48( hqwrqfhv ho srufhqwdmh gh dxphqwr gho äuhd hv D+48,1
51714
Surjudpdflöq Olqhdo
Hq orv vljxlhqwhv sureohpdv gh prghodflöq pdwhpäwlfd/ vh frqvlghudq ixqflrqhv gh grv yduldeohv8 gho wlsr } @ i+{> |, @ d{ . e|
vxmhwdv d uhvwulfflrqhv gh ghvljxdogdghv1
Hmhpsor 445 Xqd lqgxvwuld pdqxidfwxuhud surgxfh elflfohwdv | prwrqhwdv/ fdgd xqd gh odv fxdohv gheh surfhvduvh d wudyìv gh grv fhqwudohv gh pätxlqdv D | E1 Od fhqwudo D wlhqh xq pä{lpr gh 453 krudv glvsrqleohv | od fhqwudo E srvhh xq pä{lpr gh 4;3 krudv glvsrqleohv1 Od pdqxidfwxud gh xqd elflfohwd uhtxlhuh 9 krudv hq od fhqwudo D | 6 krudv hq od fhqwudo E> od pdqxidfwxud gh xqd prwrqhwd uhtxlhuh gh 7 krudv hq od fhqwudo D | 43 krudv hq od fhqwudo E1 Vdelhqgr txh ho ehqhflr sru elflfohwd hv gh ' 78 | sru prwrqhwd hv gh ' 88 irupxohprv xq prghor pdwhpäwlfr txh shuplwd rewhqhu ho qýphur gh elflfohwdv | gh prwrqhwdv txh vh ghehuäq ideulfdu sdud rewhqhu xq ehqhflr pä{lpr1 Vroxflöq= Vhdq { h | odv yduldeohv txh uhsuhvhqwdq odv fdqwlgdghv gh elflfohwdv | prwrqhwdv uhvshfwl0 ydphqwh | vhd } od jdqdqfld txh uhvxowd do yhqghu { elflfohwdv h | prwrqhwdv1 Ho remhwlyr hv hvfrjhu orv ydoruhv gh { h | gh pdqhud txh vh pd{lplfh od fdqwlgdg
+51:,
} @ 78{ . 88|
vxmhwd d od uhvwulfflöq fruuhvsrqglhqwh d od fhqwudo D
9{ . 7| 453
8 Xqd ixqflöq uhdo gh grv yduldeohv frq grplqlr D i G D
hohphqwrv E{> | hq
D
h{lvwh xq ýqlfr hohphqwr
}
M U wdo txh
\ U2 3< U hv xqd fruuhvsrqghqfld wdo txh sdud fdgd sdu gh }
'
i E{> |
+Yhu Dsìqglfh F,1
FKDSWHU 51 IXQFLRQHV
467 | od uhvwulfflöq fruuhvsrqglhqwh d od fhqwudo E 6{ . 43| 4;3
Frpr { h | vrq fdqwlgdghv gh yhkðfxorv hqwrqfhv ghehprv h{ljlu wdpelìq {3 a |3
Oxhjr ho prghor pdwhpäwlfr fruuhvsrqglhqwh do sureohpd hv
Pd{ } @ 78{ . 88|
vxmhwd d
9{ . 7| 6{ . 43| { |
453 4;3 3 3
+51;,
Qrwd= Hvwh hv xq sureohpd gh surjudpdflöq olqhdo1 Ho sureohpd jhqhudo gh surjudpdflöq olqhdo wudwd gh pd{lpl}du r plqlpl}du +hv ghflu/ rswlpl}du, xqd ixqflöq olqhdo gh yduldv yduldeohv oodpdgd ixqflöq remhwlyr +hfxdflöq 51: hq ho hmhpsor dqwhulru, vxmhwd d xq frqmxqwr gh hfxdflrqhv olqhdohv r ghvljxdogdghv olqhdohv oodpdgdv uhvwulfflrqhv +hfxdflrqhv h lqhfxdflrqhv hq 51; gho hmhpsor dqwhulru,1 Sdud xqd ixqflöq olqhdo gh grv yduldeohv i+{> |, @ d{ . e|
ho sureohpd srghprv uhvroyhuor jhrpìwulfdphqwh1 Hq hihfwr/ hq hvwh fdvr ho sureohpd vöor wlhqh grv yduldeohv gh ghflvlöq | sru or wdqwr vöor grv glphqvlrqhv/ sru or txh srghprv xvdu xq surfhglplhqwr juäfr sdud uhvroyhuor1 Hvwh surfhglplhqwr lqfox|h od frqvwuxfflöq gh xqd juäfd gh grv glphqvlrqhv frq { h | hq orv hmhv1 Ho sulphu sdvr hv lghqwlfdu orv ydoruhv gh +{> |, shuplwlgrv sru odv uhvwulfflrqhv> dvð/ sdud ho sureohpd dqwhulru/ vh judfdq odv uhfwdv 9{ . 7| 6{ . 43| { |
rewhqlhqgr od uhjlöq idfwleoh U hq od jxud
@ @ @ @
453 4;3 3 3
Ho sdvr qdo hv vhohfflrqdu/ ghqwur gh hvwd uhjlöq/ ho sxqwr +{> |, txh pd{lpl}d ho ydoru gh } @ 78{.88|1 Hvwh sdvr vh yxhoyh dxwrpäwlfr ghvsxìv gh xq srfr gh suäfwlfd/ shur sdud ghvfxeulu hq txì vh ixqgd0 phqwd ydoh od shqd lqwhqwdu dojxqrv ydoruhv sru suxhed | huuru1 Sru hmhpsor/ frqvlghuhprv } @ ;431 Vl vh glexmd od uhfwd 78{ . 88| @ ;43 vh sxhgh yhu +hq od jxud , txh h{lvwhq pxfkrv sxqwrv vreuh hvwd uhfwd txh hvwäq ghqwur gh od uhjlöq U1 Sru or wdqwr ghehprv lqwhqwdu xq ydoru pd|ru sdud }/ sru hmhpsor/ } @ } @ 45:8
hv ghflu/ ideulfdu 43 elflfohwdv | 48 prwrqhwdv rewhqlhqgr xq ehqhflr gh ' 45:81 Ghvsxìv gh xq srfr gh suäfwlfd/ vh sxhgh dsolfdu xq pìwrgr päv vhqfloor txh frqvlvwh hq glexmdu vöor xqd gh hvwdv uhfwdv sdud hvwdeohfhu od shqglhqwh | oxhjr pryhu ylvxdophqwh hvwd uhfwd sdudohod d vðplvpd1
Hmhuflflr 446 Xq kruwlfxowru ghvhd ph}fodu ihuwlol}dqwh txh sursruflrqh xq pðqlpr gh 48 xqlgdghv gh srwdvd/ 53 xqlgdghv gh qlwudwrv | 57 xqlgdghv gh irvidwrv1 Glvsrqh gh grv pdufdv gh ihuwlol}dqwhv1 Od pdufd 4 sursruflrqd 6 xqlgdghv gh srwdvd 4 gh qlwudwrv | 6 gh irvidwrv> vx frvwr hv gh 453 göoduhv +od xqlgdg,1 Od pdufd 5 sursruflrqd 4 xqlgdg gh srwdvd 8 gh qlwudwrv | 5 gh irvidwrv> vx frvwr hv gh 93 göoduhv +od xqlgdg,1 Irupxoh xq prghor pdwhpäwlfr txh shuplwd rewhqhu od frpelqdflöq gh ihuwlol}dqwhv gh phqru frvwr txh vdwlvidjd odv hvshflfdflrqhv ghvhdgdv1 Vroxflöq= Vhdq
{ +xqlgdghv, od fdqwlgdg gh ihuwlol}dqwh +Pdufd 4, | +xqlgdghv, od fdqwlgdg gh ihuwlol}dqwh +Pdufd 5, Hqwrqfhv ho remhwlyr hv plqlpl}du od ixqflöq=
vxmhwd d odv uhvwulfflrqhv=
} @ 453{ . 93|
41 Fruuhvsrqglhqwhv d odv qhfhvlgdghv gh srwdvd1 6{ . | 48
51 Fruuhvsrqglhqwhv d odv qhfhvlgdghv gh qlwudwrv1
{ . 8| 53
61 Fruuhvsrqglhqwhv d odv qhfhvlgdghv gh irvidwrv1 Fodudphqwh/ dghpäv Oxhjr ho prghor pdwhpäwlfr shglgr hv
6{ . 5| 57 {3 |3 Plq } @ 453{ . 93|
FKDSWHU 51 IXQFLRQHV
469 vxmhwr d
Judfdqgr odv uhfwdv
rewhqhprv od uhjlöq U gh od jxud
6{ . | { . 8| 6{ . 5| { | 6{ . | { . 8| 6{ . 5| { |
@ @ @ @ @
48 53 57 3 3 48 53 57 3 3
Sru rwud sduwh d sduwlu gh od ixqflöq remhwlyr/ frq f frqvwdqwh/ vh wlhqh txh |@
f 5{ 93
gh grqgh od shqglhqwh gh od uhfwd f @ 453{ . 93| hv p @ 51 Frqvlghudqgr od uhfwd +frq shqglhqwh p @ 5, päv fhufdqd do ruljhq/ txh sdvh sru do phqrv xq sxqwr gho frqmxqwr U/ whqhprv txh ho sxqwr öswlpr hv { @ 5> | @ <
hv ghflu/ 5 xqlgdghv gh ihuwlol}dqwh Pdufd 4 | < xqlgdghv gh ihuwlol}dqwh Pdufd 51
Hmhuflflr 447 Xqd hglwruldo sodqhd xvdu xqd vhfflöq gh vx sodqwd sdud surgxflu grv wh{wrv D | E1 Od jdqdqfld hv gh ' ; sru ho wh{wr D | ' 8 sru ho wh{wr E1 Ho wh{wr D uhtxlhuh gh 7 krudv sdud lpsulplu | gh 9 krudv sdud hqfxdghuqdu | hpsdvwdu1 Ho wh{wr E uhtxlhuh gh 8 krudv sdud lpsulplu | gh 6 krudv sdud hqfxdghuqdu | hpsdvwdu1 Vl vh glvsrqh gh 533 krudv sdud lpsulplu | gh 543 krudv sdud hqfxdghuqdu | hpsdvwdu | vl sru or phqrv vh ghehq surgxflu 53 wh{wrv1 41 Irupxoh xq prghor pdwhpäwlfr txh shuplwd rewhqhu ho ehqhflr pä{lpr1
51 Ghwhuplqh ho qýphur öswlpr gh fdgd wh{wr txh gheh surgxflu od hglwruldo | vx fruuhvsrqglhqwh jdqdq0 fld1 61 BcFxäqwdv krudv vh xvdq sdud lpsulpluB
5171 DSOLFDFLRQHV GH IXQFLRQHV/ PRGHODFLÖQ
46:
71 BcFxäqwdv krudv vh xvdq sdud hqfxdghuqdu | hpsdvwduB
81 Vl vh djuhjd od uhvwulfflöq= Od surgxfflöq gho wh{wr E gheh vhu sru or phqrv ho 83( gh od surgxfflöq gho wh{wr D1BcTxì sxhgh ghflu gho öswlprB
91 Vl vh glvsrqh gh 9 krudv päv sdud lpsulplu | od jdqdqfld sru fdgd wh{wr vh uhgxfh hq ' 3/35 Bctxì sxhgh ghflu gho öswlprB +QR frqvlghuh od uhvwulfflöq lpsxhvwd hq 8,1
Vroxflöq= 4,= Vhdq
{= qýphurv gh wh{wrv D1
|= qýphurv gh wh{wrv E1
Irupxodflöq gho sureohpd=
vxmhwr d
5, Judfdqgr odv uhfwdv
Pd{ } @ ;{ . 8| 7{ . 8| 9{ . 6| { | 7{ . 8| 9{ . 6| { |
rewhqhprv od uhjlöq U gh od jxud l}txlhugd
@ @ @ @
533 543 53 3 533 543 53 3
Frqvlghudqgr od uhfwd +frq shqglhqwh p @ ;8 , päv ohmdqd do ruljhq/ txh sdvh sru do phqrv xq sxqwr gho frqmxqwr U/ whqhprv txh ho sxqwr öswlpr hv { @ 58> | @ 53
frq } @ 633 6, | 7, Uhhpsod}dqgr od vroxflöq öswlpd hq od uhvwulfflöq gh lpsuhvlöq whqhprv txh vh xvdurq 7 58 . 8 53 @ 533 krudv sdud lpsulplu/ hv ghflu/ wrgdv odv krudv glvsrqleohv sdud wdo hihfwr1 Frqvlghudqgr od uhvwulfflöq gh hqfxdghuqdflöq | hpsdvwh/ rewhqhprv txh vh xvdurq odv 543 krudv glvsrqleohv sdud hvwd wduhd1 8, Do djuhjdu od uhvwulfflöq 5| { 3
FKDSWHU 51 IXQFLRQHV
46;
rewhqhprv od uhjlöq U4 txh pxhvwud od jxud dqwhulru ghuhfkd1 gh grqgh hv fodur txh ho öswlpr qr fdpelö/ hv ghflu/ od vroxflöq öswlpd vljxh vlhqgr { @ 58> | @ 53
frq } @ 6331 9, Vl vh glvsrqhq gh 9 krudv päv sdud lpsulplu | od jdqdqfld sru fdgd wh{wr vh uhgxfh hq ' 3/35 od irupxodflöq pdwhpäwlfd vhuðd Pd{ } @ :> | @ 55> } @ :> 3;
Oxhjr od jdqdqfld dxphqwd hq ' 4> 3;
518
Dojheud gh Ixqflrqhv
Xwlol}dqgr ixqflrqhv hohphqwdohv s |d frqrflgdv/ vh sxhgh ideulfdu qxhydv ixqflrqhv> sru hmhpsor xvdqgr odv ixqflrqhv i+{, @ vhq{/ j+{, @ { vh sxhgh frqvwuxlu od ixqflöq k+{, @ vhq{ .
s {
Xqd gh odv frvdv txh gheh hvshflfduvh dtxðhv ho grplqlr gh k+{,/ | hv qdwxudo shqvdu hq xq frqmxqwr wdo txh fdgd vxpdqgr hvwì ghqlgr hq U> dvðho grplqlr gh k+{, gheh vhu d or päv Grpi _ Grpj1 Hq ho fdvr dqwhulru vh xvö od rshudflöq . +vxpd,/ wdpelìq vh srguðd kdehu xvdgr od rshudflöq +pxo0 wlsolfdflöq,/ glylvlöq/ hwf1
Ghqlflöq 516 Hq jhqhudo vxsrqjdprv txh i | j vrq grv ixqflrqhv | vhd G @ Grpi _Grpj/ hqwrqfhv vh ghqhq qxhydv ixqflrqhv +frq grplqlr G, frqvwuxlgdv hq edvh d rshudflrqhv hqwuh vxv lpäjhqhv uhvshfwlydv=
5181 DOJHEUD GH IXQFLRQHV
46<
d, Od vxpd gh i | j ghqrwdgd sru i . j | ghqlgd sru=
i . j = G $ U { $ i+{, . j+{,
e, Od glihuhqfld gh i | j ghqrwdgd sru i j | ghqlgd sru=
i j = G $ U { $ i+{, j+{,
f, Ho surgxfwr gh i | j ghqrwdgr sru i j | ghqlgr sru=
i j = G $ U { $ i+{,j+{,
g, Ho fxrflhqwh i | j ghqrwdgr sru ij | ghqlgr sru=
grqgh
i @j = G3 $ U { $ i +{,@j+{, G3 @ i{ 5 G = j+{, 9@ 3j
s Hmhpsor 448 Vhdq i +{, @ +5 {,6 / j+{, @m { 5 m1 Ghwhuplqhprv od vxpd/ od glihuhqfld/ ho surgxfwr s | ho fxrflhqwh gh i | j1 Vroxflöq= Qrwhprv txh i +{, @ 5 { m 5 { m1 Ho grplqlr gh i hv ho lqwhuydor ` 4> 5` | ho grplqlr gh j hv U/ sru or wdqwr od lqwhuvhfflöq gh vxv grplqlrv hv ` 4> 5` | odv ixqflrqhv txh vh uhtxlhuhq hvwäq gdgdv sru s +i . j,+{, @ s5 { m 5 { m . m { 5 m { 5 +i j,+{, @ s5 { m 5 { m m { 5 m { 5 +ij,+{, @ s 5 {+5 {,5 {5 5{m5{m +i @j,+{, @ s m{5m @ 5{ {?5 Qrwh txh vl j hv xqd ixqflöq frqvwdqwh wdo txh j+{, @ f sdud wrgr { | vl i hv fxdotxlhu ixqflöq/ hqwrqfhv fi ghqrwd ho surgxfwr gh j | i / +fi,+{, @ fi +{,> ;{ 5 Grpi
Hmhpsor 449 Vhdq i | j odv ixqflrqhv txh vh ghqhq d frqwlqxdflöq ; { 4 ? 4> { . 4> 5 ? { ? 3 s 5> 4 ? { 8 i+{, @ j+{, @ {> 3{?7 = s { 4> { A 8
+yhu jxud , Ghwhuplqh i . j/ i j/ i j/ i@j | dghpäv judiðtxhodv1
FKDSWHU 51 IXQFLRQHV
473 Vroxflöq= Hv fodur sru od ghqlflöq txh
i sxhv j+{, 9@ 3 j @ Grpi _ Grpj @ @` 5> 7^_U @` 5> 7^
Grp+i . j, @ Grp+i j, @ Grp+ij, @ Grp
Sxhvwr txh i | j vrq ixqflrqhv ghqlgdv sru sduwhv ghehprv dqdol}du orv srvleohv fdvrv gh dfxhugr d odv lqwhuvhfflrqhv srvleohv gh grplqlrv1 41 Vhd { 5` 5> 3^_` 4> 4`/ hv ghflu/ { 5` 5> 4`/ hqwrqfhv oxhjr/
i +{, @ { . 4> j+{, @ 4
+i . j,+{, +i j,+{, +i j,+{, i +{, j
@ i +{, . j+{, @ { . 5 @ i +{, j+{, @ { @ i +{,j+{, @ { . 4 @ ij++{{,, @ { . 4
51 Vhd { 5` 5> 3^_` 4> 8`/ hv ghflu/ { 5` 4> 3^/ hqwrqfhv oxhjr/
i +{, @ { . 4> j+{, @ 5
+i . j,+{, +i j,+{, +i j,+{, i +{, j
@ i +{, . j+{, @ { . 6 @ i +{, j+{, @ { 4 @ i +{,j+{, @ 5{ . 5 @ ij++{{,, @ {.4 5
61 Vhd { 5 ^3> 7^_` 4> 8`/ hv ghflu/ { 5 ^3> 7^/ hqwrqfhv s i +{, @ {> j+{, @ 5 oxhjr/
Hq frqvhfxhqfld=
+i . j,+{, +i j,+{, +i j,+{, i +{, j
s @ i+{, . j+{, @ s{ . 5 @ i+{, j+{, @s { 5 @ i+{,j+{, @ 5 { s @ ij++{{,, @ 5{
; 5 ? { 4 ? {.5 { . 6 4 ? { ? 3 +i . j,+{, @ = s {.5 3{?7 ; 5 ? { 4 ? { { 4 4 ? { ? 3 +i j,+{, @ = s {5 3{?7 ; ? { . 4 5 ? { 4 5{ . 5 4 ? { ? 3 +i j,+{, @ = s 5 { 3{?7 ; ? { . 4 5 ? { 4 i {.4 4 ? { ? 3 +{, @ = s5{ j 3{?7 5
5181 DOJHEUD GH IXQFLRQHV
474
+juäfdv hq jxudv vljxlhqwhv,
51814
Frpsrvlflöq gh ixqflrqhv
Ghqluhprv xqd qxhyd ixqflöq txh vh frqvwux|h d sduwlu gh rwudv grv ixqflrqhv i | j shur/ hq hvwh fdvr/ qrv edvduhprv hq odv surslhgdghv dojheudlfdv gho frqmxqwr gh oohjdgd1 Ghqlflöq 517 Vhdq D/ E | F frqmxqwrv gh qýphurv uhdohv1 Vhd i xqd ixqflöq gh D hq E | vhd j xqd ixqflöq gh F hq G/ hv ghflu/
i
D $ E
j
a F $ G
Vl Uhfi _ F 9@ >/ hv ghflu/ ho uhfruulgr gh i | ho grplqlr gh j wlhqhq hohphqwrv hq frpýq/ hqwrqfhv hv srvleoh xvdu i | j sdud ghqlu xqd ixqflöq gh D4 D hq G Wdo ixqflöq vh ghqrplqd frpsrvlflöq gh j frq i/ | vh ghqrwd sru j i | vh ghqh sru= j i = D4 $ G { $ j+i +{,,
grqgh Grp+j i, @ D4 @ i{ 5 D = i +{, 5 Fj/ vl Uhfi _ Grpj 9@ >/ hv ghflu/
Grp+j i , @ D4 @ i{ 5 Grpi = i +{, 5 Grpj _ Uhfij
FKDSWHU 51 IXQFLRQHV
475
+yhu hq jxud ,1 9
Hq sduwlfxodu vl Uhfi Grpj/ hqwrqfhv Grp+j i , @ Grpi1
Hmhuflflr 44: Vhdq |
Ghwhuplqhprv j i 1
i = ^4> 4^$ U s {4.5 { $ j = U $ U { $ +{ 5,5
Vroxflöq= Sxhvwr txh Uhfi U @ Grpj/ hqwrqfhv Grp+j i, @ ^4> 4^ | s s +j i,+{, @ j+i +{,, @ j+ { 4 . 5, @ + { 4 . 5 5,5 @ { 4
oxhjr/
j i = ^4> 4^$ U { $ { 4
Dkrud ghwhuplqhprv i j1 Hq hvwh fdvr ghehprv ghwhuplqdu Uhfj1 Fodudphqwh Uhfj @ ^3> 4^ | Uhfj 9 Grpi / vlq hpedujr Uhfj _ Grpi @ ^4> 4^9@ > oxhjr1 Shur
oxhjr/
Dghpäv/
Grp+i j, @ i{ 5 Grpj = j+{, 5 Grpi j @ i{ 5 U = +{ 5,5 5 ^4> 4^j
+{ 5,5 5 ^4> 4^ +, +{ 5,5 4 +, m { 5 m 4 +, {6 b {4 Grp+i j, @ i{ 5 Grpj = j+{, 5 Grpi j @ ` 4> 4` ^ ^6> 4^ @ U`4> 6^
+i j,+{, @ i +j+{,, @ i++{ 5,5 , @
s +{ 5,5 4 . 5
9 Dojxqrv wh{wrv srqhq frpr frqglflöq qhfhvduld Uhfi \ Grp j sdud ghqlu od frpsxhvwd j i shur frpr vh yh hvwh hv xq fdvr sduwlfxodu gh od ghqlflöq gdgd dtxð1
5181 DOJHEUD GH IXQFLRQHV oxhjr/
476
i j = U`4> 6^$ s U +{ 5,5 4 . 5 { $
Qrwh txh i j 9@ j i/ hv ghflu/ qr vh fxpsoh od surslhgdg gh frqpxwdwlylgdg sdud od frpsrvlflöq gh ixqflrqhv1 Vlq hpedujr vh yhulfd od dvrfldwlylgdg +i j, k @ i +j k,
Hq ho hmhpsor vljxlhqwh vh yhulfd hvwd ljxdogdg hq xq fdvr s s Hmhuflflr 44; Vhdq i+{, @ { . 4 5/ j+{, @ {5 4 | k+{, @ {5 ixqflrqhv ghqlgdv hq vxv grplqlrv pä{lprv1 Ghwhuplqh +i j, k | i +j k,1
. F Vroxflöq= Fodudphqwh vh yh txh Grpk @ U/ Uhfk @ U. 3 / Grpj @`4> 4^ / Uhfj @ U3 / Grpi @ ^4> 4^1 Sdud kdfhu od frpsxhvwd hv dsursldgr kdfhu ho gldjudpd gh od jxud vljxlhqwh
Grp+j k, @ i{ 5 U = {5 5` 4> 4^F j @ i{ 5 U = {5 4j @ ` 4> 4^F s Grp+i j, @ i{ 5` 4> 4^F = s{5 4 5 ^4> 4^j @ i{ 5` 4> 4^F = {5 4 4j @ ` 4> 4^F
Grp++i j, k, @ @ @ @
Grp+i +j k,, @ @ @ @
i{ 5 Grpk = {5 5 Grp+i j,j i{ 5 U = {5 5` 4> 4^F j i{ 5 U = {5 4j ` 4> 4^F
i{ 5 Grp+j k, s = +j k,+{, 5 Grpij i{ 5` 4>s4^F = {7 4 5 ^4> 4^j i{ 5 U = {7 4 4j ` 4> 4^F
Dvð/ vh wlhqh txh wdqwr +i j, k frpr i +j k, vrq ixqflrqhv frq ljxdo grplqlr hq hvwh fdvr1 Dkrud d qlyho gh iöupxodv fdofxohprv ++i j, k,+{,1 Frpr s 5 +i j,+{, @ i+ ss{ 4, {5 4 . 4 5 @
FKDSWHU 51 IXQFLRQHV
477 whqhprv txh
+i j, k,,+{, @ +i j,+{5 , ss {7 4 . 4 5 @ Sru rwud sduwh hq ho fäofxor gh +i +j k,,+{, whqhprv
+i +j k,,+{, @ is++j k,+{,, @ t+j k,+{, . 4 5 s +k+{,,5 4 . 4 5 @ ss {7 4 . 4 5 @
Dvð/
i +j k, @ +i j, k |d txh wlhqhq ljxdo grplqlr/ uhfruulgr | uhodflöq +iöupxod, txh odv ghqh1 Hq orv gldjudpdv vljxlhqwhv vh ylvxdol}dq orv sdvrv khfkrv hq ho ghvduuroor1
` 4> 4^F $ U. ^4> 4^$ ^5> 4^ U $ U. 3 > 3 >
j
k
Sdud i +j k,
` 4> 4^F $ U. ^4> 4^$ ^5> 4^ 3 >
jk
i
i +jk,
| sdud +i j, k
` 4> 4^F $ ^5> 4^
` 4> 4^F $ ^5> 4^ U $ U. 3 >
k
i j
i j,k,
` 4> 4^F $ +
Hmhuflflr 44< Vhdq
i
i+{, @
Ghwhuplqh i j | j i 1
^5> 4^
5 s { > {A3 { 4> { A 4 j+{, @ 5{ . 4> { 3 {4 {6 >
Vroxflöq= Hq sulphu oxjdu ghwhuplqhprv +j i,+{,1 s j+ { 4,> { A 4 +j i ,+{, @ j+i+{,, @ j+{6 ,> {4 s 6 Shur od ixqflöq j wlhqh glvwlqwrv ydoruhv ghshqglhqgr vl { 4 | { vrq pd|ruhv r phqruhv txh fhur1 Frpr s { 4 A 3/ sdud wrgr { A 4 vh whqguä txh s s j+ { 4, @ + { 4,5 > vl { A 4 Shur {6 wlhqh grv frpsruwdplhqwrv hq ho lqwhuydor ` 4> 4`/ d vdehu/ {6 A 3 vl { 5`3> 4` | {6 3 vl { 5` 4> 3`1 Dvð/ ; s ? + { 4,5 > { A 4 { 5`3> 4` +j i,+{, @ +{6 ,5 > = 5+{6 , . 4> { 3 Ilqdophqwh fdofxohprv +i j,+{, {A4 i +{5 ,> +i j,+{, @ i +j+{,, @ i +5{ . 4,> { 4
Shur od ixqflöq i wlhqh wdpelìq glvwlqwrv ydoruhv ghshqglhqgr vl {5 | 5{ . 4 vrq pd|ruhv r phqruhv txh xqr1 Fodudphqwh {5 A 4 vl { A 4 | 3 ? {5 4 vl 3 ? { 41 Sru rwur odgr 5{ . 4 4 vl { 31 Frq wrgd hvwd lqirupdflöq whqhprv txh ; 5 6 3?{4 ? +{ s , > +i j,+{, @ {5 4> { A 4 = +5{ . 4,6 > { 3
5181 DOJHEUD GH IXQFLRQHV
51815
478
Ixqflrqhv Fuhflhqwhv | Ghfuhflhqwhv
D frqwlqxdflöq gduhprv xqdv ghqlflrqhv txh qrv vhuäq ýwlohv sdud uhvroyhu lqhfxdflrqhv/ |d txh hoodv wlhqhq txh yhu frq or txh vxfhgh do dsolfdu xqd ixqflöq d dperv odgrv gh xqd lqhfxdflöq/ sru hmhpsor hohydu do fxdgudgr1 Ghqlflöq 518 Xqd ixqflöq i = D $ E vh glfh fuhflhqwh hq D/ vl hv yhugdghud od vljxlhqwh lpsolfdflöq= +;x> y 5 D,+x ? y , i+x, i +y,,>
ghfuhflhqwh hq D=
+;x> y 5 D,+x ? y , i+x, i +y,,>
hvwulfwdphqwh fuhflhqwh hq D= hvwulfwdphqwh ghfuhflhqwh hq D=
+;x> y 5 D,+x ? y , i+x, ? i +y,,> +;x> y 5 D,+x ? y , i+x, A i +y,,=
Hv ghflu/ xqd ixqflöq i vhuä hvwulfwdphqwh ghfuhflhqwh vl lqylhuwh ho rughq/ hvwulfwdphqwh fuhflhqwh vl frqvhuyd ho rughq1
Hmhpsor 453 Od ixqflöq i +{, @ p{ . e hv hvwulfwdphqwh ghfuhflhqwh vl p hv qhjdwlyr1 Hq hihfwr/ vhdq x> y 5 U/ hqwrqfhv x ? y , px A py , px . e A py . e , i +x, A i+y,
Fodudphqwh/ i hv fuhflhqwh vl p A 31 Vl p @ 3 vh wlhqh od ixqflöq frqvwdqwh i +{, @ e txh hv fuhflhqwh | ghfuhflhqwh d od yh}1 Juäfdphqwh vh lqwhusuhwd frpr vljxh= Xqd ixqflöq vhuä hvwulfwdphqwh fuhflhqwh vl hvwdqgr sdudgr hq fxdotxlhu sxqwr gh od juäfd gh od ixqflöq vh frplhq}d d fdplqdu vreuh od juäfd hq od gluhfflöq srvlwlyd gho hmh R[/ qhfhvduldphqwh vh frplhq}d d vxelu hq ho juäfr1 Hq ho fdvr gh fuhflhqwh vöor vh sxhgh dvhjxudu txh qr edmd1 1
Hmhpsor 454 Od ixqflöq i+{, @ Hq jhqhudo=
s { hv hvwulfwdphqwh fuhflhqwh sxhv sdud wrgr {> | 5 U. s s { ? | , { ? |=
s Hmhpsor 455 Od ixqflöq i+{, @ q { frq q 5 Q hv hvwulfwdphqwh fuhflhqwh1 Hq hihfwr hq od vhfflöq gh qýphurv uhdohv vh sureö sru ho frurodulr 414; txh s s +;{> | 5 U. ,+{ ? | , q { ? q |, Hmhpsor 456 Od ixqflöq i+{, @ ^{` hv fuhflhqwh1 Hq hihfwr +;{> | 5 U,+{ ? | , ^{` ^|`,
shur qr hv hvwulfwdphqwh fuhflhqwh sxhv vl { @ 4> | @ 4> 8
{ ? | shur ^{` @ ^|` @ 4
FKDSWHU 51 IXQFLRQHV
479
Hmhpsor 457 Od ixqflöq i+{, @ {5 hv hvwulfwdphqwh fuhflhqwh hq U. / sxhv vl {> | 5 U. wdo txh { ? | hqwrqfhv {5 ? |5 > shur qr hv hvwulfwdphqwh fuhflhqwh hq U/ sxhvwr txh h{lvwhq { @ 5 5 U/ | @ 4 5 U wdohv txh { ? | shur {5 9? | 5 = Sru rwud sduwh/ qrwhprv txh od ixqflöq i +{, @ {5 hv hvwulfwdphqwh ghfuhflhqwh hq U. / sxhv vl {> | 5 . U wdo txh { ? | hqwrqfhv {5 ? |5 oxhjr {5 A | 5 / hv ghflu/ +;{> | 5 U. ,+{ ? | , i +{, A i+|,,
Sursrvlflöq 519 Vl i | j vrq ixqflrqhv hvwulfwdphqwh fuhflhqwhv wdohv txh Uhfi Grpj hqwrqfhv j i hv xqd ixqflöq hvwulfwdphqwh fuhflhqwh ghqlgd hq Grpi 1 Ghprvwudflöq= Hv fodur txh vl Uhfi Grpj hqwrqfhv Grpj i @ Grpi 1 Vhd {> | 5 Grpj i wdo txh { ? | hqwrqfhv/ gdgr txh i hv hvwulfwdphqwh fuhflhqwh whqhprv i +{, ? i+|,
| sxhvwr txh j wdpelìq hv hvwulfwdphqwh fuhflhqwh hqwrqfhv hq frqvhfxhqfld
j+i +{,, ? j+i +|,, { ? | , +j i,+{, ? +j i ,+|,
Hmhuflflr 458 Suxheh txh vl i hv xqd ixqflöq fuhflhqwh hq G hqwrqfhv i hv xqd ixqflöq ghfuhflhqwh hq G1
Vroxflöq= Vhd {> | 5 G wdo txh { ? | hqwrqfhv i+{, i+|, sxhv i hv fuhflhqwh/ pxowlsolfdqgr sru 4 vh wlhqh i +{, i +|,
hq frqvhfxhqfld
Sru or wdqwr i hv ghfuhflhqwh hq G1
{ ? | , i+{, i+|,
Hmhuflflr 459 Ghwhuplqh vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwltxh1 41 Vhdq i | j ixqflrqhv wdohv txh Grpi _ Grpj 9@ >/ hqwrqfhv
Uhf+i . j, @ Uhfi ^ Uhfj=
51 Vhdq i / j | k ixqflrqhv wdohv txh G @ Grpi @ Grpj @ Grpk 9@ >/ hqwrqfhv +i . j,k @ i k . jk hq G k+i . j, @ ki . kj hq G
61 Vl Uhfk _ Grp+i . j, 9@ > | G @ Grp+i . j, k/ hqwrqfhv
+i . j, k @ i k . j k hq G
71 Vhd G @ Grpj _ Grpk 9@ >1 Vl Uhf+j . k, Grpi / hqwrqfhv
+i +j . k,,+{, @ +i j,+{, . +i k,+{,> ;{ 5 G
81 Vl i | j vrq ixqflrqhv ghfuhflhqwhv hq U hqwrqfhv i j hv ghfuhflhqwh hq U1
5181 DOJHEUD GH IXQFLRQHV
47:
91 Vl i hv fuhflhqwh hq ` 4> d` | hq `d> 4^ hqwrqfhv i hv fuhflhqwh hq U
:1 Vhd i xqd ixqflöq frq grplqlr G h Lg od ixqflöq lghqwlgdg: / hqwrqfhv i Lg @ Lg i @ i
;1 Vhd i ixqflöq/ hqwrqfhv h{lvwh j ixqflöq wdo txh
j i @ Lg> ;{ 5 Grpi
Vroxflöq=
41 Idovr1 Vhdq/ sru hmhpsor/
j = ^4> 4^$ U i = U $ U s { $ 4 { $ {4
ixqflrqhv1 Uhfi @ i4j/ Uhfj @ ^3> 4^1 Oxhjr Sru rwud sduwh/ fx|r uhfruulgr hv ^4> 4^1
Uhfi ^ Uhfj @ ^3> 4^^i4j
i . j = ^4> 4^$ s U {44 { $
51 Yhugdghur1 Vhd { 5 G/ hqwrqfhv/
oxhjr/ gh grqgh Dqäorjdphqwh vl { 5 G>
oxhjr/ gh grqgh
++i . j,k,+{, @ @ @ @ @
+i . j,+{,k+{, +i+{, . j+{,,k+{, i+{,k+{, . j+{,k+{, +ik,+{, . +jk,+{, +ik . jk,+{,
++i . j,k,+{, @ +i k . jk,+{, ;{ 5 G +i . j,k @ i k . jk
+k+i . j,,+{, @ @ @ @ @
k+{,+i . j,+{, k+{,+i+{, . j+{,, k+{,i +{, . k+{,j+{, +ki,+{, . +kj,+{, +ki . kj,+{,
+k+i . j,,+{, @ +ki . kj,+{, ;{ 5 G k+i . j, @ ki . kj hq G
: Hq uhdolgdg ho vðperor Lg hvwä ghqrwdqgr ho frqfhswr gh ixqflöq lghqwlgdg / shur jhqhudophqwh vhuäq ixqflrqhv glvwlqwdv/ ghshqglhqgr gho Grplqlr gh ghqlflöq1 Dvðkd| LgU / Lgdf>o hwf1
FKDSWHU 51 IXQFLRQHV
47; 61 Yhugdghur1 Vhd { 5 G/ hqwrqfhv>
oxhjr/ gh grqgh
++i . j, k,+{, @ @ @ @
+i . j,+k+{,, i +k+{,, . j+k+{,, +i k,+{, . +j k,+{, +i k . j k,+{,
++i . j, k,+{, @ +i k . j k,+{, ;{ 5 G +i . j, k @ i k . j k hq G
71 Idovr1 Frqvlghuh odv vljxlhqwhv ixqflrqhv
i +{, @ { . 4 j+{, @ {5
k+{, @ 5{
hqwrqfhv Grp+j . k, @ Grpj _ Grpk @ U 9@ > | Uhf+j . k, U @ Grpi / dghpäv +i +j . k,,+{, @ @ @ @
i ++j . k,+{,, i +j+{, . k+{,, i +{5 . 5{, {5 . 5{ . 4
Shur +i j,+{, @ i +j+{,, @ {5 . 4 | +i k,+{, @ i+k+{,, @ 5{ . 4/ oxhjr/ sru or wdqwr
+i j,+{, . +i k,+{, @ {5 . 5{ . 5>
81 Idovr1 Odv ixqflrqhv i | j wdo txh
i +j . k, 9@ i j . i k i +{, @ j+{, @ {
vrq ghfuhflhqwhv hq U/ vlq hpedujr ixqflöq fuhflhqwh1
91 Idovr1 Vhd
i j+{, @ i+j+{,, @ i +{, @ {
i+{, @
{ {6
vl { 5` 4> 4` vl { 5`4> 4^
i+{, @ { hv fuhflhqwh hq ` 4> 4` | i +{, @ { 6 hv fuhflhqwh hq `4> 4^1 Vlq hpedujr i qr hv fuhflhqwh ` 4> 4`^`4> 4^@ U sxhv +sru hmhpsor,/ h{lvwh { @ 3 5 U/ | @ 5 5 U wdohv txh { ? | shur 3 @ i +{, 9? i +|, @ 41
:1 Yhugdghur1 Vhd { 5 G |
oxhjr/ ö
+i Lg,+{, @ i +Lg+{,, @ i +{, ;{ 5 G
+Lg i ,+{, @ Lg+i+{,, @ i +{, ;{ 5 G i LgG @ LgUhf i i @ i i Lg @ Lg i @ i
5181 DOJHEUD GH IXQFLRQHV
47<
;1 Idovr1 Vhd i xqd ixqflöq ghqlgd sru
i = U $ U { $ 4
| vxsrqjdprv txh h{lvwh xqd ixqflöq j wdo txh Uhfi Grpj | j i @ Lg/ hqwrqfhv Vl { @ 4 vh wlhqh txh Vl { @ 5/ vh wlhqh txh
+j i,+{, @ {> ;{ 5 Grpi @ U +j i,+4, @ j+i +4,, @ j+4, @ 4
+j i ,+5, @ j+i+5,, @ j+4, @ 5>
hv ghflu/ j+4, wlhqh grv ydoruhv 4 | 5 gh grqgh j qr hv ixqflöq/ or txh frqwudglfh ho vxsxhvwr1
Dqäorjdphqwh/ vl i hv ixqflöq gh D hq E qr qhfhvduldphqwh h{lvwh xqd ixqflöq j wdo txh i j @ LgE > ;{ 5 Grpj=
Vlq hpedujr/ edmr flhuwdv frqglflrqhv hv srvleoh txh hvwr rfxuud/ hv ghflu/ txh sdud xqd ixqflöq i gh D hq E h{lvwd j gh E hq D +oodpdgd od ixqflöq lqyhuvd gh i, wdo txh i j @ LgE
|
j i @ LgD
Yhuhprv päv dghodqwh txh xqd ixqflöq wlhqh lqyhuvd vl | vöor vl hv lq|hfwlyd | hsl|hfwlyd1 Ho frqfhswr gh lq|hfwlylgdg ; lqglfd txh d wrgr hohphqwr gho uhfruulgr gh i oh fruuhvsrqgh xqd ýqlfd suhlpdjhq hq ho grplqlr gh i 1 Ho gh hsl|hfwlylgdg lqglfd txh d wrgr hohphqwr gho frqmxqwr gh oohjdgd oh fruuhvsrqgh sru or phqrv xqd suhlpdjhq/ hq rwudv sdodeudv/ ho frqmxqwr gh lpäjhqhv gh i +uhfruulgr gh i , gheh vhu ljxdo do frqmxqwr gh oohjdgd1 Ghqlflöq 51: Xqd ixqflöq i = D $ E vh glfh lq|hfwlyd vl | vöor vl +;x> y 5 D,+i+x, @ i +y, , x @ y,
Hmhpsor 45: Vhd i od ixqflöq ghqlgd sru
i = U i4j $ U { $ 5{{.4 4
Surehprv txh i hv xqd ixqflöq lq|hfwlyd1
Vroxflöq= Hq hihfwr= Vhdq x> y 5 U i4j wdohv txh i +x, @ i+y,1 Vh wlhqh=
oxhjr/ ; Rwur
i +x, 5x.4 , x4 , +5x . 4,+y 4, , 5xy 5x . y 4 , x
@ @ @ @ @
i +y, 5y .4 y4 +5y . 4,+x 4, 5yx 5y . x 4 y
i +x, @ i +y, , x @ y> ;x> y 5 U i4j
qrpeuh xwlol}dgr hq od olwhudwxud frpr vlqöqlpr gh lq|hfwlyd hv 404 +vh ohh xqr d xqr ,1
FKDSWHU 51 IXQFLRQHV
483 Hmhuflflr 45; Vhd i od ixqflöq ghqlgd sru
Ghwhuplqh vl i hv xqd ixqflöq lq|hfwlyd
i = U $ U { $ {5 . { . 4
Vroxflöq= i qr hv lq|hfwlyd/ hq hihfwr i+4, @ i +5, @ 6/ shur 4 9@ 51 Hv ghflu d | @ 6 5 Uhfi oh fruuhvsrqghq grv suhlpäjhqhv= 4/051 Revhuyhprv od juäfd gh i hq ho sulphu juäfr vljxlhqwh
Qrwhprv txh do wud}du xqd uhfwd krul}rqwdo/ ìvwd sxhgh fruwdu d od fxuyd hq grv sxqwrv +h{fhswr hq | @ 6@7 d txlhq vöor oh fruuhvsrqgh od suhlpdjhq {@0425, Fodudphqwh vh revhuyd txh vl vh uhvwulqjh od ixqflöq do lqwhuydor ^4@5> 4^ ö do lqwhuydor ` 4> 4@5`/ hqwrqfhv vh rewlhqhq ixqflrqhv lq|hfwlydv/ hv ghflu/ i = ^4@5> 4^$ U { $ {5 . { . 4
|
i = ` 4> 4@5^$ U { $ {5 . { . 4
vrq dpedv ixqflrqhv lq|hfwlydv +juäfrv fhqwudo | ghuhfkr gh jxud dqwhulru uhvshfwlydphqwh,1 Ghwhuplqhprv dqdoðwlfdphqwh hq txì lqwhuydor i hv lq|hfwlyd Vhdq x> y 5 U wdo txh i+x, @ i+y,1 vh wlhqh= i +x, @ i+y, +, x5 . x . 4 @ y 5 . y . 4 +, x5 y 5 @ y x +, +x y,+x . y, . +x y, @ 3 +, +x y,+x . y . 4, @ 3 +, x@y b y.x.4@3 Qrwhprv txh x @ y a x . y . 4 @ 3/ vl x @ y @ 4@51 Exvfduhprv xq vxefrqmxqwr G gho grplqlr gh i hq ho fxdo vh fxpsod=
x @ y | x . y . 4 9@ 3 ;x> y 5 G i 45 j
Qrwhprv txh , , , ,
x @ y a x . y . 4 9@ 3 x @ y a +x . y . 4 A 3 b x . y . 4 ? 3, +x @ y a x . y . 4 A 3, b +x @ y a x . y . 4 ? 3, 5x . 4 A 3 b 5x . 4 ? 3 x A 4@5 b x ? 4@5
5181 DOJHEUD GH IXQFLRQHV
484
shur 4@5 shuwhqhfh do grplqlr gh lq|hfwlylgdg1 Oxhjr/ srghprv frqvlghudu frpr grplqlrv gh lq|hfwlylgdg= G @` 4> 4@5`
r elhq
G @ ^4@5> 4^
Dvð/
i mG = ` 4> 4@5` $ U i mG = ^4@5> 4^$ U { $ {5 . { . 4 { $ {5 . { . 4
vrq dpedv ixqflrqhv lq|hfwlydv frpr |d vh ylr juäfdphqwh< Hmhuflflr 45< Ghpxhvwuh txh i = U $ U { $ {6 hv xqd ixqflöq lq|hfwlyd1
Vroxflöq= Vhdq x> y 5 Grpi @ U wdohv txh i+x, @ i +y, vh wlhqh i +x, @ +, x6 @ 6 +, x y6 @ 5 +, +x y,+x . xy . y5 , @ +, x@y b
i +y, y6 3 3 x5 . xy . y5 @ 3
Shur ho glvfulplqdqwh gh od hfxdflöq gh vhjxqgr judgr x5 . xy . y5 hq od yduldeoh x/ hv @ y5 7y5 @ 6y 5 ? 3> ;y 5 U i3j
oxhjr/
x5 . xy . y5 A 3> ;x> y 5 U i3j
Sru or wdqwr x @ y1 Rwud irupd gh yhu txh x5 . xy . y5 9@ 3 hv
4 6 x5 . xy . y5 @ +x . y,5 . y 5 A 3 ;x> y 5 U i3j 5 7
Hmhuflflr 463 Ghwhuplqh vl
j = U $ U { $ j+{, @ {5{.8 hv lq|hfwlyd1 Hq fdvr gh qr vhuor kdfhu odv uhvwulfflrqhv qhfhvduldv sdud txh or vhd1
Vroxflöq= Vhdq x> y 5 Grpj @ U wdohv txh j+x, @ j+y, vh wlhqh vxfhvlydphqwh= j+x, x +, x5 .8 5 +, x+y . 8, +, xy5 . 8x +, xy5 yx5 +, xy+y x, +, xy+y x, 8+y x, +, +y x,+xy 8, +, x@y
@ @ @ @ @ @ @ @ b
j+y, y y5 .8 y+x5 . 8, yx5 . 8y 8y 8x 8+y x, 3 3 xy 8 @ 3
Revhuyhprv txh j qr hv 404 +lq|hfwlyd,/ sxhv j+4, @ j+8, @ 4@9 | 4 9@ 81 <
iG
lqglfd
i
uhvwulqjlgd do grplqlr
G
1
FKDSWHU 51 IXQFLRQHV
485
Dkrud ghwhuplqhprv xq vxefrqmxqwr G gho grplqlr grqgh j vhd lq|hfwlyd1 Qrwhprv txh x@y
vl
x@y@
a xy @ 8
s s 8 b x@y@ 8
Sdud ghwhuplqdu ho grplqlr gh lq|hfwlylgdg vh gheh yhulfdu x @ y a xy 9@ 8=
x @ y a xy 9@ 8 +, +, +, +, +, +,
Dkrud elhq=
x @ y a +xy A 8 b xy ? 8, +x @ y a xy A 8, b +x @ y a xy ? 8, x5 A 8sb x5 ? 8 s m x mAs 8 b m x m?s 8 s s +x A 8 b s x ? s 8, b + 8 ? s x asx ? 8, x 5` 4> 8^^` 8> 4^ b x 5` 8> 8^
s s Shur 8 | 8 shuwhqhfhq do grplqlr gh lq|hfwlylgdg/ oxhjr/ s s j4 = ^ 8> 8` $ U { $ j+{, @ {5{.8 | s s j5 =` 4> 8` ^ ^ 8> 4^ $ U { $ j+{, @ {5{.8 vrq dpedv ixqflrqhv lq|hfwlydv1
Revhuyhprv ho juäfr gh j hq
Hmhuflflr 464 Vhdq i = D $ E | j = E $ F ixqflrqhv | k @ j i1 Ghpxhvwuh txh vl i | j vrq lq|hfwlydv/ hqwrqfhv k wdpelìq hv lq|hfwlyd1 Vroxflöq= Vhdq x> y 5 Grpk wdohv txh k+x, @ k+y,1 Ghehprv ghprvwudu txh x @ y k+x, @ k+y, +, +j i,+x, @ +j i,+y, +, j+i+x,, @ j+i +y,,
Frpr j hv 404 vh wlhqh hqwrqfhv txh i+x, @ i +y,/ | vlhqgr i lq|hfwlyd/ uhvxowd txh x @ y or txh ghpxhvwud od sursrvlflöq
Hmhuflflr 465 Vhdq i = D $ E | j = E $ F ixqflrqhv | k @ j i 1 Ghpxhvwuh txh vl k hv lq|hfwlyd/ hqwrqfhv i wdpelìq or hv1 Vroxflöq= Vhdq x> y 5 Grpi @ D wdohv txh i +x, @ i +y,1 Ghehprv qxhydphqwh ghprvwudu txh x @ y1 Frpr i+x, @ i +y, | j hv ixqflöq whqguhprv txh j+i+x,, @ j+i +y,, gh grqgh k+x, @ k+y,/ shur frpr k hv 404 vh wlhqh txh x @ y1
Sursrvlflöq 51; Vl i hv xqd ixqflöq hvwulfwdphqwh fuhflhqwh r elhq hvwulfwdphqwh ghfuhflhqwh/ hqwrqfhv i hv lq|hfwlyd1
5181 DOJHEUD GH IXQFLRQHV
486
Ghprvwudflöq= Vlq sìuglgd gh jhqhudolgdg +v1s1j1,/ vxsrqjdprv txh i hv xqd ixqflöq hvwulfwdphqwh fuhflhqwh/ hv ghflu/ +;x> y 5 Grpi ,+x ? y , i +x, ? i +y,, +51 y 5 Grpi wdohv txh i +x, @ i+y, a x 9@ y
gh grqgh
xAybx?y
or txh frqwudglfh 51 hvwd ixqflöq hv lq|hfwlyd shur qr hv hvwulfwdphqwh fuhflhqwh ql hvwulfwdphqwh ghfuhflhqwh Ghqlflöq 51< Xqd ixqflöq i = D $ E vh glfh hsl|hfwlyd r vreuh|hfwlyd vl | vöor vl Uhfi @ E1 Hmhpsor 466 Frqvlghuhprv od ixqflöq i ghqlgd sru
$ U i = U. 3 s { $ i +{, @ {
Hvwd ixqflöq qr hv hsl|hfwlyd/ sxhv Uhfi @ U. 9@ U1 Vlq hpedujr/ vl vh uhvwulqjh ho frqmxqwr gh oohjdgd d 3 / hqwrqfhv U. 3 $ U. i = U. 3 3 s { $ i +{, @ { vðhv xqd ixqflöq hsl|hfwlyd1
Revhuyh txh xqd ixqflöq i = D $ Uhfi hv vlhpsuh hsl|hfwlyd1
Hmhpsor 467 Frqvlghuhprv od ixqflöq i ghqlgd sru
i = ` 4> 4^$ U { $ i+{, @
s {5 4
4
Uhvwulqmdprv grplqlr | uhfruulgr gh prgr txh i vhd lq|hfwlyd | hsl|hfwlyd1
Vroxflöq= Vhdq x> y 5 Grpi @` 4> 4^ wdohv txh i +x, @ i +y, vh wlhqh= i +x, @ i +y, s44 x5 @ s44y5 , s s +, 4 y5 @ 4 x5 +, +y x,+x . y, @ 3 +, x@y b x.y @3
Qrwhprv txh vl x @ 3 hqwrqfhv vh yhulfd txh x @ y a y . x @ 31 Hv fodur txh orv lqwhuydorv gh lq|hfwlylgdg gh i vrq ^3> 4^ | ` 4> 3`/ hv ghflu/ odv ixqflrqhv i4 = ` 4> 3` $ U { $ i+{, @
vrq dpedv ixqflrqhv lq|hfwlydv1
s {5 4
4
i5 = ^3> 4^$ U { $ i+{, @
Hmhpsor 468 Frqvlghuhprv i5 | ghwhuplqhprv vx uhfruulgr1
s {5 4
4
FKDSWHU 51 IXQFLRQHV
487 Vhd { 5 ^3> 4^ h | 5 U wdo txh | @ i +{,/ vh wlhqh=
dkrud elhq/ frpr { 5 ^3> 4^/ hqwrqfhv
shur | A 3/ oxhjr/ | 41 Dvð/
+, +, +, +, +, +,
| +, | +, | 5 +, {5
@ i +{, 4 @ s4 {5 4 @ 4{5 5 4 @ | | 5
|A3
t 5 | 4 ? 4 |5 5 | 4 3 |5 ? 4 3 | 5 4 ? |5 3 | 5 4 ? |5 3 | 5 4 a |5 4 ? | 5 4 | 5 a 4 ? 3 |5 4
frq | 5 4 3
3
i = ^3> 4^$ ^4> 4^ { $ i +{, @
hv xqd ixqflöq lq|hfwlyd | vreuh|hfwlyd
s {5 4
4
Ghqlflöq 5143 Xqd ixqflöq i = D $ E vh glfh el|hfwlyd vl | vöor vl hv lq|hfwlyd | vreuh|hfwlyd1
Hmhpsor 469 Odv ixqflrqhv vljxlhqwhv vrq el|hfwlydv
i = U. $ U 3 3 s { $ i +{, @ { $ U. j = U 3 3 { $ j+{, @ {5 k = U $ U { $ k+{, @ {6
Qrwh txh/ sdud i | j/ vh fxpsoh=
|
5 +i j,+{, @ i+j+{,, @ i +{ s, @ {5 @ m{m @ +{, @ {
hv ghflu/ Dqäorjdphqwh sdud k | o gh grqgh
o = U $ U s { $ o+{, @ 6 { ;{ 5 U 3
s s +j i,+{, @ j+i +{,, @ j+ {, @ + {,5 @ { ;{ 5 U. 3 i j @ LgU 3
a j i @ LgU. 3
+k o,+{, @ { > +o k,+{, @ { ;{ 5 U k o @ o k @ LgU
Odv ixqflrqhv txh fxpsohq frq hvwd surslhgdg vh glfhq lqyhuwleohv/ vhjýq vh h{solfd d frqwlqxdflöq1
5181 DOJHEUD GH IXQFLRQHV
488
Surslhgdg 5144 Vl i = D $ E hv el|hfwlyd/ hqwrqfhv h{lvwh xqd ýqlfd i 4 gh E hq D wdo txh d, i i 4 @ LgE | i 4 i @ LgD
e, +i 4 ,4 @ i
Od ixqflöq i 4 gh od sursrvlflöq dqwhulru hv ýqlfd | sxhgh ghqluvh gluhfwdphqwh frpr vljxh=
Ghqlflöq 5145 Vl i = D $ E hv xqd ixqflöq el|hfwlyd/ hqwrqfhv od ixqflöq lqyhuvd gh i vh ghqh frpr i 4 = E $ D | $ { @ i 4 +|,
grqgh
| @ i +{,
r
{ @ i 4 +|,
Hq ho hmhpsor dqwhulru od lqyhuvd gh i hv j | od lqyhuvd gh k hv o1 Uhfðsurfdphqwh od lqyhuvd gh j hv i | od
lqyhuvd gh o hv k1 Revhuyhprv odv juäfdv gh i | j 43 hq od jxud Sdud rewhqhu i 4 gh xqd ixqflöq i el|hfwlyd ghehprv ghvshmdu { gh od hfxdflöq | @ i +{,/ frpr vh h{solfd hq orv vljxlhqwhv hmhpsorv1 Hmhpsor 46: Hq ho hmhpsor 467 ylprv txh od ixqflöq
hv el|hfwlyd | rewxylprv txh> oxhjr/
i = ^3> 4^$ ^4> 4^ { $ i+{, @
4
4
| @ it+{, 5 4 +, { @ | | 5
{ @ i 4 +|, @ Sru or wdqwr
s {5
v
|5 4 |5
i 4 = ^4> 4^$ ^3> 4^ | $
i 4 +|, @
t 5 | 4 |5
43 Gdgd od juäfd gh i hq xqd krmd wudqvsdudqwh/ sdud frqrfhu od juäfd gh vx lqyhuvd edvwd urwdu orv hmhv hq eD frqwud uhorm | gdu yxhowd od krmd | pludu sru ghwuäv od jxud> hvwd ýowlpd fruuhvsrqghuä d od juäfd gh od lqyhuvd gh i 1
FKDSWHU 51 IXQFLRQHV
489
Hv frvwxpeuh uhvhuydu od ohwud { sdud orv hohphqwrv gho grplqlr gh xqd ixqflöq/ oxhjr/ or dqwhulru vh hvfuleh i 4 = ^4> 4^$ ^3> 4^
Hmhuflflr 46; Vhdq
ixqflrqhv1
{ $ i 4 +{, @
$ ^4> 4^s i = U. 3 { $ i +{, @ { 4
t
{5 4 {5
j = U $ U { $ j+{, @ { . 8
d, Suxheh txh i hv el|hfwlyd | ghwhuplqh i 4 1
e, Hqfxhqwuh xqd ixqflöq/ k wdo txh k i @ j1
Vroxflöq= d1l, Sureduhprv txh i hv lq|hfwlyd1 wdo txh i+x, @ i +y, hqwrqfhv= Vhd x> y 5 U. 3 s s x ys 4 s 4 @ x @ y +, +, x @ y
Oxhjr i+x, @ i +y, , x @ y> ;x> y 5 U. 3 ll, Vhd { 5 U. h | 5 ^4> 4^ wdo txh | @ i+{, hqwrqfhv 3 s | @ s{ 4 +, |.4 @ { > |.43 +, +| . 4,5 @ {
od h{suhvlöq +| . 4,5 hvwä ghqlgd sdud wrgr | hq U/ shur | . 4 3/ Sru or wdqwr i hv hsl|hfwlyd1 Dvðsru +l, | +ll, i hv el|hfwlyd1
Uhfi @ ^4> 4^
i 4 = ^4> 4^$ U. 3 { $ i 4 +{, @ +{ . 4,5
e, Sxhvwr txh Uhfi 4 @ U. Grpj hqwrqfhv h{lvwh 3 od ljxdogdg k i @ j/ hv ghflu/ ki +, k +i i 4 , +, k Lg +, k
j i 4 1 Oxhjr/ sdud ghvshmdu k dsolfdprv i 4 d @ @ @ @
Sru or wdqwr Grpk @ Grpj i 4 @ Grpi 4 @ ^4> 4^ |
j j i 4 j i 4 j i 4 =
k+{, @ j+i 4 +{,, @ j++{ . 4,5 , @ +{ . 4,5 . 8 k = ^4> 4^$ U { $ +{ . 4,5 . 8
Hmhuflflr 46< Ghwhuplqh vl odv vljxlhqwhv dupdflrqhv vrq Yhugdghudv r Idovdv
41 Vl j hv lqyhuwleoh/ hqwrqfhv j 4 @ 4j +grqgh 4 lqglfd od ixqflöq frqvwdqwh 4,1
5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV 51 Vl j = D $ E hv lqyhuwleoh/ hqwrqfhv
48:
j j4 @ j 4 j
61 Vl i = D $ E | j = E $ F vrq ixqflrqhv lqyhuwleohv/ hqwrqfhv j i hv wdpelìq lqyhuwleoh | Vroxflöq=
+j i ,4 @ i 4 j4
41 Idovr1 Od ixqflöq j ghqlgd sru hv lqyhuwleoh | vlq hpedujr/ oxhjr 51 Idovr1 Frqvlghuhprv od el|hfflöq
| vx lqyhuvd
hq fdpelr
. $ U j = U. 3 s3 { $ {
$ U. j 4 = U. 3 3 { $ {5
4 4 j +{, @ j+{, @
s{ >
j 4 9@ j=
4
4 j
;{ 5 U.
^4> 4^$ s U. 3 { $ {4
$ ^4> 4^ j 4 = U. 3 { $ {5 . 4
+j j4 ,+{, @ { ;{ 5 U. 3
+j 4 j,+{, @ { ;{ 5 ^4> 4^
oxhjr/ j j 4 9@ j4 j srutxh wlhqhq glvwlqwrv grplqlrv1
61 Yhugdghud1 Vx frpsuredflöq txhgd gh hmhuflflr sdud ho ohfwru1
519
Odv ixqflrqhv H{srqhqfldohv | Orjduðwplfdv
Frqvlghuhprv ho vljxlhqwh sureohpd= Vl xqd edfwhuld hq flhuwr fxowlyr vh gxsolfd hq fdgd krud | ho qýphur gh edfwhuldv txh kdeuðd do lqlfldu ho h{shulphqwr hv xqd/ hqwrqfhv ho qýphur i+q, gh edfwhuldv txh kd| hq ho fxowlyr ghvsxìv gh q krudv hv i +q, @ 5q
Qrwh txh od h{suhvlöq dqwhulru hv xqd ixqflöq fx|r grplqlr hv Q/ hv ghflu i = Q $ U q $ 5q
Txhuhprv h{whqghu od ixqflöq i d xqd ixqflöq päv jhqhudo/ frq grplqlr U1 Shur/ Bctxì vhqwlgr wlhqh h{whqghu d od h{suhvlöq 5{ frq { 5 UB
FKDSWHU 51 IXQFLRQHV
48; Vdehprv txh vl { 5 Q/ hqwrqfhv Vl { 5 ] / { 5 ]. / hqwrqfhv Vl { @ p q 5 T/ hqwrqfhv
5{ @ 5 5 5 +{ yhfhv, 4 5{ @ { 5
s q 5{ @ 5p
s s Shur vl { 95 T/ hv ghflu/ { hv xq luudflrqdo/ sru hmhpsor/ { @ 5/ Bctxì vljqlfdgr wlhqh 5 5 B s Frqvlghuhprv odv dsur{lpdflrqhv ghflpdohv vxfhvlydv h lqihulruhv d 5 | iruphprv ho frqmxqwr D @ i54>7 > 54>74 > 54>747 > 54>7475 > = = =j
Sxhvwr txh D hv xq s5frqmxqwr dfrwdgr/ hqwrqfhv sru d{lrpd gho vxsuhpr h{lvwh VxsD1 Vhd VxsD @= 5 1 Hq jhqhudo vl e A 3 | { 5 U/ vh ghqh |
e{ =@ Vxsie{3 >{4 > e{3 >{4 {5 > e{3 >{4 {5 {6 > = = =j
e{ =@ Lqiie{3 >{4 > e{3 >{4 {5 > e{3 >{4 {5 {6 > = = =j
vl e A 4
vl 3 e 4
grqgh {3 > {4 > {3 > {4 {5 > {3 > {4 {5 {6 > vrq odv dsur{lpdflrqhv ghflpdohv vxfhvlydv h lqihulruhv d { 5 U1 Oxhjr srghprv ghqlu od ixqflöq= i = U $ U { $ 5{
Od uhsuhvhqwdflöq juäfd gh hvwd ixqflöq hv xqd fxuyd gh wud}dgr frqwlqxr/ frpr vh yh hq ho judfr ghuhfkr vljxlhqwh
Dqäorjdphqwh/ j ghqlgd frpr
j = U $ U { $ + 45 ,{
vh uhsuhvhqwd juäfdphqwh sru od fxuyd gh wud}dgr frqwlqxr/ frpr vh sxhgh yhu hq ho juäfr ghuhfkr dqwhulru/ fx|r uhfruulgr hv U. 1 Hq jhqhudo srghprv ghqlu od ixqflöq h{srqhqfldo hq edvh e frq e A 3 frpr
5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV
48<
h{se = U $ U { $ e{
fx|d uhsuhvhqwdflöq juäfd hvwä hq od jxud
Qrwh txh vl e A 4 od ixqflöq h{se hv hvwulfwdphqwh fuhflhqwh | vl 3 ? e ? 4/ od ixqflöq h{se hv hvwulfwdphqwh ghfuhflhqwh1 Hq dperv fdvrv hv lq|hfwlyd | vx uhfruulgr hv U. 1 Revhuyhprv txh od fxuyd qr fruwd ho hmh {/ hv ghflu/ qr fruwd d od uhfwd | @ 31 Gluhprv txh od uhfwd | @ 3 hv xqd dvðqwrwd krul}rqwdo gh od ixqflöq | @ e{ 1 Hvwr vljqlfd txh vl e A 4 | { ghfuhfh lqghqlgdphqwh/ odv lpäjhqhv | @ e{ wlhqghq d fhur1 Hq hvwh fdvr hvfuleluhprv>
Vl { $ 4/ hqwrqfhv e{ $ 3
Hq fdpelr vl 3 ? e ? 4 | { fuhfh lqghqlgdphqwh/ hqwrqfhv odv lpäjhqhv | @ e{ wlhqghq d fhur1 Hq hvwh fdvr hvfuleluhprv
Hmhuflflr 473 Judtxh od ixqflöq
Vl { $ 4/ hqwrqfhv e{ $ 3 | @ 4 h{
Vroxflöq= Revhuyhprv od juäfd gh | @ h{ | gh | @ h{
FKDSWHU 51 IXQFLRQHV
493
Hq hvwh fdvr e @ 4h ? 41
Dvð/ | @ 4h{ hv xq wudvodflöq gh | @ h{ frpr vh pxhvwud hq od jxud
Qrwh txh od dvðqwrwd krul}rqwdo gh | @ 4 h{ hv dkrud | @ 4/ sxhv vl { $ 4 hqwrqfhv 4 h{ $ 4
1
5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV
494
Hmhpsor 474 Ghwhuplqh od juäfd gh od ixqflöq | @ de{ . f sdud orv glihuhqwhv ydoruhv srvleohv gh d/ e | f hq U
Vroxflöq= Frqvlghuduhprv glvwlqwrv fdvrv txh vh pxhvwudq hq odv juäfdv 10 -4
10 2 x
-2
4
-4
2 x
-2
0
0
-10
-10
-20
-20
-30
-30
d ? 3> e A 4> f A 3
d ? 3> e A 4> f ? 3
4
FKDSWHU 51 IXQFLRQHV
495
Surslhgdg 5146 Vl d 5 U. / e 5 U. | {> | 5 U/ hqwrqfhv l, d{ d| @ d{.| ll, d{ e{ @ +de,{ lll, +d{ ,| @ d{| { { ly, de @ de{
Hmhpsor 475 Uhvroydprv odv vljxlhqwhv hfxdflrqhv h{srqhqfldohv 41 8{ 6{4 @ 6{.4 8{4 51 5{.4 . 7{ @ ;3
Vroxflöq= 41
8{ 6{4 @ 6{.4 8{4
oxhjr/ V @ i5j1
+, +, +, +,
{4 8{ . 8{4 @ 6{.4 . 64 4 { { 8 4. 8 @6 6. 6 8{ 9 @ 6{ 43 8{8 865 @ 65 6{ 8 { 8 5 @ 6 +, 6 +, { @ 5
51 Frqvlghuhprv ho fdpelr gh yduldeohv x @ 5{ / hqwrqfhv 5{.4 . 7{ @ ;3 +, 5{.4 . 7{ ;3 @ 3 +, 5 5{ . +5{ ,5 ;3 @ 3 +, x5 . 5x ;3 @ 3 +, +x . 43,+x ;, @ 3 +, x @ 43 b x @ ;
5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV
496
Iljxuh 514= | @ vlqk+{, / | @ frvk+{, oxhjr/ V4 @ i43> ;j/ shur x A 3/ oxhjr/ V @ i;j/ oxhjr/ gh grqgh { @ 6
5{ @ ;
Od edvh h{srqhqfldo gh xvr päv iuhfxhqwh hv h1 Sru vx xvr wdq h{whqvr/ d od ixqflöq i +{, @ h{ vh oh oodpduä ixqflöq h{srqhqfldo1 Hq edvh d hvwd ixqflöq srghprv ghqlu od vljxlhqwhv ixqflrqhv= Ghqlflöq 5147 Vh ghqh od ixqflöq vhqr klshueöolfr frpr>
| frvhqr klshueöolfr frpr>
vhqk = U $ U { { { $ h 5h frvk = U $ U { { { $ h .5h
Odv juäfdv fruuhvsrqglhqwhv vrq odv txh vh pxhvwudq hq od jxud Khprv ylvwr txh vl e 9@ 4 od ixqflöq h{se hv xqd ixqflöq lq|hfwlyd frq uhfruulgr U. / oxhjr/ od ixqflöq h{se = U $ U. { $ e{
4 hv el|hfwlyd1 Sru or wdqwr h{lvwh vx lqyhuvd h{s e txh ghqrplqduhprv od ixqflöq orjdulwpr hq edvh e | hvfuleluhprv 4 h{s e @ orje hv ghflu/ od ixqflöq orjdulwpr hq edvh e vh ghqh frpr
grqgh
orje = U. $ U { $ orje +{, @ | | @ orje +{, +, { @ e|
FKDSWHU 51 IXQFLRQHV
497 Hmhpsor 476
41 orj5 +7, @ 5 sxhv 55 @ 71
51 orj5 + 47 , @ 5 sxhv 55 @ 47 1
61 orj 45 +7, @ 5 sxhv + 45 ,5 @ 71
Od uhsuhvhqwdflöq juäfd gh od ixqflöq orjdulwpr hv od txh pxhvwud od jxud Revhuydflöq= 41 Vl e A 4/ hqwrqfhv orje hv xqd ixqflöq hvwulfwdphqwh fuhflhqwh1
51 Vl 3 ? e ? 4/ hqwrqfhv orje hv xqd ixqflöq hvwulfwdphqwh ghfuhflhqwh1 Dghpäv gh od juäfd hv fodur txh 61 Vl e A 4 | { A 4/ hqwrqfhv orje +{, A 31
71 Vl e A 4 | 3 ? { ? 4/ hqwrqfhv orje +{, ? 31
81 Vl 3 ? e ? 4 | 3 ? { ? 4/ hqwrqfhv orje +{, A 31 91 Vl 3 ? e ? 4 | { A 4/ hqwrqfhv orje +{, ? 3
Hvwd hv xqd exhqd lqirupdflöq sdud uhvroyhu dojxqdv lqhfxdflrqhv | judfdu ixqflrqhv uhodflrqdgdv frq ixqflrqhv orjduðwplfdv1 Sru hmhpsor/ ho frqmxqwr vroxflöq gh od lqhfxdflöq s orj 45 +{, orj5 +{, . { 4 +5143, hv ho lqwhuydor ^4> 4^1 Hq hihfwr/ orv wìuplqrv hq + 5143, wlhqhq vljqlfdgr hq U vl { A 3 | { 4/ hv ghflu/ { 5 ^4> 4^ | frpr> s orj5 +{, 3 | { 4 3
sdud { 4/ hqwrqfhv Sru rwud sduwh
orj5 +{, .
s { 4 3 ;{ 5 ^4> 4^ orj 45 +{, 3
sdud { 4/ hq frqvhfxhqfld/ od ghvljxdogdg 5143 vh fxpsoh sdud wrgr { 5 ^4> 4^ oxhjr V @ ^4> 4^
5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV
498
Revhuyhprv txh hq { @ 3 od ixqflöq orje +{, qr hvwä ghqlgd1 Gluhprv txh od uhfwd { @ 3 hv xqd dvðqwrwd yhuwlfdo sdud od ixqflöq1 Xqd dvðqwrwd yhuwlfdo { @ {3 qrv lqglfd txh sdud { fhufdqrv d {3 odv lpäjhqhv i+{, fuhfhq r ghfuhfhq lqghqlgdphqwh1 Hq dojxqrv fdvrv/ sdud judfdu dsur{lpdgdphqwh od fxuyd txh uhsuhvhqwd d xqd ixqflöq/ hv vxflhqwh ghwhuplqdu ho grplqlr/ odv dvðqwrwdv/ odv lqwhuvhfflrqhv frq orv hmhv gh frrughqdgdv/ | hq txì lqwhuydorv od ixqflöq hv srvlwlyd +qhjdwlyd,1 Sru hmhpsor Judtxhprv od ixqflöq | @ i+{,/ grqgh Vroxflöq= Ghwhuplqhprv ho grplqlr
Grpi
Oxhjr
| @ orj 56 mm { . 5 m 8 m @ i{ 5 U =mm { . 5 m 8 mA 3j @ i{ 5 U =mm { . 5 m 8 m9@ 3j @ i{ 5 U = { 9@ 6 a { 9@ :j Grpi @ U i:> 6j
Odv dvðqwrwdv yhuwlfdohv gh od ixqflöq sdvdq sru dtxhoorv sxqwrv grqgh ho dujxphqwr gho orjdulwpr hv fhur/ oxhjr/ odv dvðqwrwdv vrq odv uhfwdv { @ 6> { @ :
Dkrud ghwhuplqhprv odv lqwhuvhfflrqhv frq orv hmhv frrughqdgrv Vl | @ 3/ hqwrqfhv orj 56 mm { . 5 m 8 m@ 3
gh grqgh
or txh htxlydoh d
mm { . 5 m 8 m@ 4
m { . 5 m 8 @ 4 +, m { . 5 m@ 9 +, { . 5 @ 9 b { . 5 @ 9 +, { @ 7 b { @ ;
b m { . 5 m 8 @ 4 b m { . 5 m@ 7 b { . 5 @ 7 b { . 5 @ 7 b { @ 5 b { @ 9
oxhjr/ od fxuyd sdvd sru orv sxqwrv S4 +7> 3,/ S5 +;> 3,/ S6 +5> 3,/ S7 +9> 3,1 Vl { @ 3/ hqwrqfhv | @ orj 56 m 6 m@ orj 56 6 oxhjr/ od fxuyd sdvd sru S8 +3> orj 56 6,1 Qrv idowd ghwhuplqdu orv lqwhuydorv grqgh od ixqflöq hv srvlwlyd | grqgh hv qhjdwlyd1 Odv lpäjhqhv gh od ixqflöq vhuäq srvlwlydv vl or txh hv htxlydohqwh d
orj 56 mm { . 5 m 8 mA 3 3 ?mm { . 5 m 8 m? 4 +, 7 ?m { . 5 m a m { . 5 m? 9 a { 9@ 6 a { 9@ : +, { 5 +` ;> 9^^`5> 7^, i:> 6j
Odv lpäjhqhv gh od ixqflöq vhuäq qhjdwlydv vl
orj 56 mm { . 5 m 8 m? 3 +, mm { . 5 m 8 mA 4
+, { 5` 9> 5^^`7> 4^^` 4> ;^
Hq edvh d od lqirupdflöq rewhqlgd vh wlhqh od vljxlhqwh juäfd gh od jxud
FKDSWHU 51 IXQFLRQHV
499
| @ orj 56 mm { . 5 m 8 m Gdgr txh orje hv od ixqflöq lqyhuvd gh h{se hqwrqfhv od frpsrvlflöq uhvxowd vhu od ixqflöq lghqwlgdg= hv ghflu/
orje h{se @ LgU
a h{se orje @ LgU.
{ @ orje +h{se +{,, @ orje +e{ , ;{ 5 U { @ h{se +orje +{,, @ eorje { ;{ 5 U.
oxhjr vh wlhqhq odv vljxlhqwhv surslhgdghv ixqgdphqwdohv=
Surslhgdg 5148 Vhd e xq qýphur uhdo srvlwlyr glvwlqwr gh 4 hqwrqfhv orje e{ @ { eorje { @ {
;{ 5 U
;{ 5 U.
Hmhpsor 477 Ghwhuplqh sdud txì ydoruhv gh d 5 U od lqhfxdflöq wlhqh frpr frqmxqwr vroxflöq d U1 Vroxflöq=
8{.5.{5 5 4 6 4^ { $ h{ 4
51 Ho uhfruulgr gh
i =` 4> 9` $ U { $ orj4@5 +{ . 5, orj5 +4@49,
hv 11111111111111 s 61 Vl i+{, @ { 5 | j+{, @ orj5 +< m{m, . 4 vrq ixqflrqhv hqwrqfhv h{lvwh i j frq grplqlr 1111111111 | +i j,+{, @ 1111111111
hv 111111111111 71 Od ixqflöq lqyhuvd gh od ixqflöq el|hfwlyd i = U $ ` 5> 4^ { $ +4@5,{4 5
Vroxflöq=
41 Grpi @ i{ 5 U = h{ 4 6j shur
oxhjr { 5` 4> oq 7`1 Dvð/ Grpi @` 4> oq 7`
h{ 4 6 +, h{ 7 +, { oq 7 +, { oq 7
51 Vhd { 5` 4> 9` h | 5 U wdo txh
shur { 5` 4> 9` oxhjr
| @ orj4@5 +{ . 5, orj5 +4@49, +, | @ orj4@5 +{ . 5, . 7 +, | 7 @ orj4@5 +{ . 5, 4 +, + ,+|7, @ { . 5 5 4 +, { @ + ,+|7, 5 5
hq frqvhfxhqfld Uhfi @ ^4> 7^1
4 4 ? + ,+|7, 5 9 @ . 5 5 4 +, 4 ? + ,+|7, ; @ orj4@5 5 +, 3 A | 7 6 +, 7 A | 4
FKDSWHU 51 IXQFLRQHV
49; 61 Grpj
@ @ @ Grpi @ Grp+i j, @ @
shur
i{ 5 U = < m{m A 3j i{ 5 U = m{m ? 4^ $ U { $ orj4@5 +{ . 5, . 4
Uhvxhoyd od lqhfxdflöq
Vroxflöq= Uhvwulqjlhqgr whqhprv
orj 46 +orj8 +{5 7,, A 5
{5 7 A 3 a orj8 +{5 7, A 3 +, {5 A 7 a {5 7sA 4 +, m{m A 5 a m{m A 8
s s Oxhjr { 5` 4> 8^^` 8> 4^/ dvð/
s s U @` 4> 8^^` 8> 4^
Dkrud uhvroyhprv od lqhfxdflöq frq { 5 U
shur { 5 U1
+, +, +, +,
orj 46 +orj8 +{5 7,, A 5 5 orj8 +{5 7, ? 46 {5 7s ? 8< m {sm? 8< . 7 s 8< . 7 ? { ? 8< . 7
5191 ODV IXQFLRQHV H[SRQHQFLDOHV \ ORJDUÐWPLFDV
49<
Hmhuflflr 479 Oxhjr ho frqmxqwr vroxflöq hv s s s s s s V @` 8< . 7> 8< . 7^_U @` 8< . 7> 8^^` 8> 8< . 7^ Hmhuflflr 47: Uhvxhoyd od hfxdflöq
Vroxflöq= Od lqhfxdflöq htxlydoh d
{ +{4, 5: orj 7 7 @ < ; orj ; 5 5{ 5 6+{4, 65 5{ 65 6+4{,
@
+, @ 6 6 +, 5{ . 6+4 {, @ 4 +, { @ 5
5 orj 5 6 orj 5
5 6
Surslhgdg 5149 Vhdq e 5 U. i4j/ {> | 5 U. / u 5 U/ hqwrqfhv= l, orje +{|, @ orje { . orje | l, orje {| @ orje { orje | l, orje {u @ u orje {
Ghprvwudflöq= Vhd x @ orje {/ y @ orje |/ hqwrqfhv
ex @ {> ey @ |
dvð/ oxhjr/ Sru rwud sduwh/ oxhjr/ Ilqdophqwh/ sru or wdqwr/
orje +{|, @ orje +ex ey , @ orje ex.y @ x . y orje +{|, @ orje { . orje |
{ @ orje ex ey @ orje exy @ x y orje | { orje @ orje { orje | |
orje {u @ orje +ex ,u @ orje eux @ ux orje {u @ u orje {
or txh suxhed qdophqwh od surslhgdg1 Uhvxhoyd od hfxdflöq orj 5 . orj+7{5 . sru or wdqwr { 5` 4> 4^
U6 orj 45 +{5 4, orj 45 +{ . 4, A 3> oxhjr ghehprg uhvroyhu od vljxlhqwh lqhfxdflöq
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Hmhpsor 493 Od vxfhvlöq +q,4 q@4 qr hv dfrwdgd sxhv qr h{lvwh n hq U wdo txh mqm N +Uhfrugdu txh ho frqmxqwr gh orv qýphurv qdwxudohv Q qr hv dfrwdgr,1 Vlq hpedujr hv fodur txh +q, hv hvwulfwdphqwh fuhflhqwh/ sxhv ;q 5 Q > q ? q . 4
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q5 4 Hmhpsor 495 +{q ,4 q@4 @ + q.6 ,q@4 hv xqd vxfhvlöq qr dfrwdgd vxshulruphqwh vöor lqihulruphqwh/ {q hv hvwulfwdphqwh fuhflhqwh1
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Sru or wdqwr sru +d, | +e, whqhprv
? m{ 5m ? 4
m{5m
?
6 5
4 6 , m{5m ? 55% , m{ 4m ? 6
mi+{, 5m ? %> vl 3 ? m{ 4m ?
LL, Yhugdghur1 Sru or dqwhulru= Gdgr % @ 47 / h{lvwh @ plqi 46 > 56% j @ plqi 46 > 49 j @ | frpr @
4 433
?
4 9
3 ? m{ 4m ? , m
hqwrqfhv
3 ? m{ 4m ?
LLL, Idovr1 Vl @ 46 whqhprv
{6 5m ? % {5
8 7
4 9
wdo txh
4 {6 4 ? ,m 5m ? % 433 9 {5
3 ? m{ 4m ? +, 3 ? m{ 4m ?
dkrud vl % @ 47 / h{lvwh { @ sxhv
7 6
4 5 7 +, { 5` > ^i4j 6 6 6
6 5` 56 > 76 ^i4j shur m {{ 9 % 5 5m ? 8@7 6 4 @ 9? 4 5 8@7 5 6 7
7141 OÐPLWHV GH IXQFLRQHV
556
.4 @ Hmhpsor 583 Ghprvwudu txh olp{$4@6 57{{ 4
4 8
Vroxflöq= Qhfhvlwdprv exvfdu ho wìuplqr +idfwru, m{ . 4@6m +r htxlydohqwhphqwh ho wìuplqr 7{ . 4 4 5{ 4 8 = \d txh
m6{.4m , hq 6
7{ . 4 4 53{ . 8 5{ . 4 4;{ . 9 9m6{ . 4m @ 8+5{ 4, @ 8m5{ 4m 5{ 4 8 @ 8+5{ 4,
edvwduä dfrwdu vxshulruphqwh
m5{4m / sru xq qýphur srvlwlyr n/ sxhvwr txh vl 4
4 n m5{ 4m
vh whqguðd
7{ . 4 4 9m7{ . 4m 9n 4;n m6{ . 4m 5{ 4 8 @ 8m5{ 4m 8 m6{ . 4m @ 8 6 8% dvð/ edvwduðd wrpdu @ 4; n | vh rewhqguðd m{ . 4@6m ? 986%n a 5{44 n , m{ . 4@6m m5{44m ? ,
m{ m m { m
9 6 .4 85 4
?%
n6 n @ 96
9
%
8
%
8
.4 4 , 57{{ 8 ? % 4
Yhdprv vl hv srvleoh dfrwdu vxshulruphqwh
4
m5{4m
sdud { hq xqd yhflqgdg gh 4@61
m{ . 4@6m ? 4 +, 4 ? { . 4@6 ? 4 +, 7@6 ? { ? 5@6 +, 5+7@6, 4 ? 5{ 4 ? 5+5@6, 4 +, 44@6 ? 5{ 4 ? 4@6 ? 44@6 , m5{ 4m ? 44@6
Dvð/ vl m5{ 4m 9@ 3 vh wlhqh m5{44m A 6@44 qr hv dfrwdgr vxshulruphqwh1 Ho sureohpd hq hvwh fdvr hv txh od yhflqgdg wrpdgd hq wruqr do 4@6 lqfox|h do ydoru { @ 4@5 txlìq dqxod d m5{ 4m1 Od yhflqgdg gheh h{foxlu d 4@5 | fhufdqr d ìo1 Sru hmhpsor od yhflqgdg gdgd sru 4@43 m{ . 4@6m ? 4@43 , 4@43 ? { . 4@6 ? 4@43 , 5;@48 ? 5{ 4 ? 55@48 , 55@48 ? m5{ 4m ? 5;@48 , 48@5; ? m5{44m ? 48@55
oxhjr vl m{ . 4@6m ? 4@43 , m5{44m ? 48 55 4 Gh hvwd irupd/ gdgr % A 3 edvwd wrpdu @ plqi 4;+8%48 , > 43 j |d txh vl ? 55
m{ . 4@6m ? ?
4 4 48 , ? 43 m5{ 4m 55
4 43
vh wlhqh
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
557 | dghpäv m{ . 4@6m ?
, m{ . 4@6m ? ,
m{.4@6m m5{4m
?
%
8 4;+ 48 55 ,
% @
8 48 55 4;+48 55,
4 4 , 75{{ 4 8 ? %
Ho dfrwdplhqwr uhdol}dgr hq hvwh sureohpd frqvlvwlö uhdophqwh hq ghwhuplqdu hq txì frqmxqwr vh hq0 fxhqwud ho uhfruulgr gh od ixqflöq j+{, @ m5{44m sdud ydoruhv fhufdqrv d 4@61 Dqdol}dqgr ho juäfr gh j+{,/ sdud xqd yhflqgdg dsursldgd gho 4@6 vh wlhqh qdwxudophqwh txh j+{, vhuä dfrwdgd> vl od yhflqgdg hv judqgh frpr sdud frqwhqhu d 4@5 qr vhuä dfrwdgd vxshulruphqwh j+{, |d txh hv lqqlwdphqwh judqgh fhufd gh 4@5 +yhu jxud ,
j+{, @
4
m5{4m
Hmhpsor 584 Ghpxhvwuh txh olp 8{+{ 6, @ 43 {$5 Vroxflöq= Qxhvwur remhwlyr hv exvfdu m{ 5m hq m8{+{ 6, . 43m1 Frpr m8{+{ 6, . 43m @ 8m{ 4mm{ 5m>
whqguhprv txh dfrwdu m{ 4m hq xqd yhflqgdg gho sxqwr 51 Yhdprv hvwr juäfdphqwh hq od jxud txh vljxh
7141 OÐPLWHV GH IXQFLRQHV
558
j+{, @ m{ 4m
1 grqgh j+{, @ m{ 4m Qdwxudophqwh txh j+{, hv dfrwdgr/ wrpdqgr xqd yhflqgdg gh 5 gh udglr 4 hv ghflu/ 4 ? { ? 6 vh wlhqh sru ho juäfr txh j+4, ? j+{, ? j+6, +, 3 ? j+{, ? 5
hv ghflu/ vl Y @ i{ 5 U = m{5m ? 4j/ hqwrqfhv vh wlhqh txh= ;{ 5 Y> m{4m ? 5 | m8{+{6,.43m ? 85m{4m1 Gh hvwd irupd/ gdgr % A 3/ h{lvwh ? 43% | ? 4 wdo txh m{ 5m ?
, m{ 4m ? 5 +|d txh ? 4 , { 5 Y ,
m{ 5m ?
, m{ 4m ? 5 a m{ 5m ? 43%
m{ 5m ? Dvð/
Sru or wdqwr
71417
%, , m{ 4m ? 43% +|d txh ? 43
m{ 4mm{ 5m ? 5 43% @ 8% , 8m{ 4mm{ 5m ? % , m8{+{ 6, . 43m ? %
; % A 3> +m{ 5m ? , m8{+{ 6, . 43m ? %,
Whruhpdv Eävlfrv vreuh Oðplwhv
D frqwlqxdflöq yhuhprv xq whruhpd txh hv xwlol}dgr sdud suredu od qr h{lvwhqfld gh oðplwhv/ ho whruhpd gh xqlflgdg Whruhpd 719 +gh xqlflgdg, Ho oðplwh gh xqd ixqflöq +hq xq sxqwr, hv ýqlfr Vhd D U> i = D $ U ixqflöq > d 5 D3 1 Vl h{lvwh ho oðplwh gh i fxdqgr { $ d hqwrqfhv olp{$d i+{, hv ýqlfr1 Hv ghflu/ vl olp i+{, @ O4 | olp i +{, @ O5 hqwrqfhv O4 @ O5 {$d {$d
Ghprvwudflöq= Sru ho devxugr1 Vxsrqjdprv O4 9@ O5 / hqwrqfhv srghprv wrpdu yhflqgdghv glvmxqwdv YO4 | YO5 / hv ghflu YO4 _ YO5 @ !1 Frpr olp{$d i+{, @ O4 / h{lvwh xqd yhflqgdg gh d/ Y4 / wdo txh i +Y4 idj _ D, YO4 / sru rwud sduwh vh wlhqh olp{$d i +{, @ O5 / hqwrqfhv h{lvwh xqd yhflqgdg gh d/ Y5 wdo txh i +Y5 idj _ D, YO5 1
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
559
Hv fodur txh E @ D _ +Y5 idj, _ +Y4 idj, 9@ ! +hv lpsrvleoh txh vhd ydfðd od lqwhuvhfflöq |d txh dpedv vrq yhflqgdghv gh d | d hv xq sxqwr gh dfxpxodflöq gh D i+E, YO4
sru or wdqwr
shur i +E, 9@ ! frqwudglflöq1 Dvð O4 @ O5
a
i +E, YO5
i+E, YO4 _ YO5 @ !
Whruhpd 71: 5 Vhd E D U> s 5 E 3 +sru or wdqwr s 5 D3 ,1 Vhd i = D $ U | j @ imE 1 Vl olp{$s i +{, @ O hqwrqfhv olp{$s j+{, @ O1 Uhflsurfdphqwh1 Vl E frqwlhqh xq lqwhuydor delhuwr txh frqwlhqh d s vh wlhqh txh vl olp{$s j+{, @ O hqwrqfhv olp{$s i+{, @ O1
Ghprvwudflöq= Od ghprvwudflöq gh od sulphud sduwh hv edvdgd hq whruðd gh frqmxqwr1 Vl olp{$s i +{, @ O vh wlhqh txh sdud wrgd yhflqgdg YO gh O h{lvwh xqd yhflqgdg Ys gh s wdo txh i++Ys isj, _ D, YO
sru or wdqwr wdpelìq vh wlhqh |d txh
j++Ys isj, _ E, YO
j++Ys isj, _ E, i ++Ys isj, _ D, YO
oxhjr
olp j+{, @ O {$s
Od rwud sduwh gho whruhpd uhdo}d ho frqfhswr gh oðplwh hq ho vhqwlgr txh edvwd dqdol}du orv sxqwrv fhufdqrv d s1
Hmhpsor 585 Fodudphqwh vh yh txh olp{$6 {5 5 @ :1 Dtxð vh hvwä shqvdqgr hq od ixqflöq i = U $ U> i+{, @ {5 51 Ho whruhpd dqwhulru qrv glfh txh i m`6>4^ @ j =`6> 4^$ U> j+{, @ {5 5 fxpsoh txh olp j+{, @ : {$6
5m Hmhpsor 586 Vhd i +{, @ m{{ 5 > { 9@ 5/ Surehprv olp{$5 i+{, qr h{lvwh +Sru ho devxugr,1 Vxsrqjdprv txh olp{$5 i+{, h{lvwh/ hv ghflu/ olp{$5 i +{, @ O1 frq O 5 U oxhjr/ vl frqvlghudprv
hqwrqfhv vhjýq ho whruhpd dqwhulru1
j @ i m`5>4^
olp j+{, @ O {$5
| k @ im`lqiw|>5^ |
olp k+{, @ O {$5
shur fodudphqwh j+{, @ 4 sdud wrgr { 5 Grpj | k+{, @ 4 sdud wrgr { 5 Grpk1 Sru or wdqwr olp j+{, @ 4 | olp k+{, @ 4 {$5 {$5
or txh frqwudglfh 714 Sru frqvljxlhqwh olp{$5 i +{, qr h{lvwh1 5 Vl i G D
i
E < U E
hv od ixqflöq i uhvwulqjlgd d xq grplqlr E | hv oodpdgd uhvwulfflöq gh i d E 1
+714,
7151 OÐPLWHV ODWHUDOHV
55:
Hmhpsor 587 Hq irupd dqdorjd vh suxhed od qr h{lvwhqfld gh olp{$3 i +{,/ grqgh i +{, @ vhq+ {4 ,/ { 5 U i3j Vhd 4 4 = q 5 Qj> E @ i = q 5 Qj D@i 5q 5q . @5 dperv wlhqhq d 3 frpr sxqwr gh dfxpxodflöq Vxsrqjdprv txh olp{$3 i +{, @ O/ sru or wdqwr
olp i m @ O+sru whruhpd 71:, {$3 E @ vhq +5q, @ 3> ;q 5 Q Shur imD +d, @ vhq + d4 ,> d 5 D/ | vhq+ d4 , @ vhq 44 5q sru or wdqwr olp i m @ olp 3 @ 3> dvð O @ 3 {$3 D {$3 4 Sru rwud sduwh/ imE +e, @ vhq+ 4e ,> e 5 E | frpr e @ 5q. @5 /hqwrqfhv olp i m @ O {$3 D
vhq
a
+715,
4 @ vhq +5q . @5, @ vhq@5 @ 4= e
Dvð i mE +e, @ 4> ;e 5 E gh grqgh O @ 4/ or txh frqwudglfh 715
715
Oðplwhv Odwhudohv
Ghqlflöq 71; Ho oðplwh odwhudo ghuhfkr gh i hq d vh hqwlhqgh frpr ho oðplwh vljxlhqwh olp{$d im`d>4^ | vh dqrwd olp{$d. i+{,1+vl d hv sxqwr gh dfxpxodflöq gh `d> 4^_Grpi 9@ !,6 Dqäorjdphqwh vh ghqh oðplwh odwhudo l}txlhugr frpr olp{$d im`4>d^ | vh ghqrwd frpr olp{$d i +{,1+vl d hv sxqwr gh dfxpxodflöq gh ` 4> d^_Grpi 9@ ! Sru whruhpd dqwhulru vh wlhqh txh= Vl i hvwä ghqlgd hq xq lqwhuydor delhuwr txh frqwlhqh d d +qr qhfhvduldphqwh ghqlgd hq d, hqwrqfhv l, Vl olp{$d i +{, h{lvwh | hv ljxdo d O/ hqwrqfhv dperv oðplwhv odwhudohv h{lvwhq | vrq ljxdohv1 Sru frqvljxlhqwh xvdqgr od htxlydohqfld s , t +, t , s whqhprv=
ll, Vl orv oðplwh odwhudohv qr h{lvwhq r vrq glihuhqwh hqwrqfhv olp{$d i +{, qr h{lvwh1 +lqvlvwlpr hq txh vlhpsuh txh i hvwä ghqlgd hq xq lqwhuydor delhuwr txh frqwlhqh d d,
Hmhpsor 588 Vhd
+
{5 .9{.; { A 7 {.7 {5 49 {5 .:{.45 { ? 7 Fdofxohprv orv oðplwhv odwhudohv hq { @ 71 Qrwhprv txh i hvwä ghqlgd hq U i7j1 i+{, @
{5 49 olp 5 d, olp i+{, @ {$7 {$7 { . :{ . 45 +{ . 7,+{ 7, @ olp {$7 +{ . 6,+{ . 7, {7 @ olp {$7 { . 6 ; @ 4 @ ;
6 Hq fdvr frqwudulr/ qr h{lvwh ho oðplwh +r qr vh sxhgh fdofxodu,1
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
55;
{5 . 9{ . ; +{ . 5,+{ . 7, e, olp i +{, @ olp @ olp . @ olp . { . 5 @ 5 {.7 {.7 {$7. {$7. {$7 {$7 Sru d, | e, vh wlhqh olp{$7 i+{, 9@ olp{$7. i +{, oxhjr olp{$7 i+{, qr h{lvwh1 s Hmhpsor 589 Frqvlghuhprv od ixqflöq i +{, @ { s Khprv ylvwr hq xq hmhpsor dqwhulru txh olp{$3 { h{lvwh | vx ydoru hv 31 Vlq hpedujr qr srghprv dupdu txh dperv oðplwhv odwhudohv h{lvwdq1 Hq hvwh fdvr/ gho Grpi @ U. 3 / oxhjr i qr hvwd ghqlgd hq wrgr xq lqwhuydor delhuwr txh frqwhqjd d 31 Revhuydprv od ghprvwudflöq gho hmhpsor 57; hv fodur txh s s olp { @ olp. { @ 3 {$3 {$3 Vreuh ho oðplwh odwhudo l}txlhugr vh wlhqh txh od sursrvlflöq
s +; % A 3,+ 3^_Grpi @ !1 Sru or wdqwr ho oðplwh odwhudo l}txlhugr hq { @ 3 qr vh sxhgh fdofxodu +qr wlhqh vhqwlgr od h{suhvlöq,1
Revhuydflöq= Hq wìuplqrv gh > % od ghqlflöq gh oðplwh odwhudo ghuhfkr | odwhudo l}txlhugr hv= olp i+{, @ O +, {$d. +; % A 3,+ i> j> k = D $ U1 Vhd Y xqd yhflqgdg gh s wdo txh 4, ;{ 5 +Y isj, _ D vh wlhqh= i+{, j+{, k+{, | dghpäv 5, olp{$s i+{, @ olp{$s k+{, @ O/ hqwrqfhv olp j+{, @ O {$s
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
565
Ghprvwudflöq= +yhu jxud dqwhulru, Gdgr % whqhprv sdud i | k or vljxlhqwh + wdq+{,, Sru ho khfkr txh od flufxqihuhqfld wlhqh udglr 4 | ho äqjxor { hvwä gdgr hq udgldqhv/ vh wlhqh txh ho dufr gh flufxqihuhqfld gh S d V> g S V hv od orqjlwxg {1 Qrwd= hq od jxud ho äqjxor { vdwlvidfh 3 { ? 5 / shur fodudphqwh vh sxhgh yhu txh vl { hv qhjdwlyr | pd|ru txh 5 / vöor kd| txh whqhu fxlgdgr frq ho vljqr hq odv uhodflrqhv txh d frqwlqxdflöq gdprv sdud {/ { 5` 5 > 5 ^1 g S V @ m{m SF @ m vhq+{,m
m vhq+{,m m{m
Sru or wdqwr/ sdud { 5` 5 > 5 ^ vh wlhqh od ghvljxdogdg
|d txh S F g SV
m{m vhq+{, m{m
Hmhpsor 596 Sru or dqwhulru vh wlhqh txh olp{$3 vhq { @ 31 Hq hihfwr= Frpr m{m vhq { m{m > { 5` > ^ 5 5 | olp{$3 m{m @ olp{$3 m{m @ 31 hqwrqfhv sru whruhpd gho dfrwdplhqwr vh wlhqh olp vhq { @ 3 {$3
+718,
Wdpelìq xvdqgr od jxud vh sxhgh yhu txh fxdqgr { wlhqgh do äqjxor fhur/ od klsrwhqxvd gho wuläqjxor RFS / RS wlhqgh do fdwhwr dg|dfhqwh RF/ gho plvpr wuläqjxor/ shur frpr RS hv vlhpsuh ljxdo d 4 | RF @ frv+{, vh wlhqh txh olp frv+{, @ 4> {$3 oðplwh txh mxvwlfduhprv gh rwud irupd pdv dghodpwh1 7 Qrwdflöq
DE
lqglfd od glvwdqfld hqwuh
D E |
r orqjlwxg gho wud}r gh uhfwd gh
D E d
1
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
567 Hmhpsor 597 Surehprv txh
vhq { @4 olp {$3 { Sdud uhvroyhu hvwr xwlol}duhprv xqd uhodflöq h{lvwhqwh hqwuh grv äuhdv gh od jxud äuhd gh vhfwru flufxodu RVS äuhd gho wuläqjxor RVG
Fdofxodqgr fdgd äuhd8 vh wlhqh
4 { wdq+{, > 5 5
vl { srvlwlyr
Pxowlsolfdqgr hvwd ýowlpd ghvljxdogdg d dperv odgrv sru ho qýphur srvlwlyr frv+{,
vhq+{, {
{ rewhqhprv
5 frv+ ,
{
Xwlol}dqgr ho khfkr txh frv+{, hv xqd ixqflöq sdu +frv+{, @ frv+{,, | frpr vhq+{, {/ srghprv ideulfdu od vljxlhqwh fdghqd dsursldgd sdud xwlol}du ho whruhpd gho dfrwdplhqwr
vhq+{, 4 { ghvljxdogdg yäolgd sdud { 9@ 3 | { 5` 5 > 5 ^ + |d txh od ixqflöq vhq+{, | { vrq ixqflrqhv lpsduhv | vx fxrflhqwh hv sdu,1 Gh hvwd irupd sru whruhpd 7144 | xwlol}dqgr txh frv+{, $ 4 fxdqgr { $ 3 rewhqhprv frv+{,
olp {$3
Hmhuflflr 598 Fdofxodu Vroxflöq=
vhq { 5{ olp {$3 6{ . 7vhq{ vhq { 5{ O @ olp {$3 6{ . 7vhq{
Frpr
vhq+{, @4 {
vhq { 5{ 4@{ olp {$3 6{ . 7 vhq { 4@{ vhq { 5 @ olp { vhq { {$3 6 . 7 { @
vhq { vhq { olp 6 . 7 @ olp 6 . 7 olp @ 6 . 7 @ : 9@ 3= {$3 {$3 {$3 { {
|
olp {$3
vhq { vhq { 5 @ olp olp 5 @ 4 5 @ 4= {$3 { {$3 {
hqwrqfhv sru dojheud gh oðplwhv O @
4 1 :
Hmhpsor 599 Fdofxohprv olp{$3. { vhq {d grqgh d 5 U1 Vroxflöq= Vdehprv txh
m vhq
Pxowlsolfdqgr sru m{m od hfxdflöq 719 rewhqhprv 8 uhfxhugh
txh
{ hvwä
hq udgldqhv1
d m4 {
d m{ vhq m m{m {
+719, +71:,
7151 OÐPLWHV ODWHUDOHV
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Dvð/ 3
d m{m { vhq + , m{m= { Frpr olp{$3 m{m @ 3 | olp{$3 m{m @ 3 hqwrqfhv sru whruhpd gho dfrwdplhqwr vh wlhqh olp{$3. m{ vhq {d m @
Hvwh ýowlpr fdvr vh sxhgh jhqhudol}du d ixqflrqhv dfrwdgdv1 Uhfrughprv txh xqd ixqflöq i vh glfh dfrwdgd hq D vl h{lvwh n 5 U. wdo txh= mi+{,m n
Whruhpd 7145 +Fhur Dqltxlod, Vhd i xqd ixqflöq dfrwdgd hq xqd yhflqgdg gh s> Ys +qr qhfhvduld0 phqwh ghqlgd hq s,1 Vl olp{$s j+{, @ 3 hqwrqfhv olp{$s j+{,i+{, @ 3
Ghprvwudflöq= Frpr mi+{,m n vh wlhqh hv ghflu/
mj+{,i+{,m nmj+{,m
nmj+{,m j+{,i+{, nmj+{,m
| sru whruhpd gh dfrwdplhqwr vh rewlhqh or uhtxhulgr1 Hmhpsor 59: Ghprvwudu
5 @3 +71;, olp +{ 4, frv {$4 {4 5 , hv dfrwdgd/ oxhjr sru whruhpd fhur dqltxlod/ Hq hihfwr olp{$4 { 4 @ 3 | od ixqflöq i +{, @ frv+ { 4 vh wlhqh 71;1 Hmhuflflr 59; Fdofxodu
olp {$6
4 vhq+ { , 6 4
{6 4 , vhq + { 6 @ olp 4 Vroxflöq= O @ olp{$6 4 { $ 6 +{ 6, vhq {6 1 {6 4 Frpr olp{$6 { 6 @ 3 | i+{, @ vhq { hv dfrwdgd/ hqwrqfhv sru whruhpd fhur dqltxlod O @ 31 6 Hmhuflflr 59< Ghprvwudu xvdqgr ho whruhpd dqwhulru txh
olp vhq +{, @ vhq +d, {$d
Vroxflöq= Edvwd ghprvwudu txh olp{$d +vhq { vhq d, @ 31 Xvdqgr od lghqwlgdg wuljrqrpìwulfd vhq { vhq d @ 5 vhq+
{.d {d , frv+ , 5 5
d , hv dfrwdgr | olp {d {d | ho khfkr txh frv+ {. {$d vhq+ 5 , @ olpx$3 vhq x @ 3 grqgh 5 @ x1 Vh wlhqh sru 5 whruhpd fhur dqltxlodgru txh= olp +vhq { vhq d, @ 3 {$d Hq irupd dqäorjd vh sxhgh suredu txh olp frv { @ frv d {$d
Hmhuflflr 5:3 Vhdq d A 3> e A 3/ Ghprvwudu txh
e { e @ olp. d {$3 d {
D|xgd= Xvdu od vljxlhqwh surslhgdg gh od sduwh hqwhud
{ 4 ^{` { > ;{ 5 U=
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
569 Vroxflöq= Sru surslhgdg dqwhulru whqhprv
e e e 4 { { {
pxowlsolfdqgr sru {d frq d A 3> { A 3 vh wlhqh { e {e { e 4 d { d { d{
r htxlydohqwhphqwh
e { { e e 4 d d d { d sxhvwr txh olp{$3. de {d @ olp{$3. de @ de hqwrqfhv sru whruhpd gh dfrwdplhqwr e { e @ olp. d {$3 d {
Hmhuflflr 5:4 Ghwhuplqh vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 41 olp{$3
@{, @{ @ 41
vhq +4 4
51 olp{$7 +{ . 7,^{` @ 31 61 olp{$3 vhq{ { @ 31 71 olp{$ vhq{ { @ 41 81 olp{$3 frv{ { @ 31 91 Qr h{lvwh olp{$3 frv{ { 1 :1 olp{$3 +{ vhq+4@{, .
{ { , qr h{lvwh1
frv+ ,
;1 Vl olp{$d ij h{lvwh | olp{$d i h{lvwh hqwrqfhv olp{$d j h{lvwh1
6^
| olp{$7 +{ . 7, @ 3 oxhjr sru whruhpd fhur dqltxlod
olp +{ . 7,^{` @ 3= {$7 61 Yhugdghur1 olp{$3 vhq{ { @ olp{$3 +vhq+4{,,@{ @ 71 Idovr1 olp{$ vhq{+{, @ vhq+, @ 3 3 81 Yhugdghur1 olp{$3 frv{ { @ frv+3, @3
4 4
@ 41
7151 OÐPLWHV ODWHUDOHV
56:
{, 91 Yhugdghur1 Vxsrqguhprv txh olp{$3 frv+ { h{lvwh | vx ydoru hv O1 Gdgr txh olp{$3 { h{lvwh sru {, dojheud gh oðplwhv olp{$3 frv+ { { gheh h{lvwlu shur olp {$3
| sru dojheud gh oðplwhv olp {$3
frv +{, { @ olp frv+{, @ 4 {$3 {
frv+{, frv+{, { @ olp olp { @ O 3 @ 3 {$3 { {$3 {
or fxdo hv xqd frqwudglfflöq sru od xqlflgdg gho oðplwh1 Sru or wdqwr ho ydoru O qr h{lvwh1
:1 Yhugdghur19 Vdehprv sru +4, txh> olp{$3 { vhq +4@{, h{lvwh +| vx ydoru hv fhur, | olp{$3 frv{ { qr h{lvwh sru +9, Oxhjr
frv { olp +{ vhq 4@{ . , qr h{lvwh= {$3 { |d txh gh h{lvwlu wdpelìq ghehuðd h{lvwlu +sru oðplwh gh od vxpd,
| hvwr ýowlpr hv idovr sru +9,1
olp +{ vhq 4@{ . {$3
frv { { vhq+4@{,, {
;1 Idovr1 Vhdq i +{, @ vhq {> j+{, @ {4 > olp{$3 i+{,j+{, h{lvwh | vx ydoru hv 4/ dghpäv olp{$3 i +{, h{lvwh | vx ydoru hv fhur/ shur olp{$3 j+{, qr h{lvwh1
{$7 {$7 {$7
7151 OÐPLWHV ODWHUDOHV
56<
hqwrqfhv sru dojheud gh oðplwhv vh wlhqh O@
5 4 olp{$7 5 s @ @ ;9 57 olp{$7 +{ . 7, olp{$7 +6 . 5{ . 4
Revhuydflöq= Vh sureduä hq od vhfflöq gh frqwlqxlgdg txh vl q 5 Q> t+s, A 3 hqwrqfhv s s olp q t+{, @ q t+s, {$s
Hmhuflflr 5:8 Fdofxodu
s 6 5 6 +{ . d,5 s d > vl d 9@ 3= olp {$3 {
Vroxflöq= Uhfrughprv txh d6 e6 @ +d e,+d5 . de . e5 ,1 s 6 5 6 +{ . d,5 s d O @ olp {$3 { s 6 5 s 6 55 6 +{ . d,5 s 6 +{ . d,5 5 . s 6 +{ . d,5 d5 . s d d @ olp s s s 5 6 {$3 6 +{ . d,5 . 6 +{ . d,5 d5 . d5 5 { +{ . d,5 d5 @ olp s 6 55 {$3 {+ 6 +{ . d,5 5 . s 6 +{ . d,5 d5 . s d , @
{5 . 5d{ s = s olp s {$3 {+ 6 +{ . d,7 . 6 +{ . d,5 d5 . 6 d7 ,
s s s s 5 6 5 6 Gdgr txh olp{$3 { . 5d @ 5d | olp{$3 @ olp{$3 6 +{ . d,5 . 6 +{ . d,5 d5 . d5 @ 6 d7 9@ 3/ hqwrqfhv sru dojheud gh oðplwhv vh wlhqh 5d O@ s 6 6 d7
Hmhuflflr 5:9 Ghwhuplqdu d 5 U gh prgr txh h{lvwd olp{$5 i +{, grqgh + s s {s {s ; { A 5 { 5 i +{, @ = d{ . 4 { ? 5
Vroxflöq= Gdgr txh Grpi @ U i5j1 Ho oðplwh h{lvwlud vl vöor vl olp i+{, @ olp i +{, {$5. {$5
olp i +{, @ {$5. @ @ @ @ @ @
s s { { ; s olp s {$5. { 5 s s s s +{ { ;,+ { 5 s s olp s s {$5. + { 5,,+ { . 5, s s s s {5 . { 5{ ; { 49 olp {5 {$5. s s 5 +{ 7, . +{ 5{ 5 5{, olp {5 {$5. s +{ 5,+{ . 5, . +{ 5,+ 5{, olp {5 {$5. s +{ 5,+{ . 5 . 5{, olp {5 {$5. s olp { . 5 . 5{= {$5.
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
573 Frpr olp{$5. { @ 5> olp{$5 5 @ 5> olp{$5
s s 5{ @ 5 5 @ 5/ hqwrqfhv sru dojheud gh oðplwh vh wlhqh
olp i+{, @ 5 . 5 . 5 @ 9= {$5.
Sru rwud sduwh olp{$5 i+{, @ olp{$5 d{ . 4 @ 5d . 4/ oxhjr olp{$5 i+{, h{lvwh vl | vöor vl 5d . 4 @ 9/ gh grqgh d @ 85 1 Hmhuflflr 5:: Ghpxhvwuh txh> Vl olp{$d i+{, h{lvwh | olp{$d j+{, qr h{lvwh hqwrqfhv olp{$d +i . j,+{, qr h{lvwh1
Vroxflöq= Sru pìwrgr gho devxugr
++s , t, vl | vöor vl +s a t,
Vxsrqjdprv txh olp{$d +i . j, h{lvwh/ oxhjr sru dojheud gh oðplwhv/ gdgr txh olp{$d i h{lvwh hqwrqfhv olp{$d +i . j, i h{lvwh/ shur i . j i @ i or txh frqwudglfh od qr h{lvwhqfld gh olp{$d j1 Hmhuflflr 5:; Ghpxhvwuh txh orv vljxlhqwhv oðplwhv qr h{lvwh1 41 olp{$3 {4
51 olp{$3 +{ . {4 ,1
Vroxflöq=
Hmhuflflr 5:< 41 Vxsrqjdprv txh olp{$3 {4 h{lvwh | vx ydoru ydoru hv O1 Gdgr txh olp{$3 { h{lvwh/ hqwrqfhv sru dojheud gh oðplwhv/ h{lvwh ho oðplwh gho surgxfwr/ hv ghflu/ olp{$3 {4 { h{lvwh 4 4 olp { @ olp olp { @ O 3 @ 3= {$3 { {$3 { {$3
Sru rwud sduwh
4 olp { @ olp 4 @ 4> {$3 { {$3 oxhjr sru xqlflgdg gho oðplwhv vh wlhqh txh 3 @ 4 or txh hv xqd frqwudglfflöq1
51 Frpr olp{$3 { h{lvwh | olp{$3 {4 qr h{lvwh/ hqwrqfhv sru hmhuflflr 5:: vh wlhqh 4 olp { . {$3 {
qr h{lvwh
Revhuydflöq= Hq jhqhudo/ vh sxhgh ghprvwudu sru ho pìwrgr gho devxugr txh= Vl olp{$d i+{, @ f 9@ 3 | olp{$d j+{, @ 3 hqwrqfhv olp{$3 ij++{{,, qr h{lvwh1 Hq hvwh fdvr vh glfh txh ho oðplwh olp{$d ij++{{,, hv gh od irupd 3f Hmhpsor 5;3 Orv vljxlhqwhv oðplwhv vrq gh od irupd 3f m{ . 5m olp > {$4 { 4
oxhjr orv oðplwhv qr h{lvwh1
{.5 olp {$5 {5 7
Hmhpsor 5;4 Frqvlghuhprv od ixqflöq i +k, @
^{` olp {$6@5 6 5{
d5 . d . 6dk5 5 k5
olp {$4
Ghwhuplqhprv sdud txì ydoru gh d 5 U/ ho oðplwh olpk$3 i +k, h{lvwh1
4 frv5 { {4
7151 OÐPLWHV ODWHUDOHV
574
5 5 5 sxhgh h{lvwlu vl: olpk$3 d5 . d . 6dk5 5 @ 3 Gdgr txh olpk$3 k5 @ 3/ hqwrqfhv olpk$3 d .dk6dk 5 5 .d.6dk5 5 d 5 5 qr h{lvwh,1 +vl olpk$3 d . d . 6dk 5 9@ 3 hqwrqfhv olpk$3 k5 Dkrud>
olp d5 . d . 6dk5 5 @ 3 +, d5 . d 5 @ 3 k$3 +, +d . 5,+d 4, @ 3 +, d @ 5 b d @ 4 5 9k5 dk5 5 @ olp Vl d @ 5 hqwrqfhv olpk$3 d .d.6 5 k$3 k5 @ 91 k 5 dk5 5 @ olp 6k5 Vl d @ 4 hqwrqfhv olpk$3 d .d.6 k$3 k5 @ 61 k5 5 5 dk 5 h{lvwh vl d 5 i5> 4j1 Sru or wdqwr olpk$3 olp d .d.6 k5 s s Hmhpsor 5;5 Prvwudu txh olp{$d { @ d/ vl d 31 s s Vroxflöq= Vl d A 3 edvwd suredu txh olp{$d + { d, @ 3 shur s s s s s s {. d s { d @ { d s {. d {d s @ s {. d 4 s +{ d, @ s {. d
|
4 4 s ?s 3? s {. d d
| |d txh olp{$d { @ d sru whruhpd fhur Dqltxlod s s olp { d @ 3 {$d
71516
Whruhpd gh Vxvwlwxflöq
Whruhpd 7146 +Vxvwlwxflöq, Vhdq [> \ U | i = [ $ U/ j = \ $ U ixqflrqhv wdohv txh i +[, \ 1 Vxsrqjdprv txh olp{$s i+{, @ e | txh h{lvwh xqd yhflqgdg gh s> Ys wdo txh i+{, 9@ e1 sdud wrgr { 5 Ys isj [1 Vl olpx$e j+x, @ O hqwrqfhv olp{$s j+i+{,, @ O1 hv ghflu> olp j+i +{,, @ olp j+x, @ O= {$s x$e
Hmhpsor 5;6 Fdofxohprv
olp {$
vhq+{ , {
Vroxflöq= Frqvlghuhprv od vxvwlwxflöq r fdpelr gh yduldeoh x @ { 1 Hv fodur txh olp{$ { @ 3/ hv ghflu/ vl { $ hqwrqfhv x $ 3/ gh grqgh { 9@ 3 hq Y ij/ oxhjr sru whruhpd gh vxvwlwxflöq +gdgr txh olpx$3 vhqx x @ 4,/ vh wlhqh vhq+{ , vhq x olp @ olp @4 {$ { x$3 x
Hmhpsor 5;7 Fdofxohprv
: Frqglflöq
qhfhvduld shur qr vxflhqwh1
6{ olp {$3 vhq+7{,
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
575
Vroxflöq= Vhd x @ 7{1 Vl { $ 3 hqwrqfhv x $ 3/ +dghpäv 7{ 9@ 3 sdud { hq xqd yhflqgdg gho fhur vlq wrpdu ho fhur, hqwrqfhv/ 6 6{ 6 x @ olp @ olp olp {$3 vhq 7{ x$3 7 vhq x 7 x$3
sru whruhpd gh vxvwlwxflöq/ dvð
6 olpx$3 4 6 vhq x @ = x @ 7 olp 7 x$3 x x
4
vhq
6{ 6 x 6 olp @ olp @ x$3 vhq+7{, x$3 7 vhq x 7
Frurodulr 7147 Vhdq [> \ U | i = [ $ U/ j = \ $ U | i +[, \ 1 Vhd s 5 [ 3 a e 5 \ _ \ 3 a olp{$s i +{, @ e a olp{$e j+{, @ j+e, hqwrqfhv olp{$s +j i ,+{, @ j+e,=
Ghprvwudflöq= Vhd j+e, @ f
[ s
$ i +[, \ :$ e
$ U : $ j+e,
Sdud xqd yhflqgdg gh f> Yf / sxhvwr txh olp{$e j+{, @ f/ h{lvwh xqd yhflqgdg gh e> Ye / wdo txh j+Ye iej, Yf 1 Sru rwud sduwh/ frpr olp{$s i +{, @ e sdud od yhflqgdg Ye h{lvwh xqd yhflqgdg gh s wdo txh i+Ys isj, Ye 1 Sru or wdqwr j+i+Ys isj,, j+Ye ,1 Dvð/ sdud xqd yhflqgdg Yf / h{lvwh xqd yhflqgdg gh s> Ys wdo txh +j i ,+Ys isj, Yj+e, / hv ghflu/ olp +j i,+{, @ j+e,= {$s
Hvwh whruhpd vh xwlol}d hq or txh vh vxhoh oodpdu fdpelr gh yduldeoh r vxvwlwxflöq
Hmhpsor 5;8
s t s +71 {$3 {$3 s s s sxhvwr txh= Vl wrpdprv j+{, @ { | i +{, @ {5 . 7 | gdgr txh olp{$7 j+{, @ olp{$7 { @ 7 @ 5 | olp{$3 i +{, @ olp{$3 {5 . 7 @ 7/ hqwrqfhv sru ho frurodulr dqwhulru vh wlhqh 71< Hmhpsor 5;9 Vdelhqgr txh olp{$d h{ @ hd 1 Surehprv txh vl h{lvwh olp{$e i+{, hqwrqfhv
olp hi +{, @ holp{$e i +{, {$e Vroxflöq= Vhd j+{, @ h{ frpr +j i,+{, @ hi +{, | sxhvwr txh olp{$d h{ @ hd / hqwrqfhv olp{$e +j i ,+{, @ j+olp{$e i +{,, @ holp{$e i +{, 1 Dvð whqhprv sru hmhpsor= s6 {4 olp h {4 @ h4@6 {$4 s 6 Hq hihfwr1 Vhd x @ {1 Vl { $ 4 hqwrqfhv x $ 4 | s 6{4 x4 @ olp 6 olp {$4 { 4 x$4 x 4 x4 @ olp x$4 +x 4,+x5 . x . 4, 4 @ olp 5 x$4 x . x . 4 4 @ = 6
7151 OÐPLWHV ODWHUDOHV Oxhjr
576
s6 {4 s6 {4 olp h {4 @ holp{$4 {4 @ h4@6 = {$4
Sdud dsolfdu ho whruhpd dqwhulru/ ghehprv dvhjxuduqrv txh vh fxpsodq od klsöwhvlv1 Gh qr kdfhuor srghprv rewhqhu frqwudglfflrqhv1 Hmhpsor 5;: Vhd d 5 ]
olp ^{` 9@ ^ olp {` {$d {$d
Hq hihfwr> Vdehprv txh olp{$d ^{` qr h{lvwh sxhv orv oðplwhv odwhudohv vrq glvwlqwrv olp ^{` @ d {$d.
>
olp ^{` @ d 4 {$d
Vlq hpedujr> gdgr txh olp{$d { @ d/ hqwrqfhv vh wlhqh ^olp{$d {` @ ^d` @ d1
Revhuydflöq= Qrwhprv txh1 Vl d 95 ] hqwrqfhv olp{$d ^{` @ ^olp{$d {`=
Hmhuflflr 5;; Suredu txh vl olp{$d mi +{,m @ 3 hqwrqfhv olp{$d i +{, @ 31
Vroxflöq= Vxsrqjdprv txh olp{$d mi+{,m @ 3 hqwrqfhv olp{$d mi+{,m @ 3 dghpdv mi +{,m i+{, mi +{,m | sru whruhpd gh dfrwdplhqwr whqhprv txh olp{$d i +{, @ 31
Hmhuflflr 5;< Fdofxodu orv vljxlhqwhv oðplwhv d, e, f, Vroxflöq= d, olp {$3
s 4 . {5 4 {
s 4 . {5 4 olp {$3 s{ 4 6 4 5{ olp {$3 { s s 4 . {5 6 4 5{ olp {$3 { s s 4 . {5 4 4 . {5 . 4 s { 4 . {5 . 4 5 4.{ 4 @ olp s {$3 {+ 4 . {5 . 4, { @ olp s {$3 4 . {5 . 4 @ 3
@
olp {$3
e, Vhd x6 @ 4 5{1 Vl { $ 3 hqwrqfhv x $ 4 oxhjr/ s 4 6 4 5{ 4x @ olp olp {$3 x$4 +4 x6 ,@5 { 4x @ 5 olp x$4 +4 x,+4 . x . x5 , 4 @ 5 olp x$4 4 . x . x5 5 @ 6
577
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV f, olp {$3
s
s 4 . {5 6 4 5{ {
@ @ @ @
s
s 4 . {5 4 . 4 6 4 5{ olp {$3 { s s 4 . {5 4 4 6 4 5{ olp{$3 . { { 5 3. 6 5 = 6
Hv qhfhvdulr uhfrugdu/ sdud ho fäofxor gh oðplwh wuljrqrpìwulfrv dojxqdv lghqwlgdghv1 41 frv5 { . vhq5 { @ 4
51 frv+5{, @ 4 5 vhq5 {
61 frv+5{, @ 5 frv5 { 4
71 vhq+{ . |, @ vhq { frv | . vhq | frv {
81 vhq+{ |, @ vhq { frv | vhq | frv { 91 frv+{ . |, @ frv { frv | vhq { vhq | :1 frv+{ |, @ frv { frv | . vhq { vhq |
| , vhq+ {| , ;1 vhq { vhq | @ 5 frv+ {. 5 5
| , vhq+ {.| ,
583
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
Vl sdud od yhflqgdg dqwhulru gh i+6,/ h{lvwlhud xqd yhflqgdg gh 6> `6 %> 6 . %^ hvwd ghehuðd fxpsolu i +`6 > 6 . ^ Yi +6,> % >
hv ghflu/ shur frpr i hv fuhflhqwh/ whqhprv
i+`6 > 6 . ^`9 %> 9 . %^
i+`6 > 6 . ^@`5+6 ,> 5+6 . ,^@`9 5> 9 . 5^
Sru or wdqwr/ gheh fxpsolu
`9 5> 9 . 5^`9 %> 9 . %^
Sdud % gdgr/ gheh fxpsolu hv ghflu/
9 % ? 9 5
a
a
5 ? %
9 . 5 ? 9 . %
>
5 ? % >
Sru or wdqwr/ sdud od yhflqgdg Yi +6,> % h{lvwh do phqrv xqd yhflqgdg gh 6 |d txh edvwd wrpdu 5 ? % +hv ghflu/ ? %@5, wdo txh i +Y6> , Y9> % 1 Yhu jxud dqwhulru Hmhpsor 638 Vhd
Ghwhuplqdu vl hv frqwlqxd hq { @ 61
i +{, @
5 {6 5=34 { A 6
Vroxflöq= Gdgd xqd yhflqgdg gh i +6,> Y5> % / ghehuðd h{lvwlu xqd yhflqgdg gh 6> Y6> / + hv ghflu/ h{lvwh A 3 , wdo txh i +Y6> , Y5> % Or dqwhulru hv htxlydohqwh d ; % A 3> i +Y6> , Y5> %
7161 FRQWLQXLGDG
584
ö Dvð/ ghehuðd rfxuulu
i+`6 > 6 . ^, `5 %> 5 . %^ ; % A 3> , @ i5> 5=34j `5 %> 5 . %^>
shur sdud % @ 3=334 r phqru hv lpsrvleoh rewhqhu od lqfoxvlöq dqwhulru1 Frqfoxvlöq= Frpr sdud wrgr % A 3 qr hv srvleoh hqfrqwudu xqd yhflqgdg gh 6> Y6> wdo txh i +Y6> , Y5> % vh wlhqh txh i qr hv frqwlqxd hq 61 Vh sxhgh yhu fodudphqwh txh i hv frqwlqxd hq orv rwurv sxqwrv Hmhpsor 639 Vhd i +{, @ 8/ +ixqflöq frqvwdqwh,1 Hood hv frqwlqxd hq wrgrv orv uhdohv1
Vroxflöq= Yhdprv od frqwlqxlgdg hq xq sxqwr duelwudulr s1 Qrwhprv txh= ; A 3 i+Ys> , @ i8j
Dvð/ ; % A 3> i+Ys> , Yi +s,> % |d txh i+Ys> , @ i8j `8 %> 8 . %^ > ; % A 3
71614
Htxlydohqfldv gh Frqwlqxlgdg
D frqwlqxdflöq gduhprv fxdwur fdudfwhul}dflrqhv gh frqwlqxlgdg 41 Vl s hv xq sxqwr gh dfxpxodflöq gh D | s 5 D1
51 Vl s hv xq sxqwr gh dfxpxodflöq gh D | D frqwlhqh xqd yhflqgdg gh s1
61 Vl s qr hv xq sxqwr gh dfxpxodflöq | s 5 D1 71 Fdudfwhul}dflöq sru vxfhvlrqhv1
41 Vl s hv sxqwr gh dfxpxodflöq gh D | s 5 D hqwrqfhv i hv frqwlqxd hq s vl | vöor vl olp{$s i+{, @ i +s, Hvwd ýowlpd ljxdogdg glfh txh
l, olp{$s i+{, h{lvwh ll, Ho oðplwh dqwhulru hv ljxdo d i+s, h lpsoðflwdphqwh hvwdprv glflhqgr txh i +s, h{lvwh1
51 Vl D frqwlhqh xqd yhflqgdg gh s +sru or wdqwr s hv xq sxqwr gh dfxpxodflöq, whqhprv or vljxlhqwh= i hv frqwlqxd hq s vl | vöor vl Hmhpsor 63: Vhd i +{, @
d, olp{$s. i +{, h{lvwh e, olp{$s i +{, h{lvwh f, orv oðplwhv dqwhulruhv vrq ljxdohv d i +s,
s { 4 BcHv frqwlqxd hq 4 B
Vroxflöq= Qrwhprv txh Grpi @ D @ ^4> 4^/ | dtxð ylhqh qdwxudophqwh od htxlyrfdgd frqfoxvlöq i qr hv frqwlqxd hq s @ 4 |d txh olp{$s i +{, qr h{lvwh%/ frpr vh h{ljh hq 5f1
Revhuydflöq= Qr srghprv xvdu od fdudfwhul}dflöq 51 sxhv qr vh fxpsoh od klsrwìvlv gh txh D frqwhqjd xqd yhflqgdg gh 41 Hq fdpelr/ vl srghprv xvdusod fdudfwhul}dflöq 4/ sxhv ssxqwr gh dfxpxodflöq gh D | 4 5 D/ s 4 hv xqd hqwrqfhv gdgr txh olp{$4 { 4 h{lvwh | olp{$4 { 4 @ 4 4 @ 3 @ i +4,1 vh wlhqh txh i hv frqwlqxd hq 41
585
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV Surehprv i hv frqwlqxd hq 4 hq wìuplqrv gh yhflqgdghv1
Wrphprv xqd yhflqgdg duelwuduld gh i +4, @ 3> Y3> % qhfhvlwdprv hqfrqwudu xqd yhflqgdg gh 4> Y4> / wdo txh rfxuud= +7143,
i+Y4> , Y3> %
Yhdprv txì frqglflöq gheh fxpsolu sdud txh hvwd ýowlpd lqfoxvlöq vhd yäolgd1
Frpr Y4> @`4 > 4 . ^ | i hv fuhflhqwh +yhu jxud, y
2
1.5
1
0.5
0 0
1.25
i +{, @ vh wlhqh txh=
3.75
s {4
5 x
s i +`4 > 4 . ^, @ i +^4> 4 . ^, @ ^3> ^
Dvð/ od lqfoxvlöq hq 7143 qrv txhgd Sru or wdqwr gheh fxpsolu
2.5
s ^3> ^` %> %^
s ? % r htxlydohqwhphqwh ? %5 1
Uhvxplhqgr1 Sdud wrgd yhflqgdg Y3> % h{lvwh xqd yhflqgdg gh 4> Y4> +edvwd txh vhd phqru txh %5 wdo txh i +Y4> , Y3> % =
Oxhjr i hv frqwlqxd hq s @ 41
61 Vl s 5 D qr hv xq sxqwr gh dfxpxodflöq +s hv xq sxqwr dlvodgr, vh wlhqh txh i hv frqwlqxd hq s1
Ghprvwudflöq= Frpr s qr hv xq sxqwr gh dfxpxodflöq/ h{lvwh xqd yhflqgdg gh s> Ys> +hv ghflu/ h{lvwh A 3, wdo txh Ys> _ D @ isj r htxlydohqwhphqwh
`s > s . ^_D @ isj>
sru or wdqwr/ i +Ys> , @ i+isj, @ ii +s,j `i +s, %> i+s, . %^
Od lqfoxvlöq ýowlpd rfxuuh sdud wrgr % A 3> gh hvwd irupd whqhprv od vljxlhqwh dupdflöq1
Sdud wrgd yhflqgdg gh i +s,> Yi +s,> % / h{lvwh xqd yhflqgdg gh s> Ys> +od wrpdgd do lqlflr, wdo txh i +Ys> , Yi +s,> %
Or txh txlhuh ghflu txh i hv frqwlqxd hq s1
7161 FRQWLQXLGDG
586
Hmhpsor 63; Wrgdv odv vxfhvlrqhv {q = Q $ U vrq frqwlqxdv/ |d txh wrgr sxqwr gh Q hv sxqwr gh dlvodgr1 Hq od suäfwlfd/ qr vh hqfxhqwudq pxfkrv grplqlrv frq sxqwrv dlvodgrv/ shur vl dsduhfhq hq vlwxd0 flrqhv hvshfldohv | sru hvh prwlyr or khprv ghvduuroodgr hq ghwdooh dtxð/ dpìq txh ohv idflolwduä ho/ hqwhqglplhqwr gho frqfhswr gh frqwlqxlgdg hq yduldv yduldeohv1 71 Ghqlflöq wudglflrqdo1
i hv frqwlqxd hq s> s 5 Grpi @ D vl
; % A 3 m{ sm ? a { 5 Grpi , mi +{, i +s,m ? %
Qrwh txh m{ sm ? | mi+{, i+s,m ? % kdeodq gh odv yhflqgdghv gh s | i +s, uhvshfwlydphqwh | od lpsolfdflrq +,, vh uhhuh d od lqfoxvlöq gh frqmxqwrv> od phqflöq h{soðflwd gh { 5 Grpi hv sdud txh whqjd vhqwlgr od qrwdflöq i +{,1
81 Vhd i = D $ U> s 5 D1 i hv frqwlqxd hq s vl | vöor vl
Sdud wrgd vxfhvlöq43 +{q ,/ wdo txh {q frqyhujh d s vh wlhqh txh od vxfhvlöq i+{q , frqyhujh d i +s,1 Hvwh fulwhulr hv pdv ýwlo sdud suredu od qr frqwlqxlgdg hq xq sxqwr txh od frq|lqxlgdg1 Or dqwhulru vh sxhgh hvfulelu frpr
i hv frqwlqxd hq { @ 3 +, ;+{q ,> {q $ s , i +{q , $ i+s,
Dvð/ od qhjdflöq hv= i qr hv frqwlqxd hq s vl | v*ror vl
Hmhpsor 63< Vhd
{q $ s a i +{q , 9$ i +s, +i+{, qr frqyhujh d i+s,, i+{, @
vhq + {4 , { 9@ 3 4 {@3
BcHv frqwlqxd hq 3 B 4 Vroxflöq= Frqvlghuhprv od vxfhvlöq +{q , frq {q @ 5q Fodudphqwh vh yh txh {q $ 3/ shur
i +{q , @ vhq+
Dvð/ i +{q , qr frqyhujh d 4
4 , @ vhq+5q, @ 3 ;q 5 Q 4@5q
Frqfoxvlöq=
i qr hv frqwlqxd hq 3 |d txh h{lvwh xqd vxfhvlöq +{q , frq {q $ 3 a i+{q , 9$ i+3, @ 4
Wdpelìq srghprv ghflu txh i qr hv frqwlqxd hq { @ 3 sxhv olp{$3 i +{, qr h{lvwh +yhu hmhpsor 636,1 43 Qdwxudophqwh
od qrflöq ghe h vhu gh hohphqwrv gh
D
1
587
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
71615
Frphqwdulrv
41 Frphqwdulrv lqwxlwlyrv44
l, i hv frqwlqxd hq s vl odv lpdjhqhv gh sxqwrv fhufdqrv d s hvwäq fhufdqrv d i +s, | sru px| h{ljhqwh txh vh srqjd +; % A 3, sdud shglu hvwdu fhufd gh i+s,/ vlhpsuh kd| xq prqwöq gh sxqwrv fhufd gh s txh hvwäq vxshu fhufd gh i+s,1
ll, Xqd ixqflöq frqwlqxd hq s qr sxhgh whqhu xq vdowr hq s/ qr sxhgh hvwdu fruwdgd hq s1
51 Frphqwdulr Uhgxqgdqwh
l, Fodudphqwh i qr hv frqwlqxd hq xq sxqwr s vl s 95 Grpi
ll, Gh od iudvh Xqd ixqflöq hv glvfrqwlqxd hq xq sxqwr s% qr vh sxhgh frqfoxlu txh qr wlhqh oðplwh |d txh shuihfwdphqwh od glvfrqwlqxlgdg sxhgh ghehuvh d txh d, s 95 Grpi r d txh
e, i+s, 9@ olp{$s i+{,1
5 Hmhpsor 643 i +{, @ {{44 / i+4, qr h{lvwh shur olp{$4 i +{, @ 51 Hvwd ixqflöq frqvwlwx|h xq hmhpsor sdud ho fdvr +d, Hmhpsor 644
i+{, @
{6, {6 8
vhq+
{ 9@ 6 {@6
Hq hvwd ixqflöq vh wlhqh i +6, @ 8 9@ olp{$6 i +{,1Hvwd ixqflöq frqvwlwx|h xq hmhpsor sdud ho fdvr +e,1
lll, Vl olp{$s i +{, qr h{lvwh/ qr hv frqwlqxd i hq s1 Od qr h{lvwhqfld gho oðplwh vh sxhgh ghehu d yduldv fdvrv/ hq ho hmhpsor 645 whqhprv ho fdvr hq txh fdgd oðplwh odwhudo h{lvwh/ shur olp{$s. i +{, 9@ olp{$s i+{,
Hmhpsor 645
i +{, @ Frpr
; A A ?
{5 < {6 9
A A = {5 9{.< {6 5:
{A6 {@6 {?6
olp i+{, @ 9 @ i +6, 9@ olp i+{, @ 3 {$6. {$6 Sru or wdqwr qr hv frqwlqxd hq 6
Ho vljxlhqwh whruhpd hv ho txh vluyh sdud prvwudu frq fodulgdg odv revhuydflrqhv +frphqwdulrv, dqwhulruhv
Whruhpd 714; l, Wrgd uhvwulfflöq gh xqd ixqflöq frqwlqxd hv frqwlqxd1 Glfkr gh rwud irupd Vhd i = [ $ U frqwlqxd hq ho sxqwr s 5 U1 Vl s 5 \ [ | j @ 45 i m\ hqwrqfhv wdpelìq j = \ $ U hv frqwlqxd hq s ll, Hq ho fdvr hvshfldo gh txh \ @ L _ [/ frq L xqd yhflqgdg gh s hqwrqfhv od uhfðsurfd vh fxpsoh1 Glfkr gh rwud irupd1 Vhd j @ im\ frqwlqxd hq ho sxqwr s +frq \ @ Ys> _ [ hqwrqfhv i = [ $ U hv frqwlqxd hq s
Hvwd ýowlpr qrv glfh txh od frqwlqxlgdg hq ho sxqwr s ghshqgh vöor gh orv sxqwrv txh hvwäq fhufd/ | hvwr hv or txh vh vxhoh oodpdu frqfhswr orfdo1 Dvð frqwlqxlgdg hv xq frqfhswr orfdo% do ljxdo txh oðplwh1 Orv vljxlhqwhv whruhpdv vrq xwlohv sru ho khfkr txh qrv vluyhq sdud ideulfdu qxhydv ixqflrqhv frqwlqxdv r sdud ylvxdol}du txh xqd ixqflöq hv frqwlqxd1 4 4 Hvwr
qr vluyh sdud qdgd hq odv ghprvwudflrqhv/ s hur vl sdud ylvxdol}du1
4 5 Uhfxhugh
txh
i\
hv od ixqflöq uhvwulfflöq gh
[ \ d
7161 FRQWLQXLGDG
588
Whruhpd 714< +Dojheud gh ixqflrqhv frqwlqxdv, Vhd i = [ $ U > j = \ $ U frqwlqxdv fdgd xqd hq s 5 [ _ \ hqwrqfhv l, i . j> i j> i j vrq frqwlqxdv hq s ll, Vl j+s, 9@ 3 hqwrqfhv i@j hv frqwlqxd hq s Hq ho fdvr txh s vhd xq sxqwr gh dfxpxodflöq gh [ _ \ / od ghprvwudflöq vh rewlhqh idflophqwh gho dojheud gh oðplwhv1 Hq fdvr frqwudulr hv gluhfwd1 Frq hvwh ýowlpr whruhpd whqhprv/ sru hmhpsor/ txh orv srolqrplrv s+{, @ d3 . d4 { . d5 {5 . dq {q >
frq dl frqvwdqwhv uhdohv
vrq ixqflrqhv frqwlqxdv1 Uhfrugdqgr orv uhvxowdgr gh oðplwhv whqhprv txh odv vljxlhqwhv ixqflrqhv vrq frqwlqxdv+hq wrgr vx grplqlr, 41 vhq {1
51 frv {1
61 wdq { vl { 9@ 5 . n> 71 vhq { . 5 frv {1
n 5 ]1
81 6 frv { . fvf {1
91 frw { . : vhq +{, vhf {1
:1 vhq5 {1 ;1 frv5 {1
| pxfkdv rwudv frpelqdflrqhv gh ixqflrqhv wuljrqrpìwulfdv | srolqrplfdv txh vhd rshudflrqhv dojh0 eudlfdv hqwuh ixqflrqhv eävlfdv frpr vhq +{,> frv +{, | d{1
Whruhpd 7153 +Od frpsxhvwd gh ixqflrqhv frqwlqxdv hv frqwlqxd, Vhdq i = [ $ U frqwlqxd hq s/ j = \ $ U frqwlqxd hq t @ i+s, wdo txh i+[, \ hqwrqfhv j i = [ $ U hv frqwlqxd hq s Hvwh whruhpd qrv d|xgd d prvwudu txh odv vljxlhqwhv ixqflrqhv vrq frqwlqxdv+hq wrgr U,1
41 vhq {5 |d txh hv od frpsxhvwd gh vhq { | {5
51 frv +6{8 . :, |d txh hv od frpsxhvwd gh frv { | 6{8 . : 61 vhq +5 frv {, frpsxhvwd gh vhq { frq 5 frv {
71 6 vhq5 {6 . : frv {6 frpsxhvwd 6 vhq5 +{, . : frv +{,> {6
Hmhuflflr 646 Frqvlghu od ixqflöq i =` 4> 4^$ U ghqlgd sru ; 5 {?3 ? { frv+4@{, 3 {@3 i+{, @ = {6 3 ? {?4 5 vhq+{ , Hvwxglh od frqwlqxlgdg gh i1
Vroxflöq=
Hmhuflflr 647 l, Vhd { ? 3/ hqwrqfhv i +{, @ {5 frv+4@{,1 Sru frpsrvlflöq | surgxfwr gh ixqflrqhv frqwlqxd/ i hv frqwlqxd hq ` 4> 3^1
589
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
ll, Vhd 3 ? { ? 4/ hqwrqfhv I +{, @ frqwlqxd hq `3> 4^1
{6 sru frpsrvlflöq | fxrflhqwh gh ixqflrqhv frqwlqxd/ i hv {5 ,
vhq+
lll, Yhdprv od frqwlqxlgdg gh i hq { @ 31 Fodudphqwh 3 5 Grpi | i +3, @ 31 Frpr 3 hv sxqwr gh dfxpxodflöq gho Grpi / ghehprv yhu vl h{lvwh olp{$3 i +{,1 Sdud hvwr/ fdofxohprv orv oðplwhv odwhudohv
{6 {5 @ olp. { olp. i+{, @ olp. @34@3 5 vhq {5 {$3 {$3 vhq+{ , {$3 sru fhur dqltxlod1
olp i+{, @ olp {5 frv+4@{, @ 3 {$3 {$3
hv ghflu/ olp{$3 i +{, h{lvwh dghpäv
oxhjr i hv frqwlqxd hq { @ 31
olp i +{, @ olp i +{, @ 3 {$3. {$3 olp i +{, @ i+3, @ 3 {$3
Sru frqvljxlhqwh i hv frqwlqxd hq wrgr vx grplqlr
71616
Ixqflrqhv frqwlqxdv vreuh frpsdfwr
Vl i hv xqd ixqflöq frqwlqxd hq xq lqwhuydor fhuudgr | dfrwdgr +frpsdfwr, ^d> e` vh ghehuðd whqhu txh 41 Hq qlqjxq sxqwr gh ^d> e` kd| vdowrv/ |d txh vl orv kxelhud vdowr orv olplwhv odwhudohv vrq glvwlqwrv jxud gh pdv dedmr
r elhq ho oðplwh gh od ixqflöq hq ho sxqwr qr frlqflgh frq ho ydoru gh od ixqflöq hq ho sxqwr yhu od vljxhqwh jxud
7161 FRQWLQXLGDG
58:
51 Wdpsrfr od ixqflöq sxhgh lu d lqqlwr hq dojýq sxqwr gh ^d> e` / yhu jxud txh vljxh/ |d txh hq hvh fdvr oðplwh qr h{lvwh
61 Wdpsrfr sxhgh rvflodu hq xq sxqwr {3 +yhu jxud txh vljxh, |d txh ho oðplwh qr h{lvwluld1
58;
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
Dvð whqhprv +lqwxlwlydphqwh, txh od juäfd gh xqd ixqflöq frqwlqxd hq xq lqwhuydor fhuudgr | dfrwdgr ^d> e` gheh vhu gh wdo irupd txh xqr od sxhgh judfdu vlq ohydqwdu ho oäsl} hpsh}dqgr hq +d> i+d,, | whuplqdqgr hq +e> i+e,,1 Hq ho irqgr xqd ixqflöq frqwlqxd hq ^d> e` hv vöor xqd ghvirupdflöq gho lqwhuydor ^d> e`/ vlhqgr vx juäfd vrodphqwh xqd fxhugd qr lqqlwd txh xqh orv sxqwrv +d> i+d,, | +e> i+e,,1 Or dqwhulru qrv shuplwh fuhhu orv vljxlhqwhv whruhpdv1
Whruhpd 7154 gho Ydoru Lqwhuphglr Vhd i = ^d> e` $ U frqwlqxd1 Vl hv xq qýphur uhdo txh fxpsoh i +d, ? ? i +e, hqwrqfhv wlhqh do phqrv xqd suhlpdjhq hq ho lqwhuydor `d> e^1 +hv ghflu/ < f 5`d=e^ wdo txh i+f, @ , Frpr päv vh xwlol}d hvwh whruhpd/ hv hq od h{lvwhqfld gh udðfhv gh xqd ixqflöq1
Frurodulr 7155 +H{lvwhqfld gh Udlfhv, Vhd d> e 5 U wdo txh 4, i+d,i+e, ? 3 +hv ghflu/ i +d,| i+e, wlhqh glvwlqwrv vljqrv, 5, i frqwlqxd hq xq frqmxqwr txh lqfox|d do lqwhuydor ^d> e` hqwrqfhv i wlhqh xqd udl} hq ho lqwhuydor `d> e^ +hv ghflu/ h{lvwh f 5`d=e^ wdo txh i +f, @ 3 ,
Hmhpsor 648 Od ixqflöq i +{, @ {6 . {5 . 6{ . 5 wlhqh xqd udð} hq ho lqwhuydor ` 4> 3^ Hq hihfwr i+4, @ 4 . 4 6 . 5 @ 4 ? 3 | i+3, @ 5 A 3 oxhjr i +4,i+3, ? 31 Dghpäv i hv frqwlqxd hq ^4> 3` oxhjr sru frurodulr dqwhulru h{lvwh xqd udð} hq ho lqwhuydor ` 4> 3^1 Hmhuflflr 649 Suxheh txh od hfxdflöq
h{ 6{ @ 3
wlhqh do phqrv xqd udð} uhdo hq ho lqwhuydor `3> 4^1 Vroxflöq= Vhd i+{, @ h{ 6{ i+3, @ h3 6 3 @ 4 A 3 | i+4, @ h 6 ? 3 oxhjr i+3,i +4, ? 3 Dghpäv i hv xqd ixqflöq frqwlqxd hq U hq sduwlfxodu hq ^3> 4`/ oxhjr sru ho frurodulr dqwhulru/ h{lvwh f 5`3> 4^ wdo txh i+f, @ 3/ hv ghflu f hv xqd udð} r vroxflöq gh od hfxdflöq1
Frurodulr 7156 Vhd i frqwlqxd hq xq lqwhuydor fhuudgr L/ hqwrqfhv i+L, wdpelìq hv xq lqwhuydor fhuudgr1
7161 FRQWLQXLGDG
58<
Frurodulr 7157 Vhd i frqwlqxd hq xq lqwhuydor fhuudgr | dfrwdgr ^d> e`/ hqwrqfhv i +L, hv xq lqwhuydor fhuudgr | dfrwdgr/ sru or wdqwr i hv dfrwdgd1 Hmhuflflr 64:s Hvwxglh s od frqwlqxlgdg gh d, i +{, @ s { 5 . 5s { s e, i +{, @ {5 4 . 8 {+4 {, . {+4 {, D|xgd= Fxlgdgr frq orv grplqlrv1 f, i = ] $ U i+{, @ ^{` +sduwh hqwhud, g, i = U $ U i+{, @ ^{` +sduwh hqwhud,
Hmhuflflr 64; Hvwxglh od frqwlqxlgdg gh od ixqflöq hq orv sxqwrv s lqglfdgrv1 4 frv+ { , { 9@ 6 6 s @ 61 d, i +{, @ 3 {@6 6 > s @ 6 | s @ 31 e, i +{, @ m{{ 6m 4 +{ 6, frv+ { , { 9@ 6 6 s @ 6 | s @ 31 f, i +{, @ 3 {@6
Hmhuflflr 64 i hv frqwlqxd hq ho sxqwr lqglfdgr1 +{ , vhq+ {4 , { @ 9 d, i +{, @ s@ s@3 3 {@ > s @ 3> s @ 5 e, i +{, @ { f, i +{, @ {{57 s@3 s@5 Hmhpsor 653 Frqvlghuh od vljxlhqwh ixqflöq ; d oq+{ . h5 ,6 . e vhq5{7{ {?3 A A A A ? d{ . 6 {@3 i +{, @ A A A s4.e{5 4 A = 3?{? vhq {5
grqgh d | e vrq qýphurv uhdohv 4, Fdofxoh ho oðplwh odwhudo ghuhfkr gh i hq 3 5, Fdofxoh ho oðplwh odwhudo l}txlhgr gh i hq 3 6, Hqfxhwuh orv ydoruhv gh d | e txh kdfhq frqwlqxd d i hq 31 7, Ghwhuplqh sdud txì ydoru gh e h{lvwh ho oðplwh odwhudo l}txlhugr gh i hq Vroxflöq= 4,
4 . e{5 4 vhq5 { s s 4 . e{5 4 4 . e{5 . 4 s olp. 5 vhq { {$3 4 . e{5 . 4 5 +4 . e{ , 4 s olp {$3. vhq5 {+ 4 . e{5 . 4, e{5 s olp {$3. vhq5 {+ 4 . e{5 . 4, {5 4 olp. s e olp. 5 {$3 vhq { {$3 4 . e{5 . 4 4 e4 5 e 5
olp i+{, @ olp. {$3. {$3 @ @ @ @ @ olp i+{, @ {$3.
s
593
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV 5,
vhq 7{ , olp i+{, @ olp +d oq +{ . h5 ,6 . e 5{ {$3 {$3 vhq 7{ @ olp d oq +{ . h5 ,6 . olp 5{ {$3 {$3 5 vhq 7{ @ d oq +h5 ,6 . olp e {$3 5 5{ vhq 7{ @ 9d . 5e olp 7{ {$3 @ 9d . 5e 4 @ 9d . 5e olp i+{, @ 9d . 5e {$3.
6, i vhuä frqwlqxd hq 3 vl olp{$3 i +{, @ i +3, oxhjr olp{$3. i+{, @ olp{$3 @ i +3,/ hv ghflu/ e e @ 9d . 5e @6 5 5 Dvð e @ 9 | d @ 65 s e{54 710 olp{$ i+{, @ olp{$ 4. vhq {5 Vl e @ 3 hqwrqfhv i+{, @ 3 | olp{$ i +{, @ 3 Vl e 9@ 3 hqwrqfhv ho olplwh qr h{lvwh1 s Hq hihfwr> Vxsrqjdprv txh h{lvwh/ hv ghflu/ olp{$ i+{, @ O> O 5 U hqwrqfhv do fdofxodu olp{$ + 4 . e{5 4, gh grv irupdv glvwlqwdv whqhprv s 4 . e5 4 @
@ @
oxhjr hvwr hv xqd frqwudglfflöq1
@ @
s olp + 4 . e{5 4, {$ s 4 . e{5 4 olp +vhq5 {, vhq5 { {$ s 4 . e{5 4 olp olp +vhq5 {, vhq5 { {$ {$ O3 3
Hmhpsor 654 Ghwhuplqh vl odv vljxlhqwh dupdflrqhv vrq yhugdghudv r idovdv1 41 Od ixqflöq i+{, @ ^5{` hv glvfrqwlqxd vöor hq ] ^{` { 7 51 Od ixqflöq i+{, @ hv glvfrqwlqxd hq { {?7
G @ i{ 5 ] = { 7j
61 Vl i . j hv frqwlqxd hq d | i hv frqwlqxd hq d hqwrqfhv j hv frqwlqxd hq d1
71 Vhdq i> j ixqflrqhv ghqlgdv hq xqd yhflqgdg gh d/ wdo txh i hv glvfrqwlqxd d hqwrqfhv i . j hv glvfrqwlqxd hq d1
81 Vl i | j vrq glvfrqwlqxdv hq d hqwrqfhv i . j hv glvfrqwlqxd hq d1
7161 FRQWLQXLGDG
594
Vroxflöq= 41 Idovr= od ixqflöq hq h{lvwh1
4 5
hv glvfrqwlqxd |d txh olp{$ 4 . ^5{` @ 4 | olp{$ 4 ^5{` @ 3 oxhjr ho oðplwh qr 5 5
51 Idovr= i hv frqwlqxd hq 7 sxhv olp{$7. i +{, @ olp{$7. ^{` @ 7 dghpäv olp{$7 i+{, @ olp{$7 { @ 7 oxhjr olp{$7 i +{, @ 7 @ i+7,1 dghpäv hv fodur sru od judfd +yhu jxud vljxlhqwh,
txh i hv glvfrqwlqxd hq i{ 5 ] = { A 7j1
61 Yhugdghur= Sru dojheud gh oðplwhv whqhprv1 Vl i . j/ i vrq frqwlqxdv hq d hqwrqfhv +i . j, i hv frqwlqxd hq d/ hv ghflu j hv frqwlqxd hq d1
71 Idovr= H{lwhq i +{, @ ^{` | j+{, @ { ^{` ixqflrqhv ghqlgdv hq xqd yhflqgdg gh d @ 4 wdo txh i hv glvfrqwlqxd hq { @ 4 vlq hpedujr +i . j,+{, @ { hv frqwlqxd hq d @ 41 81 = Ho plvpr hmhpsor1
Hmhpsor 655 Frqvlghuh od ixqflöq i +{, @
; ? =
{5 .5{
vhq 5{ 4 oq+6{5 . h,4
@5 ? { ? 3 {@3 {A3
ÁHv i frqwlqxd hq { @ 3B Hq fdvr gh qr vhuor uhghqlu i vl hv srvleoh/ gh prgr txh i vhd frqwlqxd1 Vroxflöq= { @ 3 hv xq sxqwr gh dfxpxodflöq gho Grpi @` @5> 4^ oxhjr i vhuä frqwlqxd hq { @ 3 vl | vöor vl 4, 3 5 Grpi 5, olp{$3 i +{, h{lvwh1 6, olp{$3 i +{, @ i +3,1 Dvð= 4, Hv fodur txh 3 5 Grpi sxhv i +3, @ 4 5, Fdofxohprv olp{$3 i+{,
595
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
{5 . 5{ olp i+{, @ olp {$3 {$3 vhq +5{, {+{ . 5, @ olp {$3 5 frv { vhq +{, 4 {+{ . 5,, @ olp 5 {$3 vhq+{, frv +{, 4 { {.5 @ olp olp 5 {$3 vhq +{, {$3 frv { 5 @ 4 @ 4 5 olp i+{, @ olp. oq +6{5 . h,4 {$3. {$3
Sxhvwr txh od ixqflöq orjdulwpr hv frqwlqxd hq U. |
olp 6{5 . h @ h 5 U. {$3
olp i+{, @ olp. oq +6{5 . h,4 {$3. {$3 @ olp. oq +6{5 . h, {$3 @ oq olp. +6{5 . h, {$3 @ oq+h, @ 4
oxhjr h{lvwh olp{$3. i +{, @ 4 6, olp{$3 i +{, 9@ i+3, oxhjr i qr hv frqwlqxd hq { @ 31 Vlq hpedujr/ gdgr txh olp{$3 i +{, h{lvwh hqwrqfhv hv srvleoh uhghqlu i gh prgr txh i vhd frqwlqxd hq { @ 3/ gh od vljxlhqwh irupd= ; {5 .5{ ? vhq 5{ @5 ? { ? 3 i+{, @ 4 {@3 = {A3 oq+6{5 . h,4 Hmhpsor 656 Vhd i =`3> ^$ U od ixqflöq ghqlgd sru ; { ? frw { 5 frv { { ? @5 i +{, @ = d{ 4 { @5
BcTxì ydoru +vl h{lvwh, gheh dvljqduvh d od frqvwdqwh d sdud txh i vhd frqwlqxd hq 5 B Vroxflöq= i vhuä frqwlqxd hq @5 vl l,
olp i +{, @ olp i+{, @ i + , 5 {$ 5 {$ 5 .
{ i+{, @ olp olp frw { 5 frv { {$ 5 {$ 5 { vhq { @ olp 5 frv { {$ 5 frv { 5{ vhq { @ olp 5 frv { {$ 5
+7144,
7161 FRQWLQXLGDG
596
Vhd x @ 5 { vl { $ 5 hqwrqfhv x $ 3. l, olp olp i +{, @ {$ 5 {$ 5
5{ vhq { 5 frv {
5+@5 x, vhq +@5 x, 5 frv +@5 x, + 5x, frv x @ olp. 5 vhq x x$3 +frv x 4, 5x frv x @ olp. 5 vhq x x$3 frv x 4 x frv { @ olp olp. . x$3 5 vhq x {$3 vhq x @
sru rwud sduwh>
olp x$3.
frv x 4 olp x$3. 5 vhq x
dghpäv
frv5 x 4 olp. x$3 5 vhq x+frv x . 4, vhq5 x @ olp. x$3 5 vhq x+frv x . 4, vhq x @ olp. x$3 5+frv x . 4, @ 3 @
x frv x x olp @ olp. olp frv x @ 4 4 @ 4 x$3. vhq x x$3 vhq x x$3. Oxhjr sru dojheud gh olplwh olp {$ 5
5{ vhq { frv x 4 x frv { @ olp. olp. @ 4 5 frv { x$3 5 vhq x {$3 vhq x
ll, olp{$ 5 . i +{, @ olp{$ 5 . d{ 4 @ 5 d 4 Sru +l, | +ll, | sru 7144 vh wlhqh 4 @
sru or wdqwr whqhprv d @ 31
Hmhpsor 657 Vhd
d, BcHv i frqwlqxd hq { @ 3B e, BcHv i frqwlqxd hq UB i +3, f, H{lvwh G. @ olp{$3. i +{{, 3 i +3, g, H{lvwh G @ olp{$3 i +{{, 3
d 4 @ i+ , 5 5
; 6 ? { frv + {45 , { ? 3 3 {@3 i +{, @ = s 5 { . 5{6 { A 3
Vroxflöq= d, Fdofxohprv orv oðplwh odwhudohv
4 olp i +{, @ olp {6 frv + 5 , @ 3 { {$3 {$3
sxhv olp{$3 {6 @ 3 | frv + {45 , hv dfrwdgd1
597
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
olp i+{, @ olp {$3. {$3.
s {5 . 5{6 @ 3=
Oxhjr olp{$3. i +{, @ olp{$3 i +{, @ i +3, @ 3 Sru or wdqwr i hv frqwlqxd hq { @ 3 e, Vhd i4 +{, @ {6 frv + {45 , sdud { ? 3 frpr i4 hv surgxfwr gh ixqflrqhv frqwlqxdv hq ` 4> 3^ hqwrqfhv +sru dojheud gh ixqflrqhv frqwlqxdv, i4 hv frqwlqxd1 Sru vxpd | frpsrvlflöq gh ixqflrqhv/ vh wlhqh txh i5 hv frqwlqxd hq `3> 4^1 Dghpäv sru +d, i hv frqwlqxd hq { @ 3/ oxhjr i hv frqwlqxd hq U1 f, G.
@ @ @ @ @
@
Oxhjr h{lvwh G. 1 g, G
i +{, i +3, {3 s {5 . 5{6 3 olp. > sxhv sdud { A 3 {3 {$3 s {5 +4 . 5{, olp. { {$3 s m{m 4 . 5{ olp > sxhv sdud { A 3 m{m @ { { {$3. s +4 . 5{, olp {$3. 4 olp {$3.
i +{, i +3, {3 6 4 { frv+4@{5 , 3 @ olp sxhv i +{, @ {6 frv+ 5 , sdud { ? 3 {3 { {$3 4 @ olp {5 frv+ 5 , { {$3 @ 3 @
olp {$3
sxhv j+{, @ frv+4@{5 , hv dfrwdgd | olp{$3 {5 @ 31 Oxhjr h{lvwh G 1
Hmhuflflr 658 Ghwhuplqh d> e hq U gh prgr txh i vhd frqwlqxd hq { @ 3 ; vhq d{vhq e{ {?3 ? { d{ . 6 { @3 i+{, @ = es7{5 .{6 { A 3 { Vroxflöq=
Hmhuflflr 659
41 3 5 Grpi> i 3, @ 61
51 olp{$3 i +{, h{lvwh vl | vöor vl olp{$3. i+{, @ olp{$3 i +{,1 s e 7{5 . {6 olp i +{, @ olp. { {$3. {$3 s em{m 7 . { @ olp. {A3 { {$3 s @ olp e 7 . { @ 5e {$3.
7161 FRQWLQXLGDG
598
vhq +d{, vhq +e{, { vhq +d{, vhq +e{, olp e @ olp. d d{ e{ {$3 {$3 vhq +d{, vhq +e{, e olp @ d olp. d{ e{ {$3 {$3 @ de
olp i +{, @ olp {$3 {$3
sru or wdqwr d e @ 5e/ dghpäv i +3, @ 6/ oxhjr i vhud frqwlqxd hq { @ 3 vl d e @ 6 a 5e @ 6 +, d @ 3,B
Sdud lghqwlfdu od uhfwd wdqjhqwh O d od juäfd gh xqd ixqflöq hq xq sxqwr S / edvwd hvshflfdu od shqglhqwh p gh O/ |d txh ìvwd | ho sxqwr S ghwhuplqd frpsohwdphqwh d od uhfwd1 Sdud hqfrqwudu p/ frqvlghuduhprv odv uhfwdv OST / uhfwdv vhfdqwhv txh sdvd sru S 1
Vhd pST od shqglhqwh gh OS T 1 Fxdqgr T wlhqgh d S +S mr, sru od ghuhfkd jxud
txh vljxh
7171 UHFWD WDQJHQWH D OD FXUYD
59:
| sru od l}txlhugd jxud
vh revhuyd txh pST wlhqgh d p +shqglhqwh gh od uhfwd O,1 Hvwr qrv vxjlhuh txh vl pST wlhqgh d xq ydoru mr1 fxdqgr T wlhqgh d S / hqwrqfhv hvh ydoru ghehuðdprv xvdu sdud ghqlu od shqglhqwh p gh od uhfwd wdqjhqwh O hq S 1 Hv fodur txh od shqglhqwh gh od uhfwd vhfdqwh OST hv pST @
i +{, i +d, {d
|d txh sdvd sru orv sxqwrv +d> i+d,, | +{> i+{,,1 Frq hvwd qrwdflöq srghprv vxvwlwxlu od iudvh T wlhqgh d S % sru { wlhqgh d d%/ dvð oohjdprv d od vljxlhqwh ghqlflöq
Ghqlflöq 7158 Vhd i xqd ixqflöq ghqlgd hq xq lqwhuydor delhuwr txh frqwlhqh d d1 Od shqglhqwh p gh
59;
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
od uhfwd wdqjhqwh d od juäfd gh i hq ho sxqwr +d> i +d,, hv p @ olp {$d
vlhpsuh | fxdqgr ho oðplwh h{lvwd1+Yhu jxud ,
i +{, i+d, {d
| hq hvh fdvr vh gluä txh h{lvwh od uhfwd wdqjhqwh d od juäfd gh i hq ho sxqwr +d> i+d,,1
Hmhpsor 65; Vhd i +{, @ {6 | d xq qýphur uhdo1 Hqfrqwuduhprv= d, Od shqglhqwh pd gh od uhfwd wdqjhqwh d od juäfd gh i hq ho sxqwr +d> d6 ,1 e, Od hfxdflöq gh od uhfwd wdqjhqwh d od juäfd hq ho sxqwr +4> 4,1 f, Od hfxdflöq gh od uhfwd qrupdo +uhfwd shushqglfxodu d od uhfwd wdqjhqwh, d od juäfd hq ho sxqwr +4> 4,1 Vroxflöq= d,
pd
Dvð/ od shqglhqwh hv
i +{, i +d, {d {6 d6 @ olp {$d { d +{ d,+{5 . d{ . d5 , @ olp {$d {d @ olp +{5 . d{ . d5 , @ 6d5 {$d
@
olp {$d
pd @ 6d5
e, \d txh od shqglhqwh gh od uhfwd wdqjhqwh O hq ho sxqwr +4> 4, od srghprv rewhqhu frpr ho fdvr hvshfldo hq txh d @ 4/ whqhprv txh p @ 61 Hq jhqhudo od hfxdflöq gh od uhfwd txh sdvd sru +d> i+d,, | frq shqglhqwh p hv Hq qxhvwur fdvr r
| i +d, @ p+{ d, | 4 @ 6+{ 4, | @ 6{ 5
7171 UHFWD WDQJHQWH D OD FXUYD
59<
f, Vl p> p 9@ 3 hv od shqglhqwh gh od uhfwd wdqjhqwh | pQ od shqglhqwh gh od uhfwd qrupdo d od juäfd gh | @ i+{, hqwrqfhv vh gheh fxpsolu p pQ @ 4 Dvð/ od hfxdflöq gh od uhfwd qrupdo d od juäfd gh i+{, @ {6 hq +4> 3, hv/ | i +d, @ pQ +{ d, |4 @ 46 +{ 4, | @ 46 { . 76
+, +,
Revhuyh txh vl p @ 3 sdud dojýq sxqwr hqwrqfhv od uhfwd wdqjhqwh hv krul}rqwdo/ fx|d hfxdflöq hv | @ i +d,/ | hq hvwh fdvr od hfxdflöq gh od uhfwd qrupdo hv { @ d1 Hmhpsor 65< d, Ghprvwuduhprv txh
olp {$d
vhq+{, vhq+d, @ frv+d, {d
| e, Ghwhuplqhprv od hfxdflöq gh od uhfwd wdqjhqwh d | @ vhq+{, hq S @ + 5 > 4,1
Vroxflöq= Vhd x @ { d/ vl { $ d hqwrqfhv x $ 3/ oxhjr/ xvdqgr vxvwlwxflöq/ O @ @ @ @ @
vhq+{, vhq+d, {d vhq+x . d, vhq+d, olp x$3 x vhq+x, frv+d, . vhq+d, frv+x, vhq+d, olp x$3 x frv+x, 4 vhq+x, olp +frv+d, . vhq+d, , x$3 x x vhq+x, frv+x, 4 olp frv+d, . olp vhq+d, x$3 x$3 x x olp {$d
Fdofxodqgr sru vhsdudgr orv oðplwh whqhprv |
olp x$3 olp vhq+d, x$3
vhq+x, frv+d, vhq+x, @ frv+d, olp @ frv+d, x $ 3 x x
frv+x, 4 x
@
@ @ Oxhjr sru dojheud gh oðplwhv
frv+x, 4 x +frv+x, 4,+frv+x, . 4, vhq+d, olp x$3 x+frv+x, . 4, frv5 +x, 4 vhq+d, olp x$3 x+frv+x, . 4, vhq+x, vhq+x, vhq+d, olp olp x$3 x x$3 frv+x, . 4 vhq+d, 4 3 @ 3=
@ vhq+d, olp x$3
@
vhq+x, frv+x, 4 . olp vhq+d, x$3 x x vhq+x, frv+x, . 4 @ frv+d, olp . vhq+d, olp x$3 x x$3 x @ frv+d,
O @
olp frv+d, x$3
5:3
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
e, Frpr od shqglhqwh gh od uhfwd wdqjhqwh d od juäfd gh | @ vhq+{, hq fxdotxlhu sxqwr +d> vhq+d,, hvwä gdgd sru vhq+{, vhq+d, p @ olp {$d {d | sru +d, whqhprv txh p @ frv+d,/ uhhpsod}dqgr d @ 5 rewhqhprv txh p @ frv+ 5 , @ 3 | dvð od uhfwd wdqjhqwh hv> | 4 @ p+{ 5 , +, |4@3 +, |@4
Revhuyh txh od hfxdflöq gh od qrupdo hq S @ + 5 > 4, hv { @ 5 1
Hmhuflflr 663 Ghwhuplqh od shqglhqwh gh od uhfwd wdqjhqwh d od sduäerod | @ d{5 . e{ . f hq ho sxqwr +{3 > |3 , Vroxflöq=
p @ @ @ @ @ @ @
@
i+{, i+{3 , { {3 d{5 . e{ . f +d{53 . e{3 . f, olp {${3 { {3 d{5 d{53 . e{ e{3 olp {${3 { {3 5 d+{ {53 , . e+{ {3 , olp {${3 { {3 d+{ {3 ,+{ . {3 , . e+{ {3 , olp {${3 { {3 +{ {3 ,+d+{ . {3 , . e, olp {${3 { {3 olp d+{ . {3 , . e {${3 5d{3 . e olp {${3
Hmhuflflr 664 Xq htxlsr gh lqjhqlhurv gh fdplqrv glvhôd xq wudpr gh fduuhwhud txh gheh frqhfwdu xqd dxwrslvwd krul}rqwdo frq rwud gh shqglhqwh p @ 4@8 1 Ho hqodfh gheh/ uhdol}duvh vreuh xqd glvwdqfld krul}rqwdo gh ;33 slhv xvdqgr xqd fxuyd sdudeöolfd sdud xqlu orv sxqwrv D | E1 Rewhqjd xqd hfxdflöq gho wlsr | @ d{5 . e{ . f sdud od sduäerod uhvshfwlyd | ghwhuplqh odv frrughqdgdv gh E1
Vroxflöq= Odv frrughqdgdv gh D vrq +3> 3, | orv gh E vrq +;33> |3 , grqgh |3 @ d+;33,5 .e+;33,.f/ dghpäv frpr +3> 3, wdpelìq shuwhqhfh d od sduäerod/ gheh vdwlvidfhu od hfxdflöq gh hood/ dvð3 @ d 3 . e 3 . f/ gh grqgh f @ 31 sru hmhuflflr dqwhulru/ rewxylprv txh od shqglhqwh gh | @ d{5 . e{ . f hq +{3 > |3 , hv p @ 5d{3 . e/ oxhjr hq {3 @ ;33 hv p @ 5d ;33 . e | hq {3 @ 3 hv p @ e1 Dghpäv sdud frqhfwdu od dxwrslvwd gh shqglhqwh 4@8 frq od sduäerod/ gheh rfxuulu txh 5d ;33 . e @ 4@8
+7145,
e@3
+7146,
| sdud frqhfwdu od dxwrslvwd krul}rqwdo +gh shqglhqwh p @ 3, frq od sduäerod> Dvð/ gh 7145 | 7146 whqhprv
4 @d ;333
7171 UHFWD WDQJHQWH D OD FXUYD
5:4
4 > e @ 3> f @ 31 Sru rwud sduwh E @ +;333> |3 , shuwhqhfh d od sduäerod/ oxhjr Hq frqvhfxhqfld d @ ;333 vdwlvidfh vx hfxdflöq/ hv ghflu 4 +;33,5 @ ;3 |3 @ ;333
Sru or wdqwr odv frrughqdgdv gh E vrq +;33> ;3,1 Hmhpsor 665 Gdgd od ixqflöq
i +{, @
s {
; A A A A ?
{5 {
A A A A = s 6 ;{6 . {8 vhq+ ,
{
3?{? {3
d, BcHv i xqd ixqflöq frqwlqxd hq 3B e, BcH{lvwh od uhfwd wdqjhqwh d od fxuyd | @ i+{, hq ho sxqwr S @ +3> 3,B f, BcH{lvwh od uhfwd wdqjhqwh d od judfd gh | @ i+{, hq orv sxqwrv +7/5, | +04/05,B Vroxflöq= d, Gdgr txh i hvwä ghqlgd hq xqd yhflqgdg gh { @ 3/ | 3 5 Grpi vh gheh fxpsolu txh= olp i+{, @ olp i+{, @ i +3, {$3. {$3
Frpr
{5 olp i +{, @ olp {$3. {$3. vhq+{, { @ olp. { vhq+{, {$3
{ olp. { olp. {$3 {$3 vhq+{, @ 3 4 @ 3= @
| ho rwur oðplwh odwhudo hv
olp i +{, @ olp {$3 {$3
s 6 6 ;{ . {8 =
| dperv vrq ljxdohv d i+3, vh wlhqh txh
olp i+{, @ olp i+{, @ i +3, {$3. {$3
| sru or wdqwr i hv frqwlqxd hq { @ 3 e, Gdgr txh i hvwä ghqlgd hq xqd yhflqgdg gh { @ 31 Yhdprv vl h{lvwh i+{, i+3, p @ olp {$3 {3
Sdud srghu fdofxodu hvwh oðplwh/ or kduhprv fdofxodqgr orv oðplwhv odwhudohv olp {$3.
i +{, i +3, {5 { @ olp. @ olp @ 4> {3 {$3 { vhq+{, {$3. vhq+{,
5:5
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
sru rwud sduwh i +{, i +3, olp {3 {$3
@ @ @ @ @
s 6 ;{6 . {8 olp { {$3 s 6 {6 +; . {5 , olp { {$3 s { 6 ; . {5 olp { {$3 s 6 ; . {5 olp {$3 s 6 ;
Gdgr txh orv oðplwhv odwhudohv vrq glvwlqwrv whqhprv txh qr h{lvwh p @ olp {$3
i+{, i+3, > {3
oxhjr qr h{lvwh uhfwd wdqjhqwh d od fxuyd | @ i+{, hq ho sxqwr S s @ +3> 3,1 f, i hvwä ghqlgd hq xqd yhflqgdg gh { @ 7 sxhv i +{, @ { sdud { | ? 7> dghpäv +7> 5, s shuwhqhfh d od fxuyd | @ i +{,/ sxhv 7 @ 51 Frpr od shqglhqwh exvfdgd hv s 4 i+{, i+7, {5 {7 s @ olp @ olp @ olp {$7 {$7 { 7 {$7 +{ 7,+ { . 7, 7 {7
h{lvwh od uhfwd wdqjhqwh d od fxuyd hq S @ +7> 5,/ | vhd hfxdflöq hv= 4 | 5 @ +{ 7, 7
Hq ho fdvr gho sxqwr +4> 5, whqhprv txh ìo qr shuwhqhfh d od fxuyd | @ i +{, sxhv s s i+{, @ 6 ; 4 @ 6 < 9@ 5
oxhjr qr h{lvwh uhfwd wdqjhqwh d od fxuyd | @ i+{, hq ho sxqwr +4> 5,1
Hmhuflflr 666 Xvdqgr dojheud gh oðplwh | lghqwlgdghv wuljrqrphwulfdv fdofxodu ho oðplwh gh odv vljxlhqwhv ixqflrqhv hq d 41 i+{, @ frw {
51 i+{, @ vhf {
61 i+{, @ fvf {
d 9@ q > q 5 Q
d 9@ 5 . q > q 5 Q d 9@ q > q 5 Q
71 i+{, @ vhq +{ . 6 ,
81 i+{, @ wdq +5{ . ,
91 i+{, @
5 {
4 frv
{
d5U
d@3
d dsursldgr
Hmhuflflr 667 Vhd [ U> s 5 [ 3 > i> j = [ $ U ixqflrqhv wdohv txh olp{$s i +{, @ O | olp{$s j+{, @ P frq O ? P pxhvwuh txh h{lvwh xqd yhflqgdg gh s> Ys wdo txh i +{, ? j+{,
>
;{ 5 +Ys isj, _ [
D|xgd= Frpr O ? P vh sxhgh wrpdu % @ P 5O | hqfrqwudu sdud hvh % od yhflqgdg Ys
7171 UHFWD WDQJHQWH D OD FXUYD
5:6
Hmhuflflr 668 Vhd olp{$s i+{, @ O a O A 31 Pxhvwuh txh ;{ 5 +Ys isj, _ [ D|xgd= Xvdu uhvxowdgr dqwhulru
Hmhuflflr 669 Vhd i +{, j+{,> ;{ 5 [> { 9@ s a olp{$s i +{, @ O a olp{$s j+{, @ P 1 Pxhvwuh txh OP D|xgd= Xvdu whruhpd gho dsuhwöq d +j i ,+{,
Hmhuflflr 66: Vhd D U> i = D $ U> s 5 D | vxsrqjdprv txh h{lvwh olp{$s i+{, hqwrqfhv ;{ 5 +Ys isj, _ D +hv ghflu/ i hv dfrwdgd hq xqd yhflqgdg gh s, Hmhuflflr 66; Vhd
i +{, @
{ {5T 3 {5 9 T
Prvwudu txh olp{$3 i+{, @ 3 | olp{${3 i+{, qr h{lvwh vl {3 9@ 3 D|xgd= Sdud od vhjxqgd sduwh xvdu hmhuflflr 668 | sdud od sulphud sduwh xvdu mi+{,m m{m
5:7
FKDSWHU 71 OÐPLWHV GH IXQFLRQHV
Fkdswhu 8
Ghulydgd 814
Ghulydgd
Ghqlflöq 814 Vhd L U> i = L $ U | vhd s xq sxqwr gh dfxpxodflöq Gluhprv txh i hv ghulydeoh hq ho sxqwr s fxdqgr ho vljxlhqwh oðplwh h{lvwh olp {$s
i +{, i +s, {s
Hq ho fdvr txh ho oðplwh h{lvwd/ ho ydoru gh ho vh ghqrwd sru i 3 +s,/ lqglfdqgr gh hvwd irupd txh ho ydoru ghshqgh gho sxqwr s1 Ho ydoru i 3 +s, hv oodpdgr od ghulydgd gh i hq ho sxqwr s | vh xvdq odv vljxlhqwh qrwdflrqhv1 gi gi gi i 3 +s,> +s,> m{@s > ms g{ g{ g{ Revhuydflöq= 4, i 3 +s, sxhgh vhu fdofxodgr gh rwud irupd |d txh= d, Wrpdqgr k @ { s vh wlhqh i 3 +s, @ olp k$3
i+k . s, i +s, k
txh hv rwud gh odv irupdv txh dsduhfh od ghqlflöq gh od ghulydgd hq dojxqrv oleurv1 e, Xvdqgr xqd fdudfwhul}dflöq gh oðplwh yld vxfhvlrqhv vh wlhqh/ txh i hv ghulydeoh hq s vl vöor vl sdud fxdotxlhu vxfhvlöq +{q ,> +{q , 5 L isj frq olpq$4 {q @ s vh wlhqh txh h{lvwh ho vljxlhqwh oðplwh i 3 +s, @ olp q$4
i +{q , i+s, {q s
f, Jhqhudol}dqgr or dqwhulru/ whqhprv txh wdpelìq whqhprv i 3 +s, @ olp |$e
i +j+|,, i+s, j+|, s
grqgh j = [ $ L/ olp|$e j+|, @ s +reyldphqwh e 5 [ 3 , | j+|, 9@ s ;|> | 9@ e 5, i hv ghulydeoh hq s vl sdud od juäfd gh i h{lvwh od uhfwd wdqjhqwh> lqyhuvdphqwh vh sxhgh ghflu= Vl od ixqflöq hv ghulydeoh hq s vh wlhqh txh od uhfwd wdqjhqwh d od juäfd gh i hq +s> i +s,, h{lvwh1 Vl Xg1 frpsdud odv ghqlflrqhv gh uhfwd wdqjhqwh | ghulydgd/ vh sxhgh shufdwdu txh hv or plvpr1 6, Sru surslhgdg gh oðplwh vh wlhqh txh vl h{lvwh i 3 +s,/ ìo hv ýqlfr/ sru hvwr vh sxhgh xvdu ho duwðfxor od hq od ghulydgd/ rod uhfwd wdqjhqwh1 Gh od revhuydflöq 5 vh frqfox|h txh |d whqhprv ydulrv hmhpsorv gh ixqflrqhv ghulydeohv/ hoodv vrq odv gh orv hmhpsorv | hmhuflflrv gh uhfwdv wdqjhqwhv/ dghpäv srghprv qrwdu iäflophqwh txh od shqglhqwh p gh od uhfwd wdqjhqwh d i+{, hq ho sxqwr s4 vdwlvidfh 4 Fruuhfwdphqwh
glfkr hq ho sxqwr
Es> i Es1
5:8
5:9
FKDSWHU 81 GHULYDGD
p @ i 3 +s,
Sru or wdqwr> vl i hv ghulydeoh hq s vh wlhqh txh od hfxdflöq gh od uhfwd wdqjhqwh d od juäfd gh i hq ho sxqwr s hv | i +s, @ i 3 +s, {s | @ +i 3 +s,,{ . i+s, i 3 +s,s
Xvdqgr wuljrqrphwuðd vh wlhqh txh ho äqjxor txh irupd od uhfwd wdqjhqwh d od juäfd gh i hq s +hv ghflu/ hq +s> i+s,,,frq ho hmh gh odv { hvwd gdgr sru=
81414
@ dufwdq+i 3 +s,,
Hmhpsorv gh ixqflrqhv ghulydeohv
Hmhpsor 66< Yhuhprv txh od ixqflöq i+{, @ 6{ . 8 hv ghulydeoh hq ho sxqwr s @ :1
Vroxflöq=
i +: . k, i+:, k 6+: . k, . 8 +6 : . 8, @ olp k$3 k 6k @ 6= @ olp k$3 k Ho uhvxowdgr dqwhulru qr gheh vhu h{wudôr |d txh qdwxudophqwh vh wlhqh txh od uhfwd wdqjhqwh d od juäfd gh i +od uhfwd | @ 6{ . 8, hv hood plvpd/ sru or wdqwr odv shqglhqwhv ghehq frlqflglu1 BcFxäo hv od ghulydgd gh i hq rwurv sxqwrvB Sru ho dujxphqwr juäfr txh khprv gdgr hq ho säuudir dqwhulru ghehuðdprv whqhu txh i 3 +s, @ 6 ;s 5 U1 Frpsuxhehor Xg1 xvdqgr od ghqlflöq gh ghulydgd1 i 3 +:, @
olp k$3
Hmhpsor 673 Od ixqflöq j+{, @ 7 +ixqflöq frqvwdqwh, wlhqh frpr juäfd od uhfwd | @ 7 txh hv xqd uhfwd krul}rqwdo +sdudohod do hmh {, vlwxdgd do qlyho 7> reyldphqwh vx shqglhqwh hv fhur/ sru or wdqwr od ghulydgd hq wrgr sxqwr s gheh vhu qxod1 Frpsurehprv hvwr yðd od ghqlflöq1 Vhd s xq qýphur uhdo mr
j+{, j+s, {s 77 @ olp {$s { s @ olp 3 @ 3 {$s s Hmhpsor 674 Fdofxohprv dkrud od ghulydgd gh k+x, @ x hq ho sxqwr s @ 71 j3 +s, @
Vl hv txh h{lvwh k3 +7, hvwd gheh vhu ljxdo d k3 +s, @ @ @ @ @
olp {$s
k+x, k+7, x7 s s x 7 olp x$7 x 7 s s s s x 7 x. 7 s olp s x$7 x 7 x. 7 x7 4 s s olp x$7 x 7 x. 7 4 4 s @ 7 5 7 olp x$7
8141 GHULYDGD
5::
Vl uhshwlprv orv sdvrv dqwhulruhv xq sxqwr mr s +shur duelwudulr, gh U. rewhqguhprv txh 4 4 k3 +s, @ olp s s @ s x$s x . s 5 s
Xwlol}dqgr od juäfd gh k> yhu jxud txh vljxh/ qrwhprv txh od shqglhqwh gh O5 hv pd|ru txh od shqglhqwh gh O4 | d phglgd txh wrphprv s päv fhufd gho ruljhq rewhqguhprv shqglhqwhv/ fdgd yh} pd|ruhv/ khfkr txh srghprv frpsuredu wdpelìq hq od ghulydgd gh k txh k3 +s, hv xqd ixqflöq ghfuhflhqwh hq s +s 5 U. ,/ k3 +s, hv od ixqflöq txh do vhu hydoxdgd hq s qrv gä od shqglhqwh gh odv uhfwdv wdqjhqwhv d od juäfd gh od udð} fxdgudgd1
Ho hmhpsor dqwhulru qrv lqgxfh d ghqlu od ixqflöq ghulydgd% sru dexvr gh ohqjxdmh vh dfrvwxpeud oodpdu vlpsohphqwh od ghulydgd1
Ghqlflöq 815 gh i frpr
5
Vhd i = L $ U xqd ixqflöq ghulydeoh hq wrgr sxqwr gh M1 Vh ghqh od ixqflöq ghulydgd
Gh od ghqlflöq whqhprv txh M L1
i3 = M s
$ U :$ i 3 +s,
Hq orv hmhpsorv dqwhulruhv whqhprv fdofxodgdv odv ixqflrqhv ghulydgdv i +{, @ 6{ . :> L @ U> M @ U i 3 +{, @ 6 j+{, @ 7> L @ U> M @ U j3 +{, @ 3= s 4 . k3 +{, @ s k+{, @ {> L @ U. 3 > M @ U 5 {
Qrwh txh hq ho ýowlpr fdvr k+3, h{lvwh/ shur k3 +3, qr h{lvwh/ hv ghflu/ M L1
Hmhpsor 675 Vhd i +{, @ vhq+{, fdofxohprv od ixqflöq ghulydgd gh i 3 +{, Hq orv hmhpsorv gh fäofxor gh uhfwd wdqjhqwh sdud s 5 U rewxylprv i 3 +s, @ olp k$3
5 Xq
vhq+k . s, vhq+s, @ frv+s, k
s r fr pdv dghodqwh hq yh} gh ghflu od ixqflöq ghulydgd gh
i/
gluhprv vhqfloodphqwh od ghulydgd gh
i1
5:;
FKDSWHU 81 GHULYDGD
Dvð/ od ixqflöq ghulydgd gh i+{, @ vhq+{, hv i 3 +{, @ frv+{,1 Ho vljxlhqwh hmhpsor pxhvwud xqd ixqflöq txh qr hv ghulydeoh hq wrgr sxqwr/ d shvdu gh vhu frqwlqxd hq wrgr U1 Hmhpsor 676 Vhd i +{, @ m{ 7m1 Fdofxohprv vx ghulydgd hq wrgr sxqwr txh h{lvwd1
Fäofxor gh od ghulydgd hq {3 1 Vl { 7 ? 3 vh wlhqh txh i +{, @ { . 7 | hq fdvr frqwudulr i +{, @ { 7/ gh hvwd irupd= l, Vl {3 ? 7/ i 3 +{3 , @
olp {${3
@
olp {${3
@
olp {${3
i +{, i +{3 , { {3 { 7 +{3 . 7, { {3 { . {3 @ 4 { {3
ll, Vl {3 A 7/ i 3 +{3 , @ 4 +fäofxor dqäorjr d orv uhflhq khfkrv,1 lll, Vl {3 @ 7 i +{, i +{3 , i 3 +7, @ olp {${3 { {3 Shur frpr i wlhqh glvwlqwr frpsruwdplhqwr fhufd gho 7 sdud fdofxodu hvwh ýowlpr oðplwh ghehprv wrpdu oðplwhv odwhudohv | yhulfdu txh hoorv vrq ljxdohv i+{, i+7, {73 olp @ olp. @4 {7 {7 {$7. {$7 i +{, i +7, { . 7 3 @ olp @ 4 olp {7 {7 {$7 {$7
Frpr orv oðplwhv odwhudohv txh ghqhq i 3 +7, vrq glvwlqwrv qr whqhprv od h{lvwhqfld gh i 3 +7,/ dvð od ixqflöq ghulydgd gh i hv i 3 = U i7j $ U :$
{
i 3 +{, @
4 {A7 4 { ? 7
Qrwdu txh i hv frqwlqxd hq { @ 71 Revhuydflöq= Gh or dqwhulru/ srghprv dupdu txh vl i hv frqwlqxd hq xq sxqwr s/ qr qhfhvduldphqwh i hv ghulydeoh hq d1 Vlq hpedujr ho vljxlhqwh whruhpd qrv dupd txh vl i hv ghulydeoh hq xq sxqwr s/ hqwrqfhv i hv frqwlqxd hq s1 Whruhpd 816 Vl i hv ghulydeoh hq d/ hqwrqfhv i hv frqwlqxd hq d1 Ghprvwudflöq= Ghehprv ghprvwudu txh
olp i +{, @ i +d, {$d
r htxlydohqwhphqwh qrv edvwduðd prvwudu txh= Frpr
olp +i+{, i+d,, @ 3 {$d
+{ d, i+{, i+d, olp i+{, i+d, @ olp +i+{, i+d,, @ olp olp +{ d, @ i 3 +d, 3 @ 3 {$d {$d {$d +{ d, {$d {d
whqhprv txh hihfwlydphqwh i hv frqwlqxd hq d1
8141 GHULYDGD
5:<
Revhuyh txh ho whruhpd dqwhulru hv gh od irupd s , t/ od ixqflöq i +{, @ m{ 7m txh ylprv hq ho hmhpsor 676 txh qr hv ghulydeoh hq { @ 7/ vlq hpedujr hood hv frqwlqxd hq { @ 7/ qrv vluyh gh frqwudhmhpsor sdud od dupdflöq txh= frqwlqxlgdg hq xq sxqwr lpsolfd ghulydelolgdg hq ho sxqwr1 Qrwdflrqhv= Od ghulydgd gh i hq d hv ho oðplwh i +d . k, i+d, k i + d . k , i +d, hv oodpdgd ghulydgd odwhudo l}txlhugd hq d ho oðplwh odwhudo l}txlhugr gh i 3 +d,/ hv ghflu/ olpk$3 k 3 +d,1 Wdpelìq vh ghqh od ghulydgd odwhudo ghuhfkd hq d txh fruuhvsrqgh do oðplwh odwhudo | vh ghqrwd sru i 3 +d,1 ghuhfkr | vh ghqrwd sru i. i 3 +d, @ olp k$3
Hmhuflflr 677 Fdofxodu od ghulydgd gh i+{,/ grqgh ; +{ 5,6 {A5 A A ? 5 { 7{ . 7 4 ? { 5 i+{, @ < { @ 4 A A = vhq+{ . 4, { ? 4
Fodudphqwh orv sxqwrv frq lfwlyrv vrq { @ 5 | { @ 4 |d txh hq yhflqgdghv gh hoodv/ i wlhqh glvwlqwr frpsruwdplhqwr | hvwh khfkr wudmr glfxowdghv hq ho hmhpsor dqwhulru1 d, Yhdprv hq sulphu oxjdu vl i hv ghulydeoh hq s @ 5/ sdud hoor qhfhvlwdprv fdofxodu ho vljxlhqwh oðplwh +vl hv txh h{lvwh,= i +{, i +5, @ i 3 +5, olp {$5 {5 Hvwh oðplwh ghehprv fdofxoduor phgldqwh orv oðplwhv odwhudohv
3 +5, @ olp i+{, i+5, @ olp +{ 5,6 3 @ 3 i. {5 {5 {$5. {$5. i+{, i+5, {5 7{ . 7 3 @ olp @3 olp {5 {5 {$5 {$5 Gh hvwd irupd whqhprv txh i hv ghulydeoh hq 5 | txh i 3 +5, @ 31 e, Dkrud yhdprv vl i hv ghulydeoh hq ho sxqwr { @ 41 Frpr i+{, i +4, {5 7{ . 7 < olp . @ olp . @ olp 5 { 8 @ 9 {.4 {.4 {$4 {$4 {$4 | frpr ho rwur oðplwh odwhudo1 olp {$4
i +{, i +4, vhq+{ . 4, < @ olp {.4 {.4 {$4
qr h{lvwh6 vh wlhqh txh i qr hv ghulydeoh hq 4 f, Yhdprv dkrud vl i hv ghulydeoh hq orv sxqwrv d txh vrq pd|ruhv txh 51 Hq hvwh fdvr od ixqflöq wrpd od irupd i+{, @ +{ 5,6 1 i +{, i +d, {d +{ 5,6 +d 5,6 @ olp {$d {d ^+{ 5, +d 5,`^+{ 5,5 . +{ 5,+d 5, . +d 5,5 @ olp {$d {d @ olp +{ 5,5 . +{ 5,+d 5, . +d 5,5 @ 6+d 5,5 {$d
i 3 +d, @
6 Sdud frq f
' f
olp {$d
q
suredu hvwr edvwd wrpdu od vxfhvlöq {
1
'
3 n 2q
q
{
< 3
q
> i E{
glyhujh1 Qrwdu txh ho oðplwh hv gh od irupd
f
f
5;3
FKDSWHU 81 GHULYDGD g, Fdofxohprv dkrud od ghulydgd hq ho sxqwr d/ d 5` 4> 5^ olp {$d
i +{, i +d, {5 7{ . 7 +d5 7d . 7, @ olp @ 5d 7 {$d {d {d
h, Ilqdophqwh/ fdofxohprv i 3 +d, fxdqgr d ? 4 i 3 +d, @ @ @ @ @
@
hv
vhq+{ . 4, vhq+d . 4, {d vhq+x . d . 4, vhq+d . 4, olp x$3 x vhq+x, frv+d . 4, . vhq+d . 4, frv+x, vhq+d . 4, olp x$3 x vhq+x, frv+x, 4 olp +frv+d . 4, . vhq+d . 4, , x$3 x x vhq+x, frv+x, 4 +frv+d . 4,, olp . vhq+d . 4, olp , x$3 x x$3 x frv+d . 4, olp {$d
Gh hvwd irupd/ od ixqflöq ghulydgd +txh frqvhjxlprv sru sduwh | dqdol}dprv hq orv sxqwrv frq lfwlyrv, ; {A5 6+{ 5,5 A A ? 3 {@5 i 3 +{, @ 5{ 7 4 ? {?5 A A = frv+{ . 4, { ? 4
Dkrud fdofxohprv dojxqdv ghulydgdv gh ixqflrqhv eävlfdv/ hq ho vhqwlgr txh srvwhulruphqwh qrv d|xgduäq d rewhqhu od ghulydgd gh rwudv ixqflrqhv1 Hmhpsor 678 Phgldqwh ho fäofxor gh odv ixqflrqhv ghulydgdv1 Mxvwltxh ho vljxlhqwh olvwdgr1 41 Vl i+{, @ n frqvwdqwh hqwrqfhv i 3 +{, @ 3 51 Vl i+{, @ { hqwrqfhv i 3 +{, @ 4
61 Vl i+{, @ vhq+{, hqwrqfhv i 3 +{, @ frv+{,
71 Vl i+{, @ frv+{, hqwrqfhv i 3 +{, @ vhq+{, s 81 Vl i+{, @ { hqwrqfhv i 3 +{, @ 5s4 { / vl { A 3
91 Vl i+{, @ {q hqwrqfhv i 3 +{, @ q{q4 / vl q 5 Q :1 Vl i+{, @ h{ hqwrqfhv i 3 +{, @ h{
Vroxflöq= Hq wrgrv orv fdvrv d lqglfduä xq hohphqwr jhqìulfr gho grplqlr gh od ixqflöq fruuhvsrqglhqwh1 41
i +{, i +d, {d nn @ olp {$d { d 3 @ olp @3 {$d { d
i 3 +d, @
Dvð/ i 3 +{, @ 3 hq ho fdvr i +{, @ n
olp {$d
8141 GHULYDGD
5;4
51
i +{, i +d, {d {d @ olp {$d { d @ olp 4 @ 4 {$d
i 3 +d, @
olp {$d
Sru or wdqwr od ixqflöq lghqwlgdg i +{, @ { wlhqh frpr ixqflöq ghulydgd i 3 +{, @ 41
61 Fdofxodgd dqwhulruphqwh/ hq orv hmhpsorv gh ixqflrqhv ghulydeohv1 71
i 3 +d, @ @ @ @
@
i+d . k, i+d, k frv+d . k, frv+d, olp k$3 k frv+d, frv+k, vhq+d, vhq+k, frv+d, olp k$3 k frv+d,+frv+k, 4, vhq+d, vhq+k, olp k$3 k k +frv+d,,3 +vhq+d,,4 @ vhq+d, olp k$3
g frv+{, @ vhq+{, Gh hvwd irupd g{
81 Fdofxodgd dqwhulruphqwh frpr hmhpsor gh ixqflrqhv ghulydeohv/ gh hvwd irupd s 4 + {,3 @ s 5 {
91 i 3 +d, @ @ @ @ @ @
vl { A 3
i +d . k, i +d, k +d . k,q dq olp k$3 k S q dq . qn@4 dqn kn dq n olp k$3 k Sq q q 4 kqd . n@5 dqn kn n olp k$3 k q [ q q 4 . olp qd dqn kn4 n k$3 n@5 qdq4 . 3 olp k$3
g {q @ q{q4 vl q 5 Q Dvð/ g{ Hvwd iöupxod wdpelìq hv yäolgd sdud srwhqfldv glvwlqwdv gh xq qýphur qdwxudo shur od ghprvwudflöq khfkd hv vöor yäolgd sdud q 5 Q
5;5
FKDSWHU 81 GHULYDGD
:1
i+d . k, i+d, k hd.k hd @ olp k$3 k hk 4 d @ h olp k$3 k @ hd 4 @ hd
i 3 +d, @
{ d Gh hvwd irupd gh g{ m{@d @ h
81415
olp k$3
Dojheud gh Ghulydgdv
Whruhpd 817 Vl i hv ghulydeoh hq ho sxqwr d/ hqwrqfhv od ixqflöq i frq 5 U mr wdpelìq hv ghulydeoh hq d | +i ,3 +d, @ ++i ,3 +d,,
Ghprvwudflöq=
gi m{@d g{
i+d . k, i+d, k i +d . k, i+d, , @ olp + k$3 k i +d . k, i +d, @ olp k$3 k @
olp k$3
Frpr i hv ghulydeoh/ ho ýowlpr oðplwh hv vlpsohphqwh od ghulydgd gh i hq ho sxqwr d1 Dvð/ gi gi m{@d @ m{@d g{ g{
txh hv or plvpr txh
+i ,3 +d, @ ++i ,3 +d,,
Hmhpsor 679 Xvdqgr hvwh whruhpd | orv uhvxowdgrv dqwhulruhv whqhprv txh 41 +6 vhq+{,,3 @ 6+vhq+{,,3 @ 6 frv+{,
g +:{7 , @ : g +{7 , @ :+7{6 , @ 5;{6 51 g{ g{
g + frv+{,, @ g +frv+{,, @ + vhq+{,, @ vhq+{, 61 g{ g{
g + i +{, @ :{6 . {/ i hv ghulydeoh hq U. | Uhfi U @ Grpj sru or wdqwr/ sdud d 5 U.
|d txh j3 +{, @ vhq { vh wlhqh
+j i ,3 +d, @ j3 +i+d,,i 3 +d, s 4 @ j3 +:d6 . d, +54d5 . s , 5 d
s 4 d,, +54d5 . s , 5 d s 5 Hmhuflflr 686 Vhd i 3 +{, @ {6 h{ / i+4, @ 3 | j+{, @ +{ . 4,5 . 6 Fdofxoh +j i ,3 +4,1 +j i,3 +d, @ + vhq+:d6 .
Vroxflöq= Sru uhjod gh od fdghqd Oxhjr
+j i ,3 +{, @ j 3 +i +{,, i 3 +{,
+j i ,3 +4, @ j 3 +i +4,, i 3 +4, 5 shur i 3 +4, @ 46 h4 @ h | j3 +{, @ s5+{.4,5 1 5 +{.4, .6 5 4 @ 1 gh grqgh j 3 +i+4,, @ j 3 +3, @ 5s 5 7 Sru or wdqwr/ uhhpsod}dqgr hq 814 +j i ,3 +4, @
+814,
4 h h@ 5 5
Hmhpsor 687 Vhd j+{, @ vhq+vhq {, grqgh j @ i i frq i +{, @ vhq { hqwrqfhv j3 +{, @ i 3 +i+{,,i 3 +{, @ frv+vhq {, frv {1 s Hmhpsor 688 Vhd i +{, @ 6{7 . frv+{, . { 5 U
Frpr i sxhgh vhu hvfulwd frpr od frpsxhvwd s gh grv ixqflrqhv päv vlpsohv/ frpr vljxh= i +{, @ k+6{7 . frv+{, . }, @ |5 vhq { . } 6 Ci C{ Ci C| Ci C}
@ | 5 frv { . 3 @ 5| vhq { . 3 @ 3 . 6} 5
Hmhpsor 698 Hq ho fdvr gho yroxphq gho flolqgur Y @ u5 o CY @ 5uo Cu
CY @ u5 Co
Hmhpsor 699 Vhd j+{> |> }, @ 6{5 . :{6 vhq | hqwrqfhv whqhprv txh Cj C{ Cj C| Cj C}
@ 9{ . 54{5 vhq | @ 3 . :{6 frv | @ 3
Wrphprv dkrud od ixqflöq i @ 8{5 .vhq {1 Fodudphqwh whqhprv/ sru or dqwhulru/ txh Ci C{ @ 43{.frv { Ci | txh C| @ 3 |d txh hq hvwh ýowlpr fdvr ghehprv shqvdu txh od yduldeoh hv |/ shur i vh frpsruwd frpr frqvwdqwh uhvshfwr d hvwd yduldeoh dvð txh vx ghulydgd hv fhur Ci i +| . k, i+|, 8{5 . vhq { +8{5 . vhq {, @ olp @ olp @3 k$3 k$3 C| k k
71 Qrwdflrqhv uhjod gh od fdghqd
Vhdq | @ i +x,> x @ j+{, vxsrqjdprv txh dpedv wlhqh ghulydgdv | txh hv srvleoh kdfhu od frpsxhvwd | @ i +j+{,,1 +hv ghflu/ vl gäq odv klsrwhvlv gh od uhjod gh od fdghqd,1 Hqwrqfhv C| Cy C| @ C{ Cy C|
Qrwdflrqdophqwh ho wìuplqr l}txlhugr gh od ljxdogdg vh uhhuh d od ghulydgd gh | fdofxodgd ghvsxhv gh kdehu khfkr od frpsxhvwd gh i | j +i j,/ hq fdpelr ho sulphu idfwru gho wìuplqrv ghuhfkr C| vh uhhuh d od ghulydgd gh | dqwhv gh kdfhu od frpsxhvwd1 Cy
Hmhpsor 69: Vhd | xqd ixqflöq gh {/ ghulydeoh frq uhvshfwr d {1 Fdofxohprv odv ghulydgdv txh vh lqglfdq
g +{7 | 6 , @ {7 6| g| . 7{6 |6 41 g{ g{ g +{ vhq |, @ vhq | . { frv | g| 51 g{ g{ g| , g 5 61 g{ +wj {|, @ +vhf +{|,,+| . { g{ g +5{| 6 . { vhq+{|,, @ 5| 6 . 9{| 5 g| . vhq +{|, . { frv+{|,+| . { g| , 71 g{ g{ g{
8141 GHULYDGD
g +s|, @ 81 g{ g +s|, @ 91 g|
5 e^ hqwrqfhv i . j hv fuhflhqwh hq `d> e^1
51 Vl i hv ghulydeoh hq j+d, | j hv frqwlqxd hq d shur qr ghulydeoh hq d hqwrqfhv i j qr hv ghulydeoh hq d1
Vroxflöq=
41 Yhugdghur Vl i | j vrq ixqflrqhv fuhflhqwh | ghulydeohv hq `d> e^ hqwrqfhv i 3 +{, 3 | j3 +{, 3 hq `d> e^/ oxhjr +i . j,3 +{, @ i 3 +{, . j 3 +{, 3 ;{ 5`d> e^ Sru or wdqwr i . j hv fuhflhqwh hq `d> e^1
51 Idovr1 Vhdq j+{, @ m{m/ i+{, @ {5 | d @ 3 j hv frqwlqxd hq 3 shur qr ghulydeohv hq 3 | i hv ghulydeoh hq j+3, @ 3/ shur +i j,+{, @ m{m5 @ {5 od fxdo hv ghulydeoh hq 3
Hmhpsor 69< Hq txì sxqwrv/ h{lvwh od ghulydgd gh od ixqflöq ; {3 ? frv { i+{, @ = s 6 4 5{ { ? 3 Vroxflöq=
d, Vl { A 3/ i +{, @ frv {1 Vdehprv txh frv { hv ghulydeoh sdud wrgr { 5 U hq sduwlfxodu sdud { A 3> i 3 +{, @ vhq+{,1 s e, Vl { ? 3> i +{, @ 6 4 5{1 Hq hvwh fdvr i hv od frpsxhvwd gh ixqflrqhv ghulydeohv | vx ghulydgd i 3 +{, @ 46 +4 5{,5@6 +5,/ sru uhjod gh od fdghqd1
f, Vl { @ 3/ ghehprv dqdol}du sru ghqlflöq od ghulydgd gh i/ |d txh i hvwä ghqlgd gh irupd glvwlqwd sdud { A 3 | sdud { ? 3 i +3, Yhdprv vl h{lvwh i 3 +3, @ olp{$3 i +{{, 3 1 Sdud hvwr fdofxohprv
3 +3, @ i. @ @ @ @
@
i+{, i+3, {3 frv { 4 olp { {$3. frv5 { 4 olp {$3. {+frv { . 4, vhq5 { olp. {$3 {+frv { . 4, vhq { vhq { ,+ , olp + frv { . 4 {$3. { +4,+3, @ 3 olp {$3.
5 i+{, @ oxhjr
4 4 5 , . {6 + vhq+ 5 ,,+ 6 , 5 { { {
s {5 . 5{6 hv ghulydeoh sru frpsrvlflöq gh ixqflöqhv ghulydeohv/
4 +5{ . 9{5 , i 3 +{, @ s 5 5 { . 5{6
f, Sdud { @ 3/ ghehprv dqdol}du i 3 +3, phgldqwh od ghqlflöq/ |d txh i wlhqh glvwlqwr frpsruwdplhqwr hq wruqr do fhur/ hv ghflu/ ghehprv fdofxodu i 3 +3, @ olp {$3 i 3 +3, @ .
| ho rwur oðplwh odwhudo
i +{, i +3, {3
i+{, i +3, olp @ olp. {3 {$3. {$3
3 +3, @ olp i {$3
s {5 . 5{6 @4 {
i +{, i +3, {6 frv+4@{5 , @ olp @3 {3 { {$3
8141 GHULYDGD
5 5^ od ixqflöq i +{, @ ^{` . m{ 4m hv ghulydeohB
Vroxflöq= Vhsdudqgr ho grplqlr gh i dsursldgdphqwh rewhqhprv od vljxlhqwh ixqflöq ghqlgd sru sduwhv1 ; { 4 { ? 3 A A A A ? 4{ 3{?4 i +{, @ A A A A = { 4{?5
Fdofxohprv dkrud vx ghulydgd
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3 +4, +vl h{lvwh, txh sru ghqlflöq hv od ghulydgd gh i hq e, Hq { @ 4/ vöor wlhqh vhqwlgr ghwhuplqdu i. 41 i +{, i +4, {.4 { 4 @ olp @ 4 {$4. { . 4
3 +4, @ i.
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Hmhuflflr 6:5 Frqvlghuh od ixqflöq i+{, @
4
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grqgh d> e 5 U Ghwhuplqh d | e gh prgr txh i vhd ghulydeoh hq { @ 41 i +d, 3 3 Vroxflöq= i vhuä ghulydeoh hq { @ 4 vl | vöor vl olp{$4 i +{{, d h{lvwh +hv ghflu i.+4, @ i+4, ,1 i +{, i +4, {4 6 . d{ +6 . d, @ olp {4 {$4 d+{ 4, @ olp {$4 { 4 @ olp d @ d {$4
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i +{, i +4, {4 {5 . e +6 . d, @ olp. {4 {$4
3 +4, @ i.
olp {$4.
shur hvwh oðplwh srguld h{lvwlu vöor vl olp{$4. {5 . e +{ . d, @ 3 shur olp{$4 {5 . e +6 . d, @ 4 . e +6 . d,1 3 +4, srguld h{lvwlu vöor vl 4 . e +6 . d, @ 3 gh grqgh e @ 5 . d sru or wdqwr> oxhjr i.
{5 . e +6 . d, {4 5 { . 5 . d +6 . d, @ olp {4 {$4. {5 4 @ olp {$4. { 4 @ olp. { . 4 @ 5 {$4 3 +4, @ i3 +4, +, d @ 5/ oxhjr e @ d . 5 @ 71 Dvð i.
3 +4, @ i.
olp {$4.
5 d, / hv gh od irupd f Revhuydflöq= Vl olp{$4. {5 . e +6 . d, @ f frq f 9@ 3 hqwrqfhv olp{$4 { .e{+6. 4 3 oxhjr ho oðplwh qr h{vlwh Rwud irupd gh uhvroyhu ho sureohpd/ hv ghwhuplqdu frqglflrqhv sdud txh i vhd frqwlqxd +Vl qr or ixhud i qr vhuðd ghulydeoh,/ gh grqgh Rewhqhprv> olp i +{, @ olp i +{, @ i+4, {$4. {$4 or txh htxlydoh d ghwhuplqdu od frqglflöq
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Hmhuflflr 6:6 Vhd
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Vroxflöq= Sulphur ghwhuplqhprv frqglflrqhv sdud txh i vhd frqwlqxd hq { @ 3 olp i+{, @ olp i+{, @ olp i+{, @ i +3, {$3 {$3. {$3
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Sru or wdqwr D . E @ 9 Fdofxohprv i 3 +3, sru ghqlflöq i 3 +3, @ olp {$3
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8141 GHULYDGD
5 sxhv olp i 3 +{, @ d> olp. i 3 +{, @ 5 {$4 {$4
hqwrqfhv> i vhuä ghulydeoh hq { @ 4 vl
olp i 3 +{, @ olp i 3 +{, {$4. {$4
hv ghflu/ d @ 5 | frpr e @ 5 . d/ hqwrqfhv e @ 71 Dqdorjdphqwh hq ho hmhuflflr 6:6 {8 . {5 . 9 {?3 i+{, @ D+frv+6{, . 6 vhq {, . E+{ . 4, { 3
i hv frqwlqxd hq { @ 3 vl D . E @ 91 Dghpäv 8{7 . 5{ 3 i +{, @ D+6 vhq+6{, . 6 frv {, . E Orv oðplwhv olp{$3. i 3 +{, | olp{$3 i 3 +{, h{lvwhq sxhv
|
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olp i 3 +{, @ olp. D+6 vhq+6{, . 6 frv {, . E @ 6D . E {$3. {$3
oxhjr i vhuä ghulydeoh vl gh grqgh D @ 6 | E @ grqgh d @ i 4 +e, g{ i +d,
Revhuydflöq= Vxsrqjdprv txh i | j vrq ixqflrqhv ghulydeohv | txh +i j, @ Lg hqwrqfhv qhfhvduldphqwh vxv ghulydgdv vrq qr qxodv1 Sru uhjod gh od fdghqd vh wlhqh i 3 +j+{,,j 3 +{, @ 4
| frpr vh wudwd gh od pxowlsolfdflöq gh grv qýphurv uhdohv/ dperv ghehq vhu qr qxorv/ pdv dýq/ xqr gheh vhu ho lqyhuvr pxowlsolfdwlyr gho rwur1 Uhfrughprv txh vhqr hv xqd ixqflöq gh U hq U txh qr hv lq|hfwlyd | wdpsrfr hv vreuh|hfwlyd/ frpr vroxflöq d hvwr ýowlpr srguðdprv frqvlghuduod frpr vhq = U $ ^4> 4`/ shur ho sureohpd gh od lq|hfwlylgdg vhq dýq shupdqhfh1 Hq ghqlwlyd qr vh sxhgh shqvdu txh vhq whqjd lqyhuvd gh ^4> 4` hq U/ shur gg{ +{,m{@ @ 4 txh hv glvwlqwr gh 3/ oxhjr ho whruhpd gh od ixqflöq lqyhuvd qrv jdudqwl}d od h{lvwhqfld gh xqd lqyhuvd1 BcFxäo hv od frqwudglfflöq hqwuh hvwdv ýowlpdv grv iudvhvB Qlqjxqd/ sxhvwr txh ho whruhpd gh od ixqflöq lqyhuvd qr kdeod gh xqd ixqflöq lqyhuvd gh vhq frq grplqlr hq ^4> 4` | uhfruulgr U/ vlqr txh qrv jdudqwl}d od h{lvwhqfld gh xqd lqyhuvd orfdo1 Hvwr vljqlfd txh h{lvwh M yhflqgdg gh i +d, | K yhflqgdg gh d wdo txh i = K $ M wlhqh lqyhuvd1 Hq ho fdvr gh i+{, @ vhq { h{lvw M yhflqgdg gh fhur grqgh i+, @ 3 | K yhflqgdg gh wdo txh vhq = K $ M wlhqh lqyhuvd1 Frq ho whruhpd gh od ixqflrq lqyhuvd srghprv orjudu yduldu lqyhuvdv orfdohv |d txh edvwd wrpdu sxqwrv {3 txh vdwlvidjdq od klsrwhvlv g vhq { m{3 9@ 3 g{ Hq orv sxqwrv grqgh vh dqxod od ghulydgd gh vhqr +sru hmhpsor hq @5, hv lpsrvleoh hqfrqwudu xqd lqyhuvd orfdo gh hood/ |d txh sdud wrgr lqwhuydor delhuwr txh frqwhqjd ho sxqwr/ vhqr vlhpsuh vhuä qr lq|hfwlyd1 \d txh khprv kdeodgr gh od ixqflöq i +{, @ vhq {/ fdofxohprv od ghulydgd gh i 4 +{, @ dufvhq {/ sdud { dsursldgr hq ho hmhuflflr txh vljxh Hmhuflflr 6:: Fdofxodu od ghulydgd gh i 4 +{, @ dufvhq {
8151 GHULYDGD GH RUGHQ VXSHULRU
636
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dufvhq3 { @ s
4 4 {5
Hq fdpelr vl frqvlghudprv vhq =`@5> 6@5^$` 4> 4^ hqwrqfhv frv d ? 3 vl d 5`@5> 6@5^ | +i 4 ,3 @ shur s s frv d @ 4 vhq5 d @ 4 vhq+dufvhq e,,
4 frv
d
sru or wdqwr vl frqvlghudprv i 4 sru dufvhq {/ vh whqguä
4 +dufvhq,3 +{, @ s 4 {5
frqvlghudqgr vhq =`@5> 6@5^$` 4> 4^= Revhuydflöq= Vh dfrvwxpeud d uhvhuydu dufvhq d od lqyhuvd gh od ixqflöq vhqr ghqd hq vx udpd sulqflsdo/ hv ghflu vhq =` @5> @5^$` 4> 4^ |
4 +dufvhq,3 +{, @ s 4 {5
Dqäorjdphqwh vh fdofxod odv ghulydgd gh odv lqyhuvdv gh frvhqr/ frwdqjhqwh/ vhfdqwh | frvhfdqwh/ ghqlgd hq vx udpd sulqflsdo rewhqlhqgr=
815
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Ghulydgd gh Rughq Vxshulru
Frqvlghuhprv od ixqflöq i+{, @ 6{7 . 8 vhq5 {/ txh frpr |d khprv ylvwr vx ghulydgd hv gi +{, @ 45{6 . 43 vhq { frv { @ j+{, g{
Srghprv qrwdu txh j wdpelìq hv ghulydeoh | wlhqh frpr ghulydgd d gj +{, @ 69{5 . 43 frv5 { 43 vhq5 { @ k+{,/ qxhydphqwh srghprv fdofxodu od ghulydgd gh hvwd | rewhqhu g{ gk +{,= g{ gi od ixqflöq k hv od ghulydgd gh vhjxqgr rughq Vh glfh txh j hv od ghulydgd gh sulphu rughq | vh dqrwd g{ 5 gi / gk fruuhvsrqgh d od ghulydgd gh whufhu rughq gh i | vh dqrwd gi 6 1 gh i | vh dqrwd g{ 5 g{ g{6
gi vx ghulydgd/ vh ghqh od ghulydgd gh Ghqlflöq 8145 Vhd i xqd ixqflöq ghulydeoh hq `d> e^ | vhd g{ vhjxqgr rughq gh i hq f 5`d> e^ frpr gi +f . k, gi +f, g{ olp g{ k$3 k
637
FKDSWHU 81 GHULYDGD
5 | vh ghqrwd hvwh oðplwh sru gg{i5 +f,1 Vh glfh txh i wlhqh ghulydgd gh rughq 5 hq `d> e^ vl i wlhqh ghulydgd gh vhjxqgr rughq hq fdgd sxqwr gh `d> e^1 H{lvwlhqgr od ghulydgd gh vhjxqgr rughq gh i hq `d> e^ vh sxhgh ghqlu hq irupd dqdorjd od ghulydgd gh whufhu rughq hq f 5`d> e^ frpr ho vljxlhqwh oðplwh +vl hv txh h{lvwh ,1
g5 i +f . k, g5 i +f, 5 g{ g{5 olp k$3 k
dqäorjdphqwh vh ghqh od ghulydgd gh rughq q . 4 frpr ho oðplwh txh lqyroxfud od ghulydgd gh rughq q | q.4 vh dqrwduä sru gg{q.4i +f,1
Qrwdflrqhv=
G5 i +f, @ i 33 +f, @
g5 i +f,> g{6
Vh ghqhq orv vljxlhqwhv frqmxqwrv F 3 `d> e^ F 4 `d> e^ F 5 `d> e^ F q `d> e^
@ @ @ @
ii ii ii ii
=`d> e^$ U =`d> e^$ U =`d> e^$ U =`d> e^$ U
= = = =
G6 i +f, @ i 333 +f, @ i i i i
hv hv hv hv
g6 i +f,> g{6
Gq i+f, @ i +q, +f, @
gq i +f,= g{q
frqwlqxd hq `d> e^j ghulydeoh hq`d> e^ frq ghulydgd frqwlqxdj grv yhfhv ghulydeoh hq`d> e^ frq ghulydgd frqwlqxd j q yhfhv ghulydeohv hq`d> e^frq ghulydgd frqwlqxdj
Frq hvwrv frqmxqwrv srghprv ghqlu od vljxlhqwh ixqflöq/ oodpdgd rshudgru glihuhqfldeoh1
Dvð
G = F q `d> e^ $ F q4 `d> e^ gi +{, i :$ Gi +{, @ g{
41 vl i +{, @ 6{5 . vhq {/ Gi +{, @ 9{ . frv { s 4 51 vl j+{, @ oq+{ . 4, . { . 4/ Gj+{, @ {.4 . s 6 61 vl k+{, @ {5 . {6 / Gk+{, @ 56 {4@6 . 6{5 1
s4{.4 1
5
Qrwhprv txh od ixqflöq j 5 F 4 `3> 5^ shur Gj 95 F 4 ` 5> 5^1 Dqdorjdphqwh vh sxhgh ghqlu ho rshudgru glihuhqfldo gh rughq n/ Gn
Dvð
Gn = F q `d> e^ $ F qn `d> e^ > q n n i :$ Gn i+{, @ gg{in +{,
41 vl i +{, @ vhq 8{/ hqwrqfhv G7 i +{, @ 87 vhq 8{
51 vl i +{, @ h{ / hqwrqfhv G53 i +{, @ h{ 1
61 vl i +{, @ {8 / hqwrqfhv G: i +{, @ 3 | G8 i +{, @ 8$1
71 vl i +{, @ h5{ hqwrqfhv G: i+{, @ 5: h5{ 1
Frpr G3 > G4 > G5 > G6 > G7 > G8 > ===/ vrq ixqflrqhv/ vh sxhgh ghqlu od vxpd | pxowlsolfdflöq sru hvfdodu gh hoodv rewhqlhqgrvh rwud ixqflrqhv/ sru hmhpsor1
;G3 . 6G . :G8 7G6 = F q `d> e^ $ F q4 `d> e^ 8 6 gi i :$ ;i . g{ +{, . : gg{i6 +{, gg{i6 +{, Dtxð vh hqwlhqgh txh G3 = F q `d> e^$ F q `d> e^ | G3 i @ i 1
8151 GHULYDGD GH RUGHQ VXSHULRU
638
Hmhuflflr 6:; Vhd O @ 6G5 . G . 7G3 | i+{, @ {6 . :{ vhq {/ j+{, @ 8{5 . frv {/ k+{, @ oq { . wj {1 Fdofxoh Oi> Oj> Ok1 41
Oi Oi
g5 i gi . . 7i g{5 g{ g @ 6^9{ . +: vhq { . :{ frv {,` . 6{5 . :vhq{ . :{ frv { . 7{6 . 5;{ vhq { g{ @ 4;{ . 6^: frv { . : frv { :{ vhq {` . : vhq { . :{ frv { . 7{6 . 5;{ vhq { @ 4;{ . 7{6 . +75 . :{, frv { . +: . :{, vhq {
@ 6
51 Oj
Oj
61
g5 j gj . . 7j g{5 g{ @ 6^43 frv {` . 43{ vhq { . 53{5 . 7 frv { @ 63 . 53{5 . frv { . 43{ vhq {
@ 6
g5 k gk . 7k . g{5 g{ g 4 4 Ok @ 6 + . vhf5 {,` . . vhf5 { . 7 oq { . 7 wj { g{ { { 4 6 5 @ 5 . 9 vhf { wj { . . vhf5 { . 7 oq { . 7 wj { { {
Ok @ 6
Hmhuflflr 6:< Vhd O @ G . 6G3 / P @ G . 5G3 | Q @ G5 . 8G . 9G3 1 41 Vhd i +{, @ 8{7 . 7 vhq {> j+{, @ wj { vhf {> k+{, @ dufvhq {
Fdofxoh O P+i , @ O+P +i ,,> P O+i ,> Q+i, Frpsduh orv wuhv uhvxowdgr | gljd vl vrq ljxdohv r glvwlqwrv1 Uhslwh or dqwhulru shur hvwd yh} xvh odv ixqflrqhv j | k1
51 Wrph xqd ixqflöq jìqhulfd i 5 F 5 `d> e^ | suxheh txh
61 Vhd K @ G . 5/ suxheh txh K K K @ G6 . 9G5 . 45G . ;G3 1
Hmhuflflr 6;3 Frqvlghuh od hfxdflöq
g5 i .i @3 g{5 Xqd vroxflöq gh hood hq `d> e^/ hv xqd ixqflöq i 5 F 5 `d> e^ txh vdwlvidfh od ljxdogdg/ sru hmhpsor od ixqflöq i+{, @ vhq{ hv vroxflöq gh hood |d txh g5 vhq { . vhq { @ vhq { . vhq { @ 3 g{5
Pxhvwuh txh odv ixqflrqhv j+{, @ frv {/ k+{, @ 6 frv {/ o+{, @ : vhq { | hq jhqhudo fxdotxlhu ixqflöq q+{, @ vhq { . frv {/ grqgh > 5 U/ vrq vroxflrqhv gh od hfxdflöq Hmhuflflr 6;4
41 Pxhvwuh txh hd{ | {hd{ vrq vroxflrqhv gh od hfxdflrqhv g5 i gi 5d . d5 i @ 3 5 g{ g{
d frqvwdqwh
639
FKDSWHU 81 GHULYDGD
51 Pxhvwuh txh k+{, @ hd{ . {hd{ hv wdpelìq vroxflöq gh od hfxdflöq +fxdotxlhu vhd > 5 U,1
Hmhuflflr 6;5 hfxdflöq
41 Pxhvwuh txh 5{6 oq {/ {6 > {5 +5 oq { 4, ixqflrqhv gh F 5 `3> 4^ vrq vroxflrqhv gh od {5
gi g5 i 8{ . 4^/ 6{> {5 oq 44. { 4/ 5 vrq vroxflrqhv gh od hfxdflöq +4 {5 ,G5 i 5{Gi . 5i @ 3
Hmhuflflr 6;6 Pxhvwuh txh i +{, @ +5 . 5{ . {5 . fh{ ,4 hv vroxflöq gh od hfxdflöq gh Ehuqrxool gi . i @ +{i,5 g{
Hmhuflflr 6;7 Pxhvwuh txh | @ fwj { hv vroxflöq gh od hfxdflöq gh Ulffdwl | 3 vhq5 {| 5 .
| . frv5 { @ 3 vhq { frv {
Hmhuflflr 6;8 Pxhvwuh txh wrgdv odv ixqflrqhv fxelfdv | @ +{ . f,6 vrq vroxflrqhv gh od hfxdflöq
|3 @ 6| 5@6 s s Hmhuflflr 6;9 Pxhvwuh txh orv dufrv gh flufxorv |4 @ f5 {5 h |5 @ f5 {5 vrq vroxflrqhv gh od hfxdflöq g| { @ g{ |
816
Ixqflrqhv ghqlgdv lpsoðflwdphqwh
Hq jhrphwuðd dqäolwlfd/ khprv ylvwr txh od hfxdflöq
{5 | 5 @4 7 <
wlhqh frpr uhsuhvhqwdflöq juäfd xqd klsìuerod txh wlhqh frpr hmh sulqflsdo ho hmh { | yìuwlfhv +3> 5, | +3> 5,1 Vl uhfrugdprv od pdwhuld gh ixqflrqhv srghprv dupdu txh wrpdqgr dojxqrv vhfwruhv gh od klsìuerod vh sxhgh ghqlu ixqflrqhv/ sru hmhpsor/ u {5 |@6 4 7
frq { 5 ^5> 4^ r od plvpd h{suhvlöq shur wrpdqgr { hq ho frqmxqwr ` 4> 5` od fxdo vh rewlhqh wdpelìq do ghvshmdu | shur hvwd yh} wrpdqgr ho ydoru qhjdwlyr +edmr ho hmh { , | ho lqwhuydor ` 4> 5` +udpd l}txlhugd gh od klsìuerod,1 Dvð frpr hq ho fdvr gh od ixqflöq lqyhuvd/ ho remhwlyr gh hvwd sduwh qr hv ghwhuplqdu hq irupd wdq h{soðflwd od r odv srvleohv ixqflrqhv txh vh sxhghq ghqlu d sduwlu gh xqd uhodflöq txh lqyroxfuh grv yduldeohv +frpr ho fdvr gh od klsìuerod, |d txh pxfkdv yhfhv qr hv srvleoh frqvhjxlu xq ghvshmh gh xqd gh odv yduldeohv sdud yhuod hq irupd h{soðflwd frpr ixqflöq ghshqglhqwh gh od rwud1 Ho whruhpd gh od ixqflöq lpsolflwd gä uhvsxhvwd d odv suhjxqwdv= 410 BcGdgd xqd uhodflöq hqwuh { h |/ I +{> |, @ 3/ hv srvleoh hqfrqwudu xqd ixqflöq |+{,B 510 BcVh sxhgh jdudqwl}du od h{lvwhqfld gh wdo ixqflöq |+{,1 Fxäo hv od ghulydgd gh hoodB Qrwh txh od sulphud suhjxqwd qr glfh txh |+{, vh rewlhqh h{soð flwdphqwh hq wìuplqrv gh {/ vöor kdeod gh od h{lvwhqfld/ hq fdpelr od vhjxqgd suhjxqwd qrv shuqlwh fdofxodu vxv ghulydgd1
8161 IXQFLRQHV GHILQLGDV LPSOÐFLWDPHQWH
63:
Kdvwd ho prphqwr khprv wudedmdgr frq ixqflrqhv gh od irupd | @ i+{,/ hv ghflu/ | frpr ixqflöq gh {% grqgh od yduldeoh ghshqglhqwh | hvwä ghvshmdgd txhgdqgr xqd h{suhvlöq hq wìuplqrv gh od yduldeoh lqghshqglhqwh {/ hvwr vh dfrvwxpeud d ghflu txh | hv xqd ixqflöq h{soðflwd hq od yduldeoh {%1 Shur hq od suäfwlfd vh gäq fdvrv gh ixqflrqhv qr gdgdv h{soð flwdphqwh1 Yhdprv ho vljxlhqwh hmhpsor Xqd lqgxvwuld ghvhd ideulfdu wduurv gh frqvhuyd frq yroxphq mr gh 533 fp6 BcFýdohv vrq odv glphqvlrqhv gho wduurB Uhvsxhvwd= Qdwxudophqwh ho wduur hv xq flolqgur1 Vl wrpdprv frpr k od dowxud | u ho udglr gh wdo flolqgur whqhprv txh ku5 @ 533 Hv fodur txh ghshqglhqgr gho udglr u txh ghflgd wrpdu ho k txhgd wrwdophqwh ghwhuplqdgr/ | hq irupd ýqlfd/ dvð whqhprv od ixqflöq u $ k+u,> ;u 5 ^3> 4^1 Vxsrqjdprv dkrud txh ho suhflr gh od hwltxhwd hvwd gdgr sru S @ 83
gk . 43u gu
Hqwrqfhv qhfhvlwdprv fdofxodu gk gu 1 Hq hvwh fdvr hv srvleoh ghvshmdu h{soðflwdphqwh k ghvgh od irupxod gho yroxphq/ shur hvwr qr vlhpsuh rfxuuh Yhdprv xqr gh hvwrv fdvrv Frqvlghuhprv 5 hvshflhv gh shfhv txh frpsduwhq xq plvpr kdelwdw +xq odjr gho vxu gh Fkloh,1 Vxsrqhprv txh od hvshflh { +phglgd hq wrqhodgdv, vh dolphqwd gh od hvshflh päv shtxhôd | +phglgd hq wrqhodgdv,/ dghpäv txh dperv frpsduwhq orv qxwulhqwhv gho odjr1 Vxsrqhprv txh ho dsruwh gh od hvshflh shtxhôd yduðd qrwruldphqwh vlhqgr rvflodwrulr hq ho dôr1 Frpr ho odjr wlhqh xqd fdsdflgdg sdud dolphqwdu 433 ploorqhv gh wrqhodgdv whqhprv txh { . | @ 43;
| frpr od { vh dolphqwd gh od hvshflh shtxhôd | ho dsruwh gh hvwd yduðd rvflodwruldphqwh/ whqhprv txh { vdwlvidfh { @ dhe vhq+f|{, . n{
grqgh d> e> f | n vrq frqvwdqwhv1 Sru or wdqwr odv elrpdvdv hvwdq uhodflrqdgd hq od hfxdflöq
dhe vhq+f|{, . n{ . | @ 43;
+817,
Ho hpsuhvdulr h{sorwdgru gh od elrpdvd gho odjr/ prwlydgr sru orv suhflrv hq ho phufdgr ghvhd frqrfhu g| > g{ 1 odv wdvd gh fuhflplhqwr uhodwlyr gh odv sreodflrqhv g{ g| Vh yh hq od hfxdflöq 817 txh hv lpsrvleoh rewhqhu | frpr ixqflöq h{soðflwd gh { r { frpr ixqflöq h{soðflwd gh |1 g| B BcFöpr rewhqhu hqwrqfhv g{ Yhdprv hvwr xq xq frqwh{wr sxudphqwh pdwhpäwlfr1 Vxsrqjdprv txh whqhprv xqd ixqflöq uhdo gh grv yduldeohv ghqlgd hq U5 1 Sru hmhpsor= I =
# U U5 +{> |, :$ I +{> |, @ | 5 { 4
Frqvlghuhprv od hfxdflöq I +{> |, @ 3/ hv ghflu/ od hfxdflöq
|5 { 4 @ 3
fx|d juäfd hv xqd sduäerod frq hmh gh vlpìwuld ho hmh {1 Qrwhprv txh do ghvshmdu | hq wìuplqrv gh {/ rewhqhprv sru hmhpsor odv vljxlhqwhv ixqflrqhv fx|dv juäfdv hvwäq hq od jxud l}txlhugd | fruuhvsrqgh d glvwlqwrv vhfwruhv gh od sduäerod1 | @ i +{, @
s {.4
{ 5 ^4> 4^
63;
FKDSWHU 81 GHULYDGD
y
y
1.25
1.25
0 -2
0 0
2
4
-2
0
2
x
x
-1.25
-1.25
-2.5
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i +{, @
s {.4
s | @ j+{, @ { . 4 | @ k+{, @
4
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s j+{, @ { . 4
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k+{, | i +{, Dxqtxh hvwdv ixqflrqhv vrq glihuhqwhv wrgdv vh fdudfwhul}dq sru vdwlvidfhu od hfxdflöq I +{> |, @ 3 |d txh/ i +{,5 { 4 @ 3 j+{,5 { 4 @ 3 k+{,5 { 4 @ 3
{ 5 ^4> 4^ { 5 ^4> 4^ { 5 ^4> 4^
Odv ixqflrqhv i> j> k vh glfhq hvwdu ghqlgdv lpsoðflwdphqwh sru od hfxdflöq I +{> |, @ 3
Ghqlflöq 8146 Xqd hfxdflöq I +{> |, @ 3 ghqh lpsoðflwdphqwh d | frpr ixqflöq gh { vl | vöor vl h{lvwh xq lqwhuydor delhuwr7 L wdo txh I +{> i +{,, @ 3 ;{ 5 L Grpi 7 Qr qhfhvduldphqwh gheh frqvlghuduvh xq lqwhuydor shur sdud qxhvwur remhwlyr hvwd frqglflöq vlpsolfd orv fdvrv od h{ljlu txh od ixqflöq lpsoð flwd vhd ghulydeoh1
8161 IXQFLRQHV GHILQLGDV LPSOÐFLWDPHQWH
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Od hfxdflöq | 5 { 4 @ 3 ghwhuplqd pxfkdv ixqflrqhv lpsoðflwdv shur qrv lqwhuhvdq dtxhoodv glihu0 hqfldeohv hq dojýq lqwhuydor/ sru hmhpsor i | j ghqlgdv hq 818 vrq ixqflrqhv lpsoðflwdv glihuhqfldeohv hq wrgr lqwhuydor delhuwr frqwhqlgr hq vx grplqlr/ hq fdpelr k hv xqd ixqflöq ghqlgd lpsoðflwd shur qr hv frqwlqxd hq { @ 3 oxhjr qr hv glihuhqfldeoh hq fxdotxlhu lqwhuydor delhuwr txh frqwhqjd do { @ 31 Edmr txì frqglflrqhv h{lvwh xqd ixqflöq | @ i +{, glihuhqfldeoh ghqlgd lpsoðflwdphqwh sru xqd hfxdflöq I +{> |, @ 3
| BcFýdo hv vx ghulydgdB Ho whruhpd gh od ixqflöq lpsoðflwd qrv hqwuhjd frqglflrqhv sdud jdudqwl}du od h{lvwhqfld gh wdo ixqflöq glihuhqfldeoh/ shur dqwhv frqvlghuhprv ho sureohpd gh hqfrqwudu od ghulydgd gh i edmr ho vxsxhvwr txh vx ghulydgd h{lvwd1 Hmhpsor 6;: Vxsrqjdprv txh od hfxdflöq
+819,
|5 { 4 @ 3
ghqh lpsoðflwdphqwh d xqd ixqflöq | @ i +{, glihuhqfldeoh hq dojýq lqwhuydor1 Ghwhuplqduhprv | 3 ghulydqgr frq uhvshfwr d { dperv odgrv gh od hfxdflöq 819 frqvlghudqgr d | frpr ixqflöq gh {1 +,
+,
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m
g g{
5i +{, i 3 +{, 4 @ 3 4 vlhpsuh txh i+{, 9@ 3 i 3 +{, @ 5i+{,
Vhjxluhprv od frvwxpeuh | hq yh} gh kdfhu or dqwhulru vxehqwhqghprv txh | hv ixqflöq gh { qr h{solflwdqgr hvwh khfkr | txh | 3 lqglfd od ghulydgd gh hvd ixqflöq/ dvð srghprv ghulydu gh od vljxlhqwh irupd> +,
,
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m
g g{
Hvwh surfhglplhqwr vh ghqrplqd ghulydflöq lpsoðflwd1
Qrwh txh or fuxfldo hq hvwd ghulydflöq lpsoðflwd hv vxsrqhu txh | hv ghshqglhqwh gh {/ gh qr vhu dvð g|5 @ 3/ shur hq hvwh fdvr | ghshqgh gh {/ oxhjr ghehprv dsolfdu od uhjod gh od fdghqd g{ g| 5 g| @ 5| = g{ g{
Sru or wdqwr/ vl txhuhprv ghwhuplqdu od hfxdflöq gh od uhfwd wdqjhqwh d od sduäerod fxuyd | 5 { 4 @ 3 hq ho sxqwr S @ +3> 4,/ ghehprv 410 Yhulfdu txh ho sxqwr S shuwhqhfh d od fxuyd/ hv ghflu/ txh fxpsod frq od hfxdflöq I +{> |, @ 31 g| m @ i 3 +{ , grqgh S @ 510 Hydoxdu od ghulydgd | 3 hq ho sxqwr S sdud rewhqhu od shqglhqwh> p @ g{ S 3 +{3 > |3 , @ +3> 4, Dvð od uhfwd wdqjhqwh d od fxuyd 819 hq ho sxqwr S @ +3> 4, hvwä gdgd sru od hfxdflöq | |3 @ p+{ {3 ,
g| m @ 4 / dvð od hfxdflöq exvfdgd hv | 4 @ 4 {1 grqgh p @ g{ S 5 5
Hmhpsor 6;; Vxsrqjdprv txh | @ i+{, hq xqd ixqflöq ghqlgd lpsoðflwdphqwh sru od hfxdflöq I +{3 > |3 , @ 3 | txh i hv glihuhqfldeoh hq dojýq lqwhuydor1 g| sdud fdgd xqd gh odv vljxlhqwhv hfxdflrqhv | sdud ydoruhv { h | dsursldgrv Ghwhuplqhprv | 3 @ g{
643
FKDSWHU 81 GHULYDGD
41 {| 7 . {5 | . 6{6 | . 5 @ 3 51 | 6 @ oq+{|, . wdq+| 5 , . 4
61 | dufwdq+{, @ h{| . 5 5
71 | { @ frv+{5 . |, 5| . 6 81 {| @ vhq+{5 , . |
91 {| . |{ @ 5
Vroxflöq=
41 Ghulydqgr dperv odgrv uhvshfwr d { shqvdqgr txh | ghshqgh gh { {|7 . {5 | . 6{6 | . 5 @ 3
m
g g{
h lqglfdqgr frq sduhqwhvlv od ghulydgd gh fdgd surgxfwr/ whqhprv
+| 7 . {7|6 |3 , . +5{| . {5 |3 , . |, @ vhq+{5 , . | {| 91 I +{> |, @ {| . | { 5
g| fruuhvsrqgh do fxrflhqwh gh odv ghulydgdv sdufldohv Wdpelìq srghprv qrwdu txh od ghulydgd uhtxhulgd g{ gh od ixqflöq I pxowlsolfdgd sru 41 CI g| C{ @ CI g{ C| Yhdprv hq ho fdvr 41
CI C{ CI C|
Sru or wdqwr
51
@ | 7 . 5{| . |3 , @ 3
51 I hv xqd ixqflöq frqwlqxdphqwh glihuhqfldeoh hq xqd yhflqgdg gho sxqwr +{3 > |3 , 5 U5 + sru hmhpsor xq glvfr vlq rulood txh frqwhqjd d +{3 > |3 ,,1 61 CI C| +{3 > |3 , 9@ 31
Hqwrqfhv h{lvwh xqd ixqflöq lpsoðflwd | @ i +{, ghqlgd hq xq lqwhuydor delhuwr L txh frqwlhqh d {3 | fx|r uhfruulgr frqwlhqh d |3 wdo txh |3 @ i +{3 , | sru or wdqwr I +{3 > i +{3 ,, @ 3/ päv dýq I +{> i+{,, @ 3 sdud wrgr { 5 L +hv ghflu qr wdq vöor +{3 > |3 , vdwlvidfh I +{> |, @ 3 vlqr txh wdpelìq orv lqqlwrv sxqwrv +{> i +{,,> { 5 L1 \ qdophqwh/ od phqflrqdgd ixqflöq i hv ghulydeoh hq L | vx ghulydgd hvwä gdgd sru od h{suhvlöq
CI g| C{ @ CI g{ C|
Od ghprvwudflöq qr fruuhvsrqgh d xq fxuvr gh hvwh qlyho/ vlqr txh sdud xqr fruuhvsrqglhqwh d fäofxor hq yduldv yduldeohv1 Sdud fodulfdu xq srfr odv klsöwhvlv/ h{solfduhprv hvwd hq od revhuydflöq | hmhpsor txh vljxh1 Revhuydflöq= Ho frqfhswr gh glihuhqfldelolgdg sdud ixqflrqhv gh grv yduldeohv hv xq srfr päv frpsolfdgr txh kdvwd dkrud | vhjxur txh or yhuhprv hq xq fxuvr gh fäofxor hq yduldv yduldeohv1 Frpr hmhpsor gh ixqflrqhv gh grv yduldeohv frqwlqxdphqwh glihuhqfldeohv phqflrqduhprv odv vljxlhqwhv ixqflrqhv 41 Odv ixqflrqhv srolqrpldohv hq grv yduldeohv vrq frqwlqxdphqwh glihuhqfldeohv
51 Odv ixqflrqhv wdohv frpr vhqr/ frvhqr/ h{srqhqfldo/ orjdulwpr | udl} q0 ìvlpd vrq frqwlqxdphqwh glihuhqfldeohv hq vxv uhvshfwlyrv grplqlrv1
61 Od frpsxhvwd r dojxqd frpelqdflöq dojheudlfd +vxpd/ uhvwd/ surgxfwr/ r glylvlöq gh hoodv, wdpelìq vrq frqwlqxdphqwh glihuhqfldeoh hq vxv grplqlrv gh ghqlflöq1
Hmhpsor 6 |, @ {6 | . 7{5 | . |5 { . 4
51 I +{> |, @ vhq+{5 |, . h{| . |{ . 4 s 5 61 I +{> |, @ vhq+h{| , . {6 | . | 5 { . 4 71 I +{> |, @ { oq |
649
FKDSWHU 81 GHULYDGD
Hmhpsor 6 |, @ |5 vhq { h{.| . 4
hv frqwlqxdphqwh glihuhqfldeoh hq U5 / hv ghflu/ vdwlvidfh klsöwhvlv +5, gho whruhpd gh od ixqflöq lpsoðflwd
Ho sxqwr +{3 > |3 , @ +3> 3, vdwlvidfh od hfxdflöq I +{> |, @ 31 Ghwhuplqh vl h{lvwh xqd ixqflöq | @ i+{, ghqlgd lpsoðflwdphqwh sru od hfxdflöq I +{> |, @ 3 | glihuhqfldeoh hq xq lqwhuydor delhuwr gh {3 @ 3 Vroxflöq= Frqvlghudqgr od hfxdflöq I +{> |, @ 3/ hv ghflu/ |5 vhq { h{.| . 4 @ 3 | ghulydqgr lpsoðflwdphqwh frq uhvshfwr d { vh wlhqh>
5||3 vhq { . | 5 frv+{, +4 . |3 ,h{.| @ 3
or txh lpsolfd
g| h{.| |5 frv { @ g{ 5| vhq { h{.| CI CI Sxhvwr txh C| +3> |, @ 4/ |d txh C| +{> |, @ 5| vhq { h{.| | I +3> 3, @ 3 sru ho whruhpd gh od ixqflöq lpsoðflwd/ h{lvwh xqd ixqflöq | @ i +{, glihuhqfldeoh hq X +xq lqwhuydor delhuwr gh { @ 3,/ ghqlgd lpsoð flwdphqwh sru od hfxdflöq I +{> |, @ 3 | i +3, @ 31 |3 @
Hmhpsor 6 3,1 Vroxflöq= Sxhvwr txh S @ +3> 3, vdwlvidfh od hfxdflöq I +{> |, @ 3 | g| g{ hydoxdgd hq S @ +3> 3, h{lvwh/ hqwrqfhv h{lvwh od uhfwd wdqjhqwh hq ho sxqwr +3> 3,/ fx|d hfxdflöq hv | |3 @ p+{ {3 , g| grqgh +{3 > |3 , @ +3> 3, | p @ g{ m+3>3, @ 4/ hv ghflu/ | @ {
Hmhpsor 6 |, @ { . { oq | vhq+{|, . | 5
5
hv frqwlqxdphqwh glihuhqfldeoh hq xqd yhflqgdg gho sxqwr + 5 > 4, +or hv sru revhuydflöq dqwhulru,1 Ghwhuplqh vl h{lvwh xqd ixqflöq { @ i +|, +{ hq ixqflöq gh |, ghqlgd lpsoðflwdphqwh sru od hxdflöq I +{> |, @ 3 glihuhqfldeoh hq xq lqwhuydor delhuwr gh | @ 41 Vroxflöq=
8
Ghulydqgr lpsoðflwdphqwh frq uhvshfwr d | whqhprv
g @3 m 5 g| 4 g{ g{ g{ . oq | . { frv+{|,+| . {, . 5| @ 3 , g| g| | g| { g{ @ { frv+{|, 5| , +4 . oq | frv+{|,, g| | { 5| { frv+{|, g{ | @ , g| 4 . oq | | frv+{|, CI Sxhvwr CI C{ + 5 > 4, @ 4 9@ 3/ |d txh C{ +{> |, @ 4 . oq | | frv+{|, | I + 5 > 4, @ 5 . 5 oq 4 vhq+ 5 , . 4 5 @ 3 hqwrqfhv h{lvwh i = X U $ U | :$ { @ i +|, { . { oq | vhq+{|, . | 5
ixqflöq glihuhqfldeoh hq X +xq lqwhuydor delhuwr gh | @ 4,/ ghqlgd lpsoðflwdphqwh sru I +{> |, @ 31 8 Qrwh
txh dkrud vh slgh {
'
i E| / s ru or wdqwr ghe h vdwlvidfhuvh od kls öwhvlv
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8161 IXQFLRQHV GHILQLGDV LPSOÐFLWDPHQWH
64:
Hmhpsor 6 5, sdvh sru ho ruljhq/ vdelhqgr txh | @ i +{, hv xqd ixqflöq glihuhqfldeoh hq xq lqwhuydor txh frqwlhqh d { @ 4/ ghqlgd lpsoðflwdphqwh sru | 5 . n vhq+ |, @ {| . 5 5
Vroxflöq= Qrwhprv txh lqghshqglhqwh gho ydoru gh n/ S @ +4> 5, vdwlvidfh od hfxdflöq/ 55 . n+vhq+ 5, @ 4 5 . 5 5
Ghulydqgr lpsoðflwdphqwh frq uhvshfwr d {
g | 5 . n+vhq+ |, @ {| . 5 5 g{ , 5||3 . n frv+ |, |3 @ | . {| 3 5 5 n , +5| . frv+ |, {,| 3 @ | 5 5 | g| @ , |3 @ n g{ 5| . 5 frv+ 5 |, {
hydoxdqgr hvwd ghulydgd hq ho sxqwr S @ +4> 5, whqhprv od shqglhqwh gh od uhfwd wdqjhqwh d od juäfd hq ho sxqwr +4> 5,1 5 g| mS @ p@ g{ 7 n 4 5
Oxhjr hfxdflöq gh od uhfwd wdqjhqwh d od ixqflöq | @ i +{,/ ghqlgd lpsoðflwdphqwh/ hvwä gdgd sru |5@
5 +{ 4, 6 n 5
Hvwd uhfwd sdvd sru ho ruljhq vl | vöor vl ho sxqwr +3> 3, vdwlvidfh vx hfxdflöq/ hv ghflu/ 5 @
gh grqgh rewhqhprv ho ydoru gh n1
5 +4, 6 n 5
n@
7
Hmhpsor 733 Ghwhuplqh ho +r orv, sxqwr+v, grqgh od juäfd gh od uhodflöq {5 . | 5 @ 9{ 7| . 8
wlhqh wdqjhqwh+v, sdudohod+v, d od uhfwd | @ 7 . { +juäfdu od vlwxdflöq, Vroxflöq= Ghulydqgr lpsoðflwdphqwh dperv odgrv gh od hfxdflöq
frq uhvshfwr d {/ whqhprv>
{5 . | 5 @ 9{ 7| . 8
5{ . 5|| 3 +5| . 7,| 3 |3
@ 9 7| 3 @ 9 5{ 9 5{ > frq 5| . 7 9@ 3 @ 5| . 7
+8145,
64;
FKDSWHU 81 GHULYDGD
Od uhfwd wdqjhqwh d od juäfd hv sdudohod d od uhfwd | @ { . 7 vl | vöor vl | 3 +{3 , @ 4 sdud dojýq +{3 > |3 , shuwhqhflhqwh d od fxuyd1 Frpr +, +,
9 5{3 @4 5|3 . 7 9 5{3 @ 5|3 . 7 |3 @ 4 {3 | 3 +{3 , @
| ho sxqwr +{3 > |3 , gheh fxpsolu od hfxdflöq 8145/ whqhprv sru or wdqwr Uhhpsod}dqgr |3 @ 4 {3 whqhprv
{53 . |35 @ 9{3 7|3 . 8=
{53 . +4 {3 ,5 @ 9{3 7+4 {3 , . 8
gh grqgh
{3 @ 3 ö {3 @ 9
Fxdqgr {3 @ 3/ ho ydoru gh |3 hv 4 | fxdqgr {3 @ 9 vh wlhqh |3 @ 8 Sru or wdqwr orv sxqwrv shglgrv +yhu jxud BB, vrq S4 @ +3> 4,
Qrwhprv txh od uhodflöq
>
S5 @ +9> 8,
{5 . | 5 @ 9{ 7| . 8
hv htxlydohqwh d>
+{ 6,5 . +| . 5,5 @ 4;
oxhjr/ juäfdphqwh od uhodflöq hv xqd flufxqihuhqfld frq fhqwur hq +6> 5, | odv uhfwdv wdqjhqwhv sdudohodv d | @ 7 . { wlhqhq hfxdflöq |4@{ > |.8@{9 Hmhuflflr 734 Yhultxh vl od ixqflöq | @ i +{, ghqlgd lpsoðflwdphqwh sru +4 . {,h|@{ @ { hv vroxflöq gh {6 |% @ +{| 3 |,5 1
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+4 . {,h|@{
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@
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ghulydqgr qxhydphqwh hq irupd lpsoðflwd rewhqhprv
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+8147,
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gh grqgh {6 |% @ +|3 { |,5 1 Hq frqvhfxhqfld od ixqflöq ghqlgd lpsoðflwdphqwh sru +4 . {,h|@{ @ { hv vroxflöq gh od hfxdflöq glihuhqfldo {6 |% @ +{| 3 |,5
817
Uhjod gh O*Krslwdo
Vh glfh txh ho fxrflhqwh ij++{{,, hv gho wlsr 33 hq { @ s + r txh ho oðplwh hv gho wlsr 33 , vl dpedv ixqflrqhv i | j wlhqh oðplwh 3 fxdqgr { wlhqgh d s1 Vl orv oðplwhv gh i+{, | j+{, vrq 4 r 4 fxdqgr { wlhqgh d s/ vh glfh txh ij++{{,, wlhqh od irupd 4 lqghwhuplqdgd 4 4 r ho oðplwh hv gho wlsr 4 Whruhpd 8148 +Uhjod gh O*Krslwdo, Vhd Ys xqd yhflqgdg gh s1 Vhdq i | j ixqflrqhv ghulydeohv hq Ys isj1 Vl ij++{{,, wlhqh od irupd lqghwhuplqdgd 33 r elhq 4 4 hq { @ s hqwrqfhv i +{, i 3 +{, olp @ olp 3 {$s j+{, {$s j +{,
3 vlhpsuh txh olp{$s ij3 ++{{,, h{lvwd r elhq olp{$s
i 3 +{, @ 41 j3 +{,
Revhuydflöq= Od uhjod gh O*Krslwdo wdpelìq sxhgh dsolfduvh d orv oðplwhv odwhudohv1 Hmhpsor 735 Fdofxohprv xvdqgr O*Krslwdo Vroxflöq= Hvwh oðplwh hv gh od irupd oxhjr fdofxohprv O @ olp{$3
Sru or wdqwr
3{ {3
3 3
sxhv
olp {$3
olp vhq+{, @ 3 {$3
vhq+{, { >
olp { @ 3 {$3
vhq + , + ,
O @ olp {$3
frv+{, @ frv+3, @ 4 4
vhq+{, vhq3 +{, olp @4 @ olp {$3 +{, {$3 +{,3 3 Revhuydflöq= Vl olp{$s ij++{{,, hv gh od irupd 33 | olp{$s ij3 ++{{,, qr h{lvwh/ qr srghprv dupdu txh olp{$s ij++{{,, qr h{lvwh/ frpr yhuhprv hq ho hmhpsor txh vljxh1
653
FKDSWHU 81 GHULYDGD
Hmhpsor 736 Vhd i+{, @ {5 frv+4@{, > olp{$3 ij++{{,, hv gh od irupd 33 shur i 3 +{, olp 3 {$3 j +{,
j+{, @ {1 Frpr olp{$3 i +{, @ 3 | olp{$3 { @ 3 hqwrqfhv
5{ frv+4@{, . {5 + vhq+4@{,,+4@{5 , olp {$3 4 @ olp 5{ frv+4@{, . vhq+4@{, {$3
@
hvwh oðplwh qr h{lvwh/ sxhv olp{$3 5{ frv+4@{, h{lvwh shur olp{$3 vhq+4@{, qr h{lvwh +yhu hmhuflflr 636 fdsðwxor gh oðplwh | frqwlqxlgdg,1 3 Vlq hpedujr/ +d shvdu gh txh olp{$s ij3 ++{{,, qr h{lvwh,/ olp{$s ij++{{,, h{lvwh/ sxhv sru whruhpd gh fhur dqltxlod1 {5 frv+4@{, olp @ olp { frv+4@{, @ 3= {$3 {$3 {
Hmhpsor 737 Fdofxodu
O @ olp { oq { {$3.
{1 Vroxflöq= O @ olp{$3. oq 4@{ { hv Frpr olp{$3. oq { @ 4 +yhu juäfd gh orjdulwpr, | olp{$3. {4 @ 4 hqwrqfhv ho oðplwh olp{$3. oq 4@{ gh od irupd 4 4 oxhjr/ fdofxohprv +vl hv txh h{lvwh, +oq {,3 olp. {$3 +4@{,3
Sru or wdqwr
Hmhpsor 738 Fdofxodu
+oq {,3 olp. { oq { @ olp. @3 {$3 {$3 +4@{,3 orj { O @ olp. {$3 orj+vhq+{,,
Vroxflöq= Hvwh oðplwh hv gho wlsr 4 4 / sxhv olp{$3. orj { @ 4/ olp{$3. orj vhq+{,, @ 4 +orj {,3 olp . {$3 +orj vhq {,3
oxhjr Hmhuflflr 739 Fdofxodu
+4@{, olp {$3. +4@{5 , @ olp { @ 3 {$3.
@
@
olp {$3.
4
{ oq 43 4 frv{ vhq { oq 43 vhq { 4 @ olp @4 {$3 { frv {
orj+{, olp @4 {$3. orj+vhq+{,, d, olp {$3
dufvhq { {
e, olp {$3
dufvhq { 4@{5 {
Vroxflöq= d, Frpr olp{$3 dufvhq { @ 3 | olp{$3 { @ 3 oxhjr olp{$3 olp {$3
+dufvhq {,3 4 @ olp s @4 {$3 4 {5 {3
dufvhq
{
{ hv gh od irupd 3 / fdofxohprv 3
8171 UHJOD GH O*KRSLWDO
654
Sru od uhjod gh O*Krslwdo whqhprv
olp {$3
e, Fdofxohprv
dufvhq { @4 {
O @ olp h {5 oq+ { , {$3 dufvhq { qrwhprv txh olp{$3 oq+ { , @ oq 4 @ 3 | olp{$3 {5 @ 3/ oxhjr 4
olp {$3
hv gh od irupd 33 / yhdprv/ O4
@
olp {$3
@
olp {$3
dufvhq {
{ oq+ dufvhq { , {5
{ 3 +oq+ dufvhq { ,, +{5 ,3 s { dufvhq { 4{5 { dufvhq { {5 5{
s { 5 dufvhq { { olp olp 4{ {$3 dufvhq { {$3 5{6 { s 5 dufvhq { @ olp 4{ {$3 5{6
@
ho fxdo qxhydphqwh hv gh od irupd
3 3
olp {$3
4 s olp {$3 9+4 {5 , 4 {5
@
olp {$3 4 @ 9
s4{{5
sru +d,
dufvhq {
+5{6 ,3 s4{5 . s { 4{5 4{5 9{5
Dvð sru dojheud gh oðplwh | sru od uhjod gh O*Krslwdo vh wlhqh txh O4 @
3
s44 {5
4 9
| sru frqvljxlhqwh
O @ h4@9
Revhuydflöq= Od uhjod gh O*Krslwdo wdpelìq hv yäolgd vl { $ .4 r { $ 4 | dghpäv qrwh txh olp i+4@{, @ olp i+{, {$4 {$3. olp i+4@{, @ olp i+{, {$4 {$3
Whruhpd 8149 Vhdq i | j ixqflrqhv glihuhqfldeohv sdud wrgr { A Q / grqgh Q 5 U. | vxsöqjdprv txh sdud wrgr { A Q> j 3 +{, 9@ 31 hqwrqfhv d, Vl olp{$4 i +{, @ 3 | olp{$4 j+{, @ 3 +hv ghflu/ ij++{{,, wlhqh od irupd lqghwhuplqdgd 33 , | vl 3 olp{$4 ij3 ++{{,, @ O/ hqwrqfhv olp{$4 ij++{{,, @ O1 Dqäorjdphqwh vl olp{$4 i+{, @ 4 | olp{$4 j+{, @ 4 | 3 vl olp{$4 ij3 ++{{,, @ O/ hqwrqfhv olp{$4 ij++{{,, @ O1 e, Vl olp{$4 i+{, @ 3 | olp{$4 j+{, @ 3 +hv ghflu/ ij++{{,, wlhqh od irupd lqghwhuplqdgd 33 , | vl 3 olp{$4 ij3 ++{{,, @ O/ hqwrqfhv olp{$4 ij++{{,, @ O1
655
FKDSWHU 81 GHULYDGD
Dqäorjdphqwh vl olp{$4 i +{, @ 4 | olp{$4 j+{, @ 4 | vl 3 olp{$4 ij3 ++{{,, @ O/ hqwrqfhv olp{$4 ij++{{,, @ O1 Ho whruhpd wdpelìq hv yäolgr vl O vh vxvwlwx|h sru .4 r 4
Hmhpsor 73: Fdofxodu
olp {$4
5 dufwj { h6@{ 4
Vroxflöq= olp{$4 5 dufwj { @ 3 sxhv olp{$4 dufwj { @ 5 1 olp{$4 h6@{ 4 @ h3 4 @ 3/ oxhjr ho oð plwh hv gh od irupd 33 / dsolfdqgr O*Krslwdo olp {$4
gdgr txh
+ 5 dufwj {,3 +h6@{ 4,3
@
@
5 4 5 olp 6@{4.{6 {$4 h + {5 ,
5 {5 olp 6@{ 6 {$4 h +4 . {5 ,
{5 4 @ olp @4 olp {$4 4 . {5 {$4 {45 . 4
| olp{$4 h64@{ @ 4 hqwrqfhv sru dojheud gh oðplwh/
{5 olp 6@{ @4 {$4 h +4 . {5 ,
hq frqvhfxhqfld/ sru O*Krslwdo
olp {$4
Hmhpsor 73; Fdofxodu
5 5 dufwj { @ 6 h6@{ 4 {5 olp { {$4 h
Vroxflöq= \d txh olp{$4 {5 @ 4 | olp{$4 h{ @ 4/ dsolfdqgr od uhjod gh O*Krslwdo/ rewhqhprv shur olp{$4 5h{{ hv gh od irupd
sru or wdqwr
4 4
oxhjr
+{5 ,3 5{ olp { 3 @ olp { {$4 +h , {$4 h
+5{,3 5 olp { 3 @ olp { @ 3 {$4 +h , {$4 h {5 olp { @ 3 {$4 h
Revhuyhprv txh hq dojxqrv fdvrv ghehprv dsolfdu päv gh xqd yh} od uhjod gh O*Krslwdo1 Hmhpsor 73< Fdofxodu Hvwh oðplwh hv gho wlsr 44
olp +h{ {,4@{ {$3
5
8171 UHJOD GH O*KRSLWDO
656
Vroxflöq=
Fdofxohprv
5 olp oq+h{ {,4@{ {$3 4 @ olp 5 oq+h{ {, {$3 {
@
O4
hv gho wlsr
+8148,
5 { 4@{5 O @ olp +h{ {,4@{ @ olp hoq+h {, {$3 {$3
3 3
hqwrqfhv yhdprv vl h{lvwh
h{ 4 { h @ olp { {$3 5{ h{ 4 @ olp { {$3 +h {,5{
+oq+h{ {,,3 olp {$3 +{5 ,3 hv gho wlsr 33 / shur +h{ 4,3 olp {$3 +5{h{ 5{5 ,3 Sru or dqwhulru/ whqhprv
h{ olp { {$3 5{h . 5h{ 7{ 4 @ 5
@
5 4 O4 @ olp oq+h{ {,4@{ @ {$3 5
oxhjr hq 8148 xvdqdgr od frqwlqxlgdg gh od ixqflöq h{srqhqfldo vh wlhqh O @ holp{$3 oq+h
Sru or wdqwr
{ {,4@{5
@ h5 4
olp +h{ {,4@{ @ h4@5 {$3 5
Hmhpsor 743 Fdofxodu
olp + 5{,frv { {$+@5,
Hvwh oðplwh hv gho wlsr 33 Vroxflöq=
O@ Fdofxohprv
hoq+5{, olp + 5{,frv { @ olp {$+@5, +@5, $3 O4
olp oq+ 5{,frv { {$+@5, @ olp frv { oq+ 5{, {$+@5, oq+ 5{, @ olp {$+@5, vhf { @
frv {
+8149,
657
FKDSWHU 81 GHULYDGD
hv gho wlsr 4 4 Fdofxohprv olp {$+@5,
+oq+ 5{,,3 +vhf {,3
5
5{ olp {$+@5, vhf { wdq { 5 frv5 { @ olp {$+@5, + 5{,vhq{
@
hv gho wlsr 33 / oxhjr dsolfdqgr O*Krslwdo qxhydphqwh
Sru or wdqwr
+5 frv5 {,3 olp {$+@5, ++ 5{, vhq {,3
oxhjr hq 8149 vh wlhqh gh hvw irupd oohjdprv d txh Hmhpsor 744 Fdofxodu Hvwh oðplwh hv gho wlsr 43 Vroxflöq=
Fdofxohprv
O4 @
7 frv { vhq { olp {$+@5, + 5{, frv { 5 vhq { @ 3 @
olp oq+ 5{,frv { @ 3 {$+@5,
O @ holp{$+@5, oq+5{,
frv {
@ h3
olp + 5{,frv { @ h3 @ 4 {$3 olp +{ . h{ ,4@{ {$4
O @ olp +{ . h{ ,4@{ @ olp hoq+{.h {$4 {$4
{ ,4@{
@
olp oq+h{ . {,4@{ {$4 4 @ olp oq+{ . h{ , {$4 {
O4
shur hvwh oðplwh hv gho wlsr + 4 4 ,/ oxhjr dsolfdqgr O*Krslwdo +oq+{ . h{ ,,3 olp {$4 +{,3 hv gho wlsr
4 4
4.h{ { { @ olp .h {$4 4 4 . h{ @ olp {$4 { . h{
h , | vhuä ljxdo d olp{$4 ++4. {.h{ ,3 vlhpsuh txh h{lvwd hvwh ýowlpr/ {
3
+4 . h{ ,3 olp {$4 +{ . h{ ,3
h{ olp {$4 4 . h{ 4 @ olp {$4 4 . 4@h{ @ 4 @
Hq frqvhfxhqfld/ sru uhjod gh O*Krslwdo vh wlhqh txh
O4 @ olp oq+{ . h{ ,4@{ @ 4 {$4
8171 UHJOD GH O*KRSLWDO oxhjr hq BB rewhqhprv
658
O @ holp{$4 oq+{.h
Sru or wdqwr/
{ ,4@{
@ h4
olp +{ . h{ ,4@{ @ h4 {$4
41 Hmhuflflr 745 Fdofxoh ho ydoru gh orv vljxlhqwhv oðplwhv +O*Krslwdo, { {, 51 olp{$3 h {+4. 5 61 olp{$4 oq{{ q 71 olp{$4 {h{
81 olp{$3. { oq { 91 olp{$4 {+h4@{ 4,
:1 olp{$4 +4 . {5 ,{ ;1 olp{$3. {{ s 3,1
Hmhpsor 747 Xvdqgr ghulydgd hqfrqwudu ho yìuwlfh gh od sduäerod | @ {5 . 7{ ;
Vroxflöq= Hq ho yìuwlfh od wdqjhqwh wlhqh shqglhqwh fhur g| @ 5{ . 7 @ 3 g{
Sru or wdqwr ho yìuwlfh hvwä hq ho sxqwr +5> 45,1
+,
{ @ 5
Hmhpsor 748 Xvdqgr ghulydgd pxhvwuh txh hq xqd flufxqihuhqfld gh fhqwur +k> n, | udglr u1 +{ k,5 . +| n,5 @ u5
od uhfwd wdqjhqwh hq ho sxqwr S @ +{3 > |3 , hv shushqglfxodu d od uhfwd txh xqh S frq +k> n, +uhfwd udgldo,
Vroxflöq= Vl |3 @ n odv uhfwdv wdqjhqwhv vrq shushqglfxoduhv do hmh { frq hfxdflöq { @ u . k r { @ u k | od uhfwd udgldo hv | @ n sru or wdqwr vrq shushqglfxoduhv1 Vl |3 9@ n hqwrqfhv +,
5+{ k, . 5+| n, 5+{ k, g| @ g{ 5+| n,
g| @3 g{
k 1 Shqglhqwh gh uhfwd wdqjhqwh d od flufxqihuhqfld hq ho sxqwr S @ +{3 > |3 , hv {|33 n Od hfxdflöq gh od uhfwd wdqjhqwh d od flufxqihuhqfld hq S @ +{3 > |3 , hv= | |3 @
Od hfxdflöq gh od uhfwd udgldo hq ho sxqwr s hv
| fodudphqwh vrq shushqglfxoduhv1
|n @
+{3 k, +{ {3 , |3 n |3 n +{ k, {3 k
8181 DSOLFDFLRQHV GH OD GHULYDGD
81815
65:
Dsolfdflrqhv Iðvlfdv
Frqvlghuhprv ho sureohpd gh hqfrqwudu od yhorflgdg hq fxdotxlhu lqvwdqwh gh xq remhwr txh vh pxhyh vreuh xqd wud|hfwruld uhfwd +prylplhqwr uhfwlolqhr,1 Xqd iöupxod px| xwlol}dgd hq od uhvroxflöq gh sureohpdv dfhufd gho prylplhqwr gh xqd sduwðfxod vreuh xqd oðqhd uhfwd hv g y@ w grqgh y hv od yhorflgdg phgld gxudqwh xq lqwhuydor gh wlhpsr/ w od pdjqlwxg gho lqwhuydor gh wlhpsr | g hv od glvwdqfld wrwdo uhfruulgd1 Hmhpsoltxhprv hvwd vlwxdflöq1 Vxsrqjdprv txh xq dxwrpöylo vdoh gh od flxgdg D d odv 44=33 DP | yldmd d or odujr gh xqd fduuhwhud uhfwd kdvwd xqd flxgdg E txh vh hqfxhqwud d 453 np gh D/ oohjdqgr d odv 4=33 SP1 Dsolfdqgr od iöupxod y @ gw frq g @ 453> w @ 5/ yhprv txh od yhorflgdg phgld gho dxwrpöylo hv yphg @
453 @ 93np@kuv 5
Hvwr vljqlfd txh xq dxwrpöylo 93np@kuv txh pdqwhqjd vx yhorflgdg d 93np@kuv gxudqwh wrgr ho wlhpsr/ wrpdud 5 krudv hq uhfruuhu od glvwdqfld hqwuh odv grv flxgdghv1 Shur/ BcFxäo hv od yhorflgdg +lqvwdqwäqhd, txh pdufö ho yhorfðphwur +sru hmhpsor, d odv 44 = 63kuvB Qrwhprv txh ho dxwrpöylo sxgr kdehu hvwdgr hq uhsrvr +ghwhqlgr hq xq shdmh, r kdehu pdufdgr vx yhorfðphwur :3np@kuv x rwur ydoru1 Sdud ghwhuplqdu od yhorflgdg frq txh ho dxwrpöylo yldmdq flhuwr lqvwdqwh +hq hvwh fdvr d odv 44=63kuv,/ vh qhfhvlwd frqrfhu dojxqd lqirupdflöq vreuh od srvlflöq gho dxwrpöylo douhghgru gh hvd krud1 Sru hmhpsor/ vxsrqjdprv txh d odv 44=63 kuv/ ho dxwrpöylo hvwä d 53 npv gh D | txh d odv 4=68 krudv hvwä d 57/8 npv gh D/ hqwrqfhv hq 8 plqxwrv od glvwdqfld uhfruulgd hv 7/8 npv1 Vxvwlwx|hqgr hq od iöupxod y @ gw whqhprv txh od yhorflgdg phgld gxudqwh hvh lqwhuydor gh wlhpsr hv 8 y @ 87@>93 @ 87np@kuv Shur hvwh ydoru/ qr hv ho ydoru h{dfwr gh od yhorflgdg d odv 44=63kuv/ hv vöor xqd dsur{lpdflöq1 Re0 yldphqwh xqd phmru dsur{lpdflöq gh od yhorflgdg d odv 44=63 kuv vh rewlhqh vl frqvlghudprv lqwhuydorv gh wlhpsr päv shtxhôrv1 Hvwr qrv oohyd d frqvlghudu ho oðplwh gh od yhorflgdg phgld fxdqgr w wlhqgh d 44=63kuv1 Vxsrqjdprv txh vh frqrfh od srvlflöq v+w, gh xq sxqwr S hq wrgr lqvwdqwh hq xq lqwhuydor gh wlhpsr/ hqwrqfhv od yhorflgdg gh S hq ho wlhpsr w @ w3 yhorflgdg lqvwdqwdqhd vh ghqh frpr olp w$w3
v+w, vh frqrfh frpr od ixqflöq gh srvlflöq gh S 1
v+w, v+w3 , w w3
Ghqlflöq 814: Vl xq sxqwr S vh pxhyh vreuh xqd uhfwd gh prgr txh vx srvlflöq hq ho wlhpsr w hv v+w,/ frq w 5`w3 > w3 . ^> A 3/ hqwrqfhv od yhorflgdg phgld gh S hqwuh orv wlhpsrv w3 | w hv y@
v+w, v+w3 , w w3
| od yhorflgdg lqvwdqwäqhd y+w3 , gh S hq ho wlhpsr w3 hv y+w3 , @ olp w$w3
v+w, v+w3 , w w3
d+w3 , @ olp w$w3
y+w, y+w3 , w w3
vlhpsuh | fxdqgr ho oðplwh h{lvwh/ hv ghflu y+w3 , @ v3 +w3 ,1 Od udslgh} gh S do wlhpsr w @ w3 vh ghqh frpr my+w3 ,m | od dfhohudflöq gh S do wlhpsr w3 frpr
65;
FKDSWHU 81 GHULYDGD
hv ghflu/ xvdqgr rwud qrwdflöq
d+w3 , @ y 3 +w3 , @ v33 +w3 , d+w3 , @
gy g5 v mw@w3 @ 5 mw@w3 gw g{
d+w3 , @ y+w b 3 , @ v+w3 ,
Hmhpsor 749 Xqd dwohwd fruuh orv flhq phwurv sodqrv9 / gh pdqhud txh od glvwdqfld v+w, txh kd uhfruulgr d orv w vhjxqgrv hvwä gdgd sru 4 v+w, @ w5 . ;w phwurv 8 Fdofxodu od yhorflgdg gho fruuhgru d, hq ho prphqwr gh od vdolgd e, d orv 8 vhjxqgrv gh od vdolgd f, do fux}du od phwd Vroxflöq= d, Qrwhprv txh hq ho prphqwr gh od vdolgd w @ 3/ od srvlflöq gho dwohwd hv v+3, @ 3 phwurv1 Xvdqgr od ghqlflöq/ od yhorflgdg gho dwohwd hq w @ 31 v+w, v+3, w3 4@8w5 . ;w @ olp w$3 w @ olp 4@8w . ; w$3 @ ;p@vhj
y+3, @
olp w$3
Txh hv htxlydohqwh d fdofxodu y+w,/ ghulydqgr v+w, frq uhvshfwr do wlhpsr | oxhjr hydoxdqgr hq w @ 31 Dvð/ 5 y+w, @ w . ; , y+3, @ ; 8 e, D orv 8 vhjxqgrv gh od vdolgd/ whqhprv 5 y+8, @ 8 . ; @ 43p@vhj 8
f, Do fux}du od phwd/ hv ghflu/ d orv 433 phwurv vh wlhqh/
4 V+w, @ 433 +, w5 . ;w @ 433 +, w @ 43 8
sru or wdqwr d orv 43 vhj ho dwohwd kd fux}dgr od phwd | vx yhorflgdg hv 5 y+43, @ 43 . ; @ 45p@vhj 8
Hmhpsor 74: Grv fduuhwhudv vh fux}dq irupdqgr hq xq äqjxor gh 78 1 Vdelhqgr txh hq xq lqvwdqwh w3 grv dxwrpöylohv D | E glvwdq gho fuxfh 433 npv/ ho dxwrpöylo D dohmäqgrvh frq xqd udslgh} gh ;3 np@kuv | E dohmäqgrvh frq xqd udslgh} gh 93 np2kuv1 Ghwhuplqh od udslgh} frq txh vh vhsdud xqr gho rwur hq ho lqvwdqwh w3 1 +jxud BB, Vroxflöq= Vhd {+w, od glvwdqfld ghvgh ho fuxfh do dxwrpöylo D hq ho lqvwdqwh w/ |+w, od glvwdqfld ghvgh ho fuxfh do dxwrpöylo E/ | v+w, od glvwdqfld hqwuh D | E/ dghpäv yD +w3 ,> yE +w3 , yhorflgdg gh orv dxwrpöylo D/ E uhvshfwlydphqwh hq ho lqvwdqwh w3 1 Sru klsöwhvlv {+w3 , @ |+w3 , @ 433npv / yD +w3 , @ {3 +w3 , @ ;3np2kuv | yE +w3 , @ | 3 +w3 , @ 93np2kuv1 Qrvrwurv exvfdprv gv gw +w3 ,1 9 Wr grv
orv prylplhqwr vrq frqvlghudgrv uhfwlolqhr1
8181 DSOLFDFLRQHV GH OD GHULYDGD
65<
Dsolfdqgr ho whruhpd gho frvhqr do wuläqjxor DRE whqhprv
v5 +w, @ {5 +w, . |5 +w, 5{+w,|+w, frv+78 ,>
hv ghflu/
s v5 +w, @ {5 +w, . | 5 +w, {+w,|+w, 5
Frq ho remhwr gh rewhqhu gv gw ghulydprv od h{suhvlöq 814: frq uhvshfwr d w whqhprv>
+814:,
gv g{ g| s g{ s g| @ 5{ . 5| 5| 5{ = +814;, gw gw gw gw gw Hydoxdqgr hq w3 od h{suhvlöq 814: rewhqhprv od glvwdqfld hqwuh orv dxwöprylohv hq ho lqvwdqwh w3 1 t s {5 +w3 , . | 5 +w3 , 5{+w3 ,|+w3 , v+w3 , @ t s 4335 . 4335 54335 v+w3 , @ t s v+w3 , @ 433 5 5 5v
Dvð/ xvdqgr: hvwr hq 814; rewhqhprv od yhorflgdg uhtxhulgd s s 4 5 5 gv s mw@w3 @ 433 ;3 433 93, s +433 ;3 . 433 93 gw 5 5 433 5 5 s t s gv 473 :3 5 5 5 @ :3 mw@w3 @ s s gw 5 5 s s Oxhjr orv dxwrpöylohv hq ho lqvwdqwh w3 vh vhsdudq d xqd yhorflgdg gh :3 5 5np2kuv1 Rwud irupd gh gdu hvwd uhvsxhvwd hv= Oxhjr orv dxwrpöylohv hq ho lqwdqwh w3 vh vhsdudq d ud}öq gh s s :3 5 5np2kuv1
81816
Ud}öq gh Fdpelr/ wdvd gh yduldflöq
D frqwlqxdflöq yhuhprv xqd gh odv pýowlsohv dsolfdflrqhv gh od ghulydgd/ od txh uhhuh d yduldflöq r ud}öq gh fdpelr1 Sru hmhpsor/ od yhorflgdg txh kd vlgr ghqlgd dqwhulruphqwh hv od yduldflöq gh od srvlflöq v+w, frq uhvshfwr do wlhpsr +vx udslgh} gh yduldflöq,1 Od dfhohudflöq d+w, gh S do wlhpsr w hv od ud}öq gh fdpelr gh od yhorflgdg frq uhvshfwr do wlhpsr1 Frqvlghuhprv od vljxlhqwh vlwxdflöq jhqhudo> Vhd | @ i+w, grqgh i hv xqd ixqflöq ghulydeoh uhvshfwr do wlhpsr w 41 Od wdvd +r ud}öq, phgld gh yduldflöq gh | @ i +w, hq ho lqwhuydor ^w> w . k` hv i +w . k, i +w, k
51 Od wdvd +r ud}öq, gh yduldflöq gh | @ i+w, frq uhvshfwr d w hv
i +w . k, i +w, g| @ | 3 +w, @ olp k$3 gw k
Qrwdu txh hvwd ghqlflöq hv dqäorjd do frqfhswr gh yhorflgdg1
Odv xqlgdghv txh ghehq xvduvh hq odv ghqlflrqhv dqwhulruhv ghshqghq gh od qdwxudoh}d gh od fdqwlgdg uhsuhvhqwdgd sru i1 : Frpr
or päv gluhfwr | iäflo gh rewhqhu
vEwf
xvdqgr
814:/ ho hvwxgldqwh frphwh ho huuru gh kdfhu ho fäofxor/ s huglhqgr gh
ylvwd txh od hfxdflöq 814: oh s huplwh fdofxodu od ghulydgd1 Vh dfrqvhmd do ohfwru d sodqwhdu vlhpsuh orv sureohpdv gh hvwh wls r hq yduldeohv/ vlq hydoxdu kdvwd ghvsxìv gh ghulydu1
663
FKDSWHU 81 GHULYDGD
Hmhpsor 74; Ho yroxphq Y +hq slh6 , gh xqd shtxhôd uhsuhvd gxudqwh od ìsrfd gh ooxyld/ hvwä gdgr sru Y @ 8333+w . 4,5
grqgh w vh plgh hq phvhv | 3 w 61 Od wdvd gh fdpelr gho yroxphq frq uhvshfwr do wlhpsr hv ho xmr lqvwdqwäqhr kdfld od uhsuhvd1 Fdofxoh ho xmr hq orv wlhpsrv w @ 3 | w @ 5 Bc Fxäo hv ho ydoru gho xmr fxdqgr ho yroxphq hv gh 44=583slh6 B
Vroxflöq= Sru ghqlflöq od wdvd gh fdpelr gho yroxphq frq uhvshfwr do wlhpsr hv gY gw 1 Hq hvwh sureohpd gY uhsuhvhqwd ho xmr kdfld od uhsuhvd/ hqwrqfhv ghulydqgr od ixqflöq Y uhvshfwr gho gw wlhpsr whqhprv gY @ 43=333+w . 4, gw hydoxdqgr hq w @ 3 | w @ 5 rewhqhprv ho xmr> gY mw@3 @ 43=333slh6 @phvhv gw gY mw@5 @ 63=333slh6 @phvhv gw
Vl ho yroxphq Y @ 44=583 slh6 hqwrqfhv gh grqgh w @ 3=8 phvhv
44=583 @ 8=333+w . 4,5 gY mw@3=8 @ 48=333slh6 @phvhv gw
Hmhpsor 74< Od Whpshudwxud W +hq judgrv Fhovlxv, gh xqd vroxflöq do wlhpsr w +hq plqxwrv, hvwä gdgd sru 6 W +w, @ 43 . 7w . w.4 sdud 4 w 43
41 Fdofxoh dsur{lpdgdphqwh od wdvd phgld gh fdpelr r yduldflöq gh od whpshudwxud gxudqwh ho lqwhuydor gh wlhpsr ^5> 5=68`1
51 Fdofxoh od wdvd gh yduldflöq r ud}öq gh fdpelr gh W +w, frq uhvshfwr do wlhpsr w hq w @ 5 | w @ k @ 3=68 hq od ghqlflöq vh rewlhqh txh od wdvd phgld gh fdpelr gh W hq ^5> 5=68` hv W +5=68, W +5, 6=: 3=68 51 Sru ghqlflöq od wdvd gh yduldflöq gh W +w, frq uhvshfwr do wlhpsr hv
Dvð/
W 3 +w, @ 7 W 3 +5, @ 7
6 +5 . 4,5
6 +w . 4,5
W 3 + 41 Od surixqglgdg hq od sduwh päv krqgd hv gh 7 slh +4jdo 3=466:slh6 ,1
51 Od surixqglgdg hv gh 9 slh1 61 Od surixqglgdg hv gh 8 slh1
Vroxflöq=
41 Hq ho hmhpsor 43: gho fdsðwxor gh ixqflrqhv rewxylprv txh od ixqflöq yroxphq hq wìuplqr gh k hv 3?k?8 933k . 453k5 Y +k, @ 933k . 6333 . 4533+k 8, 8 k <
grqgh k hv ho qlyho gho djxd hq ho odgr päv surixqgr1 Oxhjr sdud fdofxodu od udslgh} gh fdpelr gho qlyho gho djxd frq uhvshfwr do wlhpsr fxdqgr k @ 7 slh frqvlghudprv od ixqflöq yroxphq sdud 3 ? k ? 8> Y +k, @ 933k . 453k5 Dvð/ ghulydqgr frq uhvshfwr do wlhpsr/ sru +uhjod gh od fdghqd, whqhprv
sru or wdqwr
gk gk gY @ 933 . 453 5k gw gw gw gY 4 gk @ gw gw 933 . 573k
665
FKDSWHU 81 GHULYDGD 6 grqgh k @ k+w,/ shur gY gw @ 833 3=466:slh @plq | k @ 7slh oxhjr 4 gk mk@7 @ 833 3=466: 3=375
D @ 7u5
grqgh u hv ho udglr1 Hv fodur txh vl ghulydprv Y frq uhvshfwr d u whqhprv gY 7 @ 6u5 @ 7u5 @ D> gu 6
hv ghflu/ od wdvd gh fdpelr gho yroxphq hv ljxdo do äuhd gh od vxshuflh1 Hmhpsorv wðslfrv
Sureohpd gh od vrpeud pwv > ho irfr Sdxolqd/ gh 4> 5 phwurv gh dowr/ fruuh hq od qrfkh dohmdqgrvh gh xq srvwh d od yhorflgdg gh 8 vhj gho srvwh txh loxplqd d Sdxolqd hvwd d :> 8 phwurv gh dowxud1 D phglgd txh vh dohmd gho srvwh od vrpeud gh ohood fuhfh +Áhv yhugdgB r Áuhdophqwh ghfuhfhB,1 Frqwhvwhprv odv vljxlhqwhv suhjxqwdv d10 ÁD txì yhorflgdg fdpeld ho odujr gh od vrpeud gh Sdxolqd B e10 Ho h{wuhpr gh od vrpeud vh dohmd wdpelìq gho srvwh/ od yhorflgdg gh hvwh dohmdplhqwr Á hv od plvpd yhorflgdg gh od qlôdB r Áhv od plvpd yhorflgdg gdgd frpr uhvsxhvwd d od suhjxqwd dqwhulruB1
667
FKDSWHU 81 GHULYDGD
Vhd=
{+w, od glvwdqfld gh od qlôd do srvwh hq ho lqvwdqwh w> v+w, ho odujr gh od vrpeud gh od qlôd hq ho wlhpsr w
Hqwrqfhv od yhorflgdg gh od qlôd hv
g{+w, | hq wìuplqr gh hvwdv yduldeohv/ or txh exvfdprv sdud frqwhvwdu gw
gv+w, gw Or txh ghehprv kdfhu hv hqfrqwudu xqd uhodflöq hq grqgh dsduh}fd v+w, | ghvsxìv ghulydu1 Frpr orv wuläqjxorv DEF | GHE vrq vhphmdqwhv +wlhqhq orv plvprv äqjxorv,/ vh gd od vljxlhqwh uhodflöq hqwuh orv odgrv
od sulphud suhjxqwd hv
| gh hvwd irupd whqhprv
Txh hv or plvpr txh
ghulydqgr d dperv odgrv
DE DF @ GH GE
{+w, . v+w, :> 8 @ 4> 8 v+w, :> 8v+w, @ 4> 8{+w, . 4> 8v+w, 9v+w, @ 4> 8{+w, v+w, @ 3> 58{+w, g{+w, pwv pwv gv+w, @ 3> 58 @ 3> 58 +8, @ 4> 58 gw gw vhj vhj
Od suhjxqwd e vh uhhuh d od ghulydgd gh od vxpd {+w, . v+w, | frpr frqrfhprv fdgd ghulydgd/ od pw = uhvsxhvwd hv qr hv ljxdo d qlqjxqd gh odv grv vxjhulgdv/ od yhorflgdg gho h{wuhpr gh od vrpeud hv 9> 58 vhj np > Vxsrqjdprv dkrud txh Shgulwr hv ho txh fruuh/ shur d 53 ku ÁFxäo fruuh päv uäslgr/ Shgulwr r od qlôdB
8181 DSOLFDFLRQHV GH OD GHULYDGD
668
Vl Shgulwr plgh 468fp> ÁD txì yhorflgdg ho h{wuhpr gh od vrpeud vh dohmd gho srvwhB Sureohpd gh od hvfdohud Xqd hvfdohud gh phwdo frq uxhgdv hq vx edvh hvwd dsr|dgd hq xqd pxudood gh wdo irupd txh vx h{wuhpr vxshulru hvwd d 6pwv gho vxhor1 Ho eortxhr gh vxv uxhgdv oh lpslgh pryhuvh1 Od hvfdohud plgh 7pwv gh odujr | hv shvdgd gh prgr wdo txh fxdqgr vh ghveortxhdq odv uxhgdv od hvfdohud vh ghvol}d hq od pxudood pw = Fxdqgr hvwd d xq phwur gho vxhor Ád txì prylhqgrvh vx h{wuhpr vxshulru d od yhorflgdg gh 433 vhj yhorflgdg vh pxhyhq odv uxhgdv hq hvh lqvwdqwhB Sdud frqwhvwdu hvd suhjxqwd qhfhvlwdprv ghqlu dojxqdv yduldeohv | kdfhu xq gldjudpd gho sureohpd Ho gldjudpd gho sureohpd hv=1
Vhdq=
u+w, od glvwdqfld gh odv uxhgdv d od pxudood | k+w, od glvwdqfld gho h{wuhpr vxshulru gh od hvfdohud do vxhor1
gk pw gu +w, @ 433 | or txh exvfdprv hv hq w3 txh hv ho lqvwdqwh hq txh od hvfdohud hvwä gw vhj gw d 4phwur gho vxhor/ hv ghflu/ k+w3 , @ 4 phwur1 Frpr vh yh/ od pxudood frq od hvfdohud | ho vxhor hq wrgr wlhpsr w irupdq xq wuläqjxor uhfwäqjxor dvl whqhprv ho whruhpd gh Slwäjrudv/ Sru or wdqwr
+k+w,, . +u+w,, @ +7, > 5
5
5
ghulydqgr hq dperv odgrv gh od ljxdogdg +| xvdqgr od uhjod gh od fdghqd, whqhprv gu+w, gk+w, . 5u+w, @3 gw gw k+w, gk+w, gu+w, @ gw u+w, gw
5k+w,
| sru or wdqwr hq ho lqvwdqwh w3
k+w3 , gk 4 gu pw +w3 , @ +w3 , @ 433 gw u+w3 , gw u+w3 , vj
Xvdqgr srghprv rewhqhu u+w3 ,
Q
669
FKDSWHU 81 GHULYDGD
Iljxuh 815= +k+w3 ,, . +u+w3 ,, @ +7, / +4, . +u+w3 ,, @ 49 s u+w3 , @ 48 5
5
5
5
5
Sru or wdqwr odv uxhgdv vh ghvol}dq d od yhorflgdg gh
pw 4 pw gu +w3 , @ s 433 58> ;53 gw vj vj 48 ÁSru txì qrv glö xqd uhvsxhvwd qhjdwlydB
Sru ho vlvwhpd gh uhihuhqfld hohjlgr> frpr od glvwdqfld k+w, glvplqx|h/ od yhorflgdg gh fdlgd gh od hvfdohud pw pw ghehuðd vhu qhjdwlyd/ hv ghflu/ hq yh} gh kdehu sxhvwr hq Q 433 ghehuldprv kdehu sxhvwr 433 = vj vj Sureohpd gh od Vrpeud 5
Hq ho fuxfh gh grv fdoohv/ txh wlhqhq dqfkr 44 phwurv fdgd xqd/ kd| xq srvwh gh 6 phwurv gh dowr hq pw xqd gh odv hvtxlqdv1 Hq od qrfkh Shgulwr/ gh 4> < phwurv/ fdplqd d4> 6 dohmdqgrvh gh od hvtxlqd sru od vj yhuhgd frqwuduld do srvwh | d xq phwur gh od rulood gh od fdooh1
ÁD txì yhorflgdg vh dodujd od vrpeud gh Shgulwr fxdqgr hvwh vh hqfxhqwud d 46 phwurv gh od rulood gh od fdooh txh uhflhq fux}öB Frpr srghprv yhu hq ho gldjudpd hvwh vh sxhgh vlpsolfdu d xq wuläqjxor uhfwäqjxor frpr vljxh
8181 DSOLFDFLRQHV GH OD GHULYDGD
Vhd
66:
s+w, glvwdqfld gh Shgulwr d od edvh gho srvwh +s+w, @ DF, v+w, odujr gh od vrpeud gh Shgulwr +v+w, @ FG, h+w, glvwdqfld gh Shgulwr d od fdooh wudqvyhuvdo d od fdooh grqgh fdplqd +h+w, @ FI ,
Vhd w3 ho lqvwdqwh txh Shgulwr hvwd d 46 phwurv gv ÁTxì exvfdprv B/Ho ydoru gh +w3 ,/ gw gh pw ÁTxì frqrfhprvB +w3 , @ 4> 6 gw vj Frpr whqhprv grv wuläqjxorv uhfwäqjxorv vhphmdqwhv
s+w, . v+w, 6 @ 4> < v+w, +6 4> < gs+w, @ gw +6 4> :5: 6 +w3 , gw gw
Sru or wdqwr qhfhvlwdprv frqrfhu
ghulydgr
gs +w3 ,= Frpr DI F irupd xq wuläqjxor uhfwäqjxor/ whqhprv gw
+s+w,,5 @ +44 . 4,5 . +h+w,,5
5s+w, dvl/ hq ho lqvwdqwh w3
I
gs gh @ 5h+w, gw gw
h+w3 , gh 46 pw gs +w3 , @ +w3 , @ 4> 6 gw s+w3 , gw s+w3 , vj
Hq ho lqvwdqwh w3 > xvdqgr I whqhprv
66;
FKDSWHU 81 GHULYDGD
Gh hvwd irupd orjudprv
Ho jorer gh Shgulwr
5 @ 477 . 49< @ 646 +s+w3 ,,5 @ 477 . +h+w3 ,,s s+w3 , @ 646
46 gs pw +w3 , @ s 4> 6 > | dvð/ gw vj 646
pw 46 pw gv 4> 6 +w3 , @ 4> :5: 6 s 4> 983 3 gw vj vj 646
np > hq ho lqwhulru yd Shgulwr frq xq jorer frq kholr1 Shgulwr vxhowd ku pw ho jorer | hvwh vxeh hq irupd shushqglfxodu d 43 = Yhdprv d txì yhorflgdg vh vhsdud Shgulwr gho jorer vj ghvsxìv gh : vhjxqgrv1 Hvtxhpd gho sureohpd Hq xqd fduuhwhud xq dxwr yd d hq sduwlfxodu +w3 , @ 43 > ÁTxì whqhprvB1 Uhvsxhvwd = +w, @ 43 gw vhj gw vhj gd +w,> ho fxdo hvwd hq nloöphwurv sru krud/ dvl txh ghehprv fdpelduod d phwurv vhjxqgrv gw np 4333phwurv phwurv gd +w, @ d+:, pw gj +w, @ 43 > ghvsxìv gh : vhjxqgr ho jorer hvwd d 43 : @ :3 phwurv gw vhj phwurv gd +w, @ 58 ghvsxìv gh : vhjxqgr ho dxwr hvwä d 58 : @ 4:8 phwurv | xwlol}dqgr } hq w @ : gw vhjxqgrv {5 +:, @ j5 +:, . d5 +:, @ :35 . +4:8, @ 68 858 5
Gh hvwd irupd whqhprv wrgd od lqirupdflöq sdud srqhuod hq u 4 pw pw g{ +:, @ :3 43 . 4:8 58 gw u 68 858 vhj vhj pw 478 pw 4 g{ +:, @ +:3 43 . 4:8 58 , @ s gw 68 858 vhj 8 vhj
Od Whohwöq Ho fðufxor gh od jxud uhsuhvhqwd xq hvfhqdulr uhgrqgr/ gh udglr 6 phwurv/ xelfdgr do fhqwur gho Hvwdglr Qdflrqdo | txh jlud d 3> 8 uhyroxflrqhv sru plqxwrv1 Grq Iudqflvfr fdplqd ghvgh ho fhqwur gho hvfhqdulr +sxqwr R, hq irupd uhfwd kdvwd ho sxqwr P xelfdgr d od rulood gho hvfhqdulr Od uhfwd RP irupd xq äqjxor @ frq od uhfwd txh xqh R frq od xelfdflöq gh Sdxod txh hvwd hq od jdohuld +sxqwr S , d grflhqwrv phwurv 7 gho fhqwur1 pw B1 ÁD txh yhorflgdg yduld od glvwdqfld hqwuh Grq Iudqflvfr | Sdxod vl Grq Iudqflvfr fdplqd d 3> 5 vhj Vhd {+w, od glvwdqfld hqwuh Grq Iudqflvfr | Sdxod hq ho wlhpsr w> pw gp +w, @ 3> 5 p+w, od glvwdqfld gh Grq Iudqflvfr do fhqwur gho hvfhqdulr hq ho wlhpsr w> dvl gw vhj +w, ho äqjxor txh irupd od uxwd gh Grq Iudqflvfr hq ho hvfhqdulr frq od uhfwd RS1
g{ Or txh exvfdprv hv +w3 ,> grqgh w3 hv ho lqvwdqwh txh Grq iudqflvfr oohjd d P= gw Frpr od xelfdflrq gh Grq Iudqflvfr/ Sdxod | ho fhqwur gho hvfhqdulr hq wrgr wlhpsr w irupdq xq wuläqjxor/ xvduhprv od oh| gho frvhqr {5 +w, @ p5 +w, . 5335 5p+w, 533 frv++w,,
Ghulydqgr hq uhvshfwr do wlhpsr qrv txhgd
g{ gp gp g +w, @ 5p+w, +w, . 3 5 +w,533 frv++w,, . 5p+w, 533 vlq++w,, +w, gw gw gw gw g{ 4 gp gp g +w, @ p+w, +w, 533 +w, frv++w,, . 533p+w, vlq++w,, +w, K gw {+w, gw gw gw 5{+w,
Frpr hq w3 > p+w3 , @ 6 phwurv | +w3 , @
=> xwlol}dqgr orjuduhprv {+w3 , 7
673
FKDSWHU 81 GHULYDGD {5 +w3 , @ p5 +w3 , . 5335 5p+w3 , 533 frv++w3 ,, @ 65 . 5335 5 6 533 frv+ , 7 u 4s {+w3 , @ 73 33< 4533 5 5
Gh hvwd irupd K qrv txhgd 4 gp gp g g{ 6 +w3 , @ u +w3 , 533 +w3 , frv+ , . 933 vlq+ , +w3 , gw gw gw 7 7 gw 4s 73 33< 4533 5 5 g{ 4 gp gp g 6 +w3 , @ u +w3 , 533 +w3 , frv+ , . 933 vlq+ , +w3 , gw gw gw 7 7 gw 4s 73 33< 4533 5 5 4s 4 4s g{ 6 3> 5 533 5 3> 5 . 933 5 3> 8 +w3 , @ s s 4 gw 5 5 493 369 7;35 5 5 s g{ 4 pw +w3 , @ s = 3=9 . 463 5 3= grqgh w3 hv ho lqvwdqwh txh vh oohqd ho gw hvwdqtxh / hv ghflu/ k+w3 , @ K= Vl Y +w, ghqrwd ho yroxphq gh od duhqd hq ho wlhpsr w> frpr ho yroxphq wrwdo gh duhqd hq ho wlhpsr w hv grv yhfhv ho yroxphq gho frqr uhfwr gh dowxud k+w, | udglr u+w,> vh wlhqh Vl vh ghmd fdhu D
4 Y +w, @ 5 k+w,u5 +w, 6
g5k Gh hvwd uhodflöq vh sxhgh rewhqhu +w3 , yld ghulydflöq/ shur wdpelìq dsduhfhuðd od ghulydgd gh u+w,/ gw frpr vh yh dfrqwlqxdflöq 4 gY 4 gk gu +w, @ 5 +w,u5 +w, . 5 k+w,5u+w, +w, v gw 6 gw 6 gw Shur vl yhprv ho wuläqjxor glexmdgr do odgr gho frqr +frpr prghor pdwhpäwlfr,/ yhprv grv wuläqjxorv vhphmdqwhv/ sru or wdqwr whqhprv od vljxlhqwh uhodflöq hqwuh vxv odgrv K U @ u+w, k+w,
675
FKDSWHU 81 GHULYDGD | sru or wdqwr u@
U k | K
gu U gk +w, @ +w, gw K gw
uhhpsod}dqgr hq v whqhprv 4 gk D@5 +w, 6 gw
% & 5 5 5 4 7 U U gk gk U U . 5 k+w,5u+w, k+w, +w, @ +w, k+w, . k+w,u+w, K 6 K gw gw 6 K 6 K gk D 4 D +w3 , @ @ 5 gw 5 U5 7 U U 5 K . KU 6 K 6 K
ÁFxdqgr ho qlyho vxeh päv uäslgr/ fxdqgr od dowxud gh od duhqd hv päv srfd fxdqgr hvwä fhufd gh KB
819
Pä{lpr | Pðqlpr
Khprv ylvwr dojxqdv dsolfdflrqhv gh od ghulydgd1 Dkrud yhuhprv txh od ghulydgd dghpäv qrv gd lqirupdflöq vreuh ho fuhflplhqwr | ghfuhflplhqwr gh od ixqflrqhv/ | qrv shuplwduä hqfrqwudu orv sxqwrv grqgh kd| pä{lprv r pðqlprv1 Kd| grv wìfqlfdv gluhfwdv sdud hqfrqwudu wdohv sxqwrv/ hvwdv wìfqlfdv jhqhudophqwh vh ghqrplqdq fulwhulr gh od sulphud ghulydgd | fulwhulr gh od vhjxqgd ghulydgd1 Sdud hqwudu hq pdwhuld qhfhvlwdprv ghqlu frq suhflvlöq txh hqwhqghuhprv sru pä{lpr/ pðqlpr/ hwf1 Ghqlflöq 814< Vhd i = D U $ U/ gluhprv txh i +d, hv ho ydoru pä{lpr gh od ixqflöq i +ö Ydoru Pä{lpr devroxwr gh i r vlpsohphqwh ho pä{lpr gh i, vl sdud wrgr hohphqwr { 5 D vh fxpsoh i +{, i +d, Hq irupd wrwdophqwh dqäorjd vh ghqh pðqlpr devroxwr r pð qlpr1 i +d, hv ho ydoru pðqlpr devroxwr hq D vl i +d, i+{,
;{ 5 D
Qrwh txh qhfhvduldphqwh ho ydoru pä{lpr +ho ydoru pðqlpr, hv od lpdjhq gh xq sxqwr d 5 D @ Grpi> Vh glfh hq d hvwä ho pä{lpr gh i +hq d hvwä ho pðqlpr gh i%,1 Vl i qr hv lq|hfwlyd/ sxhgh h{lvwlu ydulrv rwurv hohphqwrv gh D grqgh i dofdqfh ho ydoru pä{lpr r ho ydoru pðqlpr/ vlq hpedujr ho ydoru pä{lpr r ho ydoru pðqlpr devroxwr vl h{lvwhq vrq ýqlfrv1 Sru hmhpsor od ixqflöq i +{, @ vhq { qr hv lq|hfwlyd/ ho ydoru pä{lpr gh i hv 4 | ho ydoru pðqlpr gh i hv 4 | hvwrv ydoruhv vrq dofdq}dgrv hq päv gh xq sxqwr1 Kd| rwur frqfhswr gh pä{lpr +r pðqlpr, gh i txh hv phqrv h{ljhqwh/ hvwh hv pä{lpr orfdo r uhodwlyr +pðqlpr orfdo r uhodwlyr,1 Hq ho juäfr gh yhqwdv gh xq surgxfwr hq ho dôr/ xqd pxowlwlhqgd wlhqh grv judqghv phvhv gh yhqwdv Pdu}r | Glflhpeuh +yhu juäfr vljxlhqwh,
8191 PÄ[LPR \ PÐQLPR
676
shur fodudphqwh dperv qr vrq pä{lpr devroxwrv1
Ghqlflöq 8153 d, Vhd i = D U $ U1 Gluhprv txh hq 5 D kd| xq pä{lpr uhodwlyr r pä{lpr orfdo vl h{lvwh xqd yhflqgdg gh > Y wdo txh i +, hv ho pä{lpr devroxwr gh od ixqflöq i mY / hv ghflu/ i+, i+{, ;{ 5 Y / dqäorjdphqwh vh ghqh pðqlpr orfdo
e, Vhd i = D U $ U1 Gluhprv txh i wlhqh xq pä{lpr uhodwlyr hvwulfwr hq 5 D vl h{lvwh xqd yhflqgdg gh > Y wdo txh i +, A i +{, ;{ 5 Y @ Y ij1 Dqäorjdphqwh vh ghqh pð qlpr uhodwlyr hvwulfwr
Hmhpsor 758 Vhd i = ^d> e` $ U xqd ixqflöq frqwlqxd frq juäfd gdgd hq od jxud
1 Fodudphqwh i +g, hv ho pä{lpr devroxwr/ d glihuhqfld gh orv ydoruhv i +d,> i +o, txh vrq vrodphqwh pä{lpr uhodwlyr | gh orv ydoruhv i +h,> i+e, txh vrq pðqlprv uhodwlyrv xqlfdphqwh +i +g,> i +f, vrq wdpelìq uhodwlyrv,
Ghqlflöq 8154 Gluhprv txh xq ydoru i +d, frq d 5 D hv xq ydoru h{wuhpr gh i hq D vl hv ho pä{lpr r pðqlpr gh dojxqd hvshflh hq D | sxqwr h{wuhpr do sxqwr d grqgh dofdq}d ho ydoru h{wuhpr1
677
FKDSWHU 81 GHULYDGD
Hmhpsor 759 Vhd i +{, @ 6{5 . 5> { 5 ^7> 4` 1 Od juäfd gh i hv xq wur}r gh xqd sduäerod | fodudphqwh vh sxhgh qrwdu phgldqwh vx juäfd txh i +3, @ 5 hv ho pðqlpr devroxwr | txh i +7, hv ho pä{lpr devroxwr +wdpelìq uhodwlyr, | i +4, pä{lpr devroxwr hvwulfwr1 Hq hvwh fdvr i wlhqh wuhv ydoruhv h{wuhprv i+3,> i +7,> i +4,1 Hmhpsor 75: Vhd i +{, @ h5{ > { 5 ^ oq+8,> oq+:,` 1 Frpr od ixqflöq hv hvwulfwdphqwh ghfuhflhqwh i wlhqh vöor grv ydoruhv h{wuhprv i+ oq+8,, @ 58 txh hv ho pä{lpr devroxwr hq D1 4 txh hv ho pðqlpr devroxwr hq D1 i+oq+:,, @ 7<
Hmhpsor 75; Vhd i +{, @ 6{ 5> { 5`5> 8`1 Fodudphqwh i hv hvwulfwdphqwh fuhflhqwh/ sru or wdqwr ho pä{lpr vh hqfxhqwud hq 8/ | hvwh hv i+8, @ 461 Shur hvwd ixqflöq qr wlhqh xq pðqlpr/ |d txh 5 95 Grpi / vlq hpedujr i+{, hv dfrwdgr lqihulruphqwh sru 7/ sxhv 7 ? i +{, > ;{ 5`5> 8`
Qr srghprv ghflu txh 7 hv pðqlpr devroxwr ql orfdo |d txh ìo gheh vhu lpdjhq gh xq hohphqwr gho grplqlr txh vh hvwä xwlol}dqgr/ hq hvwh fdvr `5> 8`
Hq hvwrv grv ýowlprv hmhpsorv vh sxhgh ylvxdol}du or ýwlo txh uhvxowd whqhu od lqirupdflöq uhodwlyd do ghfuhflhqwr r fuhflhqwr gh xqd ixqflöq1 Frqvlghuhprv od jxud I
Iljxud I
Vxsrqjdprv txh hood uhsuhvhqwd od juäfd gh xqd ixqflöq frqwlqxd hq ^d> e` | glihuhqfldeoh hq `d> e^ Qrwhprv txh hq ho lqwhuydor `d> f^/ grqgh od ixqflöq hv fuhflhqwh/ od shqglhqwh gh od wdqjhqwh d od juäfd gi +x, A 3 Hvwr rfxuuh sdud fxdotxlhu x 5`d> f^/ dvð i 3 +{, A 3 gh i hq +x> i +x,, hv srvlwlyd/ sru or wdqwr g{ sdud wrgr { hq `d> f^1 Or plvpr rfxuuh hq ho lqwhuydor `g> e^/ hv ghflu/ i hv fuhflhqwh dkð | wdpelìq i 3 +{, A 3 hq `g> e^1 Vlq hpedujr hq ho lqwhuydor `f> g^ od ixqflöq hv ghfuhflhqwh | od uhfwd wdqjhqwh wud}dgd hq +y> i+y,, wlhqh shqglhqwh qhjdwlyd/ dvð i 3 +{, ? 3 vl { 5`f> g^1 Ilqdophqwh/ xvdqgr jxud I vh wlhqh txh odv wdqjhqwhv d od juäfd hq orv sxqwrv h{wuhprv f | g wlhqhq shqglhqwh fhur/ hv ghflu> i 3 +f, @ 3 a i 3 +g, @ 3
8191 PÄ[LPR \ PÐQLPR
678
Qrwh txh i +d,> i+e, vrq wdpelìq h{wuhprv shur odv ghulydgdv ghuhfkdv h l}txlhugd; %/ gh h{lvwlu qr vrq qxodv1 Frpr frqfoxvlöq gho fdvr dqwhulru whqhprv txh orv h{wuhprv vh sxhghq hqfrqwudu grqgh od ghulydgd vh dqxod/ grqgh qr h{lvwh od ghulydgd r hq orv h{wuhprv gho lqwhuydor1 Yhdprv d frqwlqxdflöq dojxqrv hmhpsorv grqgh hvwrv fdvrv rfxuuh1
Hmhpsor 75< Vhd i = U $ U / od ghulydgd vh dqxod hq 3/ hv ghflu/ i 3 +3, @ 3 | i wlhqh xq pðqlpr { :$ {5 hq 3 +yhu jxud > juäfr l}txlhugr1,1
Iljxud
Hmhpsor 763 Vhd i = U $ U / od ghulydgd qr h{lvwh hq 3/ vlq hpedujr i wlhqh xq pðqlpr hq 3 { :$ m{m +yhu jxud > juäfr fhqwudo1,1
Hmhpsor 764 Vhd d 9@ 3 | vhd i = ^d> e` $ U / od ghulydgd gh i qr vh dqxod hq d/ hv ghflu/ i 3 +d, 9@ 3 { :$ {6 | i wlhqh xq pðqlpr hq d1+yhu jxud > juäfr ghuhfkr,1
Whruhpd 8155 Vhd i ghulydeoh hq d 5 D Grpi / D yhflqgdg gh d | i +d, xq pðqlpr uhodwlyr hqwrqfhv i 3 +d, @ 31
Ghprvwudflöq= Frpr i+d, hv xq pðqlpr uhodwlyr vh wlhqh txh
i +d, i +d . k, sdud k 5` %> %^ + dojýq %,
Gh hvwd irupd
i +d . k, i +d, 3
Dvð/ vl k A 3
i +d . k, i +d, 3 k
| vl k ? 3
; od
ghulydgd ghuhfkd gh
i
hq d hv
i +d . k, i +d, 3 k
i Ednk3i Ed in Ed ' *4k e` | ghulydeoh hq `d> e^1 Sru or wdqwr/ vl i hv xq ixqflöq ghulydeoh hq `d> e^| frqwlqxd hq ^d> e` srghprv dvhjxudu txh i dofdq}d ho pä{lpr | ho pðqlpr hq dojýq sxqwr f 5`d> e^grqgh i 3 +f, @ 3 r hq orv erughv { @ dr { @ e1 Dvð/ sdud ghwhuplqdu fxäo hv ho ydoru pä{lpr | fxäo hv ho ydoru pðqlpr edvwd hydoxdu od ixqflöq hq orv h{wuhprv gho lqwhuydorv { @ d r { @ e/ | orv sxqwrv grqgh od ghulydgd vh dqxod | frqvlghudu ho ydoru pä{lpr | ho ydoru pðqlpr hqwuh hvwrv ydoruhv1 Hmhpsor 765 Vhd i = ^4> 6` $ U = { :$ {5 4 Whqhprv txh i hv frqwlqxd hq ^4> 6` | ghulydeoh hq ` 4> 6^1 Sxhvwr txh i 3 +{, @ 5{ @ 3 +, { @ 3 5`4> 6^ hqwrqfhv hydoxdqgr i hq orv sxqwrv { @ 3> { @ 4> { @ 6 rewhqhprv> i +3, @ 4 i +4, @ 3 i+6, @ ; Dvð ho pä{lpr devroxwr gh i hv ; | ho pðqlpr devroxwr hv 41
Hmhpsor 766 Vh ghvhd frqvwuxlu xq rohrgxfwr gh xq sxqwr D d rwur sxqwr E txh glvwdq 8np | vh hq0 fxhqwudq hq ulehudv rsxhvwdv gh xq uðr gh fdxfh uhfwr gh 6np gh dqfkr1 Ho rohrgxfwr luä edmr ho djxd gh D d xq sxqwr S hq od ulehud rsxhvwd | oxhjr vreuh ho vxhor gh S d E1 Ho frvwr sru nloöphwur gh wxehuðd edmr ho djxd hv ho fxäguxsoh gho frvwr vreuh wlhuud1 Fdofxoh od srvlflöq gh S txh plqlpl}duä ho frvwr +ghvsuhflh od shqglhqwh gh ohfkr gho uðr,1 Vroxflöq= Vhd {od glvwdqfld +hq npv, gho sxqwr Rdo sxqwr S yhu jxud
8191 PÄ[LPR \ PÐQLPR
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Sru Slwäjrudv hq ho wuläqjxor DREvh wlhqh
oxhjr RE @ 7/ gh grqgh S E @ 7 {1 Dghpäv/ hq ho wuläqjxor DRS / vh wlhqh
5
jxud
5
RE . 65 @ 85
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, DS @
s
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Vhd f ho frvwr sru np gh wxehuðd vreuh wlhuud/ hqwrqfhv ho frvwr sru np gh wxehuðd edmr ho djxd hv 7f Oxhjr od ixqflöq txh uhsuhvhqwd ho frvwr gh frqvwuxflöq gho rohrgxfwr hv s F+{, @ 7f < . {5 . f+7 {,
grqgh 3 { 71 Frpr F+{,hv xqd ixqflöq frqwlqxd hq ^3> 7`| ghulydeoh hq `3> 7^ hqwrqfhv F+{,dofdq}d vx pä{lpr hq dojýq sxqwr d grqgh i 3 +d, @ 3r hq orv erughv gho lqwhuydorv ^3> 7`1
s
F 3 +{, @ s
7f{ f < . {5
Frpr F 3 +{, @ 3 +, { @ 848s5`3> 7^/ hydoxdqgr od ixqflöq F+{, hq { @ 848 | hq orv sxqwrv { @ 3 | { @ 7 vh wlhqh> s s 48 F+3, @ 49f F+7, @ 53f F+ , @ 6 48f . 7f 8 s Frpr f A 3 | +6 48 . 7,f ? 49f ? 53f/ s hqwrqfhv ho pðqlpr frvwr hv +6 48 . 7,f | od srvlflöq gho sxqwr S gheh vhu 7
s48 8 npv gho sxqwr E1
Hmhpsor 767 Xq fdpsr shwurohur wlhqh rfkr sr}rv txh surgxfhq xq wrwdo gh 4933 eduulohv gh fuxgr do gðd1 Sru fdgd sr}r qxhyr txh vh wlhqh/ od surgxfflöq phgld sru sr}r glvplqx|h hq 43 eduulohv gldulrv BcFxäqwrv sr}rv dglflrqdohv vh ghehq irupdu sdud rewhqhu od pd|ru surgxfflöq gh fuxgr do gðdB
Vroxflöq= Vhd {= ho qýphur gh sr}rv dglflrqdohv> { 3> { 5 ]1 Vl vh wlhqh ; sr}rv txh surgxfhq xq wrwdo gh 4933 eduulohv hqwrqfhv od surgxfflöq phgld sru sr}r hv 533 eduulohv gldulrv1 Sru fdgd sr}r qxhyr txh vh wlhqh od surgxfflöq phgld sru sr}r glvplqx|h hq 43 eduulohv oxhjr/ od surgxfflöq phgld sru sr}r hv 533 43{
67;
FKDSWHU 81 GHULYDGD
sru or wdqwr od surgxfflöq gh fuxgr do gðd hv S +{, @ +; . {,+533 43{,
Frpr { 3 | S +{, 3/ hqwrqfhv 533 43{ 3 +gh grqgh 3 { 53,1 Hq frqvhfxhqfld od ixqflöq d pd{lpl}du hv> S +{, @ +; . {,+533 43{,/ frq { 5 ^3> 53`1
Frpr S hv xqd ixqflöq frqwlqxd hq ho lqwhuydor fhuudgr ^3> 53` | ghulydeoh hq `3> 53^/ S dofdq}d vx pä{lpr hq dojýq sxqwr d grqgh i 3 +d, @ 3 r hq orv erughv gho lqwhuydor1 oxhjr>
S 3 +{, @ +533 43{, . +; . {,+43, S 3 +{, @ 3 +, { @ 9 5`3> 53^=
Dvð do hydoxdu od ixqflöq S gh surgxfflöq hq orv sxqwrv { @ 3> { @ 9> { @ 53 whqhprv> S +3, @ 4933
S +9, @ 4 S +{, @ +57 . 6{,+933 45{,=
Frpr { 3> { 5 ] | S +{, 3/ hqwrqfhv 933 45{ 3 +gh grqgh { 83,1 Dvð> S +{, @ +57 . 6{,+933 45{, frq { 5 ^3> 83` Oxhjr
S 3 +{, @ 6+933 45{, . +57 . 6{,+45, S 3 +{, @ 3 +, { @ 54=
Gdgr txh S hv frqwlqxd hq ^3> 83` | ghulydeoh hq `3> 83^ hqwrqfhv hydoxdqgr hq orv sxqwrv { @ 3> { @ 54> { @ 83 whqhprv S +3, @ 47733 S +54, @ 635:9 S +83, @ 3 oxhjr od pä{lpd surgxfflöq gh pdq}dqd vh rewlhqh fxdqgr vh sodqwdq 96 äuerohv päv sru dfuh hv/ hv ghflu/ ;: äuerohv sru dfuh1
8191 PÄ[LPR \ PÐQLPR
81914
67<
Whvw gh od Sulphud Ghulydgd
Revhuydflöq= Hq orv hmhpsorv dqwhulruhv vöor edvwö hydoxdu hq orv sxqwrv grqgh i 3 +s, @ 3 | hq orv erughv gho lqwhuydor ^d> e`/ shur hq jhqhudo vl txhuhprv ghwhuplqdu orv h{wuhprv gh xqd ixqflöq qhfhvlwdprv lqirupdflöq vreuh ho fuhflplhqwr | ghfuhflplhqwr gh od ixqflöq i hq xqd yhflqgdg gh s1 Dvð/ sdud A 3 vh rewlhqh orv vljxlhqwhv fdvrv= d, Vl i hv hvwulfwdphqwh fuhflhqwh hq od yhflqgdg l}txlhugd gh s/ `s > s^ | hvwulfwdphqwh ghfuhflhqwh hq od yhflqgdg ghuhfkd gh s/ `s> s . ^ vh wlhqh txh i+s, hv xq pä{lpr orfdo hvwulfwr1 +jxud I l}txlhugd,1
Iljxud I
e, Dqäorjdphqwh vl i hv hvwulfwdphqwh ghfuhflhqwh hq `s > s^ | i hv hvwulfwdphqwh fuhflhqwh hq `s> s . ^ hqwrqfhv i +s, hv xq pðqlpr uhodwlyr hvwulfwr +jxud I juäfr ghuhfkr,1 f, Vl qr rfxuuh dojxqd gh odv grv dowhuqdwlydv dqwhulruhv/ i+s, qr hv ydoru h{wuhpr1
Gh or dqwhulru hv fodur txh qhfhvlwdprv xqd irupd gh ylvxdol}du fxdqgr i hv fuhflhqwh | fxdqgr hv ghfuhflhqwh1 Frq hvh remhwr hqxqflduhprv orv whruhpdv vljxlhqwhv1
Ghqlflöq 8157 Xq sxqwr f 5 Grpi/ vh glfh txh hv xq sxqwr fuðwlfr gh i vl rfxuuh xqd gh odv vljxlhqwhv srvlelolgdghv> i 3 +f, @ 3 ö elhq i 3 +f, qr h{lvwh Hmhpsor 769 Vhd i +{, @ 6+{ 5,5 1 { @ 5 hv xq sxqwr fuðwlfr gh i/ sxhv i 3 +5, @ 31
Hmhpsor 76: Vhd i +{, @ m{ 8m oxhjr whqhprv txh vx ghulydgd hv= i 3 +{, @
Sxhvwr txh 8 95 Grpi 3 / { @ 8 hv xq sxqwr fuð wlfr
Hmhpsor 76; Vhd i +{, @ sxqwr fuð wlfr1
s 6
4 {A8 4 { ? 8
4 {5 / hqwrqfhv whqhprv txh i 3 +{, @ 56 { 6 vl { 9@ 31 Sru or wdqwr { @ 3 hv xq
683
81915
FKDSWHU 81 GHULYDGD
Whruhpd gho ydoru phglr | Urooh
Whruhpd 8158 +Whruhpd gh Urooh, Vhd i hv xqd ixqflöq frqwlqxd hq ho lqwhuydor ^d> e` | ghulydeoh hq ho lqwhuydor delhuwr `d> e^/ wdo txh i+d, @ i +e,/ hqwrqfhv i wlhqh do phqrv xq sxqwr fuðwlfr hq `d> e^/ hv ghflu/ h{lvwh f 5`d> e^ wdo txh i 3 +f, @ 31+yhu jxud KI juäfr l}txlhugr,1
Iljxud KI Ghprvwudflöq=
<
d, Vl od ixqflöq hv frqvwdqwh hq ^d> e`/ vx ghulydgd hv fhur hq `d> e^/ hv ghflu i 3 +{, @ 3> ;{ 5`d> e^
e, Vl qr rfxuuh or dqwhulru/ i qr hv frqvwdqwh | whqhprv grv fdvrv/ +l, h{lvwh xq lqwhuydor grqgh od ixqflöq fuhfh | oxhjr ghfuhfh r elhq +ll, i ghfuhfh hq xq lqwhuydor sdud oxhjr fuhfhu +|d txh i+d, @ i +e,,1 Hq ho sulphu fdvr vh rewlhqh xq pä{lpr orfdo i +f, | hq ho vhjxqgr xq pðqlpr orfdo i +f, hq dperv fdvr od wdqjhqwh hq hvrv sxqwrv vhuä gh shqglhqwh fhur i 3 +f, @ 31 Gh hvwd irupd f hv xq sxqwr fuð wlfr 1 Xqd jhqhudol}dflöq hq hvwh whruhpd hv ho whruhpd gh ydoru phglr txh vh sxhgh prwlydu gh od vljxlhqwh irupd1 Frqvlghuhprv xq dxwr +xq mhhs sdud hvwdu hq rqgd, txh sduwh gh Ylôd d odv 44=63 krudv | oohjd d xqd sduwh fhufdqd d Vdqwldjr d odv 46=33 krudv1 Vl uhfruuh 463 nlorphwurv/ hqwrqfhv od yhorflgdg surphglr r phgld hv +463 3,nloöphwurv 463 @ @ ;9> 9np@k= +44 = 63 46 = 33,krudv 4> 8
Hvwr vljqlfd txh do phqrv dojýq prphqwr gh vx yldmh ho yhorfðphwur pdufö ;9> 9np@k/ hv ghflu/ od yhorflgdg lqvwdqwäqhd hq dojýq prphqwr hqwuh 44=63 | odv 46=33 krudv ixh gh ;9> 9np@k1 Hq od jxud KI vh kd judfdgr od uhfwd O txh lqglfd od yhorflgdg phgld +uhfwd vhfdqwh d od juäfd gh od ixqflöq v+w, txh qrv gä od glvwdqfld uhfruulgd sruho pöylo hq ho wlhpsr w 5`44 = 63> 46 = 33^, < Frpr i hv frqwlqxd/ od juäfd gh i hv frpr xqd fxhugd txh xqh orv sxqwrv Ed> i Ed> Ee> i Ee1 Frpr hvwrv sxqwrv hvwdq do plvpr qlyho/ r elhq od fxhugd hvwd whqvd +frqvwdqwh, r elhq wlhqh rqgxodflrqhv +kd| pä{lpr |2r pðqlpr orfdohv,1
8191 PÄ[LPR \ PÐQLPR
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Odv uhfwdv O4 > O5 > O6 vrq sdudohodv d O/ oxhjr wlhqh od plvpd shqglhqwh gh od uhfwd vhfdqwh> hoodv hvwäq gdgdv sru odv ghulydgdv hq ho sxqwr/ hq hvwh fdvr rewhqhprv od shqglhqwh gh od vhfdqwh v+e, v+d, ed
txh hv ljxdo odv shqglhqwh gh odv wdqjhqwhv hq f4 > f5 / f6 / hv ghflu/ d orv ydoruhv v3 +f4 ,/ v3 +f5 ,/ v3 +f6 ,1 Dvð whqhprv od h{lvwhqfld d or phqrv gh xq sxqwr f 5`d> e^ hq hvwh fdvr +f @ f4 > f @ f5 > f @ f6 , wdo txh v3 +f, @
v+e, v+d, ed
Whruhpd 8159 +Whruhpd gho Ydoru Phglr, Vhd i xqd ixqflöq ghqlgd hq D/ ^d> e` D wdo txh 41 i vhd frqwlqxd hq ^d> e`
51 i vhd ghulydeoh hq `d> e^
Hqwrqfhv vh wlhqh txh h{lvwh do phqrv xq f 5`d> e^ wdo txh i 3 +f, @
Ghprvwudflöq= Od ixqflöq
i +e, i+d, ed
j+{, @ i+{, ^i +d, .
i+e, i+d, +{ d,` ed
i +d, vdwlvidfh odv klsöwhvlv gho whruhpd gh Urooh sru or wdqwr h{lvwh f 5`d> e^ wdo txh j3 +f, @ 3 r i 3 +f, i +ee, d @ 3 Revhuydflöq= Or txh glfh hv txh od uhfwd wdqjhqwh vhfdqwh txh sdvd sru orv sxqwrv +d> i +d,,> +e> i +e,, hv sdudohod do phqrv d xqd gh odv uhfwdv wdqjhqwhv d od juäfd gh i hqwuh orv sxqwrv phqflrqdgrv1 Od dsolfdflöq päv gluhfwd gh hvwh whruhpd hv hq od ghprvwudflöq gho vljxlhqwh whruhpd Whruhpd 815: Vhd i xqd ixqflöq ghulydeoh hq vx grplqlr1
41 Vl i 3 +{, 3 hq `d> e^/ vh wlhqh txh i hv fuhflhqwh hq ^d> e`1
685
FKDSWHU 81 GHULYDGD
51 Vl i 3 +{, A 3 hq `d> e^/ vh wlhqh i hv hvwulfwdphqwh fuhflhqwh hq ^d> e`1
61 Vl i 3 +{, 3 hq `d> e^/ vh wlhqh txh i hv ghfuhflhqwh hq ^d> e`
71 Vl i 3 +{, ? 3 hq `d> e^/ vh wlhqh txh i hv hvwulfwdphqwh ghfuhflhqwh hq ^d> e`1
Ghprvwudflöq= Prvwuduhprv vöor +7, Whqhprv txh vxsrqhu txh {> | 5 ^d> e` | ghehprv suredu txh= { ? | , i+{, A i+|, Vhdq {> | 5 ^d> e`> { ? |1 Frpr i vdwlvidfh ho whruhpd gho Ydoru Phglr/ whqhprv txh h{lvwh f 5 ^{> |` wdo txh i+|, i +{, @ i 3 +f, {| Frpr | { A 3 | i 3 +f, ? 3 vh wlhqh txh i+|, i +{, ? 31 Dvð vh kd suredgr txh +;{> | 5 ^d> e`,+{ ? | , i +{, A i +|,,
Hmhpsor 76< Vhd i xqd ixqflöq fx|d ghulydgd hv i 3 +{, @ +{ 6,+7 {,+{ 8,1 Ghwhuplqh hq txh lqwhuydorv i hv hvwulfwdphqwh ghfuhflhqwh Vroxflöq= Xvdqgr od wdeod gh vljqrv {6 7{ {8 i 3 +{,
6 3
0 . 0 .
3
. . 0 0
7 3 3
. 0 0 .
8 3 3
. 0 . 0
whqhprv i 3 +{, ? 3 hq `6> 7^/ sru or wdqwr i+{, hv hvwulfwdphqwh fuhflhqwh dool/ or plvpr rfxuuh hq ho lqwhuydor ^8> 4^1
Hmhpsor 773 Vhd i+{, @ h{ / frpr i 3 +{, @ h{ hv srvlwlyd sdud wrgr ydoru {/ vh frqfox|h txh i hv hvwulf0 wdphqwh fuhflhqwh/ vlq hpedujr j+{, @ h6{ / j3 +{, @ 6h6{ / sru or wdqwr j hv hvwulfwdphqwh ghfuhflhqwh1 Hmhpsor 774 Vhd i +{, @ orjd +{,>
{ A 3> 3 ? d 9@ 4/ frpr
g 4 orjd +{, @ g{ { oqd +{,
| od yduldeoh { hv srvlwlyd/ vh wlhqh txh oq d ghwhuplqd ho vljqr gh od ghulydgd gh i sru or wdqwr whqhprv txh= 41 Vl d A 4/ i hv hvwulfwdphqwh fuhflhqwh
51 Vl d ? 4/ i hv hvwulfwdphqwh ghfuhflhqwh
4 ,/ yhdprv hq txh lqwhuydorv i hv ghfuhflhqwh Hmhpsor 775 Vdh i +{, @ oq+ {{ .5 Ho grplqlr gh od ixqflöq hv ` 4> 5^^`4> 4^/ | vx ghulydgd od rewhqhprv xwlol}dqgr uhjod gh od fdghqd | od ghulydgd gh xq fxrflhqwh
{ . 5 +{ . 5, +{ 4, , ,+ {4 +{ . 5,5 6 +{ 4,++{ . 5,
i 3 +{, @ + @
Frpr i 3 +{, A 3 hq vx grplqlr vh frqfox|h txh i hv hvwulfwdphqwh fuhflhqwh hq od fdgd lqwhuydor +qr srghprv ghflu txh hv fuhflhqwh hq od xqlöq,1 Huuöqhdphqwh srguldprv ghflu txh i 3 +{, ? 3 hq lqwhuydor ` 5> 4^/ od ud}öq gho huuru hv txh od ixqflöq qr hvwd ghqlgd hq hvwh lqwhuydor/ dvð txh hv lpsrvleoh txh i 3 +{, hvwh ghqlgd hq xq frqmxqwr pd|ru1
8191 PÄ[LPR \ PÐQLPR
686
Hmhpsor 776 Vhd y+w, @ w6 85w . 9w . ; od yhorflgdg gh xq pöylo hq ho wlhpsr w/ w 5 ^3> ;`1 Ghwhuplqhprv orv lqwhuydorv gh wlhpsr grqgh ho pöylo hvwd dxphqwdqgr gh yhorflgdg +dfhohudqgr, | hq txh prphqwr hvwä glvplqx|hqgr vx yhorflgdg +iuhqdqgr r dfhohudflöq qhjdwlyd, 6
5
Vroxflöq= Qhfhvlwdprv ghwhuplqdu ho vljqr gh y3 +w, @ w5 8w . 9 @ +w 6,+w 5,1 Whqhprv txh y+w, ? 3> ;w 5 ^5> 6` / dvð ho pöylo hvwd iuhqdqgr hqwuh orv ydoruhv w 5 ^5> 6`/ dghpäv y3 +w, A 3 hq +` 4> 5^^`6> 4^, _ ^3> ;`1 Sru or wdqwr ho pöylo dfhohud ghvgh 3 d 5 | srvwhulruphqwh dfhohud gh 6 d ;1 BcFxdqgr vh ghwlhqh ho pöylo Revhuydflöq= Hq whruhpd dqwhulru hv vöor gh od irupd s , t |d txh od ixqflöq i +{, @ {6 hv hvwulfwdphqwh fuhflhqwh hq U shur i 3 +{, @ 6{5 qr hv srvlwlyd vlhpsuh |d txh i +3, @ 31 Edvdgrv hq ho whruhpd dqwhulru srghprv hqxqfldu ho fulwhulr gh od sulphud ghulydgd sdud pä{lprv |2r pðqlprv uhodwlyrv hvwulfwr1 Whruhpd 815; +Fulwhulr gh od sulphud ghulydgd, Vhd i frqwlqxd hq ^d> e` @ D | ghulydeoh hq `d> e^ qr qhfhvduldphqwh hq s 5 D
d, Vl h{lvwh A 3 wdo txh i 3 +{, A 3 hq `s > s^ | i 3 +{, ? 3 hq `s> s . ^ hqwrqfhv i wlhqh xq pä{lpr uhodwlyr hvwulfwr hq { @ s
e, Vl h{lvwh A 3 wdo txh i 3 +{, ? 3 hq `s > s^ | i 3 +{, A 3 hq `s> s . ^ hqwrqfhv i wlhqh xq pðqlpr uhodwlyr hvwulfwr hq { @ s Jhqhudophqwh/ odv klsöwhvlv hq +d,/ sdud xq pä{lpr uhodwlyr/ vh hvfuleh uhvxplgdphqwh dvð { i 3 +{, i+{,
s . (
s
s. 0 )
{ i 3 +{, i+{,
s 0 )
s
s. . (
| odv klsöwhvlv sdud xq pðqlpr uhodwlyr vh hvfuleh>
Hmhpsor 777 Vhd i +{, @ 6{5 . :{/ frpr i 3 +{, @ 9{ . :
Oxhjr
i 3 +{, @ 9{ . : A 3 +, { A :9 i 3 +{, @ 9{ . : ? 3 +, { ? :9 {
i 3 +{, i +{,
0 )
:@9 3
. (
gh grqgh i hv hvwulfwdphqwh fuhflhqwh hq ` :9 > 4^ | hvwulfwdphqwh ghfuhflhqwh hq ` 4> :9 ^/ dvð sru ho fulwhulr gh od sulphud ghulydgd srghprv frqfoxlu txh i + :9 , hv xq ydoru pðqlpr uhodwlyr hvwulfwr/ txh dghpäv hv devroxwr1 Hvwh uhvxowdgr wdpelìq or srghprv rewhqhu judfdqgr od sduäerod txh uhsuhvhqwd d | @ i+{, @ 6{5 . :{ frq yìuwlfh hq + :9 > i + :9 ,, Hmhpsor 778 Frqvlghuh xqd ixqflöq i ghulydeoh wdo txh Ghwhuplqdu orv h{wuhprv gh i1
i 3 +{, @ +{ 6,+{ . 5,+{ 8,=
687
FKDSWHU 81 GHULYDGD
Vroxflöq= Ghwhuplqhprv hq txì lqwhuydorv i 3 +{, A 3 | grqgh i 3 +{, ? 31 Sdud hvwr iruphprv xqd wdeod sdud od ixqflöq i 3 txh qrv lqgltxh uhvxplgdphqwh grqgh hv srvlwlyd | grqgh hv qhjdwlyd1 0 0 0
{6 {.5 {8
5 3
0 . 0
6 3
. . 0
8 3
. . .
Gh hvwd irupd whqhprv lqirupdflöq vreuh i uhvxplgdv hq ho vljxlhqwh fxdgur> { i 3 +{, i +{,
0 )
05 3 3
. (
6 3 3
0 )
8 3 3
. (
Gho uhvýphq vh sxhgh frqfoxlu txh i +5,> | i +8, vrq pðqlpr orfdohv hvwulfwr | i +6, hv xq pä{lpr orfdo hvwulfwr1 Hmhpsor 779 Vhd i +{, @ +{ 4,5@6 1 Ghwhuplprv orv h{wuhprv gh i1
5 Vroxflöq= Grpi @ U | i 3 +{, @ 56 +{ 4,4@6 @ 6 s { 5 U i4j qrwhprv txh i qr hv ghulydeoh 6 {4 > hq { @ 4/ vlq hpedujr hv frqwlqxd | vl { A 4> i 3 +{, A 3 vl { ? 4> i 3 +{, ? 3
hv ghflu/
{ i+{, i 3 +{, i+{,
0 )
4 3 9<
. (
oxhjr sru ho fulwhulr gh od sulphud ghulydgd i wlhqh xq qðqlpr hq { @ 4 txh hv wdpelìq devroxwr +qrwdu txh ho Grpi @ U qr hv dfrwdgr,/ i qr wlhqh pä{lpr1
Hmhpsor 77: Ghwhuplqhprv vl odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwlfduhprv qxhv0 wudv uhvsxhvwd1 d, Vhd A 3/ od ixqflöq i+{, @ h{ / ghqlgd hq vx grplqlr pä{lpr wlhqh vöor xq ydoru h{wuhpr1 vhq5 +4@{, { 9@ 3 e, Od ixqflöq i+{, @ qr wlhqh h{wuhprv1 3 {@3 5
f, Od ixqflöq i+{, @ {6 h5{ hv hvwulfwdphqwh ghfuhflhqwh hq ho lqwhuydor ` 4> 6@5^ s g, Od ixqflöq i+{, @ { 6 qr wlhqh pä{lpr | qr wlhqh pðqlpr1 5 { frv5 +4@{, 4 { ? 3 h, Od ixqflöq i+{, @ wlhqh pä{lpr | pðqlpr1 vhq { 3{4 i, Od ixqflöq i+{, @ h{ 5h{ ghqlgd hq ` 4> 8` wlhqh xq pä{lpr vl A 51
j, Od ixqflöq i+{, @ {6 ghqlgd hq ^4> 5^ qr wlhqh h{wuhprv1
Vroxflöq=
8191 PÄ[LPR \ PÐQLPR
688
d, Yhugdghur1 Grpi @ U i 3 +{, @ 5{h{ > { 5 U/ frpr h{ A 3> ;{ 5 U hqwrqfhv> i 3 +{, A 3 +, 5{ A 3 +, { ? 3 sxhv A 3 5
Gh hvwd irupd
{ i+{, i 3 +{, i+{,
5
. . (
3 4 3
. 0 )
Sru fulwhulr gh od sulphud ghulydgd i wlhqh xq pä{lpr uhodwlyr hvwulfwr txh hv wdpelìq devroxwr +sxhv od ixqflöq vxeh | edmd vöor xqd yh},/ | qr wlhqh pðqlpr1
e, Idovr1 Sru ghqlflöq hv fodur txh i wlhqh xq pð qlpr orfdo devroxwr hq { @ 3/ sxhv sru or wdqwr i+{, 3>
i+{, @ vhq5 +4@{, 3>
;{ 5 Grpi1
;{ 5 Grpi @ U
f, Yhugdghur1 i 3 +{, @ 6{5 h5{ . {6 h5{ 5 @ {5 h5{ +6 . 5{, dqdol}dqgr orv vljqrv gh i 3 +{,/ rewhqhprv { i +{, i 3 +{, i +{,
0 0 )
6@5 3
Oxhjr i hv hvwulfwdphqwh ghfuhflhqwh hq ` 4> 6@5^1
0 . (
3 3 3
. . (
Revhuydflöq= Qrwdu txh sru fulwhulr gh od sulphud ghulydgd srghprv dupdu txh i wlhqh xq pðqlpr hq { @ 6@5/ päv dýq ìo hv ho pðqlpr devroxwr1
g, Idovr1 Grpi @ ^6> 4^ dghpäv i 3 +{, @ 5s{4 6 A 3 ;{ 5 ^6> 4^/ hv ghflu/ i hv hvwulfwdphqwh fuhflhqwh | ghqlgd hq xq grplqlr dfrwdgr lqihulruphqwh sru { @ 6 5 Grpi / oxhjr i wlhqh xq pðqlpr hq { @ 61 Hvwh uhvxowdgr wdpelìq hv fodur judfdqgr od ixqflöq i =
h, Yhugdghur1 Qrwhprv txh i hv xqd ixqflöq frqwlqxd ghqlgd hq ho lqwhuydor fhuudgr ^4> 4`/ oxhjr sru ho whruhpd gh orv ydoruhv h{wuhprv i wlhqh xq pä{lpr | pðqlpr txh vrq devroxwrv1 i, Yhugdghur1 Qrwhprv txh i 3 +{, @ i +{, @ + 5,h{ 1 Vl A 5 hqwrqfhv i 3 +{, A 3 ;{ 5`4> 8`/ oxhjr i hv fuhflhqwh hq vx grplqlr dfrwdgr vxshulruphqwh sru { @ 8 5 Grpi gh hvwd irupd i wlhqh xq pä{lpr +devroxwr, txh hv i +8,1
j, Idovr1 i 3 +{, @ 6{5 3 ;{ 5 ^4> 5^ oxhjr i hv ghfuhflhqwh ghqlgd hq xq grplqlr dfrwdgr lqihulruphqwh sru { @ 4 5 ^4> 5^/ dvð i wlhqh frpr pä{lpr i +4,> fodudphqwh srghprv yhu txh i qr wlhqh pðqlpr1
Hq edvh d od lqirupdflöq txh khprv rewhqlgr kdvwd ho prphqwr/ srghprv judfdu dojxqdv ixqflrqhv frqvlghudqgr> grplqlr/ grplqlr gh frqwlqxlgdg/ lqwhufhswrv frq orv hmhv frrughqdgrv/ lqwhuydorv gh fuhf0 lplhqwr | ghfuhflplhqwr/ pä{lprv |2r pðqlprv1 Hmhpsor 77; Juätxhprv i+{, @ 6{6 {1 Vroxflöq=
d, Grpi @ U1
e, i hv frqwlqxd hq wrgr U +ixqflöq srolqrpldo,1
f, Lqwhufhswrv frq ho hmh | S3 @ +3> s3, s Lqwhufhswrv frq ho hmh { S4 @ + 4@6> 3,> S5 @ + 4@6> 3,
689
FKDSWHU 81 GHULYDGD
g, Lqwhuydorv gh fuhflplhqwr | ghfuhflplhqwr= i 3 +{, @ 4^/ | i 3 +{, ? 3+hq frqvhfxhqfld hvwulfwdphqwh ghfuhflhqwh, hq ho lqwhuydor ` 4@6> 4@6^1 Uhvxplhqgr whqhprv od vljxlhqwh lqirupdflöq gh i> { i+{, i 3 +{,
(
4@6 5@< 3
)
4@6 5@< 3
(
Sru ho fulwhulr gh od sulphud ghulydgd i wlhqh xq pä{lpr uhodwlyr hvwulfwr hq { @ 4@6 | xq pðqlpr uhodwlyr hvwulfwr hq { @ 4@61 Dvð od juäfd gh i hv dsur{lpdgdphqwh frpr od gh od jxud
Hmhpsor 77< Vhd i +{, @ vhq5 +{,1 Ghwhuplqhprv orv sxqwrv h{wuhprv | hvshflfduhprv gh txh wlsr vrq1
Vroxflöq= i 3 +{, @ 5 vhq+{, frv+{, orv sxqwrv fuðwlfrv vrq odv vroxflrqhv gh od hfxdflöq vhq+{, frv+{, @ 3
txh hv htxlydohqwh d od xqlöq gh odv vroxflrqhv gh odv hfxdflrqhv vhq+{, @ 3
Dvð/ orv sxqwrv fuðwlfrv vrq { @ q> { @ 5 . q>
frv+{, @ 3
q 5 ]1
8191 PÄ[LPR \ PÐQLPR
68:
Xvdqgr ho vljqr gh odv ixqflrqhv vhq+{, | frv+{, hq orv fxdgudqwhv gh U5 / whqhprv txh= Hq ho sulphu | whufhu fxdgudqwhv dpedv wlhqh ho plvpr vljqr sru or wdqwr od ghulydgd hv srvlwlyd1 hv ghflu/ | hq orv rwurv fxdgudqwhv vh wlhqh
6 i 3 +{, A 3 vl { 5`3> ^^`> ^ 5 5
6 i 3 +{, ? 3 vl { 5` > ^^` > 5^ 5 5
Xvdqgr od shulrglflgdg gh vhq+{, | frv+{, vh rewlhqh ho vljqr gh i 3 +{, hq wrgd od uhfwd uhdo1 Vl 6 { 5 ^q5] +`5q> . 5q^^`+5q . 4,> . 5q^,> i 3 +{, A 3 5 5
| hq fdvr frqwudulr i 3 +{, 31 Gh hvwd irupd vh wlhqh pä{lprv orfdohv hq orv sxqwrv 5 . 5q | 65 . 5q> q 5 ]/ | pðqlpr orfdohv orv sxqwrv q> q 5 ]1 Sru rwud sduwh 6 i +3, @ 3> i+5, @ 3> i + , @ 4> i + , @ 4> i +, @ 3 5 5
Frq od lqirupdflöq txh khprv rewhqlgr/ srghprv kdfhu od juäfd gh i hq irupd dsur{lpdgd1 Vdehprv txh od ixqflöq hv fuhflhqwh hq ho lqwhuydor `3> 5 ^ | txh 3 | 4 vrq pðqlpr | pä{lpr uhodwlyrv uhvshfwlydphqwh/ frpr i hv xqd ixqflöq frqwlqxd/ od juäfd hq ^3> 5 ` fruuhvsrqgd d xqlu orv sxqwrv +3> i +3,, | + 5 > 4, hq irupd fuhflhqwh/ d glihuhqfld gh od xqlöq gh orv sxqwrv + 5 > 4, | +> 3, txh hv hq irupd ghfuhflhqwh Ho juäfr vh sxhgh kdfhu frpsohwr/ sru od shulrglflgdg edvwd vdehu or txh rfxuuh hq ^3> `1
y
y
1
1
0.75
0.75
0.5
0.5
0.25
0.25
0 0 -7.5
-5
-2.5
0
2.5
i+{, @ vlq { 5
5
7.5
0 x
0.5
1
1.5
2
i+{, @ vlq {
2.5
3 x
5
Ho vljxlhqwh sureohpd gh dsolfdflöq hv qhfhvdulr rewhqhu suhyldphqwh xqd ixqflöq txh vh exvfduä ho pä{lpr r pðqlpr | oxhjr dsolfdu dojýq fulwhulr gh rewhqflöq gh pä{lpr r gh pðqlpr1 Hmhpsor 783 Vh lqwhqwd fhufdu xq fdpsr uhfwdqjxodu frq 7 kheudv gh dodpeuh | ghvsxìv vxeglylglu ho fdpsr hq grv sduwhv +frpr hq od jxud , frq xqd fhufd sdudohod d xqr gh orv odgrv/ xwlol}dqgr 5 kheudv gh dodpeuh1 Ghwhuplqdu odv glphqvlrqhv gho whuuhqr gh äuhd pä{lpd1
68;
FKDSWHU 81 GHULYDGD
Vroxflöq= Vhdq { h | odv glphqvlrqhv gho whuuhqr/ hqwrqfhv ho äuhd hvwä gdgd sru od iöupxod D @ {|
shur do glvwulexlu orv 933 phwurv gh dodpeuh vh wlhqh od hfxdflöq> gh grqgh
933 @ 43{ . ;|
| @ :8 8{@7 oxhjr>
8 D @ {+:8 {, 7 8{ shur { A 3 | | A 3/ hv ghflu/ { A 3 | :8 7 A 3 Sru or wdqwr od ixqflöq txh vh qhfhvlwd pd{lpl}du hv> D+{, @ :8{
ghqlgd hq ho lqwhuydor delhuwr `3> 93^1 Dkrud dsoltxhprv ho fulwhulr gh od sulphud ghulydgd> oxhjr
Uhvxplhqgr whqhprv
D3 +{,
8{5 7
@
:8
{
43
7
D3 +{, A 3 +, :8 85{ A 3 D3 +{, A 3 +, 63 A { 63 ? { D3 +{, ? 3 +,
{ D+{,, D3 +{,, D+{,
. . (
oxhjr ho äuhd pä{lpd vh dofdq}d fxdqgr { @ 63 1 Dvð odv glphqvlrqhv gho whuuhqr vrq { @ 63pwv
>
63 4458 3
. 0 )
| @ 6:> 8pwv
Revhuydflöq= Dtxð qr vh hydoýd hq orv h{wuhprv gho lqwhuydor/ sxhv qr hvwäq frqvlghudgrv hq ho grplqlr gh od ixqflöq1
8191 PÄ[LPR \ PÐQLPR
68<
Whruhpd 815< +Fulwhulr gh od vhjxqgd ghulydgd, Vhd i xqd ixqflöq ghulydeoh hq xq lqwhuydor delhuwr txh frqwlhqh d s wdo txh i 3 +s, @ 31 +hv ghflu/ s hv xq sxqwr fuðwlfr,1 d, Vl i %+s, ? 3 hqwrqfhv i wlhqh xq pä{lpr orfdo hvwulfwr hq { @ s1 e, Vl i %+s, A 3 hqwrqfhv i wlhqh xq pðqlpr orfdo hvwulfwr hq { @ s
Revhuydflöq= +4, Hv lpsruwdqwh kdfhu qrwdu txh d glihuhqfld gho fulwhulr gh od sulphud ghulydgd hvwh fulwhulr h{ljh txh i / vhd ghulydeoh grv yhfhv hq s frq i 3 +s, @ 31 +5, Vl i%+s, @ 3 hvwh fulwhulr qr qrv gä lqirupdflöq/ | vh gheh uhfxuulu do fulwhulr gh od sulphud ghulydgd1
Hmhpsor 784 Uhvroydprv ho hmhpsor 783 xvdqgr hvwh qxhyr fulwhulr Od ixqflöq d pd{lpl}du hv= 8{5 D+{, @ :8{ > frq { 5`3> 93^ 7 Ghwhuplqhprv s 5`3> 93^ wdo txh D3 +s, @ 31 D3 +{, @ :8
43{ +, { @ 63 7
Oxhjr s @ 63 hv xq sxqwr fuðwlfr txh ghehprv hydoxdu hq od vhjxqgd ghulydgd1 Frpr D%+{, @
43 ?3 7
;{ 5`3> 93^
hq sduwlfxodu D%+63, ? 3/ dvð srghprv dupdu txh D+63, hv ho pä{lpr exvfdgr1
Hmhpsor 785 Vh ghvhd frqvwuxlu xq wdqtxh gh dfhur frq od irupd gh xq flolqgur uhfwr | vhplhvihud hq orv h{wuhprv +yhdvh od jxud Q, sdud dopdfhqdu jdv sursdqr1 Ho frvwr gh ideulfdflöq sru slh fxdgudgr gh orv h{wuhprv hv ho greoh do gh od sduwh floðqgulfd1 Á Txì glphqvlrqhv plqlpl}dq ho frvwr/ vl od fdsdflgdg ghvhdgd gho wdqtxh hv gh 43slh6 B
Iljxud Q Vroxflöq= Vhd S ho frvwr gho slh fxdgudgr gh od sduwh floðqgulfd hqwrqfhv ho frvwr gho slh fxdgudgr gh orv h{wuhprv hv 5S 1 Sxhvwr txh hq od sduwh floðqgulfd vh wlhqh 5uk slh5 | hq orv h{wuhprv 7u5 slh5 hqwrqfhv ho frvwr gh frqvwuxfflöq hv F @ 5ukS . 7u5 5S
shur ho yroxphq hv
7 Y @ u5 k . u6 @ 43 6
693
FKDSWHU 81 GHULYDGD
hqwrqfhv | od ixqflöq frvwr hq wìuplqrv gh u hv>
k@
F+u, @ 5u+ hv ghflu>
43 76 u6 u5
43 7u ,S . 7u5 5S u5 6
8 7u5 , F+u, @ 7S + . u 6
Frpr u A 3> k A 3 hqwrqfhv t 6 63 / oxhjr 3 ? u ? gh grqgh u6 ? 63 7 7 Dvð od ixqflöq frvwr d plqlpl}du hv
k@
43 76 u6 A3 u5
8 7u5 F+u, @ 7S + . , u 6
>
u 5`3>
u 6
63 ^ 7
Dkrud ghwhuplqhprv ho pðqlpr sru ho fulwhulr gh od vhjxqgd ghulydgd +qrwhprv txh F+u, hv grv yhfhv glihuhqfldeoh hq vx grplqlr, 8 ;u F 3 +u, @ 7S + 5 . , u 6
oxhjr
8 ;u . @3 u5 6 48 +, u6 @ 7 s 6 48 +, u @ 5
F 3 +u, @ 3 +,
s6
Frpr 548 5 GrpF | F 3 +u, hv ghulydeoh srghprv dsolfdu ho fulwhulr gh od vhjxqgd ghulydgd1 Od ghulydgd gh F 3 +u, hv 43 ; F%+u, @ 7S + 6 . , u 6
s6
s6
oxhjr F%+ 548 , A 3/ gh grqgh F wlhqh xq pðqlpr hq u @ 548 5 GrpF1 Odv glphqvlrqhv txh plqlpl}dq ho frvwr gh ideulfdflöq gho wdqtxh vrq s s 6 7 6 48 73 48 > k@ s u@ 6 5 9 48
81:
Frqfdylgdg
Ghqlflöq 8163 Vhd i = L $ U/ L xq lqwhuydor
41 Gluhprv txh od ixqflöq i hv frqyh{d r frqfdyd kdfld duuled hq L vl ;d> e> { 5 L wdo txh d ? { ? e ho sxqwr +{> i +{,, +gh od juäfd gh i , hvwä dedmr gh od uhfwd vhfdqwh txh xqh orv sxqwrv +d> i+d,, | +e> i+e,, jxud^_ juäfr l}txlhugr1
81:1 FRQFDYLGDG
694
51 Gluhprv txh od ixqflöq i hv frqfdyd r frqfdyd kdfld dedmr hq L vl ;d> e> { 5 L wdo txh d ? { ? e ho sxqwr +{> i+{,, +gh od juäfd gh i , hvwä vreuh od uhfwd vhfdqwh txh xqh orv sxqwrv +d> i +d,, | +e> i +e,, jxud ^_ juäfr1 ghuhfkr
Iljxud ^_ Hmhpsor 786 Od sduäerod | @ 6{5 . 9{ . 5 hv frqfdyd kdfld duuled r frqyh{d1 Od sduäerod | @ 6{5 . 9{ . 5 hv frqfdyd kdfld dedmr r frqfdyd1
Od ixqflöq | @ vhq { hv frqfdyd kdfld dedmr hq ho lqwhuydor ^3> ` | frqfdyd kdfld duuled hq ho lqwhuydor ^> 5`1 Ghqlflöq 8164 Vhd i xqd ixqflöq | vhd { 5 Grpi 1 41 Gluhprv txh i hv frqfdyd kdfld duuled r frqyh{d hq ho sxqwr +{> i+{,, vl h{lvwh xqd yhflqgdg L @`d> e^ wdo txh { 5 L Grpi | od ixqflöq i mL hv frqfdyd kdfld duuled hq L1
51 Gluhprv txh i hv frqfdyd kdfld dedmr r frqfdyd hq ho sxqwr +{> i +{,, vl h{lvwh xqd yhflqgdg L @`d> e^ wdo txh { 5 L Grpi | od ixqflöq i mL hv frqfdyd kdfld dedmr hq L1 Revhuydflöq= 41 Gh od ghqlflöq vh sxhgh frqfoxlu txh frqfdylgdg vh uhhuh d xqd surslhgdg gh od juäfd1
Hq ho fdvr gh whqhu xqd ixqflöq hvwulfwdphqwh fuhflhqwh hq ho lqwhuydor L @ ^d> e` vh sxhgh whqhu txh i vhd frqfdyd hq L r frqyh{d hq L r hq xq vhfwru frqfdyd | hq rwur frqyh{d +yhu jxud ,/ or plvpr vh sxhgh ghflu gh xqd ixqflöq hvwulfwdphqwh ghfuhflhqwh1
695
FKDSWHU 81 GHULYDGD
Iljxud
51 Frpr od hfxdflöq gh od uhfwd vhfdqwh txh sdvd sru orv sxqwrv +d> i +d,,/ +e> i +e,, hv |@
r
|@
i +d, i+e, +{ d, . i +d, de i+d, i +e, +{ e, . i +e, de
Ghflu txh sdud d ? { ? e/ ho sxqwr hvwd ghedmr gh od vhfdqwh vljqlfd sru or wdqwr i +{, ?
| txh
i +{, ?
rshudqgr hq od ghvljxdogdg 8155
i +d, i+e, +{ d, . i +d, de i +d, i +e, +{ e, . i +e, de
i +{, i +d, ?
dkrud rshudqgr hq od ghvljxdogdg 8156
+8155, +8156,
i+d, i +e, +{ d, de
i +e, i+d, i+{, i+e, A {e ed
Xvdqgr odv ýowlpdv ghvljxdogdghv vh wlhqh txh i hv frqfdyd kdfld duuled vl | vöor vl d ? { ? e vh wlhqh i+e, i+d, i+{, i+e, i+{, i +d, ? ? {d ed {e
61 Kd| xqd fdudfwhul}dflöq jhrpìwulfd pdv fodud sdud ho fdvr gh xqd ixqflöq txh vhd ghulydeoh hq xq sxqwr {3 +d, Od juäfd gh i hv frqfdyd kdfld duuled hq ho sxqwr +{3 > i +{3 ,, vl h{lvwh xq lqwhuydor L txh frqwlhqh d {3 / wdo txh od juäfd gh imL hvwä sru hqflpd gh od uhfwd wdqjhqwh hq +{3 > i +{3 ,, jxud / juäfd l}txlhugd
81:1 FRQFDYLGDG
696
Iljxud 1 +e, Od juäfd gh i hv frqfdyd kdfld dedmr hq +{4 > i+{4 ,, vl h{lvwh xq lqwhuydor L txh frqwlhqh d {4 wdo txh od juäfd gh imL hvwd sru dedmr gh od uhfwd wdqjhqwh hq +{4 > i +{4 ,, jxud > juäfd ghuhfkd
Iljxud RK 71 Hq od jxud RK l}txlhugd whqhprv od juäfd gh xqd ixqflöq ghulydeoh frqfdyd kdfld duuled hq ho lqwhuydor L/ hq hood khprv glexmdgr glvwlqwdv uhfwdv wdqjhqwhv/ odv shqglhqwhv gh hvwdv uhfwdv dxphqwdq d phglgd txh qrv pryhprv gh l}txlhugd d ghuhfkd1 Frpr odv shqglhqwhv hvwdq ghwhuplqdgd sru odv ghulydgdv vh frqfox|h txh i 3 +{, hv fuhflhqwh1 Vl od ixqflöq i 3 +{, hv wdpelìq ghulydeoh vh frqfox|h txh i %+{, hv srvlwlyd r fhur sru ho khfkr txh i 3 +{, hv xqd ixqflöq fuhflhqwh1 Hq od jxud RK ghuhfkr khprv glexmdgr yduldv uhfwdv wdqjhqwhv d od juäfd gh i/ fx|dv shqglhqwhv vrq orv ydoruhv gh odv ghulydgdv gh i/ i 3 1 i 3 +{, hv ghfuhflhqwh |d txh odv shqglhqwhv gh odv wdqjhqwhv glvplqx|h d phglgd txh fdpeldq gh sxqwrv gh l}txlhugd d ghuhfkd hq od juäfd gh od ghuhfkd gh jxud RK1
Frpr od ixqflöq i 3 +{, hv ghfuhflhqwh/ gh vhu hood ghulydeoh/ vx ghulydgd hv qhjdwlyd r fhur/ hv ghflu/ i %+{, 3> ;{ 5 L1
Whruhpd 8165 Vhd i = L $ U/ grv yhfhv ghulydeoh hq ho lqwhuydor delhuwr L 41 i hv frqfdyd kdfld duuled hq L +, i %+{, 3 sdud wrgr { 5 L1 51 i hv frqfdyd kdfld dedmr hq L +, i %+{, 3 sdud wrgr { 5 L1
Od irupd frpr jhqhudophqwh vh dsolfd hvwh whruhpd hv= xvdqgr od lqirupdflöq gh od vhjxqgd ghulydgd vh rewlhqh od frqfdylgdg r frqyh{lgdg gh od juäfd
697
FKDSWHU 81 GHULYDGD
5 Frurodulr 8166 Vhd i = L $ U> {3 5 L> i ghulydeoh hq L | vxsrqjdprv txh h{lvwh gg{i5 +{3 ,1 5 41 gg{i5 +{3 , A 3 , od juäfd gh i hv frqfdyd kdfld duuled hq +{3 > i +{3 ,, 5 51 gg{i5 +{3 , ? 3 , od juäfd gh i hv frqfdyd kdfld dedmr hq +{3 > i +{3 ,,
Frq hvwh frurodulr txhgd delhuwr ho fdvr txh od vhjxqgd ghulydgd vh dqxoh1
Ghqlflöq 8167 Xq sxqwr +{3 > i +{3 ,, gh od juäfd gh i vh glfh txh hv xq sxqwr gh lq h{lrq vl h{lvwh xq delhuwr L wdo txh sdud wrgr { 5 L i{3 j/ i %+{, h{lvwh | dghpäv i%+{, fdpeld gh vljqr hq {3 1
Hmhpsor 787 Vhd i+{, @ 6{5 . 9{ . ; xqd sduäerod1 Frpr vh glmr dqwhulruphqwh hv ho hmhpsor päv vlpsoh gh frqfdyd kdfld duuled/ i %+{, @ 9 A 31
Hmhpsor 788 Vhd i +{, @ 9{5 . 9{ . ;/ i%+{, @ 9 txh hv qhjdwlyd/ sru or wdqwr od juäfd gh i +od sduäerod, hv frqfdyd kdfld dedmr1 Orv grv hmhpsorv dqwhulruhv oh ghehuldq shuplwlu uhfrugdu fxdo hv ho vljqr gh od vhjxqgd ghulydgd h{ljlgr sdud frqfdylgdg | fxdo hv h{ljlgr sdud frqyh{lgdg1 Hq orv grv hmhpsorv dqwhulruhv qr whqhprv sxqwrv gh lq h{löq/ d frqwlqxflöq yhuhprv xqr Hmhpsor 789 Vhd i +{, @ 5{6 . 8{5 . 47;@5:/ yhdprv vl i wlhqh sxqwrv gh lq h{löq1 i 3 +{, @ 9{5 . 43{ >
i %+{, @ 45{ . 43
sru or wdqwr orv sxqwrv gh lq h{löq wlhqh sulphud frpsrqhqwh txh suredeohphqwh vdwlvidfh od hfxdflöq i %+{, @ 3/ hq hvwh fdvrv { @ 8@91 Frpr i %+{, fdpeld gh vljqr hq 8@9 vh wlhqh txh +8@9> i+8@9,, hv ho sxqwr gh lq h{löq1 Srghprv kdfhu xqd wdeod uhvxphq gh od vhjxqgd ghulydgd | vx frqfoxvlöq { i%+{,, juäfd gh i
0 Frqfdyd
8@9
. Frqyh{d
Lq h{löq
Dkrud xvdqgr od sulphu ghulydgd srghprv kdfhu xqd wdeod uhvxphq txh qrv shuplwluä judfdu i frq pd|ru suhflvlöq i 3 +{, @ 3 +, 9{5 . 43{ @ 3
Dvð/ orv sxqwrv fuðwlfrv vrq { @ 3 | { @ 8@61 { i 3 +{, i %+{, i+{, juäfd gh i
. 0 ( Frqfdyd
8@6 3
0 0 ) Frqfdyd
8@9 3
0 . ) Frqyh{d
r xqd wdeod pdv uhvxplgd | juäfd pdv h{soðflwd hq od ýowlpd oðqhd { i 3 +{, i %+{, i +{,
. 0 ) _
8@6 3
0 0 ( _
8@9 3
0 . ( ^
3 3
. . ) ^
3 3
. . ( Frqyh{d
81:1 FRQFDYLGDG
698
Dvð/ od juäfd gh i ghehuðd vhu od xqlöq gh odv sxqwrv +8@6> i +8@6,, / +8@9> i+8@9,,/ +3> i +3,, frpr vh lqglfd hq od ýowlpd olqhd gh od wdeod/ rewhqhprv gh hvwd irupd ho juäfr gh od jxud
i+{, @ 5{6 . 8{5 . 47;@5:
Hmhpsor 78: Ho hmhpsor dqwhulru qrv srguðd oohydu d shqvdu txh orv sxqwrv gh lq h{löq vlhpsuh vh rewlhqh gh od hfxdflöq i%+{, @ 31 Do ljxdo txh hq ho fdvr gh ydoruhv h{wuhprv hvwrv qr vh rewlhqh vlhpsuh gh od hfxdflöq i 3 +{, @ 31 Hvwh hmhpsor qrv pxhvwud hq txh rwurv fdvrv dsduhfhq orv sxqwrv gh lq h{löq1
Vhd i +{, @ +{ 7,8@6 . :1 Ho grplqlr gh i +{, hv U | vxv ghulydgdv gh rughq 4 | gh rughq 5 vrq uhvshfwlydphqwh1 8 i 3 +{, @ +{ 7,5@6 6
i %+{, @
43 +{ 7,4@6 <
Fodudphqwh i%+{, @ 3 qr qrv duurmd vroxflöq/ shur vl srghprv qrwdu txh i %+{, qr h{lvwh hq { @ 7 | txh vx i%+{, fdpeld gh vljqr hq 7 sru or wdqwr +7> i +7,, @ +7> :, hv xq sxqwr gh lq h{löq { i %+{,, juäfd gh i
0 frqfdyd
7 9<
. frqyh{d
Wdeod uhvxphq sdud juäfd gh | @ +{ 7,8@6 . : hq od jxud
699
FKDSWHU 81 GHULYDGD
i +{, @ +{ 7,8@6 . : i 3 +{, @ 3 +, { @ 7
{ i 3 +{, i %+{, i +{,
0 0
7 3 i +4,,/ +5> i+5,,/ +4> i +4,, qr vrq sxqwrv gh od juäfd, shur srghprv yhu d txh ydoruhv wlhqghq fxdqgr { $ 5 / { $ 4 | fxdqgr { $ 4 olp i +{, @ 3 {$4
>
olp i +{, @ 3 {$4
Gh hvwd irupd od juäfd ghehuðd vhu frpr vh lqglfd hq jxud
>
olp i +{, @ .4 {$5
81:1 FRQFDYLGDG
69:
Dvð gh od plvpd wdeod srghprv ghflu txh i hv fuhflhqwh hq ` 4> 5^1
i hv ghfuhflhqwh hq `5> 4^1
i hv frqfdyd kdfld duuled hq ` 4> 5^ | wdpelìq hq `5> 4^1
i qr wlhqh ydoruhv h{wuhprv + 4 qr hv ydoru h{wuhpr |d txh qr shuwhqhfh d U,1
i qr wlhqh sxqwrv gh lq h{löq |d txh ho ýqlfr fdqglgdwr d wdo hud +5> i+5,,/ shur qr shuwhqhfh d od juäfd gh i + i +5, qr h{lvwh, dpìq gh txh i%+{, qr fdpeld gh vljqr hq { @ 51
81:14
Whvw gh od Vhjxqgd Ghulydgd=
Orv uhvxowdgrv gh frqfdylgdg hqxqfldgrv hq ho frurodulr dqwhulru qrv shuplwhq hq pxfkrv fdvrv fodvlfdu udslgdphqwh orv sxqwrv h{wuhprv hq pä{lpd orfdohv r pðqlprv orfdohv phgldqwh or txh vxhoh oodpduvh fulwhulr gh od vhjxqgd ghulydgd Hmhpsor 78< Gh xqd sodqfkd phwäolfd gh 433 fp 5 pwv vh ghvhd frqvwuxlu grv wxerv gh 5 phwurv gh odujr xqr frq vhfflöq wudqvyhuvdo xqd flufxqihuhqfld | rwur vhfflöq fxdgudgd BcFrpr vh gheh fruwdu od sodqfkd d or odujr gh wdo irupd txh od vxpd gh orv yroýphqhv lqwhulruhv vhd pä{lpr1 BcFxdqgr hv pðqlprB Vroxflöq= Vxsrqjdprv txh od sodqfkd vh fruwdq hq {fp5pwv | +433 {, fp5pwv/ frq od sulphud kduhprv ho flolqgur gh udglr u/ hqwrqfhv 5u @ { | frq od vhjxqgd ho wxer frq vhfflöq fxdgudgd/ hv ghflu/ fdgd odgr hv gh + 4337{ ,fpv1 Ho yroxphq gho flolqgur hv { u5 5333fp6 @ + ,5 5333fp6 5 Ho yroxphq gho rwur wxer hv
+
433 { 5 , 5333fp6 7
69;
FKDSWHU 81 GHULYDGD
Od vxpd gh orv yroýphqhv gh orv wxerv hq wìuplqr gh { hv
{5 +433 {,5 . ,> frq 3 { 433 7 gY 5{ 533 5{ @ 833+ + ,, g{ 7 Sxqwrv fuðwlfrv vrq vroxflöq gh od hfxdflöq ;{ 533 . 5{ @ 3 7 +; . 5,{ @ 533 433 {3 @ 7. Y +{, @ 833+
5 5 vh rewlhqh grv Frpr gg{Y5 @ 833+ 5 . 45 ,/ gg{Y5 +{3 , A 3> vh wlhqh gh hvwd irupd txh frq ho fruwh hq 433 7. wxerv fx|d vxpd gh orv yroxphqhv hv or pðqlpr srvleoh gh rewhqhu1 Frpr qr vh rewxyr rwur sxqwr fuðwlfr yld od ghulydgd/ ho r orv pä{lprv ghehq hqfrqwuduvh hq od iurqwhud gho lqwhuydor ^3> 433`1
d, Vl kdfhprv xq wxer floðqgulfr { @ 4331
Y +433, @
8+43,9 833+433,5 @
e, Vl vöor kdfhprv xq wxer frq vhfflöq fxdgudgd { @ 3
833+433,5 8+43,9 @ 7 7 Frpr ? 7 vh wlhqh txh fxdqgr vöor kdfhprv xq wxer floðqgulfr rewhqhprv pä{lpr yroxphq1 Y +3, @
Hvwh hmhpsor qrv pxhvwud xqd yh} päv txh orv h{wuhprv vh exvfdq qr wdq vöor grqgh od ghulydgd vh dqxod/ vlqr txh wdpelìq hq orv h{wuhprv gho lqwhuydor fhuudgr grqgh od ixqflöq hvwä ghqlgd1 Whvw gh od Vhjxqgd Ghulydgd= Vhd i xqd ixqflöq frqwlqxd hq xq lqwhuydor fhuudgr43 ^d> e` | vxsrqjdprv txh hv ghulydeoh hq `d> e^ = g5 i gi +, h{lvwh vh wlhqh +, @ 3 , | Vl 5 `d> e^ hv xq sxqwr fuðwlfr gh i +l1h1/ vx ghulydgd vh dqxod/ g{ g{5 5 g i grv srvlelolgdghv ghshqglhqgr gho vljqr gh +, g{5 5 g i +, A 3 vh frqfox|h txh hq kd| xq pðqlpr uhodwlyr/ hvwh hv i +, 410 Vl g{5 5 g i 510 Vl +, ? 3 vh frqfox|h txh hq kd| xq pä{lpr uhodwlyr/ hvwh hv i +, g{5 g5 i Frpr sxhghv qrwdu vl +, @ 3> qr vh glfh qdgd/ srguðd rfxuulu fxdotxlhu frvd frpr vh sxhghq yhu g{5 hq odv ixqflrqhv i+{, @ {6 hq xq lqwhuydor fhuudgr fhufd gho fhur qr wlhqh pä{lpr ql pðqlpr hq fhur/
i+{, @ {7 > wlhqh xq pðqlpr hq fhur |
i+{, @ 5{7 wlhqh xqd pä{lpr hq fhur
Revhuydflöq= Hvwr gh srqhu xq lqwhuydor fhuudgr hv vöor sdud jdudqwl}du od h{lvwhqfld gh ydoruhv h{0 wuhprv/ shur ho whvw hv dsolfdeoh hq rwurv fdvrv |d txh vh gheh fxpsolu odv klsöwhvlv hq xqd yhflqgdg gho sxqwr fuðwlfr1 Uhfxhugh txh wdpelhq sxhgh kdehu pä{lpr r pðqlprv hq oxjduhv txh od ixqflöq qr hv ghulydeoh/ dvl phqrv vhuä grv yhfhv ghulydeoh | hq hvh fdvr vöor qrv txhgd ho whvw gh od sulphud ghulydgd1 43
Sru or wdqwr |d vdehprv txh hood wlhqh pä{lpr | plqlpr
81:1 FRQFDYLGDG
81:15
69<
Oðplwh Lqqlwr
Frpr xq hohphqwr päv txh qrv shuplwluä judfdu i frq pd|ru suhflvlöq hvwxglduhprv orv vljxlhqwhv wlsrv gh oðplwhv 4, olp{$4 i +{, 5, olp{$4 i +{, 6, olp{$d i+{, @ 4 7, olp{$d i +{, @ 4 Od ud}öq gh hvwxgldu dsduwh hvwh wlsr gh oðplwhv hq irupd vhsdudgd d orv rwurv vh gheh do khfkr txh .4 | 4 qr vrq qýphurv uhdohv/ sru or wdqwr qr wlhqhq vhqwlgr odv h{suhvlrqhv 4. 4/ .4 4 | hq jhqhudo rshudflrqhv txh hvwdq ghqlgd hqwuh qýphurv uhdohv1 Sru or dqwhulru qr srghprv kdeodu gh yhflqgdg 4 frpr vh ghqlö dqwhulruphqwh |d txh gh khfkr qr kd| vxefrqmxqwrv gh U txh frqwhqjd d lqqlwr/ shur sdud dpsoldu qxhvwud ghqlflöq gh oðplwhv d hvwh fdvr/ gdprv od ghqlflöq vljxlhqwh1 Ghqlflöq 8168 =
41 Wrgr lqwhuydor delhuwr gh od irupd `Q> 4^> Q 5 U hv xqd yhflqgdg gh lqqlwr/ xqd yhflqgdg gh 4 hv fxdotxlhu lqwhuydor delhuwr gh od irupd ` 4> P ^> P 5 U 51 Gluhprv txh oðplwh gh i fxdqgr { wlhqgh d lqqlwr hv O/ O 5 U1 Vl sdud wrgr yhflqgdg gh O> YO> % / h{lvwh xqd yhflqgdg gh lqqlwr YQ wdo txh i +YQ , YO> %
Lqwurgxflhqgr od qrwdflöq dqwhulru/ qdwxudo gh hvwh fdvr/ h lqglfdqgr or txh vljqlfd odv yhflqgdghv whqhprv olp i+{, @ O +, +; % A 3,+ YO> % / h{lvwh xqd yhflqgdg gh phqrv lqqlwr/ Y4>P / wdo txh Ghqlflöq 8169
i+Y4>P , YO> %
Hv dvð txh
olp i +{, @ O +, +; % A 3,+ t+{, @ e3 .e4 {. .eo {o srolqrplr gh judgr q | o uhvshfwlydphqwh hqwrqfhv dq s+{, eq q @ o olp @ {$4 t+{, 3 q?o Ghprvwudflöq= Fdvr +4, vxsrqjdprv txh q @ o1
d3 . d4 { . . dq {q 4@{q olp {$4 e3 . e4 { . . eq {q 4@{q d3 . d4 . . d q q {q4 @ olp {e3 = {$4 q . qe44 . . eq { {
s+{, olp {$4 t+{,
@
Sru hmhpsor dqwhulru vh wlhqh txh orv q sulphurv oðplwhv gho qxphudgru | orv q sulphurv oðplwhv gho ghqrplqdgru vrq qxorv/ dvð dq s+{, olp @ {$4 t+{, eq
Fdvr+5, q ? o s+{, olp {$4 t+{,
Hmhpsor 796 Fdofxohprv Vroxflöq= 4@6
Hmhpsor 797 Fdofxodu
Vroxflöq=
Hmhpsor 798 Fdofxohprv
d3 . d4 { . . dq {q 4@{o olp {$4 e3 . e4 { . . eo {o 4@{o d3 . d4 . . dq o o4 {oq @ olp { e3 { e4 {$4 o . o4 . . eo { { 3.3..3 @ @3 3 . 3 . . eo
@
6{5 . 8{6 . :{ . 5 olp {$4 7{5 . 9{ . 48{6 . < s 6 ;{9 . :{5 . 5 olp {$4 8{5 . 9{ . : s 6 ;{9 . :{5 . 5 olp {$4 8{5 . 9{ . :
@
olp {$4
s6 {9 {5 5 ;
.:
.5
{ {5 .9{.: 5 t { 6 ; {9 . : {5 . 5 {9 {9 {9 @ olp 9 {$4 8 . { . {: s 6;.3.3 5 @ @ 8.3.3 8
olp {$4
frv {6 {6
8
81:1 FRQFDYLGDG
6:4
Frpr frv {6 hv dfrwdgd hqwuh 4 | 4 vh wlhqh txh
frv {6 4 4 6 {6 {6 {
| frpr dghpäv olp{$4 {46 @ 3 xvdqgr dfrwdplhqwr vh rewlhqh olp {$4
frv {6 @3 {6
Hmhpsor 799 Xq pdwhuldo udglr dfwlyr hq ho wlhpsr w wlhqh xqd pdvd p+w, @ nhw grqgh n> 5 U. 1 Pxhvwuh txh od pdvd wlhqgh d ghvdsduhfhu fxdqgr w $ 4 Vroxflöq= olpw$4 nhw @ olpw$4 hnw shur frpr hw 4 . w 3>
Dvð/
81:16
Dvðqwrwdv
;w 5 U.
n n 3 olp w olp @3 w$4 h w$4 4 . w
Dvlqwrwdv Krul}rqwdohv
Vhd i ghqlgd hq U. | frq olp{$4 i +{, @ O1 Frpr olp{$4 i +{, @ O vh wlhqh txh sdud wrgr % A 3/ h{lvwh Q 5 Q wdo txh vl { A Q> mi+{, Om ? %/ shur hvwr ýowlpr hv htxlydohqwhphqwh d O % ? i+{, ? O . %/ ;{ A Q 1 Wrpdprv od uhfwd | @ O/ uhfwd krul}rqwdo/ vh wlhqh txh d phglgd txh { vh dfhufd do lqqlwr i vh dfhufd d hvwd uhfwd frpr od jxud1
Dvðqwrwd Krul}rqwdo
6:5
FKDSWHU 81 GHULYDGD
Hq ho fdvr gh olp{$4 i+{, @ O rfxuuh or plvpr sdud ixqflrqhv ghqlgdv hq grplqlr txh frqwhqjd lqwhuydor gh od irupd ` 4> Q `/ |d txh +; % A 3,+ | @ :7 > hv xqd dvðqwrwd krul}rqwdo1 Frpr olp{$4 i+{, @ :7 > | @ :7 vh rewlhqh od plvpd dvðqwrwdv krul}rqwdohv1
Hmhpsor 79; Fdofxoh odv dvðqwrwdv krul}rqwdohv gh i +{, @ Vroxflöq=
s 8 5{5 . 7 {.6
t s 8 4 . {75 s 8 5.3 olp i +{, @ olp @ @8 5 { .6 {$4 {$4 4.3 {
81:1 FRQFDYLGDG
6:6
s oxhjr | @ 8 5 hv xqd dvðqwrwd krul}rqwdo1
t s 8 4 . {75 s 8 5 . 3 olp i+{, @ olp @ @ 8 5 { .6 {$4 {$4 4.3 {
s +Uhfxhugh txh {5 @ m{m,1 s Sru or wdqwr | @ 8 5 hv rwud dvðqwrwd krul}rqwdo1 Dkrud hvwxglduhprv oðplwhv txh ydq d lqqlwr/ frpr hv ho fdvr gh Ghqlflöq 816;
4 olp {$3 m{m
r
4 olp {$< m{ Q / h{lvwh xqd yhflqgdg gh s> Ys> wdo txh i +Ys> , Y4> %
Kdflhqgr od wudgxfflöq gh odv yhflqgdghv whqhprv
olp i +{, @ 4 +, +;Q,+Q h{lvwh xqd yhflqgdg gh s> Ys> wdo txh i +Ys> , Y4> %
Kdflhqgr od wudgxfflöq gh odv yhflqgdghv whqhprv
olp i +{, @ 4 +, +;Q ,+ n lpsdu 4
l, olp{$3 m{4m @ 4 | frpr m{4m m{4mn vl m{m ? 4 | n 4/ vh wlhqh
ll, Wrpdqgr x @ { d/ vh wlhqh
4 olp @4 {$3 m{mn
4 4 olp @ olp {$s m{ smn x$3 mxmn
lll, Gdgr Q 5 U > Q 5 U. h{lvwh A 3 wdo txh Dvð Sru or wdqwr
3 9@ m{m ? ,
? { ? 3 ,
4 AQ m{m
4 AQ {
Gdgr Q 5 U > 5 U vh wlhqh 41 olp{$s +i . j,+{, @ 4
51 Sdud A 3
61 Sdud ? 3 71 olp{$s ij +{, @ 3
olp +ij,+{, @ 4 {$s olp +i j,+{, @ 4 {$s
i olp +{, @ 4 {$s j
i olp +{, @ 4 {$s j
81:1 FRQFDYLGDG
6:8
Ghprvwudflöq= +7, Vhd srvlwlyr/ frpr olp{$s j+{, @
+; % A 3,+ i j hqwrqfhv
j+{, . % i +{, Qi Dvð/ wrpdqgr Qi vxflhqwhphqwh judqgh vh rewlhqh or uhtxhulgr1 Hv dqdorjdod ghprvwudflöq sdud ho fdvr ? 31 Revhuydflöq= Ho whruhpd dqwhulru vljxh vlhqgr yäolgr sdud oðplwhv odwhudohv1 ;{ 5 Grpj@i a 3 ? m{ sm ? ,
Hmhpsor 7:5 Fdofxohprv
{5 . 7 olp 5 {$5 m{ 7m
5 .7 4 @ 4 | olp{$5 m{{.5 Vroxflöq= olp{$5 m{ 5m m @ 51 Sru whruhpd dqwhulru sduwh +5, whqhprv olp
Dvðqwrwdv Yhuwlfdohv
{5 . 7 @4 m{5 7m
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FKDSWHU 81 GHULYDGD
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Ghqlflöq 8174 Vhd i frq grplqlr txh frqwhqjd d xq lqwhuydor gho wlsr `q> 4^ r ` 4> Q ^1 Gluhprv txh od uhfwd | @ d{ . e hv xqd dvðqwrwd reolfxd gh i vl olp mi +{, d{ em @ 3 {$4
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Sdud srghu frpsuredu txh qr kd| päv dvðqwrwdv qhfhvlwdprv od vljxlhqwh sursrvlflöq
Sursrvlflöq 8175 Vhd i xqd ixqflöq wdo txh ho grplqlr gh i frqwlhqh d xq frqmxqwr gho wlsr `Q> 4^ r ` 4> Q^ hqwrqfhv Od uhfwd | @ d{ . e hv xqd dvðqwrwd gh i vl | vöor vl i +{, @ d | olp i+{, d{ @ e olp {$ 4 { {$ 4
81:1 FRQFDYLGDG
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5 { / Pxhvwuh txh | @ 8{ 5 hv xqd dvðqwrwdv reolfxd1 Hmhpsor 7:9 Vhd i +{, @ 8{ :{{.5.vhq 4
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61 Ghwhuplqdu orv lqwhuydorv gh fuhflplhqwr | ghfuhflplhqwr +vljqr gh i 3 ,1
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FKDSWHU 81 GHULYDGD
71 Ghwhuplqdu grqgh i % qr h{lvwh r grqgh i % vh dqxod1
81 Ghwhuplqdu od frqfdylgdg gh i +vljqr gh i %,
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81 Od ixqflöq i wlhqh xq pä{lpr hq { @ 31
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ghqh xqd uhodflöq H/ frq H U5 +txh qr hv ixqflöq,1 H @ i+{> |, 5 U5 =
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Ho grplqlr gh od sulphud klsìuerod hv U` d> d^ | gh od vhjxqgd hv U1 Orv uhfruulgrv gh hoodv vrq U | U` e> e^ uhvshfwlydphqwh1 Qlqjxqd gh ìvwdv hv xqd ixqflöq1
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Frrughqdgdv Sroduhv
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E15
E1514
Wudqvirupdflöq gh Frrughqdgdv
Wudvodflrqhv hq frrughqdgdv fduwhvldqdv
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| @y.n
Iljxud v Hmhpsor 7;< Vxsrqjdprv txh ho ruljhq gho vlvwhpd RX Y wlhqh frrughqdgdv +6> 8, hq ho vlvwhpd R[\ / | txh orv vlvwhpdv wlhqhq orv hmhv sdudohorv hqwuh vð1 Ho sxqwr S gh frrughqdgdv +5> :, hq ho vlvwhpd RX Y wlhqh frrughqdgdv +5 . 6> : 8, hq ho vlvwhpd R[\ Hmhpsor 7 6vhq+@7, . 7 frv+@7,, @ + 5@5> : 5@5,
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e, Gdgr ho sxqwr S frq frrughqdgdv sroduhv +u> ,/ odv frrughqdgdv fduwhvldqdv vrq +{> |, grqgh { @ u frv >
| @ uvhq
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Od uhfwd { @ | hq frrughqdgdv fduwhvldqdv wrpd xqd irupd wrwdophqwh glvwlqwd hq frrughqdgdv sroduhv1 Hq hihfwr { @ | +, u frv @ uvhq +, wj @ 4
Dvð/ orv vljxlhqwhv frqmxqwrv uhsuhvhqwd or plvpr
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E16
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ge vl f d 9@ 3 fd
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r htxlydohqwhphqwh
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d, Vlphwuðd= Sdud wrgr sdu gh sxqwrv S4 > S5 5 U5
g+S4 > S5 , @ g+S5 > S4 ,
e, Ghvljxdogdg wuldqjxodu= Sdud wrgr wuðr gh sxqwrv S4 > S5 > S6 5 U5 vh wlhqh f, Vhd S4 > S5 5 U5 hqwrqfhv
g+S4 > S6 , g+S4 > S5 , . g+S5 > S6 , g+S4 > S5 , @ 3 +, S4 @ S5
Hmhuflflr 7 e, | S5 @ +f> g, 5 U5 grv sxqwrv duelwudulrv | vhd g+S4 > S5 , @ 3/ dvð g+S4 > S5 , @
s +d f,5 . +e g,5 @ 3
sru or wdqwr +d f,5 @ 3 a +e g,5 @ 31 Gh dtxðvh wlhqh d @ f a e @ f/ hv ghflu/ +d> e, @ +f> g,
+f,++, Vl S4 @ S5 > g+S4 > S5 , @ 3 wulyldophqwh
S4 @ S5
6 e, | od uhfwd D{ . E| . F @ 3 hv mDd . Ee . Fm s D5 . E 5
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DH GE @ 3 DG @ EH
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E17
E1714
Föqlfdv
Flufxqihuhqfld
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F @ i+{> |, 5 U5 = g++{> |,> S , @ uj=
g++{> |,> S , @ u hv htxlydohqwh d
s +{ k,5 . +| n,5 @ u>
F @ i+{> |, 5 U5 = +{ k,5 . +| n,5 @ u5 j
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+E14,
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+E15,
|/ phgldqwh od uhvroxflöq gh orv elqrplrv fxdguäwlfrv vh sxhgh oohydu d xqd h{suhvlöq frpr od vljxlhqwh
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Iljxud K
Hq ho fdvr +gh od jxud K fhqwudo, txh ho irfr vhd I @ +3> s, | od uhfwd O hv | @ s/ S vh uhgxfh d s s S @ i+{> |, 5 U5 = +{ 3,5 . +| s,5 @ +{ {,5 . +| . s,5 j
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+{ 3,5 . +| s,5 @ +| . s,5 {5 @ 7s|
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wlhqh irfr I @ +k . s> n,/ gluhfwul} { @ k s | yìuwlfh Y @ +k> n,1 Ghvduuroodqgr odv h{suhvlrqhv gh od hfxdflöq +E17, vh wlhqh txh hood vh sxhgh uhgxflu d xqd hfxdflöq gho wlsr D{5 . E{ . F| . G @ 3> Dqäorjdphqwh vh wlhqh txh od hfxdflöq +E18, vh sxhgh oohydu d xqd hfxdflöq gho wlsr D| 5 . E| . F{ . G @ 3
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E1716
Holsvh
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r htxlydohqwhphqwh
{5 +n5 s5 , . n5 | 5 @ n5 +n5 s5 ,
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Iljxud
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6 34: | sdud ho frphwd Kdooh| 3> hv ghflu od öuelwd gh od wlhuud hv päv uhgrqgd% txh od gho frphwd1 Ho frphwd Kdooh| wlhqh xq shuðrgr gh :9> 5 dôrv +hv ghflu/ vh ghprud hvh wlhpsr hq uhfruuhu od holsvh txh irupd vx öuelwd,1 Od glvwdqfld pðqlpd gho frphwd do vro hv dsur{lpdgdphqwh 3> 8;:XD +xqlgdghv dvwurqöplfdv5 ,/ hvwr fruuhvsrqgh d od glvwdqfld hqwuh ho irfr | ho yìuwlfh sulflsdo uhvshfwlyr/ hv ghflu/ d s 3> 8;:
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E1717
Klsìuerod
Vhdq I4 I5 grv sxqwrv gh U5 Xqd klsìuerod K hv ho frqmxqwr gh wrgrv orv sxqwrv +{> |, gho sodqr wdohv txh od glihuhqfld gh vxv glvwdqfld d orv sxqwrv mrv I4 | I5 / oodpdgrv irfrv/ hv xqd frqvwdqwh%1 Gh hvwd irupd K @ i+{> |, 5 U5 = mg+I4 > +{> |,, g+I5 > +{> |,,m @ frqvwdqwhj
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holplqdqgr odv udðfhv vh sxhgh oohjd d od h{suhvlöq
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|5 {5 5 @4 5 d s d5
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+| n,5 +{ k,5 @4 e5 d5 wlhqh wdpelìq frpr uhsuhvhqwdflöq juäfd xqd klsìuerod shur hq hvwh fdvr ho hmh txh frqwlhqh orv yìuwlfhv | irfrv hv ho hmh R\ 1
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Ixqflrqhv Uhdohv gh Grv Yduldeohv Ghqlflöq F14 Oodpduhprv hvsdflr hxfolgldqr q0glphqvlrqdo do frqmxqwr Uq gh wrgrv orv duuhjorv rughqdgrv +q0xsodv, +{4 > {5 > {6 > > {q , gh qýphurv uhdohv1 Xq hohphqwr +{4 > {5 > {6 > > {q , 5 Uq vh glfh txh hv xq sxqwr hq Uq 1 Vh wlhqh txh +{4 > {5 > > {q , @ +|4 > |5 > > |q , vl | vöor vl {4 @ |4 > {5 @ |5 > > {q @ |q Uq @ i+{4 > {5 > > {q , = {l 5 U> l @ 4 qj
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Ghqlflöq F15 Xqd ixqflöq uhdo gh grv yduldeohv frq grplqlr D/ i = D U5 $ U hv xqd fruuhvsrq0 ghqfld wdo txh sdud fdgd sdu +{> |, hq D U5 h{lvwh xq ýqlfr hohphqwr } 5 U wdo txh } @ i+{> |,1 Qrwdflöq= Hq irupd dqäorjd/ vh ghqh ixqflöq uhdo gh q0yduldeohv
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DSSHQGL[ F1 IXQFLRQHV UHDOHV GH GRV YDULDEOHV
51 Ho grplqlr Dgh xqd ixqflöq uhdo gh grv yduldeohv
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hv xq vxefrqmxqwr gh U5 / hv ghflu/ hv xq vxefrqmxqwr gho sodqr1 Sru rwud sduwh od juäfd gh od ixqflöq itxh vh ghqh frpr Judii
hv xq vxefrqmxqwr gh U6
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61 Do ljxdo txh sdud ixqflrqhv uhdohv gh xqd yduldeoh/ vl } @ i +{> |, uhsuhvhqwd xqd ixqflöq uhdo gh grv yduldeohv/ hqwrqfhv vxehqwhqghprv txh ho grplqlr gh i hv xq vxefrqmxqwr gh U5 sdud ho fxdo od h{suhvlöq i +{> |, shuwhqhfh d U1
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736 Ho uhfruulgr hvwä irupdgr sru orv ydoruhv uhdohv txh wrpd } @ i +{> |, frq +{> |, 5 Grpi/ hv ghflu/ Uhfi @ ii+{> |, 5 U = +{> |, 5 Grpi j
s Dvðsru hmhpsor> vl } @ 4 {5 |5 hqwrqfhv } wrpd ydoruhv uhdohv qr qhjdwlyrv/ hv ghflu/ } 31 Dghpäv/ hohydqgr do fxdgudgr whqhprv } 5 @ 4 {5 | 5
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41 Grpi @ i+{> |, 5 U5 = {5 . | 5 9@ 3j @ U5 i+3> 3,j
51 Grpj @ i+{> |, 5 U5 = {5 | 5 9@ 3j
61 Grpk @ i+{> |, 5 U5 = | 7 {5 j
5 5 71 Grpm @ i+{> |, 5 U5 = {7 . |49 4j
81 Grpn
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91 Grpo @ i+{> |, 5 U5 = m{ . |m ? 7j @ i+{> |, 5 U5 = 7 ? { . | ? 7j
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;1 Grpt @ i+{> |, 5 U5 = +4 . 7{,+4 . 9|, 3 a 5{ . 6| 9@ 3j
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DSSHQGL[ F1 IXQFLRQHV UHDOHV GH GRV YDULDEOHV
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i = D U5 +{> |,
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Do ljxdo txh ixqflrqhv uhdohv gh xqd yduldeoh/ vh ghqh ho äojheud gh ixqflrqhv gh grv yduldeohv +r päv yduldeohv, Vl i | j vrq ixqflrqhv uhdohv gh grv yduldeohv wdo txh G @ Grpi _Grpj 9@ !/ hqwrqfhv h{lvwhq ixqflrqhv i . j> i j> i j frq grplqlr G | od ixqflöq i@j frq grplqlr i+{> |, 5 G = j+{> |, 9@ 3j grqgh 4, +i . j,+{> |, @ i+{> |, . j+{> |,
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LL1 Frpsohwdflöq=
41 Ho frqmxqwr vroxflöq gh od lqhfxdflöq s {5 3 hv1111111111111111111 m{ 5m m4 {m 51 Ho frqmxqwr vroxflöq gh od lqhfxdflöq s {5 3 hv 111111111111111 m6 {m m{ 5m 61 Od lqhfxdflöq
5 5 4 6 + ,8{.5{ i +{, @ + 5:4h ,6{4 hqwrqfhv ho grplqlr pä{lpr gh i hv 111111111111 541 Vhd i +{, @ orj +{, | j+{, @ {5 . e{ e ixqflrqhv/ hqwrqfhv ho frqmxqwr gh wrgrv orv e 5 U wdo txh ho grplqlr gh i j hv ydfðr hv 11111111111111111 s 551 Vhd i+{, @ orj++{ 6,5 , | j+{, @ 7 {/ hqwrqfhv ho grplqlr pä{lpr gh j i hv 111111111 561 Ho grplqlr gh od ixqflöq s s 4 4 {5 i +{, @ {5 hv 11111111111111111111 571 Od uhodflöq |5 @ 5 { ghqh xqd ixqflöq el|hfwlyd hq 11111111111111111 5 { >{ ? 3 { >{ ? 4 hqwrqfhv j i +{, hv 11111111111111111111 581 Vhd i+{, @ | j+{, @ s { >{ 4 5{ > { 3 591 Gdgd odv ixqflrqhv i | j ghqlgdv sru s {.4 { ?3 {4 4{8 i+{, @ j+{, @ {.5 { 3 { . : {A8 +d, Ho Uhfruulgr gh i hv 111111111111 +e, Ho Grplqlr gh j i hv 1111111111111111 +f, +j i ,+{, @
5:1 Grv wxehuðdv wdugdq 9 krudv hq oohqdu xqd slvflqd1 Xqd vrod od oohqduðd 8 krudv päv gh sulvd txh od rwud1 Hqwrqfhv fdgd wxehuð d wdugduðd lqglylgxdophqwh hq oohqdu od slvflqd 111111111111111 | 1111111111111111 uhvshfwlydphqwh 5;1 Vhd i = `3> 4^i4j $ U 4 { :$ { .6 4 hqwrqfhv ho uhfruulgr hv 111111111111111
G141 UHVSXHVWDV D ORV HMHUFLFLRV SURSXHVWRV
746
5 |, @
G14
oq+4
{
{.|
|,
hv ho frqmxqwr 1111111111111 fx|d juäfd hv 11111111111
Uhvsxhvwdv d orv Hmhuflflrv Sursxhvwrv
Vroxflrqhv gh od vhfflöq 41814 41
4 3 +d, Idovr1 + , 4 4 4 3 +f, Idovr1 + , 5 4 3 3 +h, Idovr1 + , 3 4
4 3 +e, Yhugdghur1 + , 3 4 3 3 +g, Idovr1 + , 3 4 3 3 +i, Idovr1 + , 4 4
Vroxflrqhv gh od vhfflöq 41815 41 < shuvrqdv1
51 Ho suhflr gho sulphu | gho vhjxqgr yroxphq hv gh ' 6=333 1 61 6 phwurv | 9 phwurv/ uhvshfwlydphqwh1
71 Ho sulphu vdfr wlhqh ;3 nlorjudprv | ho vhjxqgr 93 nlorjudprv1 81 Ho shglgr vh frpsohwduä hq 5: gðdv1
91 4: nlorjudprv gh freuh | : nlorjudprv gh }lqf1 :1 Ho qýphur exvfdgr hv 971 ;1 Ho qýphur exvfdgr hv 571
ho sulphur hvwä d 6 krudv gho hqfxhqwur hq od fduuhwhud/ ho vhjxqgr/ 5 krudv1 471
< 7
olwurv sru plqxwr> 6 olwurv sru plqxwr1
481 Ho sulphu wudedmdgru wdugduä 47 gðdv/ ho vhjxqgr 44 gðdv hq uhdol}du lqglylgxdophqwh ho wudedmr1 491 Od sulphud pätxlqd shuirud 5 phwurv gh wýqho sru gðd/ | od vhjxqgd
6 5
phwurv sru gðd1
4:1 Od sulphud fxdgulood sxhgh uhdol}du od uhsdudflöq frpsohwd hq 43 gðdv/ | od vhjxqgd hq 48 gðdv1 4;1 Ho sulphu reuhur sxhgh kdfhu ìo vror ho wudedmr hq 45 krudv> ho vhjxqgr/ hq 57 krudv1 4 od vhjxqgd/ hq 48 krudv1
747
DSSHQGL[ G1 SUREOHPDV PLVFHOÄQHRV SDUWH L
531 Od uxhgd pd|ru gd 533 usp | od phqru 933 usp1
541 Ho sulphu prwru frqvxph 93 judprv gh frpexvwleoh sru krud> ho vhjxqgr/ 7; judprv gh frpexvwleoh sru krud1
551 Od juýd gh pd|ru srwhqfld ghprud 57 krudv hq uhdol}du vrod ho wudedmr frpsohwr/ plhqwudv txh od gh phqru srwhqfld ghprud 69 krudv1 561 Od dowxud hv k 68> 89fp vx udglr u 4 4 |, @ +5> 3, S5 +{> |, @ +3> 5, 51 S4 +{> |> }, @ +5> 5> 5,
61 S4 +{> |, @ +6> 4, S5 +{> |, @ +4> 6,
71 S4 +{> |, @ +7> 6, S5 +{> |, @ +7> 6,
: > 47 , 81 S4 +{> |> , @ +3> 4> 4, S5 +{> |> , @ + 67 > 49
s s s s 91 S4 +{> |> , @ + 55 > 5> 47 , S5 +{> |> , @ + 55 > 5> 47 , s s s s S6 +{> |> , @ + 55 > 5> 47 , S7 +{> |> , @ + 55 > 5> 47 ,
:1 S @ + 46 > 46 > 46 > 4< ,
;1 S @ +7> 7> 7> 4, s s s s 6> ,> + 6> 6> , = 5 Uj
431 S4 +{> |> }> , @ + 65 > 5>
8 5
> 4, S5 +{> |> }> , @ + 65 > 5> 85 > 4,
441 S4 +{> |> }> , @ +4> 4> 3> 5, S5 +{> |> }> , @ +4> 4> 3> 5, S6 +{> |> }> , @ +3> 3> 4> 4, S7 +{> |> }> , @ +3> 3> 4> 4, s s s s 451 S4 +{> |> }> , @ + s 8@6> 8@6> s 8@6>s 8@45,s S5 +{> |> }> , @ + 8@6> 8@6> 8@6> 8@45,
Vroxflrqhv gh od vhfflöq 41817
Or lqglfdgr hqwuh sduìqwhvlv +, hv vöor xqd d|xgd sdud xqd fruuhfwd mxvwlfdflöq 41 +d, +f, +h, +j, +l, +n, 51
+d, +g, +j, +m,
I dovr=+{ @ 4> | @ 5,> +e, I dovr=+{ @ 5> | @ 4, I dovr=+{ @ 4@6> | @ 4@5,> +g, I dovr=+{ @ 4@;, Y hugdghur= +i, I dovr=+{ @ 4,> Y hugdghur= +k, I dovr=+{ @ 4> | @ 3,> I dovr=+V @ i6j,> +m, I dovr=+V @` 4> 3` ^ ^5> 4^, I dovr=+U 9@ U i5j,> +o, Y hugdghur= Yhugdghur1 Idovr1 Idovr1 Idovr1
+e, +h, +k,
Yhugdghur1 Yhugdghur1 Yhugdghur1
61 +d, ^4> 3` ^ ^4> 4^1 +e, ^3> 4`1
+f, +i, +l,
Yhugdghur1 Yhugdghur1 Idovr1
+f, ^3> 4`1 +g, +{ 5 ^4> 4^a| 5 U, b +{ @ 3 a | 5 U i3j,1
+i, `4> 4^1
G141 UHVSXHVWDV D ORV HMHUFLFLRV SURSXHVWRV
748
71 +d, V @` 4@6> s 4^1 +e, Vs@`5> 4^1 +g, V @` 4> 3^^`4@:> 4^1 s s s s s +h, V @`4@5> :9 46 ^^`4> :.9 46 ^1 +i, V @` 4> 45^^` 8> 3^^` 8> 7 44^^`45> 7 . 44^= +j, V @ ^3> 4^1 +k, V @ ^3> 4^ +l, Vs @ ! +m, V @` 4> 5^ +n, V @ ^9> 4^ +o, V @ ! +p, V @` 4> 9@46^i3j +q, V @ ^ 45 :6 > 5` ^ ^4> 6` +r, V @ ^5> 657` +s, V @ !1 +t, V @ i4j +u, V @ ! +v, V @ i5j s s 81 +d, V @ !1 +e, V @ i4j1 +f, V @ i4 6> 5j1 +g, V @ U1 +h, V @` 4> 6@5`^`4@7> 4^1
+i, V @ U`4@7> 6^1
+j, V @` 4> 5^ +k, V @` 4> 5` ^ ^3> 6=8^1
+l, V @ U i3j +m, V @` :> 4^ +n, V @ ^4@6> 6` s s +o, V @`3> 4` +p, V @` :> 5^ +q, V @ ^ 8> 8` ^ i45j
+r, V @ ^6> :`1
+s, V @ ^4> 5` +t, V @ U +u, V @ i3j
91 D|xgd= Hvfulelu frpr hfxdflöq gh vhjxqgr judgr | txh ho glvfulplqdqwh hv xqd vxpd gh fxdgudgr +s t,5 | 7d5 1
s
:1 ^ 4;43 ;7 > 4^
;1 +d, p @ :1 +e, p @ 5 4^1
431 !
441 +d, Vxsuhpr= 9 4> 9 4> 9 9 9 9 9 4@5> 9 4@51
451 +d, vxs D @ :@6 +e, ! +f, :@6
461 D|xgd= ghpxhvwuh txh { . vxs D hv frwd vxshulru gh { . D | dvð wdpelìq txh vxs+{ . D, { hv frwd vxshulru gh D1
471 Q qr hv dfrwdgr | Q ]1
q 481 D|xgd= +l, Frqvlghuh q5.6 +Surs1 Dutxlphgldqd,1
5q
q
@ 5 oxhjr 5 hv frwd vxshulru1 +ll, +; A 3,+ ;q oxhjr 3 hv frwd lqihulru1 +ll, Sru Dutxðphghv 4^> ^3> 4@5` s e, ^5> 7@6`> ^3> 4@ 6` g, s i, ^4@5> 6@5`> ^3> 5` ^4> 4^> ^3> 4^ k,
U i4j> U s i7j ^4> 4^> ^ 5> 4^ ` 4> 5` ^ ^3> 4^> U. U i6j> U i4j
51 d, ` ;> 3^ e, ^4> 5^ f, ^49> 97^ g, ^ 5934^ h, Qr h{lvwh i, ` 4> 6^^` 5> 5^^`8> 4^ 61 d, ^5> 4^ e, `3> 4@6^f, ` 4> 4^ g, ^4> 4^ h, ^3> 4^
? frq
4
Q
9
q.6
A
? % 5
6q.4
1
749
DSSHQGL[ G1 SUREOHPDV PLVFHOÄQHRV SDUWH L
Vroxflrqhv gh od vhfflöq 51:161 41 D+{, @
5{+O
7{,
6
51 D+{, @ +{ 45 ,+ 4; 56 , {
61 w+{, @
s5
.
{ .49 4;
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63
s 71 F+{, @ 6 437 +7 {, . 8 437 < . {5 ; 5=83{ 3 { 53 A A ? 5=33{ 53 ? { 83 81 | @ 4=:8{ 83 ? { 433 A A = 4=83{ 433 ? {
91 | @
{ 45
. 3=43+{ 45
{ , 45
:1 E+{, @ +533 . 53{,+;333 833{, 5333+533 . 53{, ;1 X +{, @ +4533 . 433{,+57 {,
P ^
>
; 7d5 . ;d{ A A ? 6
3 { ? 7d { @ 7d 7d ? { ? :d { @ :d
u 5`3> 7^1
461 S +{, @ +; . {,+533 43{,>
471 L+{, @ +583 . {,+; 3=34{,>
481 Plq F @ 6{ . 7| V1D1
5{. 6| 5{. 5| ;{. 5| 3{/
3
69 5; 65 |
9{. 7{. 8{. 3
3
87 7; 83 |
491 Pd{ } @63{ . 53| V1D1
6| 9| 8| {/
4:1 Pd{ } @6333{ . 5333| V1D1
{. 5| 5{. | 0{. | | 3 {/
3
9 ; 4 5 |
{ 5 ^3> 53` _ ] { 5 ^3> 783`1
G141 UHVSXHVWDV D ORV HMHUFLFLRV SURSXHVWRV
74:
4;1 d,Pd{ } @533F . 433P V1D1
F. 5P 5F. 418P F P 3 F/
3
433 454 83 78 P
Ho öswlpr F@83/ P@47> frqwulexflöq pä{lpd 44173310 e, Pd{ } @533F . 433P V1D1
F. 5P 5F. 418P F P 3 F/
3
433 463 83 78 P
Ho öswlpr gh hvwh qxhyr sureohpd hv= F@83/ P@53> frqwulexflöq pä{lpd 45133310
Vroxflrqhv gh od vhfflöq 51:17 41 Wrgdv vrq el|hfwlydv
5 51 d, D @` 4> 3^^`4> 4^ E @ U i5> 5j i 4 +{, @ {5{7 { 4 e, D @ ^3> 4^i4j E @` 4> 3^^`4> 4^ i +{, @ + {4 ,5 61 d, Grp+j i , @ ^6@5> 4^ Grp+i j, @ U. e,
f,
Grp+j i , @ ^7 4^ Grp+i j, @ ^4> 4^
Grp+j i , @` 4> 6 Grp+i j, @`4> 4^
s s 43` ^ ^6 . 43> 4^
g, Grp+j i , @ ^ 4^ qr h{lvwh od frpsxhvwd gh i r j h,
Grp+j i , @ i3j Grp+i j, @ ^4> 4^
e,
Grp+j i , @ ^7 4^
f,
Grp+j i , @` 4> 6
g, Grp+j i , @ ^ 4^ h,
Grp+j i , @ i3j
s s 43` ^ ^6 . 43> 4^
74;
DSSHQGL[ G1 SUREOHPDV PLVFHOÄQHRV SDUWH L
71 d,
j i+{, @ i j+{, @
e,
j i+{, @ i j+{, @
f,
j i+{, @
= s
{ 4
9{ . 49
5{ . 6 4 { A 4
; ? 9{ . 4:
{4
= s 5 {.4 {A4
; ?
s 9 { 47
= s s 5 {:4 ; s ? 5 6{ . : :
= ss 5 {4:
3 { 49 { A 49
:@6 { 4 {A4
; ? 6{5 . 4;{ . :
{ 5`> ^i3j
= s
5 {s . 9{ 4 { 5` s 4> ` ^ ^> 4^= grqgh @ 6 43 @ 6 . 43
i j+{, @
g, j i+{, @
h,
; ?
; ? 4` 81 j i+{, @ A A A A s = 7{ { 5`4> 4^
91 e, k+{, @ 46 +{5 . 6{ 4, | ho grplqlr hv U
f, Grp @ ^6@5> 4^ Uhf @ ^ 4^> i 4 +{, @
s 6.
f*, Grp @` 4> 6@5` Uhf @ ^ 4^> i 4 +{, @
k4 +{, @ orj5 +{5 . 7,
;1 Grpi @ U > Uhfi @ U. > Qr10
8@7^> Uhfi @ U> Grpj @ U> Uhfj @` 4> 4^
e, Grp+i j, @ U. > +i j,+{, @ orj8@7 +4 . 6h{ 7h5{ , s 8 5849+8@7,{ f, Grpi @`3> 8@;^ Uhfi @` 4> 5^> i 4 +{, @ ; s 8. 5849+8@7,{ f*, Grpi @`8@;> 8@7^ Uhfi @` 4> 5^> i 4 +{, @ ;
Vroxflrqhv gh od vhfflöq 51:18 41 +d, +e, +f,
51 +d, +e, +f,
+g, +h,
+i,
+j,
61 +d, +e, +f,
+g,
8 orj s 6 6s 85
orj sd676 5 s{5|.f orj | s 7 f{ { @ 5@6
{ @ 4:@7 { @ 4@6
Ho frqmxqwr vroxflöq hv ydfðr s { @ 8 . 6 s { @ :5 { @ 5
{ @ 3> 34 > { @ 3> 334 { @ 5 > { @ 3> 58
{ @ 5 > { @ 46 54 4.s54 4s54 { @ 43 5 > { @ 43 5
+h,
{ @ 7
+j,
{ @ 3 > { @ 4
+i,
+k, +l,
+m,
{ @ 4 > { @ 8 { @ orj: ;
{ @ orj4>38 5
{ @ orj8 7 > { @ orj8 9
+n,
{ @
+p,
{ @ 5
+o,
+q, +r,
+s,
71 +d, +e, +f,
4 orj6 +8@6, 5 s { @ 8
{ @ 5
: { @ orj89 : + 8;4: , { @ 57
s
s
` 65 6: > 4^^`7> 6.5 6: ^ t t ` 4> 87 ` ^ ^ 87 > 4^
d A 4 `6> 9` > d ? 4 ^9> 4^
74<
753
DSSHQGL[ G1 SUREOHPDV PLVFHOÄQHRV SDUWH L +g,
`3> 6^
+i,
^
+h,
+j,
+k,
+l,
`3> 4< `^` 46 > 4` ^ ^6> 4^
s
5
5
> 4^
` 4> 4^
` 4> 67 ^
` 4> orj: ;`
s
+m,
`3> 5
+o,
`6> 4^
+n, +p, +q,
5
`4@7> 4^
s
^^`5
5
> 4^
`3> 6^^^ 44 > 7^= 6
`5> 4^
4 81 +d, Frpr 3 ? { 9@ 4 | 3 ? | 9@ 4 Vl ts 9@ 4 hqwrqfhv odv vroxflrqhv vrq { @ +dt e, ts4 +e,
+f,
{ @ 83 > | @ 53
+g,
{ @ 45 orj 7< 6 > | @
+i,
{ @ 6@5 > | @ 4@5
+h,
+j,
+k,
+l,
+m,
91 +d, +e,
+f,
4 7
orj 56 +59 @68 ,
{ @ 5orj9 < > | @ orj9 < 5orj9 <
{ @ 8> | @ 9 r { @ 89 > | @ 4
Ho frqmxqwr vroxflöq hv ydfðr1 6 4 4 6 { @ d5 > | @ d5 r { @ d5 > | @ d5
{ @ + 65 ,6 > | @ + 65 ,5 r { @ 4> | @ 4 { @ 65 > | @
4 5
l, 47/5 dôrv ll, 55/8 dôrv ' 719 | 5 U,+{ . | 9@ 3 ,
s 51 +;{ 5 U,+ 7{5 @ 5m{m,
{ |
+ . ,
{ s s 54 5 , {5 .|5 { .|
61 +;{ 5 U,+{5 @ 49 +, { @ 7 r { @ 7, 4 {4 , 71 +;{,+;|,+{> | 5 U. ,+ {|
81 Vhdq d> e 5 U1 Vl d ? e hqwrqfhv de ? 4
91 i{ 5 U @
m{m m {m 4
A 4j @ i{ 5 U @ m{m A m4 {mj
:1 +;{ 5 U i3j, +{ ? 43 ,
s
4
43
? {4 ,
5 5 ;1 +;d 5 U i3j,+;e 5 U,+ d d.e 4,
| 5 U i3j, + {5m. |5 441 +;{> | 5 U i3j, +6m{m m|m ,
451 +;{> | 5 U i3j, +{5 | 5 , 461 i{ 5 U @
t
m{m m|m
4,
{ 4j @ i{ 5 U @ s{ s4 {j
{
4
s m5{m 5 45 , { .|
471 Vhdq d> e 5 U1 Vl d ? e hqwrqfhv 4 ? de
481 i{ 5 U @
m{m m {m 5
A 4j @ i{ 5 U @ m{m A m5 {mj
491 +;{ 5 U i3j, +{ : ? 3 , 4: A {4 , s 5 5 { .| 4, 4:1 +;{ 5 U i3j,+;| 5 U,+ { 4;1 Vl d=e 5 U wdo txh de 3 hqwrqfhv
4 e 5 U ,+d ? e , d5 A e5 ,
s s s de @ d e 776
777
DSSHQGL[ H1 HMHUFLFLRV SURSXHVWRV
531 Ho frqmxqwr vroxflöq gh od lqhfxdflöq
{ s ?3 { {
hv ydfðr
541 +;{ 5 U, +{5 {7 ,
551 vxsi{ 5 U@{7 . 6{5 7 @ 3j @
s
5
561 Ho lqpr gh i{ 5 U@ 5{{.8 A 4j hv 08
571 Ho vxsuhpr gho frqmxqwr D @ i4 q4 mq 5 Qj hv 31 581 Ho vxsuhpr gho frqmxqwr
i{ 5 ] = m{5 6m ? 7j hv
s :
s m{m | 591 +;{> | 5 U i3j, + m{m.|5 m{m, s s 5:1 Vl { vdwlvidfh od hfxdflöq 8{ . 4 @ 5{ 5 . 5 hqwrqfhv ho ydoru gh od h{suhvlöq LL Frpsohwdflöq=
41 Ho frqmxqwr vroxflöq gh od lqhfxdflöq s {5 3 hv1111111111111111111 m{ 5m m4 {m 51 Ho frqmxqwr vroxflöq gh od lqhfxdflöq
61 Od lqhfxdflöq
s
{5 3 hv 111111111111111 m6 {m m{ 5m
5 5 4 6 + ,8{.5{ d, ho yroxphq gho flolqgur
Yf @ u5 k Df @ uk
e, ho äuhd odwhudo gho flolqgur1
51 Uhvroyhu
Fdsðwxor 5
{5 | 5 @ 5+{| 6, 5+6 {|, @ +{5 | 5 , 5 6{ | 9{ | 6 @ 3
L10 Ghwhuplqh fxdo gh odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwltxh 41 Vl orj{ +8, ? 4 hqwrqfhv vh wlhqh txh 8 ? {1
51 Ho frqmxqwr vroxflöq gh od lqhfxdflöq
orj8 { 4 hv ^6> 4^ orj8 +4@6,
779
DSSHQGL[ H1 HMHUFLFLRV SURSXHVWRV
s 61 5 orj 5 . +4 . 54{ , orj 6 orj + { 6 . 5:, @ 3/ od hfxdflöq dqwhulru qr wlhqh vroxflöq srvlwlyd
71 Vl orjd e @ 6 | orjd f @ 5 hqwrqfhv orjf e5 @ 6 | ef @ d8 81 i{ 5 Um
{, A 4j @ i{ 5 Um 4 A orj5 +{,j
4 orj5 +
91 Sdud wrgr d> e 5 U. i4j vh wlhqh orje +d5 , orje +d6 ,1 s :1 vxsi{ 5 U@{7 . {5 9 @ 3j @ 5 ;1 +;d 5 U. , +orj+d, ? orj+d5 ,, 4^
431 +;e 5 U. i4j, +orje +{, 4 , { e,
441 Vl i hv lq|hfwlyd hqwrqfhv i hv fuhflhqwh r elhq ghfuhflhqwh s 451 Vhd i +{, @ {5 4 > j+{, @ { . 4/ hqwrqfhv ho Grp+i j, @ U 461 +;d 5 U. , +orj+d, ? orj+d6 ,,
471 +;e 5 U. i4j, +orje + {4 , 4 , 4 {e,
481 Vl i hv el|hfwlyd hqwrqfhv i hv fuhflhqwh r elhq ghfuhflhqwh s 491 Vhd i +{, @ {5 . 4> j+{, @ { 4/ hqwrqfhv ho Grp+i j, @ U
4:1 Ho ydoru pðqlpr gh od ixqflöq i+{, @ {+{ 6, hv 4^ :$ h{ 4
hqwrqfhv ho grplqlr pä{lpr gh i hv ^ oq+7,> 4^ s 4 j+{, @ oq+{5 . 44^1
531 Vl i hv xqd ixqflöq lq|hfwlyd hqwrqfhv i hv hvwulfwdphqwh ghfuhflhqwh
541 Od vroxflöq gh od lqhfxdflöq orj{ +{ . 4@5, ? 4 hv ydflr1 551 Vl i+{, @ oq+{ . 7, | j+{, @ h{ . 6 vrq ixqflrqhv hqwrqfhv +j i ,+{, @ { . : | vx grplqlr hv U
561 Vl i | j vrq ixqflrqhv el|hfwlydv hqwrqfhv ho surgxfwr i j hv el|hfwlyd 571 Vl A 3/ od ixqflöq i +{, @ oq+{5 . 4, wlhqh pðqlpr devroxwr1 LL Frpsohwdflöq=
41 Od uhfwd | @ 8{ . e qr lqwhuvhfwd d od sduäerod | @ 5{5 5{ . 6 vl e 51111111111111
51 Od sduäerod {5 5{ d| 5d . | @ 6/ frq fhqwur hq +4> 5, vh deuh kdfld duuled vl | vöor vl d 51111111111111
61 Od hfxdflöq gh od uhfwd txh sdvd sru ho fhqwur gh od holsvh 5{5 7{ . 6|5 . 9| @ 7 | hv sdudohod d od uhfwd 5{ . 6| @ 4 hv 11111111111111 71 Od uhodflöq {5 | 5 7{ . 5| . 7 @ 3 uhsuhvhqwd od föqlfd 1111111111111 frq yìuwlfh hq 11111111111111
77: 81 Vhd DEFG ho uhfwäqjxor frq odgrv sdudohorv d orv hmhv lqvfulwr hq od holsvh 49{5 . 7|5 @ 971 Hqwrqfhv od ixqflöq txh uhsuhvhqwd ho äuhd gho uhfwäqjxor hq ixqflöq gh { hv= 11111111111111111
{
91 Od uhodflöq {5 |5 . 7{ 5| . 7 @ 3 uhsuhvhqwd od föqlfd 111111111111111 frq fhqwur hq1111111111111
:1 Sdud d 511111111| e 5 111111111111 od ixqflöq +sduäerod, i+{, @ {5 . d{ . e hv ghfuhflhqwh hq ^5> 4^ s ;1 Ho grplqlr gh od ixqflöq i +{, @ orj+{5 9, 4 hv 1111111111111 s 4^i7j $ { :$
hqwrqfhv ho uhfruulgr gh od ixqflöq i hv 1111111111111
U
m{m 4
7
451 Vhd i = G $ ^h5 > h4 ^> i +{, @ + 5:4h ,6{4 hqwrqfhv ho grplqlr pä{lpr gh i hv 111111111111
461 Vhd i +{, @ orj +{, | j+{, @ {5 . e{ e ixqflrqhv/ hqwrqfhv ho frqmxqwr gh wrgrv orv e 5 U wdo txh ho grplqlr gh i j hv ydflr hv 11111111111111111 s 471 Vhd i+{, @ orj++{ 6,5 , | j+{, @ 7 {/ hqwrqfhv ho grplqlr pä{lpr gh j i hv 111111111 481 Ho grplqlr gh od ixqflöq hv 11111111111111111111 491 Frqvlghuh od ixqflöq i +{, @ e 5 111111111111111
s s 4 4 {5 i +{, @ {5
d{5 . 4 { 3 hqwrqfhv i hv lq|hfwlyd vl | vöor vl d 5 11111111111111 | {5 . e { ? 3
4:1 Od uhodflöq |5 @ 5 { ghqh xqd ixqflöq el|hfwlyd hq 11111111111111111 5 { >{ ? 3 s{ > { ? 4 hqwrqfhv j i +{, hv 11111111111111111111 4;1 Vhd i+{, @ | j+{, @ 5{ > { 3 { >{ 4 s 4 4^i4j $ { :$
hqwrqfhv ho uhfruulgr hv 111111111111111 U . 6 {4 4
541 Vhd i = U i4j $ U i4j xqd ixqflöq el|hfwlyd hqwrqfhv i 4 +{, hv ljxdo d 1111111111111111 {.4 { :$ 4{
551 od ixqflöq i+{, @ 5{ m{m ghqlgd hq ^5> 8` wlhqh xq pä{lpr devroxwr hq { @11111111111 | xq ydoru pðqlpr devroxwr ljxdo d 11111111111
561 Od ixqflöq txh uhsuhvhqwd ho yroxphq +hq ixqflöq gh k, gho flolqgur flufxodu uhfwr lqvfulwr hq xq frqr gh 53fp gh dowxud | 8fp gh udglr/ hv i +k, @1111111111111 | vx grplqlr hv 11111111111111111
571 Ho grplqlr gh od ixqflöq i+{> |, @
{|, hv ho frqmxqwr 1111111111111 {.| | od juäfd gh hvwh frqmxqwr hv 11111111111
581 Vl i+{, @ h
oq+4
s {5 1 4
+d, i hv hvwulfwdphqwh fuhflhqwh hq 111111111111111
+e, i hv hvwulfwdphqwh ghfuhflhqwh hq 111111111111111 +f, i dofdq}d vx ydoru pä{lpr hq 11111111111111
+g, Ho ydoru pðqlpr gh i hv 111111111111111
LLL10 Ghvduuroor
41 Od rulood gh xqd slvflqd irupd xq uhfwäqjxor gh 73 slhv gh odujr | 53 slhv gh dqfkr1 Vx surixqglgdg dxphqwd xqliruphphqwh gh 6 d : slhv hq xq wudpr krul}rqwdo gh 57 slhv | ghvsxìv frqwlqýd do plvpr qlyho orv uhvwdqwhv 48 slhv/ frpr vh loxvwud hq od jxud/ od fxdo pxhvwud xqd vhfflöq wudqvyhuvdo1 Vl od slvflqd vh hvwä ydfldqgr/ dofdq}dqgr xq qlyho k hq ho odgr päv surixqgr1 Ghwhuplqh ho yroxphq Y +k, gho djxd hq ixqflöq gh od dowxud k/ hvshflfdqgr vx grplqlr1
51 Od frpsdôld gh duulhqgr gh dxwrv Mlphqh}%/ duulhqgd fkhyhwwh d ' 431833 sru gðd | 73 shvrv sru nloöphwur uhfruulgr/ plhqwudv txh od frpsdôld Dudqfleld% duulhqgd ho plvpr wlsr gh dxwr d ' ; q 5333 d, dq @ 4 q > q A 5333
e, dq @
q6
s s s 71 Od vxfhvlöq idq j @ i+ q . 4 q, q . 6j frqyhujh d 4@5
4
q
> q A 5333 > q 5333
LL Frpsohwdflöq=
41 Ho ydoru gho oðplwh
olp q$4
hv 11111111111111111
51 Vhd dq @ 6q {5q hqwrqfhv q
vvl { 5 111111111111111
61 Ho ydoru gho oðplwh hv 71 Ho oðplwh
q5 q.4
q
6q . s 7 49q7 . 8
dq.4 ?4 olp q$4 dq
olp q$4
5q 5q . frv+q$, s hv 111111111111111 . 6 5q.4 . 4 q6 . 8
olp q$4
81 Ho ydoru gho oðplwh olpq$4
vhq +q,.+
91 Ho ydoru gho oðplwh olpq$4
6
4,
+5,q . frv+q, @ ============== 6q . 7
q .86q.4
q.8q
5+q.4,
mvhq
.
q 7.8q
<
q ,m
+<
hv 1111111111111111111
hv 1111111111111111111
5q 4 / hqwrqfhv ho juäfr gh i hv 1111111111 :1 Vhd i+{, @ olpq$4 {{5q .4
;1 Ho oðplwh
+ , olp h q6 4 @ ============ q$4
q5 .4
q hv 111111111111 s s 431 Ho oðplwh olpq$4 +q+ K @ U i4j1 +e, i +{, @ {{.4 6 {4 > K @ U. +f, i +{, @ vhq +{, { ?4 { + vhq +4@+{4,, { 9@ 4 4@+{4, 541 Od ixqflöq i +{, @ hv frqwlqxd hq 41 {@4 LL Frpsohwdflöq=
+{, hv 111111111111 41 Ho ydoru gho oðplwh olp{$7 +{vhq 7, frv+{,
51 Ho oðplwh olp{$8
{8,5 vhq 6+{8,5 hv 111111111111 {8, {5 43{.58
+
4.frw5 +
s
d{mdm > d ? 3 hv 111111111111 61 Ho oðplwh olp{$d d{ d5
71 Ho ydoru gho oðplwh olp{$3
{5
{, hv 11111111111111111
4.frw5 +
81 Orv ydoruhv gh D | E wdo txh od ixqflöq
; ?
vhq 5+{
4,
{4 i +{, @ D{5 . 5 = E{ . 8
>{ A 4 >{ @ 4 >{ ? 4
uhvxowd frqwlqxd hq wrgrv orv uhdohv vrq D 5 1111111111111 E 5 11111111
91 Vhd
i+{, @
{6 .5: {3 {5 < 6 frv+{ . , { A 3
Ghqlgd hq xq grqlplr dsursldgr/ vxefrqmxqwr gh U
+d, Ho pä{lpr grplqlr gh frqwlqxlgdg gh i hv 111111111
+e, i wlhqh glvfrqwlqxlgdg uhsdudeohv hq orv sxqwrv 111111111
+f, Ho frqmxqwr gh orv sxqwrv hq txh i wlhqh glvfrqwlqxlgdg luuhsdudeohv hv 111111111111111
s
em 5 :1 Vhd i+{, @ {{m e5 / frq e ? 3/ hqwrqfhv ho ydoru gho oðplwh gh i +{, fxdqgr { wlhqgh d e hv 111111111111 ; ? 6{5 . 8 { A 4 { . 5 { @ 4 ;1 Vhd i+{, @ Hqwrqfhv orv ydoruhv gh | txh shuplwhq txh od ixqflöq i +{, = 46{ . { ? 4 vhd frqwlqxd hq { @ 4 vrq @11111111111 | @ 111111111111
785
DSSHQGL[ H1 HMHUFLFLRV SURSXHVWRV
{A4 5D{ . {5 E+{ 4, . 8 > { @ 4 i+{, @ = 8{ . E > {?4
vhd frqwlqxd hq { @ 4 vrq 11111111111111
431 Vhd d 5 U wdo txh
; d+4{5 , ? vhq { i +{, @ = {.d
od ixqflöq i hv frqwlqxd hq { @ 4 vvl d @ ====================
441 Vhdq d> e 5 U wdo txh
i+{, @
; A ? A =
3?{?4 {4
vhq{+e{, 4 {5 d5 {d
vhq +d{,
{
i vhuä frqwlqxd hq { @ 3 vvl d @ ==========> e @ ===========
{?3 {@3 { A 3> { 9@ d
LLL10 Ghvduuroor ; 4frv+{@6, >{ ? 3 A { A A A ? 5{ . 5d . 4 > { @ 3 grqgh d> e vrq qýphurv uhdohv mrv= 41 i+{, @ A A A A = ds{5 .{6 .e >{ A 3 { +d, Fdofxoh ho oðplwh odwhudo ghuhfkr gh i hq 31
+e, Fdofxoh ho oðplwh odwhudo l}txlhugr gh i hq 31 +f, Fdofxoh orv oðplwh odwhudohv gh i hq 041
+g, Hqfxhqwuh orv ydoruhv gh d | e/ sdud txh i +{, vhd frqwlqxd hq 04 +h, Hqfxhqwuh orv ydoruhv gh d | e/ sdud txh i +{, vhd frqwlqxd hq 3
Fdsðwxor 8
L10 Ghwhuplqh fxdo gh odv vljxlhqwhv dupdflrqhv vrq yhugdghudv r idovdv1 Mxvwltxh
41 Vl j qr hv ghulydeoh hq d shur frqwlqxd hq d | i hv ghulydeoh hq d hqwrqfhv ho surgxfwr i j qr hv ghulydeoh hq d1
51 Vhdq A 3/ od ixqflöq i +{, @ h{ wlhqh vöor xq pä{lpr hq orv uhdohv1 5
61 Od hfxdflöq gh od uhfwd wdqjhqwh d od fxuyd ghqlgd lpsoðflwdphqwh sru hq ho sxqwr +4@5> 3, hv 5{ 5| @ 41
| @ dufvhq+{|,
71 Od ixqflöq i+{, @ +{ 4,5@6 qr wlhqh pðqlpr s 81 Od ixqflöq i+{, @ { 5 qr wlhqh pä{lpr | qr wlhqh pðqlpr1
91 Vl j+{, @ vhf+8{, . { | i 3 +{, @ 6h5{ hqwrqfhv +i j,3 +3, @ 6h5
786 :1 Vl od yhorflgdg gh fuhflplhqwr gh od edvh gh xq wuläqjxor hv gh 6 p2vhj | od dowxud 7 p2vhj/ hqwrqfhv od yhorflgdg gh fuhflplhqwr gh vx äuhd hq ho lqvwdqwh txh fxdqgr vx edvh hv gh 45 phwurv | vx dowxud gh 53 phwurv hv gh 87 p5 2vhj
;1 Wrgd ixqflöq dfrwdgd wlhqh h{wuhprv s f 5 U
; s ? +{ ,6 . 5d{ f s i+{, @ = frv+ +{7 . 5 ,@5, . e{ d
s { ? s {@ s {A
s sdud orv vljxlhqwhv ydoruhv gh d> e> f111111111111111 s +e, i hv ghulydeoh hq sdud orv vljxlhqwhv ydoruhv gh d> e> f 111111111111111 ; 6 {A3 ? { vhq+{, { @ 3 561 Vhd i +{, @ hqwrqfhv od ghulydgd gh i hv 11111111111111111111111 = {?3 7{5 +d, i hv frqwlqxd hq
788 571 Vhd 6{5 | 5 . wdq+| 6 , @ 4/ hqwrqfhv hq xq frqmxqwr dsursldgr od ghulydgd gh | frq uhvshfwr d { hv= g| g{ @1111111111111
581 Gdgd od uhodflöq vhq+ 7 frv+{6 , . | 5 7, @ gi m hqwrqfhv od ghulydgd gh g{ {@3 hv1111111111
s
5
5
hood ghqh fhufd gho fhur xqd ixqflöq ghulydeoh i /
591 Vhd K+{, @ {I 4 +{, frv+I 4 +{,,/ frq I 3 +{, 9@ 3 sdud wrgr { hqwrqfhv od ghulydgd gh K+{, hv K 3 +{, @1111111111111
5:1 Orv odgrv gh xq wuläqjxor gh shuðphwur pðqlpr frq äuhd md D/ dvð frpr wdpelìq md vx edvh e vrq 1111111111111111 5;1 Od hfxdflöq gh od uhfwd wdqjhqwh d od juäfd gh od ixqflöq gdgd sdudphwulfdphqwh sru { @ 6+w 4,5 . 8 | @ 9w6 vhq+w, > w 5 U hq ho sxqwr +;> 3, hv 1111111111111111
5 4, hv 11111111111111111 s 641 Od shqglhqwh gh od uhfwd wdqjhqwh d od juäfd gh i+{, @ { . 5 hq ho sxqwr +4> 6, hv1111111111111
651 Od ghulydgd gh xqd ixqflöq i +{, hv i 3 +{, @ 6{ . :vhq+{ 4,+{ 5, . ; | i +4, @ 5> i+5, @ 6 hqwrqfhv +i 4 ,3 +5, @111111111111111
661 Vhd i +{, @ vhq+6{, frv+5{, . 4/ frq { 5` @;> @;^/ hqwrqfhv od ghulydgd gh od ixqflöq lqyhuvd hq ho 4 sxqwr 4/ hv ghflu/ gig{ +4, hv 1111111111111111
671 vhd i +{, @ 5vhq+{, frv+{,> { 5`@7> 6@7^ hqwrqfhv od ghulydgd gh od ixqflöq lqyhuvd hq ho sxqwr 4 3/+ gig{ +3,, hv 111111111111111111 681 Ho oðplwh
hv 111111111111111111
4 olp + ,wdq+{, {$3. {
691 Ho ydoru gho oðplwh olp{$3 +4 . 6{,+4@5{, hv 1111111111
6:1 Ho ydoru gho oðplwh olp{$4 +4 {,frv+{@5, hv 11111111111111
6;1 Gdgd od fxuyd {| . | 6 {5 @ 4 hq irupd lpsoðflwd1 Fxdqwrv sxqwrv gh od ixqflöq | @ i +{, wlhqh uhfwd wdqjhqwh sdudohod d hmh { 111111111111111
6 e| f/ hqwrqfhv s+{, wlhqh xq pä{lpr orfdo hq { @ 4 | xq pðqlpr orfdo hq { @ 6 vl vxv frhflhqwhv vrq ljxdohv d 11111111111111111111
789
DSSHQGL[ H1 HMHUFLFLRV SURSXHVWRV
761 Grv wuhqhv vh dohmdq gh xq fuxfh txh irupd xq äqjxor gh 3,/ xqd gh hood hv od wdqjhqwh d rwur sxqwr1 Ho sxqwr hv 11111111111 7:1 Od juäfd gh od ixqflöq i +{, @ +{5 4,7 wlhqh sxqwrv gh lq h{löq hq 1111111111111
7;1 Od juäfd gh i +{, @ +{5 4,7 hv 11111111111111111111
5 {5 .4, hqwrqfhv 5{ 7 i 3 +{, @ +{ i%+{, @ 5+6 5 4,5 > +{5 4,6
+d, i hv fuhflhqwh hq 111111111111111
+e, i hv frqfdyd +kdfld duuled, hq 1111111111111 +f, i wlhqh dvðqwrwd yhuwlfdo hq 111111111111
+g, i wlhqh dvðqwrwd krul}rqwdo hq 1111111111111 831 Odv dvðqwrwdv gh | @
s{{55
4
vrq 1111111111
841 Od hfxdflöq gh od uhfwd wdqjhqwh d od fxuyd | @ i+{, ghqlgd lpsolflwdphqwh sru od uhodflöq s wdq+{|7 , 5 oq+ |, @ 3
hq +3> 4, hv 1111111111
851 Od hfxdflöq gh od uhfwd wdqjhqwh d od fxuyd +5{ . 6|,5 +6{ 5|,5 @ 57 hq +4> 4, hv 11111111111111111 861 Od ixqflöq i+{, @ { hv hvwulfwdphqwh fuhflhqwh hq 11111111111111111
871 Od ixqflöq | @ +vhq{,5 . { ghqlgd hq ^3> ` wlhqh xq pä{lpr hq { @ 111111111111111111111
881 Od ixqflöq i+{, @ {5 h{ hv hvwulfwdphqw ghfuhflhqwh vöor hq ho lqwhuydor 1111111111111
891 Od yhorflgdg gh ghvydorul}dflöq gh xq htxlsr sru ho xvr gho plvpr hv sursruflrqdo hq fdgd prphqwr d vx frvwr uhdo |1 Od h{suhvlöq txh uhsuhvhqwd ho hqxqfldgr dqwhulru grqgh n hv xqd frqvwdqwh srvlwlyd | w hv ho wlhpsr hv 1111111111111 LLL10 Ghvduuroor
41 Ghulyh
l, | @
s 6 {5 wdq+{5 ,
51 Fdofxodu odv ghulydgd gh
ll, | @ oq+duffrv+6{,,
orj{ +43, 43vlq+{, .
61 Xwlolfh od hfxdflöq gho Rydor gh Fdvvlql1
g| +d, Hqfxhqwuh g{
lll, | @ orj{5 +6{,
6{ . dufvlq +frv +{5 ,, s wdq +frv :{, . {
+{5 . |5 . 7,5 49{5 @ 69>
78: +e, Bc Vh s sxhgh hqfrqwudu odv hfxdflrqhv gh od uhfwd wdqjhqwh do rydor gh Fdvvlql hq orv sxqwrv +4> 5, | +5> 5,B +f, Hqfxhqwuh od uhfwd wdqjhqwh dq orv sxqwrv dqwhulruhv hq orv txh vh sxhgdq1
71 Vhdq 5 U> e 5 U1
Ghwhuplqdu vl od ixqflöq | @ hd{ . {hd{ hv vroxflöq gh od hfxdflöq glihuhqfldo g5 | g| . d5 | @ 3> 5d g{5 g{
d frqvwdqwh
81 Vh ghvhd frqvwuxlu xq wdqtxh gh dfhur frq od irupd gh xq flolqgur flufxodu uhfwr | vhplhvihudv hq orv h{wuhprv +yhu od jxud, sdud dopdfhqdu jdv sursdqr1 Ho frvwr sru slh fxdgudgr gh orv h{wuhprv hv ho greoh gho gh od sduwh floðqgulfd/ grqgh ho frvwr gho phwur fxdgudgr gho flolqgur hv S 1 BcTxì glphqvlrqhv plqlpl}dq ho frvwr vl od fdsdflgdg ghvhdgd hv gh 43 slh6 B Dh @ 7u5 D|xgd= Yf @ u5 k Yh @ 76 u6 Df @ 5uk 91 Xq fdpshvlqr wlhqh glqhur sdud frpsudu 4833 phwurv gh dodpeuh gh sxdv/ wdo fdpshvlqr ghvhd fhufdu frq 8 kheudv gh dodpeuhv xq whuuhqr uhfwäqjxodu hq vx suhglr/ hq fdvr txh { uhsuhvhqwh xq odgr gho uhfwäqjxor/ ghwhuplqh od ixqflöq D+{, äuhd gho whuuhqr fhufdgr frq hvwh dodpeuh hq whuplqrv gh {1 Phgldqwh/ od juäfd gh od ixqflöq D+{, hqfxhqwuh odv glphqvlrqhv gho whuuhqr frq pä{lpd äuhd
s
:1 Vh ghvhd frqvwuxlu xq hvwdqtxh hq irupd föqlfd +uhfwr, frq fdsdflgdg sdud ;3336 5 olwurv1 vl 4 olwurv s gh lpshuphdelol}dqwh fxeuh 6p5 gh vxshuflh1 BcFxdohv ghehq vhu odv glphqvlrqhv gho frqr sdud txh vh xvh xqd fdqwlgdg pðqlpd gh lpshuphdelol}dqwh B BcFxäqwrv olwurv vh ghehq frpsudu sdud slqwdu wdo frqrB s D|xgd= Vxshuflh gh xq frqr gh udglr u | dowxud k hv u u5 . k5 | vx yroxphq hv 46 u5 k1 Fxlgdgr frq odv xqlgdghv
;1 Grv iäeulfdv D | E txh vh hqfxhqwudq d 7 ploodv xqd gh od rwud/ hplwhq kxpr frq sduwðfxodv txh frqwdplqdq ho dluh gh od uhjlöq 1 Vxsrqjd txh ho qýphur gh sduwðfxodv suryhqlhqwh gh fdgd iäeulfd hv gluhfwdphqwh sursruflrqdo d od fdqwlgdg gh kxpr h lqyhuvdphqwh sursruflrqdo do fxer gh od glvwdqfld ghvgh od iäeulfd1 Bc Txì sxqwr hqwuh D | E whqguä od phqru frqwdplqdflöq vl od iäeulfd D hplwh ho greoh gh kxpr txh od iäeulfd E B i 3 +{, @ +{4,5 +{7, | i 33 +{, @ 45+{4, 1 i+{, @ +5+{ 5,5 +{5,6 +{5,7 Lqglfdu hq od wdeod vljxlhqwh= +d, +e, +f, +g,
Ghwhuplqh ho Grpi Orv ydoruhv gh { grqgh od juäfd fdpeld gh frpsruwdplhqwr1 Orv vhfwruhv gh fuhflplhqwr | ghfuhflplhqwr orv vhfwruhv grqgh od juäfd hv frqfdyd kdfld duuled | grqgh hv frqfdyd kdfld dedmr1 Sxqwrv Lpsruwdqwh Vljqr gh i 3 +{, Vljqr gh i 33 +{, Fuhflplhqwr | Ghfuhflplhqwr Frqfdyd | frqyh{d
+h, Odv dvðqwrwdv yhuwlfdohv vrq +i, Odv dvðqwrwdv reolfxdv vrq +j, Judtxh
471 Frqvlghuh/ sdud ho dqäolvlv | od juäfd gh od ixqflöq i/ {6 > i 3 +{, @ {5 +{6, | i 33 +{, @ 9{ 1 i+{, @ +{ 4,5 +{4,6 +{4,7 Lqglfdu hq od wdeod vljxlhqwh= +d, +e, +f, +g,
Ghwhuplqh ho Grpi Orv ydoruhv gh { grqgh od juäfd fdpeld gh frpsruwdplhqwr1 Orv vhfwruhv gh fuhflplhqwr | ghfuhflplhqwr orv vhfwruhv grqgh od juäfd hv frqfdyd kdfld duuled | grqgh hv frqfdyd kdfld dedmr1 Sxqwrv Lpsruwdqwh Vljqr gh i 3 +{, Vljqr gh i 33 +{, Fuhflplhqwr | Ghfuhflplhqwr Frqfdyd | frqyh{d
78< +h, Odv dvðqwrwdv yhuwlfdohv vrq +i, Odv dvðqwrwdv reolfxdv vrq
+j, Judtxh