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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 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 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 44 41515 D{lrpdv gh U frpr 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 1 1 1 1 1 1 1 47 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54 41517 Frqmxqwr uhvwulfflöq | frqmxqwr vroxflöq gh xqd hfxdflöq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 55 41518 Uhvroxflöq gh sureohpdv gh sodqwhr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57 416 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 67 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6< 41615 Vlvwhpdv gh Hfxdflrqhv qr Olqhdohv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 78 41616 s5 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 84 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8: 41618 Od hfxdflöq gh vhjxqgr judgr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 96 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :4 417 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;5 418 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 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 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <3 41816 Hmhuflflrv gh Vlvwhpdv gh Hfxdflrqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <5 41817 Hmhuflflrv Dfhufd gh orv 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 <5 5 Ixqflrqhv <: 514 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 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <; 516 Uhsuhvhqwdflöq Juäfifd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 437 51614 Udðfhv phgldqwh juäfifdv 1 1 1 1 1 1 1 1 1 1 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: 517 Dsolfdflrqhv gh Ixqflrqhv/ Prghodflöq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 455 51714 Surjudpdflöq Olqhdo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 466 518 Dojheud gh Ixqflrqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 46; 51814 Frpsrvlflöq gh ixqflrqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 474 51815 Ixqflrqhv Fuhflhqwhv | Ghfuhflhqwhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 478 519 Odv ixqflrqhv H{srqhqfldohv | Orjduðwplfdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 48: 51: 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:7 51:14 Hmhuflflrv gh Grplqlr | Uhfruulgr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:7 51:15 Hmhuflflrv gh Juäfifd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:8 51:16 Hmhuflflrv gh Prghodflöq Pdwhpäwlfd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:9 51:17 Hmhuflflrv gh Frpsxhvwd gh ixqflrqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:< 51:18 Hmhuflflrv gh Orjdulwprv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;3 6
7 FRQWHQWV LL Fäofxor Glihuhqfldo 4;6 6 Vxfhvlöq 4;8 614 Frqfhswrv Eävlfrv | Hmhpsorv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;8 615 Vxfhvlrqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4;: 616 Oðplwh gh xqd Vxfhvlöq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<4 617 Fäofxor gh oðplwhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<: 618 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 547 7 Oðplwhv gh Ixqflrqhv 54: 714 Oðplwhv gh Ixqflrqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54: 71414 Sxqwr gh Dfxpxodflöq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54: 71415 Ghfiqlflöq gh oðplwh gh ixqflrqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54; 71416 Ghprvwudflöq gh Oðplwh % 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54< 71417 Whruhpdv Eävlfrv vreuh Oðplwhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 558 715 Oðplwhv Odwhudohv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 55: 71514 Dojheud gh Oðplwhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 563 71515 Oðplwh gh ixqflrqhv 0 vxfhvlrqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 56: 71516 Whruhpd gh Vxvwlwxflöq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 574 716 Frqwlqxlgdg 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57< 71614 Htxlydohqfldv gh Frqwlqxlgdg 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 584 71615 Frphqwdulrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 587 71616 Ixqflrqhv frqwlqxdv vreuh frpsdfwr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 589 717 Uhfwd wdqjhqwh d od Fxuyd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 599 8 Ghulydgd 5:8 814 Ghulydgd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5:8 81414 Hmhpsorv gh ixqflrqhv ghulydeohv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5:9 81415 Dojheud gh Ghulydgdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5;5 81416 Uhjod gh od Fdghqd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5;9 81417 Ghulydgd gh xqd ixqflöq Lqyhuvd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 634 815 Ghulydgd gh Rughq Vxshulru 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 636 816 Ixqflrqhv ghfiqlgdv lpsoðflwdphqwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 639 817 Uhjod gh O*Krslwdo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 64< 818 Dsolfdflrqhv gh od Ghulydgd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 659 81814 Dsolfdflrqhv Jhrpìwulfdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 659 81815 Dsolfdflrqhv Iðvlfdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 65: 81816 Ud}öq gh Fdpelr/ wdvd gh yduldflöq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 65< 819 Pä{lpr | Pðqlpr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 675 81914 Whvw gh od Sulphud Ghulydgd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 67< 81915 Whruhpd gho ydoru phglr | Urooh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 683 81: Frqfdylgdg 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 693 81:14 Whvw gh od Vhjxqgd Ghulydgd= 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 69: 81:15 Oðplwh Lqfiqlwr 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 69< 81:16 Dvðqwrwdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6:4 81:17 Rwudv dvðqwrwdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6:9 D Uhodflrqhv 6;8 D14 Uhodflrqhv Eävlfdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6;:
FRQWHQWV 8 E Jhrphwuðd Dqdoðwlfd 6;< E14 Ho sodqr elglphqvlrqdo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6;< E1414 Frrughqdgdv Fduwhvldqdv r Uhfwdqjxoduhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6;< E1415 Frrughqdgdv Sroduhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6;< E15 Wudqvirupdflöq gh Frrughqdgdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<3 E1514 Wudvodflrqhv hq frrughqdgdv fduwhvldqdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<3 E1515 Urwdflrqhv hq frrughqdgdv fduwhvldqdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<3 E1516 Uhodflöq hqwuh frrughqdgdv sroduhv | fduwhvldqdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<4 E16 Uhfwd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<5 E1614 Glvwdqfld 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<6 E17 Föqlfdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<7 E1714 Flufxqihuhqfld 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<7 E1715 Sduäerod 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<8 E1716 Holsvh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<9 E1717 Klsìuerod 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<; E1718 Hfxdflrqhv Fxdguäwlfdv | Föqlfdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6<< F Ixqflrqhv Uhdohv gh Grv Yduldeohv 734 F14 Juäfifd gh ixqflrqhv uhdohv gh grv yduldeohv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 738 F15 Dojheud gh ixqflrqhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 73: G Sureohpdv Plvfhoäqhrv Sduwh L 73< G14 Uhvsxhvwdv d orv 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 1 1 1 1 1 746 H Hmhuflflrv Sursxhvwrv 756
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43 FKDSWHU 41 ORV QÝPHURV UHDOHV Hv fodur txh Q wdpelìq yhulfifd odv surslhgdghv dvrfldwlyd | frqpxwdwlyd/ hv ghflu/ sdud wrgr d> e> f 5 Q > +d . e, . f @ d . +e . f, | d . e @ e . d Vlq hpedujr/ qr h{lvwh xq hohphqwr hq Q txh vhd qhxwur sdud hvwd rshudflöq1 Glfkr hq irupd päv suhflvd/ qr h{lvwh xq hohphqwr h 5 Q wdo txh do vxpduor frq fxdotxlhu hohphqwr d 5 Q vh rewhqjd ho plvpr qýphur d hq Q 1 Sru rwud sduwh/ vl frqvlghudprv Q edmr od rshudflöq surgxfwr +r pxowlsolfdflöq, vh wlhqh txh vh fxpsohq odv vljxlhqwhv surslhgdghv= l, Vl d> e 5 Q / hqwrqfhv de 5 Q ll, Sdud wrgr d> e> f 5 Q vh yhulfifd +de,f @ d+ef, lll, H{lvwh 4 5 Q wdo txh sdud wrgr d 5 Q 4 d @ d 4 @ d ly, Sdud wrgr d> e 5 Q de @ ed Dvð/ hq Q frq od pxowlsolfdflöq +,/ vh yhulfifd odv surslhgdghv gh fodxvxud +l,/ dvrfldwlylgdg +ll,/ h{lvwhqfld gh qhxwur +lll, | frqpxwdwlylgdg1 Shur/ gdgr d 5 Q qr h{lvwh xq hohphqwr d3 5 Q wdo txh do pxowlsolfduor frq d vh rewhqjd ho hohphqwr qhxwur h @ 4/ hv ghflu/ qr vh yhulfifd od surslhgdg gh h{lvwhqfld gh lqyhuvr1 BcTxì frqmxqwr fxpsoluä frq wrgdv odv surslhgdghv phqflrqdgdvB Frqvlghuhprv ho frqmxqwr gh orv Qýphurv Hqwhurv ] @ i===> 6> 5> 4> 3> 4> 5> 6>===j @ Q ^ Q ^ i3j grqgh Q @ iq = q 5 Qj= Hq ] frq od vxpd +., vh yhulfifd= l, Fodxvxud= Sdud wrgr d> e 5 ] / vh wlhqh txh d . e 5 ] ll, Dvrfldwlylgdg= Sdud wrgr d> e> f 5 ] +d . e, . f @ d . +e . f, lll, H{lvwhqfld gh qhxwur= H{lvwh h @ 3 5 ] / wdo txh sdud wrgr d 5 ] 3 . d @ d .3@ d 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 ] d . e @ e . d
4151 D[LRPDV GH FXHUSR 44 Hq fdpelr/ vl frqvlghudprv ] edmr od rshudflöq surgxfwr +, vh yhulfifdq +l,/ +ll,/ +lll, +hq hvwh fdvr 4 kdfh ho sdsho gh qhxwur, | +y,1 Vlq hpedujr qr vh yhulfifd 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 Ghfiqlflöq 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 Ghfiqlflöq 415 Vhd J xq frqmxqwr qr ydfðr | xqd rshudflöq hq J1 Gluhprv txh J hv xq juxsr edmr od rshudflöq vl odv vljxlhqwhv wuhv dfiupdflrqhv vrq flhuwdv l, Dvrfldwlylgdg= Sdud wrgr {> |> } hq J/ vh fxpsoh +{ |, } @ { +| }, Hq vðperorv +;{> |> } 5 J,++{ |, } @ { +| },, ll, H{lvwhqfld gh hohphqwr qhxwur= H{lvwh h/ hohphqwr qhxwur hq J/ wdo txh sdud wrgr { hq J/ vh wlhqh { h @ h { @ { Hq vðperorv +<h 5 J,+;{ 5 J,+{ h @ h { @ {, lll, H{lvwhqfld gh hohphqwrv lqyhuvrv= Sdud wrgr { hq J/ h{lvwh {3 hq J wdo txh { {3 @ {3 { @ h Hq vðperorv4 +;{ 5 J,+<{3 5 J,+{ {3 @ {3 { @ h, Hvwr vh ghqrwd uhvxplgdphqwh> +J> , hv juxsr1 Vl dghpäv vh yhulfifd> 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 4Qrwhprv od lpsruwdqfld gho rughq gh orv fxdqwlfifdgruhv / ! +frpsduh frq ll ,1
45 FKDSWHU 41 ORV QÝPHURV UHDOHV Hmhpsor 4 Vhd P5+], @ ffl d e f g = d> e> f> g 5 ] ff ho frqmxqwr gh pdwulfhv fxdgudgdv gh rughq 5 frq frhfiflhqwhv hq ] | +., od rshudflöq ghfiqlgd sru d e f g . d3 e3 f3 g3 @ d . d3 e . e3 f . f3 g . g3 > hqwrqfhv P5+], hv xq juxsr1 Vroxflöq= Yhulfitxhprv txh P5+], fxpsoh frq odv surslhgdghv gh juxsr l, Dvrfldwlylgdg= Vhdq d e f g > d3 e3 f3 g3 > d33 e33 f33 g33 shuwhqhflhqwhv d P5+],1 Vh wlhqh vxfhvlydphqwh= @ d e f g . d3 e3 f3 g3 . d33 e33 f33 g33 @ d . d3 e . e3 f . f3 g . g3 . d33 e33 f33 g33 / sru ghfiqlflöq gh . @ +d . d3 , . d33 +e . e3 , . e33 +f . f3 , . f33 +g . g3 , . g33 @ d . +d3 . d33 e . +e3 . e33 f . +f3 . f33 g . +g3 . g33 / dvrfldwlylgdg hq ] @ d e f g . d3 . d33 e3 . e33 f3 . f33 g3 . g33 @ d e f g . d3 e3 f3 g3 . d33 e33 f33 g33 ll, H{lvwhqfld gho qhxwur= H{lvwh 3 3 3 3 5 P5+], wdo txh sdud wrgr d e f g 5 P5+], vh wlhqh 3 3 3 3 . d e f g @ d e f g . 3 3 3 3 @ d e f g lll, H{lvwhqfld gh hohphqwrv lqyhuvrv= Sdud wrgr d e f g 5 P5+], h{lvwh d e f g 5 P5+], wdo txh d e f g . d e f g @ d e f g . d e f g @ 3 3 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/ yhulfifdqgr txh wrgrv hq frqmxqwr fxpsohq odv surslhgdghv gh fodxvxud/ dvrfldwlylgdg/ h{lvwhqfld gh hohphqwr qhxwur | h{lvwhqfld gh hohphqwrv lqyhuvrv +Qrwh orv fxdqwlfifdgruhv 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 46 Wrgrv orv hohphqwrv gh J fxpsohq wrgdv odv surslhgdghv hv= 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 @ ffl d 4 e 3 = d> e 5 ] ff frq od vxpd +., ghfiqlgd hq ho hmhpsor 41 Hqwrqfhv +E> ., qr hv xq juxsr/ sxhvwr txh qr vh yhulfifd od surslhgdg gh fodxvxud1 Hq hihfwr> 4 4 4 3 5 E | 5 4 3 3 5 E 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 ghfiqlgd sru +{4> |4, +{5> |5,@+{4 {5> |4 . |5, Hqwrqfhv +F> , qr hv juxsr1 Hq hihfwr> +4> 5,>+6> 8,>+4> 7, 5 F ++4> 5, +6> 8,, +4> 7, @ +4 6> 5 . 8, +4> 7, @ +4> 44, Sru rwud sduwh +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 e = d 5 ]> e 5 ] i3j r > oodpdgr ho frqmxqwr gh orv Qýphurv Udflrqdohv1 Frqvlghuh odv rshudflrqhv vxpd | surgxfwr ghfiqlgdv sru= d e . f g @ dg . ef eg d e f g @ df eg Qrwh txh dg.ef eg 5 T | df 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 yhulfifdflöq gh odv uhvshfwlydv surslhgdghv1 Hmhpsor 8 = Vhdq J @ i+{> |, = { 5 T > | 5 ]j | od rshudflöq ghfiqlgd sru= +{> |, +{3 > |3 ,@+{{3 > | . |3 , Ghwhuplqhprv vl odv vljxlhqwhv dfiupdflrqhv vrq yhugdghudv r idovdv1
47 FKDSWHU 41 ORV QÝPHURV UHDOHV 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> +3> 3, +{> |, @ +3> |, 9@ +4> 3, hv ghflu/ +3/ 3, qr wlhqh lqyhuvr1 h, Yhugdghur/ sxhv sdud wrgr +{> |,>+{3 > |3 , 5 J> +{> |, +{3 > |3 ,@+{{3 > | . |3 , @ +{3 {> |3 . |, @ +{3 > |3 , +{> |, oxhjr +{> |, +{3 > |3 ,@+{3 > |3 , +{> |, 41515 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 ghfiqh +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 +414, ;e 5 U e . 33 @ 33 . e @ e +415, Frpr 33 5 U/ hq sduwlfxodu vh sxhgh wrpdu d @ 33 hq 414/ dvðwhqhprv txh 33 @3.33 @ 33 . 3 +416, Dqäorjdphqwh frqvlghudqgr e @ 3 hq 415 vh wlhqh 3.33 @ 33 .3@3 +417, 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 fl +s , t,> hv ghflu/ vxsrqhu txh vh yhulfifd sa fl t +419, | d sduwlu gh hood rewhqhu xqd frqwudglfflöq/ od txh qrv shuplwh dfiupdu 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 s5hv sdu/ hqwrqfhv s hv sdu1 +41:, Revhuyh txh od sursrvlflöq hq 41: hv gho wlsr u , v grqgh5 u @= s5 hv sdu | od sursrvlflöq v hv v @= s hv sdu1 5Xvduhprv ho vðperor 'G% sdud lqglfdu txh od sursrvlflöq d od l}txlhugd gh ìo hvwä ghfiqlgd sru od h{suhvlöq d vx ghuhfkd1 Dtxð u 'G s2 hv sdu vljqlfifd txh od sursrvlflöq u vh ghfiqh frpr od sursrvlflöq s2 hv sdu1
49 FKDSWHU 41 ORV QÝPHURV UHDOHV 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 @ d4e4> d 9@ 3> e 9@ 3 ly, +d4,4 @ d> d 9@ 3 y, e 3@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 +d . e,.+d,.+e,@+d . +d,, . +e . +e,, @ 3.3 @ 3 Sru or wdqwr +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 [email protected] hqwrqfhv whqhprv sru glvwulexwlylgdg e 3 @ e +3 . 3, @ e 3 . e 3 oxhjr e 3 @ e 3 . e 3 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 e 3.++e 3,, @ e 3 . e 3.++e 3,, 3 @ e 3.+e 3.++e 3,,, 3 @ e 3.3 3 @ e 3 oxhjr e 3@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 de . +d,e @ +d . +d,,e @ 3 e @ 3 oxhjr 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 +de, @ +ed, @ +e,d @ d+e, sru or wdqwr +de, @ d+e,
4; FKDSWHU 41 ORV QÝPHURV UHDOHV txh hv od rwud ljxdogdg txh txhuðdprv suredu1 yll, Frqvlghuhprv +d,+e, @ +d+e,, xvdqgr yl, @ ++de,, xvdqgr yl, @ de xvdqgr ll, oxhjr +d,+e, @ de or txh suxhed fiqdophqwh 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 +e, @ +4e,@+4,e Qrwdflrqhv= Vhdq d> e 5 U / hqwrqfhv hvfuleluhprv d . +e,@= d e d4 @ 4 d > d 9@ 3 de4 @ d e > e 9@ 3 +41;, Hmhpsor ; Fxdqgr dqrwdprv 4 {5 7 hvwdprv kdeodqgr gho lqyhuvr pxowlsolfdwlyr gh {5 71 Gh hvwd irupd/ 4 {57 vhuðd xq qýphur uhdo vl | vöor vl {5 7 9@ 3> hv ghflu/ vl { 9@ 5 | { 9@ 5/ dvðod ljxdogdg 4 {5 7 @ +{5 7,4 hvwä ghfiqlgd 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 e 4 @ e d hq hihfwr/ xvdqgr od sursrvlflöq 417 | od qrwdflöq dqwhulru> d e 4 @ +de4, 4 @ d4+e4, 4 @ d4e @ 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 e . f g @ dg . ef eg +41<, d e f g @ df eg +4143, Xvdqgr od qrwdflöq/ sursrvlflöq 417 | orv d{lrpdv gh U vh wlhqh txh dg.ef eg @ +dg . ef,+eg,4 qrwdflöq @ dg+eg,4 . ef+eg,4 glvwulexwlylgdg @ dge4g4 . efe4g4 417lll, @ d+gg4,e4 . +ee4,fg4 frqpxwdwlylgdg | dvrfldwlylgdg @ d4e4 . 4fg4 surslhgdg lqyhuvrv @ de4 . fg4 surslhgdg qhxwur @ d e . f g qrwdflöq Dghpäv= d e f g @ +de4,+fg4, @ dfe4g4 @ df+eg, 4 @ df eg Odv surslhgdghv dqwhulruhv wlhqhq od yhqwdmd gh txh qrv shuplwhq wudedmdu frq h{suhvlrqhv päv vlpsol0 fifdgdv Vxefrqmxqwrv Qrwdeohv Dojxqrv vxefrqmxqwrv qrwdeohv gh U txh phqflrqdprv do sulqflslr gho fdsðwxor vrq Ho frqmxqwr gh orv qýphurv Qdwxudohv1 Q @ i4> 5> 6> 7> 8>===j Ho frqmxqwr gh orv qýphurv Hqwhurv1 ] @ i===> 6> 5> 4> 3> 4> 5> 6> 7>===j Ho frqmxqwr gh orv qýphurv Udflrqdohv1 T @ qd e = d 5 ] a e 5 ] i3j r Rwurv frqmxqwrv qrwdeohv vrq Q3 @ Q ^ i3j @ ]. 3 ] @ iq = q 5 Qj L @ U T hvwh ýowlpr oodpdgr ho frqmxqwr gh orv qýphurv Luudflrqdohv1 Qdwxudophqwh vh wlhqh od vljxlhqwh fdghqd gh ghvljxdogdghv Q ffl Q3 @ ]. 3 ffl ] ffl T ffl U | dghpäv6 T ^bL @ U Qrwhprv txh orv vljxlhqwhv qýphurv uhdohv 4> 657> 3> 6> 4> 5 3> 69 6 lqglfd xqlöq glvmxqwd1
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4151 D[LRPDV GH FXHUSR 54 41516 Srwhqfldv Hqwhudv Ghfiqlflöq 419 Vhd d 5 U / q 5 ].1 Vh ghfiqh od srwhqfld uhdo gh edvh d | h{srqhqwh q sru= dq @ d d d d d ~} q yhfhv Xqd ghfiqlflöq päv suhflvd gh srwhqfld hv od vljxlhqwh Vhd d 5 U 1 Vh ghfiqh sru uhfxuuhqfld> l, d3 @ 4/ d 9@ 3 ll, dq.4 @ dq d/ ;q 5 Q3 Vhd q 5 ]/ d 9@ 31 Vh ghfiqh od srwhqfld uhdo gh edvh d | h{srqhqwh q sru= dq @ 4 dq @ 4 d d d d d ~} q yhfhv +4144, Hmhpsor 44 56 @ 5 5 5@;> 56 @ 4 56 @ 4 5 5 5 @ 4 ; Whruhpd 41: Vhd d> e 5 U i3j/ q 5 ] / p 5 ] / hqwrqfhv l, dq dp @ dq.p ll, dqeq @ +de,q lll, dp dq @ dpq ly, dq eq @ d 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 ghfiqlflöq 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 @ +d . e,+d e, 5, +d . e,5 @ d5 . 5de . e5 6, d6 . e6 @ +d . e,+d5 de . e5, 7, d6 e6 @ +d e,+d5 . de . e5, 8, +d . e,6 @ d6 . 6d5e . 6de5 . e6 9, +d e,6 @ d6 6d5e . 6de5 e6 :, dp ep @ +d e,+dp4 . dp5e . dp6e5 . . dep5 . ep4, ;p 5 ].
55 FKDSWHU 41 ORV QÝPHURV UHDOHV Hmhpsor 45 Vhdq d> e hq U/ wdohv txh od h{suhvlöq vljxlhqwh hvwì ghfiqlgd hq U= L @ d4 . e4 f+de,4 d5.e5f5 d5e5 . 5d4e4 # 4 4 d.e.f $ Vlpsolfitxh L1 Vroxflöq= Surfhglhqgr hq hwdsdv/ whqhprv L @ d5e5+d4 . e4 f+de,4, d5 . e5 f5 . 5d4e4d5e5 +d . e . f, @ +de5 . d5e 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 ghfiqlgd hq U/ | vlpsolfitxh1 H @ {5 5{ . 4 +{4,6 {54 Vroxflöq= Od h{suhvlöq hvwä ghfiqlgd hq U vl +{ 4,6 {5 4 9@ 3 a {5 4 9@ 3 hv ghflu/ +{ 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 {54 { 9@ 4 > { 9@ 4 @ {5 5{ . 4 +{ 4,6 +{5 4, @ 4 +{ 4,+{ 4,+{ . 4, @ { . 4 Qrwh txh { . 4 wlhqh vhqwlgr7 sdud wrgr { 5 U / qr dvðH |d txh do phqrv sdud 4 | 4 qr wlhqh vhqwlgr1 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 ghfiqlgd 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 7Txh { n whqjd vhqwlgr sdud wrgr { hq U vljqlfifd txh { n hvwä ghfiqlgr hq wrgr { M U/ hv ghflu/ { M U> { n M U1 8Vxhoh oodpäuvhoh grplqlr gh xqd h{suhvlöq1 9SE{ffi | TE{ffi vrq h{suhvlrqhv wdohv txh do phqrv xqd gh hoodv lqyroxfud od yduldeoh {1
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 ffl U1 Sru hmhpsor/ ho frqmxqwr i{ 5 U = 4 { @ {j hv ho frqmxqwr gh orv uhdohv wdohv txh fxpsohq od hfxdflöq 4 { @ {/ hv ghflu/ i{ 5 U = 4 { @ {j @ i4> 4j Uhvroyhu xqd hfxdflöq vljqlfifd hqfrqwudu vx frqmxqwr vroxflöq1 Uhvroydprv od hfxdflöq {5 5{ . 4 +{4,6 {54 @ 6 Vroxflöq= Sru ho hmhpsor 41516 hv fodur txh ho frqmxqwr uhvwulfflöq hv U4 @ U i4> 4j/ Dghpäv vlpsolfifdqgr od h{suhvlöq l}txlhugd gh od ljxdogdg/ rewhqhprv {55{.4 +{4,6 {54 @ 6 / { .4 @ 6 / { @ 5 Sru or wdqwr ho frqmxqwr vroxflöq gh od hfxdflöq hv= V @ + { 5 U = {5 5{ . 4 +{4,6 {54 @ 6, @ i5j_U4 @ i5j _ +U i4> 4j, @ i5j Sursrvlflöq 41; Vhdq d> e hq U1 l, Od hfxdflöq { . d @ e wlhqh sru ýqlfd vroxflöq ho uhdo e d1 ll, Vl d 9@ 3 od hfxdflöq d{ @ e wlhqh frpr ýqlfd vroxflöq ho uhdo e d1 Od ghprvwudflöq vh ghmduä frpr hmhuflflr sdud ho ohfwru> gheh xvduvh odv surslhgdghv gh lqyhuvr dglwlyr | pxowlsolfdwlyr uhvshfwlydphqwh1 Hmhuflflr 46 Ghwhuplqh vl odv vljxlhqwhv dfiupdflrqhv vrq yhugdghudv r idovdv1 Mxvwlfitxh1 41 q { 5 U = {.6 {.6 @ 4r @ U 51 q { 5 U = +{.6,+{.7, {.6 @ 4r @ ! 61 +;d> e 5 U,+de @ 4 , +d @ 4 b e @ 4,, 71 +;{> | 5 U,+{5 @ |5 , { @ |, Vroxflöq=
57 FKDSWHU 41 ORV QÝPHURV UHDOHV 41 Idovr1 Ho frqmxqwr uhvwulfflöq gh od hfxdflöq { . 6 { . 6 @ 4 hv U @ U i6j1 Oxhjr/ vlpsolfifdqgr od h{suhvlöq { . 6 { . 6 @ 4 rewhqhprv txh 4@4 or fxdo hv yhugdghur1 Sru or wdqwr ho frqmxqwr vroxflöq gh od hfxdflöq hv V @ ffl { 5 U = { . 6 { . 6 @ 4ff @ U _ U @ U i6j 51 Yhugdghur1 Ho frqmxqwr uhvwulfflöq gh od hfxdflöq +{ . 6,+{ . 7, { . 6 @ 4 hv U4 @ U i6j1 Vlq hpedujr do vlpsolfifdu od h{suhvlöq +{ . 6,+{ . 7, { . 6 @ 4 rewhqhprv od hfxdflöq { .7@4 gh grqgh whqhprv txh { @ 61 Oxhjr ho frqmxqwr vroxflöq gh od hfxdflöq hv V @ ffl { 5 U4 = { . 6 { . 6 @ 4ff @ i6j_U4 @ i6j _ +U i6j, @ ! 61 Idovr1 Frqvlghuhprv d @ 5@6/ e @ 6@5 hq U 5 6 6 5 @ 4 shur 5 6 9@ 4 | 6 5 9@ 4 71 Idovr1 Vhdq { @ 5/ | @ 5 +5,5 @ 55 shur 5 9@ 5 41518 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 { krudv oohqduä { : gho hvwdqtxh/ hv ghflu/ oohqduä {Y : +xqlgdghv gh yroxphq,1 Od rwud erped oohqd ho hvwdqtxh
4151 D[LRPDV GH FXHUSR 58 hq ; krudv/ hqwrqfhv hq { krudv oohqduä { ; gho hvwdqtxh/ hv ghflu/ {Y ; +xqlgdghv gh yroxphq,1 Frpr hq { krudv/ odv grv erpedv mxqwdv oohqdq do hvwdqtxh hqwrqfhv vh wlhqh {Y : . {Y ; @ Y or txh htxlydoh d { : . { ; @ 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 { 7 8=333 @ { 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,> lghqwlfifdu 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> 5=6@9=< txh vh ohh 5 hv d 6 frpr 9 hv d < 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
59 FKDSWHU 41 ORV QÝPHURV UHDOHV 41 +d, Sdudohohsðshgr gh dulvwdv {/ |/ }1 Yroxphq= Y @ {|} Duhd= D @ 5{| . 5{} . 5|} +e, Flolqgur gh udglr u | dowxud k1 Yroxphq= Y @ ffiu5k Duhd= D @ 5ffiu5 . 5ffiuk +f, Frqr uhfwr gh udglr edvdo u | dowxud k1 Yroxphq= Y @ 4 6 ffiu5k Duhd= D @ ffius u5 . k5 +g, Hvihud gh udglr u1 Yroxphq= Y @ 7 6 ffiu6 Duhd= D @ 7ffiu5 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> <u Vl od dowxud dxphqwd hq xq 45( hqwrqfhv od qxhyd dowxud kq vhuä kq @ k . 3> 45k @ 4> 45k oxhjr ho qxhyr yroxphq vhuä Yq Yq @ ffiuq 5kq @ ffi+3> <u, 5+4> 45k,@3> <3:5ffiu5k @ ffiu5k 3> 3<5;ffiu5k hvwr hv/ ho yroxphq glvplqx|h hq 3> 3<5;ffiu5k xqlgdghv> hv ghflu/ glvplqx|h hq <> 5;( e, Ho äuhd gho flolqgur vhuä Dq @ 5ffiuqkq @ 5ffi+3> <u,+4> 45k,@4> 33; 5ffiuk @ 5ffiuk . 3> 33; 5ffiuk 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 45 433{ qr uhvroylhurq ho sureohpd 65 433{ uhvroylhurq frq dojxqrv huuruhv ho sureohpd 47 = uhvroylhurq ho sureohpd fruuhfwdphqwh Oxhjr whqhprv od vljxlhqwh hfxdflöq 45 433{ . 65 433{ . 47 @ { gh grqgh 89{ @ 4733> hv ghflu/ { @ 4733 89 @ 58 oxhjr kdeðd 58 hvwxgldqwhv hq od fodvh1 Hmhuflflr 4; Xqd wlhqgd gh dqwlj hgdghv frpsuö grv pxheohv hq ' 558=333 | ghvsxìv orv yhqglö | rewxyr xq ehqhfiflr gho 73(1 ÁFxäqwr sdjö sru fdgd pxheoh vl ho sulphur ghmö xq ehqhfiflr gho 58( | ho vhjxqgr xq ehqhfiflr 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 ehqhfiflr gho 58(/ oxhjr vh yhqglö hq +{ . 3> 58{, plo shvrv> hv ghflu/ 4> 58{ plo shvrv1 Ho vhjxqgr glr xq ehqhfiflr 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 ehqhfiflr wrwdo gho 73(/ oxhjr orv grv pxheohv vh yhqglhurq hq 4> 73 558 @ 648 plo shvrv1 Dvð/ rewhqhprv od hfxdflöq 4> 58{ . 4> 83+558 {, @ 648 gh grqgh 3> 58{ @ 55> 8 oxhjr { @ <3 Ho sulphu pxheoh vh yhqglö hq ' <3=333 | ho vhjxqgr hq ' 468=3331 Revhuydflöq= Ho wdqwr sru flhqwr gh ehqhfiflr vh fdofxod hq uhodflöq frq ho suhflr gh frvwr +wrpdgr frpr 433(,1
5; FKDSWHU 41 ORV QÝPHURV UHDOHV Hmhuflflr 4< Orv gläphwurv gh grv flolqgurv vrq hqwuh vðfrpr 6=7 | vxv dowxudv frpr 8=91 BcTxì wdqwr sru flhqwr gho yroxphq gho pd|ru gh hoorv hv ho yroxphq gho phqruB Vroxflöq= Vhdq g4> g5 orv gläphwurv gh orv flolqgurv | k4> k5 vxv uhvshfwlydv dowxudv1 Sru klsöwhvlv gho sureohpd whqhprv odv vljxlhqwhv uhodflrqhv> g4 g5 @ 6 7 > k4 k5 @ 8 9 +4145, Dghpäv od iöupxod gho yroxphq gh xq flolqgur hv Y @ ffiu5k 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 Y @ ffi 7 g5k | sdud fdgd flolqgur Y4 @ ffi 7 g5 4k4 Y5 @ ffi 7 g5 5k5 Shur gh 4145 whqhprv g4 @ 6 7 g5 | k4 @ 8 9 k5 oxhjr Y4 @ ffi 7 < 49g5 5 8 9 k5 @ 78 <9 ffi 7 g5 5k5 @ 78 <9Y5 dvð/ Y4 @ 78 <9Y5 Oxhjr Y4 hv ho 79> ;:8( gh Y51 Xqd surslhgdg txh qrv vhuä px| ýwlo hq od uhvroxflöq gh hfxdflrqhv hv od vljxlhqwh1 Sursrvlflöq 41< Vhdq d> e 5 U de @ 3 +, d @ 3 b e @ 3 +4146, Ghprvwudflöq= +,,: Vxsrqjdprv d 9@ 3> hqwrqfhv h{lvwh d4 5 U wdo txh de @ 3 , d4de @ d43 , 4 e @ 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 P5+U, @ ffl d e f g = d> e> f> g 5 U ff :Sru htxlydohqfld oöjlfd hvwd lpsolfdqfld hq 4146 vh sxhgh hvfulelu Es i t j vffi/ txh hv htxlydohqwh Es2 ; t i vffi | hvwr ýowlpr hv or txh xvduhprv hq hvwd sduwh gh od ghprvwudflöq1
4151 D[LRPDV GH FXHUSR 5< edmr od rshudflöq gh surgxfwr ghfiqlgd sru d e f g d3 e3 f3 g3 @ dd3 . ef3 de3 . eg3 fd3 . gf3 fe3 . gg3 Vhdq D @ 4 3 3 3 | E @ 3 3 3 4 Hv fodur txh DE @ 3 3 3 3 shur D 9@ 3 | E 9@ 31 Or txh vðvh fxpsoh hv txh 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 {5 @ 58 hv i8> 8j Vroxflöq= Hq hihfwr> {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 Od shuvrqd SD wdugduä w4 @ 57 y krudv | SE wdugduä w5 @ 57 y5 krudv Frpr SE oohjd xqd krud päv wdugh txh SD/ hqwrqfhv> w5 @ w4 . 4 , 57 y5 57 y @ 4 , 57y 57+y 5, @ +y 5,y , y5 5y 7; @ 3 , +y ;,+y . 9, @ 3 , y @ ; b y @ 9
63 FKDSWHU 41 ORV QÝPHURV UHDOHV Shur y A 3/ dvð/ y @ ; np2ku1 Oxhjr od glvwdqfld txh uhfruuh SD hq xqd krud hv gh g4 @ y 4@; np | SE uhfruuh 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 @ ;3 y ku shur uhdophqwh ixh w5 @ ;3 y . 43 ku Sru klsöwhvlv whqhprv txh w4 w5 @ 49 93> hv ghflu/ ;3 y ;3 y.43 @ 49 93 / +y . 43,;3 ;3y @ 7 48 y+y . 43, / y5 . 43y 6=333 @ 3 / +y 83,+y . 93, @ 3 / y @ 83 b y @ 93 shur y A 3 oxhjr 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| @ 5 8{ . | @ 5 Vroxflöq= Pxowlsolfdqgr sru 6 od vhjxqgd hfxdflöq | vxpäqgrvhod d od sulphud vh rewlhqh od hfxdflöq 49{ @ 7 gh grqgh { @ 4 7 1 Vxvwlwx|hqgr hvwh ydoru hq fxdotxlhud gh odv grv hfxdflrqhv vh rewlhqh | @ 6 7 1 Oxhjr od vroxflöq gho vlvwhpd hv { @ 4 7 > | @ 6 7 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 fijxud
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/ | @ ; { @ 4; Sru or wdqwr kd| ; kdelwdflrqhv | 4; hvwxgldqwhv1
65 FKDSWHU 41 ORV QÝPHURV UHDOHV Hmhuflflr 58 Xqd fkdsd uhfwdqjxodu gh hvwdôr gh shuðphwur <9 fp vh xvd sdud kdfhu xqd fdmd vlq wdsd1 Sdud hoor vh uhfruwd xq fxdgudgr gh 7 fp gh odgr hq fdgd hvtxlqd | vh xqhq orv erughv/ +yhu fijxud dqwhulru,1 ÁFxäohv vrq odv glphqvlrqhv gh od fkdsd vl ho yroxphq gh od fdmd hv gh :9;B fp6= Vroxflöq= Vhdq { @= orqjlwxg gh od fkdsd hq fhqwðphwurv | @= dqfkr gh od fkdsd hq fhqwðphwurv/ hqwrqfhv od fdmd wlhqh odv glphqvlrqhv vljxlhqwhv +yhu fijxud yhufijxud dqwhulru,= orqjlwxg = +{ ;, fp dqfkr = +| ;, fp dowxud = 7 fp Sru klsöwhvlv/ yroxphq | shuðphwur vrq> 7+{ ;,+| ;, @ :9; 5{ . 5| @ <9 or txh htxlydoh d +{ ;,+| ;, @ 4<5 { . | @ 7; Ghvshmdqgr | hq od vhjxqgd hfxdflöq | uhhpsod}dqgr hq od sulphud rewhqhprv {5 7;{ . 845 @ 3 hv ghflu +{ 65,+{ 49, @ 3 gh grqgh { @ 65 b { @ 49 qrwh txh vl { @ 65/ hqwrqfhv | @ 49 | vl { @ 49/ hqwrqfhv | @ 651 Frpr { hv od orqjlwxg> odv glphqvlrqhv gh od fkdsd vrq 65 fp 49 fp1 Hmhuflflr 59 Xq reuhur kdfh xq flhuwr qýphur gh slh}dv lgìqwlfdv hq xq wlhpsr ghwhuplqdgr1 Vl kxelhud khfkr 43 slh}dv päv fdgd gðd/ kdeuðd whuplqdgr ho wudedmr frpsohwr hq 7 4 5 gðdv dqwhv gh or suhylvwr/ | vl kxelhvh khfkr 8 slh}dv phqrv fdgd gðd kdeuðd wdugdgr 6 gðdv päv gh or suhylvwr1 ÁFxäqwdv slh}dv kl}r | hq fxäqwr wlhpsrB Vroxflöq= Vxsrqjdprv txh ho reuhur kdfh { slh}dv hq | gðdv1 Hqwrqfhv surgxfh { | slh}dv sru gðd/ sru klsöwhvlv/ vl kxelhud uhdol}dgr { | . 43 slh}dv / kdeuðd frpsohwdgr ho wudedmr hq | 7 4 5 gðdv1 Oxhjr frpr 7 4 5 @ < 5 whqhprv { | . 43 | < 5 @ { Od rwud frqglflöq gh od hfxdflöq hv { | 8 +| . 6, @ {= Dvð/ rewhqhprv ho vljxlhqwh vlvwhpd 43| < 5 { | @ 78 8| . 6{ | @ 48 gh grqgh { | @ 83 oxhjr/ | @ 5: { @ 4=6831
4151 D[LRPDV GH FXHUSR 66 Hmhuflflr 5: Grv reuhurv wlhqhq txh kdfhu xq wudedmr frqvlvwhqwh hq phfdql}du xq orwh gh slh}dv lgìqwlfdv1 Ghvsxìv txh ho sulphur kd wudedmdgr gxudqwh : krudv | ho vhjxqgr gxudqwh 7 krudv/ kdq frpsohwdgr 8 < gho wrwdo gho wudedmr1 Vl vljxlhudq wudedmdqgr orv grv d od yh} gxudqwh 7 krudv päv/ ohv txhgduðd sru kdfhu 4 4; gho wudedmr1 Á Fxäqwr wdugduðd fdgd xqr hq kdfhu ho wudedmr frpsohwrB Vroxflöq= Vxsrqjdprv txh ho sulphu rshudulr/ wudedmdqgr vror/ hv fdsd} gh frpsohwdu ho wudedmr hq { krudv/ | ho vhjxqgr hq | krudv1 Hqwrqfhv hq xqd krud ho sulphur kdfh 4 { gho wudedmr frpsohwr | ho vhjxqgr 4 | 1 Sru klsöwhvlv : 4 { . 7 4 | @ 8 < Frpr ghvsxìv wudedmdq mxqwrv rwudv 7 krudv/ kduäq 7 { . 7 | gho wudedmr/ txh hv ljxdo d 4 8 < . 4 4; @ : 4; sru or wdqwr whqhprv od hfxdflöq 7 { . 7 | @ : 4; Uhvwäqgrod gh od sulphud/ rewhqhprv 6 { @ 6 4; gh grqgh { @ 4;1 Hqwrqfhv 4 | @ 4 57 h | @ 57 Oxhjr ho sulphur wdugd 4; krudv | ho vhjxqgr 57 krudv hq kdfhu ho wudedmr frpsohwr1 Hmhuflflr 5; Vh kd gh wudqvsruwdu 9<3 wrqhodgdv gh phufdqfðdv ghvgh xq pxhooh d xqd hvwdflöq gh ihu0 urfduulo phgldqwh 8 fdplrqhv gh 6 wrqhodgdv | 43 fdplrqhv gh xqd wrqhodgd | phgld1 Hq srfdv krudv/ orv fdplrqhv kdq wudqvsruwdgr 58279 gh odv phufdqfðdv1 Sdud frpsohwdu ho wudqvsruwh d wlhpsr/ vh kd gh oohydu odv phufdqfðdv uhvwdqwhv hq xq odsvr 5 krudv phqru txh ho |d wudqvfxuulgr1 Vh frpsohwö ho wudqv0 sruwh judfldv d txh orv frqgxfwruhv gh orv fdplrqhv frphq}durq d kdfhu xq yldmh sru krud päv txh dqwhv1 Ghwhuplqh fxäqwdv krudv wdugdurq hq wudqvsruwdu wrgdv odv phufdqfðdv | wdpelìq ho qýphur gh yldmhv sru krud txh vh kdfðd do sulqflslr vdelhqgr txh orv fdplrqhv gh xqd | phgld wrqhodgd kdfhq xq yldmh päv sru krud txh orv fdplrqhv gh 6 wrqhodgdv ; 1 Vroxflöq= Vxsrqjdprv txh od sulphud sduwh gh odv phufdqfðdv/ txh dvflhqgh d 58 799<3 @ 6:8 wrqhodgdv vh wudqvsruwd hq { krudv kdflhqgr fdgd fdplöq gh 6 wrqhodgdv | yldmhv sru krud1 Hqwrqfhv fdgd fdplöq gh xqd | phgld wrqhodgd kduä | . 4 yldmhv sru krud1 Sru klsöwhvlv/ od sduwh uhvwdqwh gh phufdqfðdv +hv ghflu/ 9<306:8@648 wrqhodgdv, vh wudqvsruwö hq { 5 krudv/ kdflhqgr orv fdplrqhv gh 6 wrqhodgdv | . 4 yldmhv sru krud | orv fdplrqhv gh xqd | phgld wrqhodgdv +| . 4, . 4 @ | . 5 yldmhv sru krud1 Rewhqhprv dvðho vljxlhqwh vlvwhpd gh hfxdflrqhv 8 6{| . 43 6 5 {+| . 4, @ 6:8 8 6+{ 5,+| . 4, . 43 6 5 +{ 5,+| . 5, @ 648 Vlpsolfifdqgr whqhprv 5{| . { @ 58 5{| . 6{ 7| @ 5: ; Vh gheh vxsrqhu txh wrgrv orv fdplrqhv ledq frpsohwdphqwh fdujdgrv hq fdgd yldmh1
67 FKDSWHU 41 ORV QÝPHURV UHDOHV gh grqgh 5{ 7| @ 5 oxhjr 5| @ { 41 Vxvwlwx|hqgr hq od sulphud hfxdflöq rewhqhprv {5 @ 58/ hv ghflu< / { @ 81 Od sulphud sduwh gh odv phufdqfðdv vh wudqvsruwö hq 8 krudv | od vhjxqgd sduwh hq 8 5@6 krudv1 Qrwd= Wrgdv odv phufdqfðdv vh wudqvsruwdurq hq ; krudv> do sulqflslr/ orv fdplrqhv gh wuhv wrqhodgdv kdfðdq 5 yldmhv sru krud | orv gh xqd | phgld wrqhodgdv/ 6 yldmhv sru krud1 Hmhuflflr 5< Xqd lqgxvwuld wlhqh xq hqfdujr gh ;43 duwðfxorv | rwud gh <33 duwðfxorv hq ho plvpr shulrgr gh wlhpsr1 Od sulphud kd frpsohwdgr ho shglgr 6 gðdv dqwhv gho sod}r suhylvwr | od vhjxqgrv 9 gðdv dqwhv1 BcFxäqwdv duwðfxorv surgxfh do gðd fdgd lqgxvwuld/ vdelhqgr txh od vhjxqgd surgxfh sru gðd 7 duwðfxorv päv txh od sulphudB Vroxflöq= Vhd { od surgxfflöq gh duwðfxorv glduld gh od sulphud lqgxvwuld/ hqwrqfhv od vhjxqgd surgxfh {.7 duwlfxorv sru gðd1 Od sulphud kd frpsohwdgr vx shglgr hq ;43 { gðdv/ oxhjr ho wlhpsr gdgr sdud fxpsolu ho shglgr kd vlgr ;43 { . 6 gðdv1 Dqäorjdphqwh whqhprv txh <33 {.7 . 9 hv ho wlhpsr dvljqdgr d od vhjxqgd lqgxvwuld/ shur ho wlhpsr dvljqdgr d dpedv odv lqgxvwuldv hv ho plvpr/ oxhjr ;43 { .6@ <33 { . 7 . 9 gh grqgh { @ 531 Hv ghflu/ od sulphud lqgxvwuld surgxfh 53 duwlfxorv | od vhjxqgd 57 duwlfxorv sru gðd1 Hmhuflflr 63 Kdoodu xq qýphur gh grv fliudv vdelhqgr txh ho qýphur gh xqlgdghv h{fhgh hq fxdwur do qýphur gh od ghfhqdv | txh ho surgxfwr gho qýphur ghvhdgr sru od vxpd gh vxv gljlwrv hv ljxdo d 8:91 Revhuydflöq= Vl xq qýphur d wlhqh q gljlwrv d3> d4> d5> dq4/ rughqdgrv gh l}txlhugd d ghuhfkd hqwrqfhv d @ dq443q4 . . d443 . d3 Vroxflöq= Vhd { ho gljlwr gh od xqlgdghv/ | ho gljlwr gh od ghfhqdv hqwrqfhv sru klsöwhvlv vh wlhqh od hfxdflöq { @ | . 7 +4147, dghpäv ho qýphur ghvhdgr hv 43| . { oxhjr +43| . {,+{ . |, @ 8:9 +4148, Sru 4147 | 4148 +44| . 7,+5| . 7, @ 8:9 +, 44+| 7,+| . :3 44,@3 +, | @ 7 b | @ :3 44 shur | A 3/ oxhjr | @ 71 Sru or wdqwr ho qýphur exvfdgr hv 7;1 416 D{lrpdv gh Rughq D frqwlqxdflöq gduhprv xq frqmxqwr gh d{lrpdv txh irupd sduwh gh od fdudfwhul}dflöq gh orv qýphurv uhdohv1 Hvwrv d{lrpdv qrv shuplwhq rughqdu orv qýphurv uhdohv +| dýq srghu judfifdu hvwh rughq hq xqd uhfwd,1 Dgplwlprv od h{lvwhqfld gh xq vxefrqmxqwr gh U txh ghqrwdprv sru U./ fx|rv hohphqwrv vh oodpdq qýphurv uhdohv srvlwlyrv | txh vh fdudfwhul}dq sru orv d{lrpdv= <Gh {2 ' 2D vh rewlhqh { ' D r { ' 3D shur { lqglfd krudv oxhjr { gheh vhu srvlwlyr1
4161 D[LRPDV GH RUGHQ 68 D{lrpd l, Fodxvxud gh od vxpd Vl d> e 5 U./ hqwrqfhv d . e 5 U. D{lrpd ll, Fodxvxud gho surgxfwr Vl d> e 5 U./ hqwrqfhv de 5 U. D{lrpd lll, Wulfrwrpðd Sdud fdgd qýphur uhdo vöor xqd gh odv vljxlhqwhv sursrvlflrqhv hv yhugdghud d 5 U.> d @ 3> d 5 U. Hq edvh d orv d{lrpdv dqwhulruhv ghprvwuduhprv odv vljxlhqwhv surslhgdghv= Sursrvlflöq 4143 Ho frqmxqwr gh orv qýphurv uhdohv fxpsoh l, d 5 U i3j , d5 5 U. ll, d 5 U. , d4 5 U. lll, d e 5 U. a e f 5 U. , d f 5 U. ly, e d 5 U. / +e . f, +d . f, 5 U. y, e d 5 U. a f 5 U. , ef df 5 U. yl, e d 5 U. a f 5 U. , df ef 5 U. Ghprvwudflöq= l, Vl d 5 U i3j hqwrqfhv sru D{lrpd lll, d 5 U. b d 5 U. Vl d 5 U. hqwrqfhv sru D{lrpd ll, d d 5 U./ hv ghflu/ d 5 U. , d5 5 U. Vl d 5 U. hqwrqfhv sru D{lrpd ll, +d, +d, 5 U. shur sru sursrvlflöq 417 yll/ +d,5 @ d5/ oxhjr d5 5 U.1 Qrwhprv txh hq sduwlfxodu vl d @ 4/ hqwrqfhv 45 @ 4 5 U. ll, Ghprvwuduhprv hvwd surslhgdg sru ho pìwrgr gho devxugr1 Vhd d 5 U.1 Vxsrqjdprv txh d4 95 U./ hqwrqfhv sru D{lrpd lll, d4 @ 3 b +d4, 5 U. Vxsrqjdprv txh d4 @ 3 hqwrqfhv d d4 @ 3 shur sru surslhgdg gh lqyhuvr d d4 @ 4> hvwr hv xqd frqwudglfflöq | sru or wdqwr d4 9@ 31 Vxsrqjdprv txh +d4, 5 U.1 Frpr d 5 U. whqhprv txh sru D{lrpd ll, txh +d4,d @ 4 5 U. txh hv wdpelìq xqd frqwudglfflöq sxhv 4 5 U.1 Frpr hq dperv fdvrv rewhqhprv xqd frqwudglfflöq/ qr vh sxhgh gdu qlqjxqd gh odv grv srvlelolgdghv/ sru or wdqwr/ d4 5 U.> dvð/ vh wlhqh txh ho lqyhuvr pxowlsolfdwlyr gh xq qýphur srvlwlyr hv wdpelìq xq qýphur srvlwlyr1 lll, d e 5 U. a e f 5 U. , +d e,.+e f, 5 U. sru D{lrpd l, , d . +e . e, f 5 U. dvrfldwlylgdg , d . 3 f 5 U. lqyhuvr dglwlyr , d f 5 U. qhxwur dglwlyr
69 FKDSWHU 41 ORV QÝPHURV UHDOHV ly, e d 5 U. / e d . 3 5 U. qhxwur dglwlyr / e d . f f 5 U. lqyhuvr dglwlyr / +e . f,.+d f, 5 U. dvrfldwlylgdg frqpxwdwlylgdg / +e . f, +d . f, 5 U. glvwulexwlylgdg y, e d 5 U. a f 5 U. , +e d,f 5 U. sru D{lrpd ll, , ef df 5 U. glvwulexwlylgdg yl, e d 5 U. a f 5 U. , +e d,+f, 5 U. sru D{lrpd ll, , +ef, +df, 5 U. glvwulexwlylgdg , df ef 5 U. Vh ghfiqh ho frqmxqwr gh orv qýphurv uhdohv qhjdwlyrv frpr U @ id 5 U = d 5 U.j Qrwhprv txh U 9@ > sxhv 4 5 U gdgr txh 4 5 U.1 Dqäorjdphqwh vh ghfiqh ho frqmxqwr gh orv qýphurv uhdohv qr qhjdwlyrv frpr U. 3 @ U. ^ i3j Qrwd= Sru D{lrpd lll, vh wlhqh od vljxlhqwh ljxdogdg grqgh orv frqmxqwrv vrq glvmxqwrv grv d grv U @ U. ^ i3j ^ U Ghfiqlflöq 4144 Gluhprv txh d hv pd|ru txh e ö e hv phqru txh d vl | vöor vl d e 5 U. +4149, | hvfuleluhprv hvwr lqglvwlqwdphqwh frpr dAe r frpr e ? d= Gluhprv txh d hv pd|ru r ljxdo txh e vl d hv pd|ru txh e r d hv ljxdo d e/ hv ghflu dAe b d @ e Hvfuleluhprv hvwr lqglvwlqwdphqwh frpr d e r e d= Dvð/ vh wlhqh d e +, dAe b d @ e Dqäorjdphqwh/ gluhprv txh d hv phqru r ljxdo txh e vl d hv phqru txh e r d ljxdo txh e1 Hvfuleluhprv lqglvwlqwdphqwh hq hvwh fdvr d e r e d r htxlydohqwhphqwh d e / d?e b d @ e
4161 D[LRPDV GH RUGHQ 6: Frq od qrwdflöq dqwhulru whqhprv txh= d 5 U. / d A 3 d 5 U / d ? 3 | odv surslhgdghv ghprvwudgdv dqwhulruphqwh vh sxhghq hvfulelu gh od vljxlhqwh irupd Frurodulr 4145 Vhdq d> e> f 5 U l, d 5 U i3j , d5 A 31 ll, d A 3 , d4 A 3 lll, dAe a eAf , dAf ly, eAd / e . fAd . f y, eAd a f A 3 , ef A df yl, eAd a f ? 3 , ef ? df Qrwdflöq= d e f / d e a e f Orv Lqwhuydorv gh U= Vhdq d> e 5 U1 frq43 d e Vh ghfiqhq ^d> e^ @ i{ 5 U = d {?ej `d> e` @ i{ 5 U = d?{ ej `d> e^ @ i{ 5 U = d?{?ej ^d> e` @ i{ 5 U = d { ej `4> d` @ i{ 5 U = { dj `4> d^ @ i{ 5 U = {?dj ^d>4^ @ i{ 5 U = { dj `d>4^ @ i{ 5 U = {Adj Qrwd= Od ghfiqlflöq gdgd qr dgplwh lqwhuydorv gho wlsr ^d> e^ frq dAe +hq wdo fdvr vh wudwduld gh xq frqmxqwr ydfðr,1 Hmhpsor 64 ^4> 3^ @ i{ 5 U = { ? 3j `4>4^ @ i{ 5 U = { A 4j `4> 3^ ^ `4>4^ @ i{ 5 U = { ? 3 b { A 4j `3> 4^ @ i{ 5 U = 3 ?{? 4j ^4> 4` @ i{ 5 U = 4 { 4j ^4> 3^ @ i{ 5 U = 4 { ? 3j ^5> 4^ @ i{ 5 U = 5 { ? 4j 4 3Dojxqrv gh hvwrv lqwhuydorv sxhghq vhu ho frqmxqwr ydfðr1
6; FKDSWHU 41 ORV QÝPHURV UHDOHV Xqr gh orv whruhpdv päv lpsruwdqwhv txh vh ghgxfh gho d{lrpd gh wulfrwrpðd/ hv ho vljxlhqwh> Whruhpd 4146 44 Sdud wrgr d> e 5 U / vh yhulfifd xqd | vöor xqd gh odv sursrvlflrqhv d A e> d @ e> d ? e Ghprvwudflöq= Vhdq d> e 5 U / hqwrqfhv sru D{ lll, vöor xqd gh odv vljxlhqwhv sursrvlflrqhv hv yhugdghud sdud ho qýphur uhdo d e> d e 5 U.> d e @ 3> +d e, 5 U. Oxhjr/ xvdqgr od qrwdflöq 4149 | wudgxflhqgr fdgd sursrvlflöq gh od iudvh dqwhulru vh rewlhqh txh d A e> d @ e> d ? e> vlhqgr vöor xqd gh hoodv yhugdghud1 Revhuydflöq= Od uhodflöq hqwuh d | e/ d e vh dfrvwxpeud d ghflu txh hv xqd uhodflöq gh rughq wrwdo |d txh hood fxpsoh= sdud d> e> f 5 U l, Uh h{lylgdg d d ll, Dqwlvlphwuðd d e a e d , d @ e lll, Wudqvlwlylgdg d e a e f , d f yl, Wulfrwrpðd d e b e d Ho khfkr txh hq U vh vdwlvidjdq orv d{lrpdv gh fxhusr | orv d{lrpdv gh rughq vh h{suhvd glflhqgr txh U hv xq Fxhusr Rughqdgr%1 Sru orv d{lrpdv gh fxhusr vdehprv txh h{lvwhq 3 5 U | 4 5 U wdo txh 4 9@ 31 Dkrud frq orv d{lrpdv gh rughq khprv ghprvwudgr txh> 4 5 U. | 4 5 U> hv ghflu/ 4 A 3 | 4 ? 3 Dvðsrghprv rughqdu wrgrv orv qýphurv uhdohv/ frvd txh kduhprv d frqwlqxdflöq sru hwdsdv= Frpr hmhpsorv lqlfldohv whqhprv 4 A 3 , 4.4 A 4.3 , 5 A 4 4 ? 3 , +4, . +4, ? 4.3 , 5 ? 4 | gh hvwd irupd whqhprv 5 ? 4 ? 3 ? 4 ? 5> hwf= Sdud rughqdu orv udflrqdohv xvduhprv od vljxlhqwh surslhgdg= Vhdq d @ s t | e @ p q frq s> p 5 ] | t> q 5 ]./ hqwrqfhv sru frurodulr 4145 y,/ vh wlhqh= d?e / s t ? p q / sq ? tp 4 4Hq dojxqrv wh{wrv vh suhvhqwd ho whruhpd 4146 frpr d{lrpd gh wulfrwrpðd +D{lrpd lll, | frurodulr 4 ly, | y, frpr D{lrpd l, | D{lrpd ll, uhvshfwlydphqwh1
4161 D[LRPDV GH RUGHQ 6< Sru hmhpsor/ rughqhprv gh phqru d pd|ru orv vljxlhqwhv qýphurv udflrqdohv d @ 4> 64 e @ 46 45 f @ 58 3>45 g @ 4> 7 h @ 45 ; H{suhväqgrorv frpr iudfflrqhv udflrqdohv | uhgxflhqgr whqhprv> d @ 464 433 e @ 46 45 f @ 458 9 g @ : 8 h @ 6 5 = Hqwuh orv qhjdwlyrv whqhprv 6 5 ? : 8 sxhv 6 8 ? 5 := Hqwuh orv srvlwlyrv 46 45 ? 464 433 sxhv 46 433 ? 45 464 | 464 433 ? 458 9 sxhv 464 9 ? 458 433> hq frqvhfxhqfld 6 5 ? : 8 ? 46 45 ? 464 433 ? 458 9 Sxhvwrv hq xqd uhfwd vh ylvxdol}d vx xelfdflöq uhodwlyd/ frpr sxhgh yhu hq od uhfwd fijxud txh vljxh= D frqwlqxdflöq revhuyduhprv txh hqwuh grv udflrqdohv vlhpsuh kd| do phqrv xq udflrqdo1 Sursrvlflöq 4147 Hqwuh grv udflrqdohv h{lvwh do phqrv xq udflrqdo1 Ghprvwudflöq= Vhdq s> t 5 T | s?t/ sru ghprvwudu txh h{lvwh u 5 T wdo txh s?u?t Xvdqgr ho frurodulr 4145 ly, whqhprv s?t , s . s?s . t , 5s?s . t , s ? s.t 5 Dghpäv vxpdqgr dkrud t d dperv odgrv rewhqhprv s?t , s . t?t . t , s . t ? 5t , s.t 5 ? t Sru or wdqwr h{lvwh u @ s.t 5 5 T wdo txh s?u?t1 41614 Udð} q0ìvlpd D frqwlqxdflöq gduhprv xqd sursrvlflöq txh jdudqwl}d od qr h{lvwhqfld gh xq qýphur uhdo frq od surslhgdg lqglfdgd päv dedmr1 Sursrvlflöq 4148 Vhdq q qýphur qdwxudo sdu | d ? 3/ hqwrqfhv qr h{lvwh xq qýphur uhdo e wdo txh fxpsod od ljxdogdg d @ eq
73 FKDSWHU 41 ORV QÝPHURV UHDOHV Ghprvwudflöq= Vxsrqjdprv txh h{lvwh e 5 U wdo txh d @ eq shur q hv sdu/ sru or wdqwr q @ 5n sdud dojýq n 5 ]. oxhjr d @ e5n @ +en, 5 Dkrud elhq1 Vl e @ 3 hqwrqfhv d @ 31 Vl e 9@ 3/ hqwrqfhv +en,5 A 3/ hv ghflu d A 31 Hq dperv fdvrv h{lvwh frqwudglfflöq sxhv d ? 31 Hq frqvhfxhqfld qr h{lvwh e 5 U wdo txh d @ eq1 Sru ho frqwudulr/ xq uhvxowdgr lpsruwdqwh txh vh rewlhqh sdud d A 3 hv ho vljxlhqwh= Sursrvlflöq 4149 Gdgr xq qýphur uhdo srvlwlyr d | xq qýphur qdwxudo q vlhpsuh h{lvwh xq ýqlfr qýphur uhdo srvlwlyr e wdo txh d @ eq Or dqwhulru vh hvfuleh päv irupdophqwh frpr= +;d 5 U.,+;q 5 Q,+<$e 5 U.,+eq @ d, Hvwh qýphur uhdo srvlwlyr vh oodpd udð} q 0ìvlpd gh d | vh ghqrwd sru e @= s q d r e @= d 4 q Dghpäv/ vl p 5 ] vh ghfiqh d p q @= s q dp sdud d 3 frq q 5 Q Vrodphqwh sdud d 3 vh sxhgh dvhjxudu txh= d p q @ +dp, 4 q @ +d 4 q , p @ + s q d, p Dkrud vl q hv xq qýphur lpsdu/ hvwd ghfiqlflöq vh sxhgh h{whqghu d edvhv qhjdwlydv/ dghpäv s q d @= s q d +414:, frq d A 3 H{dpsoh 4 Hmhpsor 65 41 H{dpsoh 5 +d, s 7@5 sxhv 7@551 Qrwh txh 55 @ +5, s 5 @ 71 Vlq hpedujr 7@5 | s7 9@ 5/ +hv ghflu/ qr vh wlhqh s7 @ 5 , +e, s6 5: @ 6 sxhv s6 5: @ s6 5: @ 6 +f, s4 { 5 U vl 4 { 3 hv ghflu vl { 41 Vlq hpedujr s6 4 { 5 U/ sdud wrgr { 5 U1 Hmhpsor 66 Ghwhuplqhprv sdud txì ydoruhv gh d od h{suhvlöq sd : . s: d d : uhsuhvhqwd xq qýphur uhdo1 Vroxflöq= Od h{suhvlöq uhsuhvhqwd xq qýphur uhdo vl> d : 3 a : d 3 a d : 9@ 3 oxhjr d : a d : a d 9@ : hv ghflu/ d 5 ^:>4^_` 4> :` _ +U i:j, @ ! hqwrqfhv sdud qlqjýq uhdo d od h{suhvlöq gh duuled uhsuhvhqwd xq uhdo1
4161 D[LRPDV GH RUGHQ 74 Hmhuflflr 67 Ghwhuplqh ho frqmxqwr uhvwulfflöq | ho frqmxqwr vroxflöq gh od hfxdflöq 6 { @ s { 4 Vroxflöq= Od h{suhvlöq 6{ hvwä ghfiqlgd sdud wrgr { hq U | s{ 4 hvwä ghfiqlgd hq U vöor sdud { 4/ oxhjr ho frqmxqwr uhvwulfflöq hv U @ ^4>4^1 Frpr s{ 4 3 hqwrqfhv sdud txh h{lvwdq { hq U txh fxpsodq frq od ljxdogdg ghehprv h{ljlu txh 6 { 3/ hv ghflu/ { 6/ shur frpr { 5 ^4>4^ hqwrqfhv sdud { 5` 4> 6` _ ^4>4^@ ^4> 6` srghprv hohydu do fxdgudgr1 +6 {,5 @ +s{ 4,5 +, < 9{ . {5 @ { 4 +, {5 :{ . 43 @ 3 +, +{ 8,+{ 5, @ 3 +, { @ 8 b { @ 5 shur { 5 ^4> 6`/ oxhjr ho frqmxqwr vroxflöq hv V @ i5> 8j _ ^4> 6` @ i5j Qrwhprv txh vl hohydprv do fxdgudgr vlq kdfhu od uhvwulflöq 6 { 3/ kxelìudprv rewhqlgr wdpelìq { @ 8 frpr vroxflöq gh BB/ | do uhhpsod}du hvwh qýphur hq od hfxdflöq/ rewhqguðdprv 6 8 @ s 8 4 5 @ s 7 or txh hv xqd frqwudglfflöq1 Sru hvr qr roylghprv txh sdud od uhvroxflöq gh xqd hfxdflöq phgldqwh od hohydflöq d dojýq h{srqhqwh sdu ghehprv kdfhu odv uhvwulfflrqhv qhfhvduldv sdud dvhjxudu txh dperv plhpeurv gh od ljxdogdg whqjdq ho plvpr vljqr1 Hmhuflflr 68 Ghwhuplqdu ho frqmxqwr uhvwulfflöq | ho frqmxqwr vroxflöq gh od hfxdflöq s 8{ .4@ s 5{ 5.5 Vroxflöq= Od h{suhvlöq s8{ . 4 hvwä ghfiqlgd hq U vl 8{ . 4 3 | s5{ 5.5 hvwä ghfiqlgd hq U vl 5{ 5 3 oxhjr ho frqmxqwr uhvwulfflöq gh od hfxdflöq hv U @ i{ 5 U = 8{ . 4 3 a 5{ 5 3j @ ^4>4^ Dkrud uhvroydprv od hfxdflöq s 8{ .4@ s 5{ 5.5 Frqvlghudqgr { hq ho frqmxqwr uhvwulfflöq hohyhprv do fxdgudgr +sxhv dperv plhpeurv gh od ljxdogdg vrq srvlwlyrv, + s8{ . 4,5 @ +s 5{ 5,5 . 7s5{ 5.7 or txh htxlydoh d 6{ 4@7s5{ 5 frpr { 4 hqwrqfhv 6{ 4 A 3 | hqwrqfhv srghprv hohydu do fxdgudgr qxhydphqwh rewhqlhqgr> +6{ 4,5 @ 49+5{ 5, +, <{5 9{ . 4 @ 65{ 65 +, <{5 6;{ . 66 @ 3 +, <+{ 6,+{ 44 < ,@3 +, { @ 6 b { @ 44 < oxhjr V @ i6> 44 < j _ ^4>4^@ i6> 44 < j
75 FKDSWHU 41 ORV QÝPHURV UHDOHV Hmhuflflr 69 Ghwhuplqdu od yhudflgdg r idovhgdg gh od vljxlhqwh sursrvlflöq= Vl { vdwlvidfh od hfxdflöq s8{ .4@ s5{ 5.5 hqwrqfhv ho ydoru qxpìulfr gh s8{ . 4 s5{ 5 hv 5 ö elhq 6 Vroxflöq= Sru ho hmhpsor dqwhulru/ vdehprv txh orv uhdohv txh vdwlvidfhq od hfxdflöq vrq { @ 6 | { @ 44 < 1 Sru rwud sduwh/ vl { @ 6 hqwrqfhv s8{ . 4 s5{ 5 @ s8 6.4 s5 6 5 @ 7 5 @ 5 Vl { @ 44 < hqwrqfhv s8{ . 4 s5{ 5 @ t 8 44 < . 4 t 5 44 < 5 @ ;@6 5@6 @ 7 hq frqvhfxhqfld od dfiupdflöq hv idovd sxhv ho ydoru qxpìulfr gh s8{ . 4 s5{ 5 hv 5 ö elhq 7 Revhuydflöq= Sdud dtxhoodv h{suhvlrqhv txh wlhqhq vljqlfifdgr hq U / odv surslhgdghv gh srwhqfldv wdpelìq vh fxpsohq sdud h{srqhqwhv udflrqdohv1 Dvðsru hmhpsor= Vl d> e 5 U. q 5 ]> p 5 ] i3j hqwrqfhv +de, q p @ d q p e q p Hv lpsruwdqwh kdfhu qrwdu txh s de @ sd s e qr vh fxpsoh45 hq jhqhudo sdud d> e 5 U1 s de wlhqh vljqlfifdgr hq U sdud de 3> hv ghflu/ +d 3 a e 3, b +d 3 a e 3, hq fdpelr sd s e wlhqh vljqlfifdgr hq U vöor vl d 3 | e 3 Dvð/ sdud uhvroyhu xqd lqhfxdflöq gho wlsr s{+{ 4, 4> ghehprv whqhu fxlgdgr gh qr frphwhu ho vljxlhqwh huuru= s{+{ 4, 4 +, s{ s{ 4 4 Hmhuflflr 6: Ghwhuplqh vl odv vljxlhqwhv dfiupdflrqhv vrq yhugdghudv r idovdv 41 i{ 5 U = s{+{ 4, 4j @ i{ 5 U = s{ s{ 4 4j 51 i{ 5 U = t { 4{ 4j @ i{ 5 U = s{ s4 {j 61 +;{ 5 U.,+ s 4 { 4, 4 5Ghwhfwh ho huuru ' I ' sE3ffiE3ffi ' EI3ffi2 ' 31
4161 D[LRPDV GH RUGHQ 76 Vroxflöq= 41 Idovr1 H{lvwh { @ 4 5 U wdo txh s4+4 4, @ s5 4 oxhjr 4 5 i{ 5 U = s{+{ 4, 4j shur 4 95 i{ 5 U = s{ s{ 4 4j sxhv 4 95 U grqgh U @ ^4>4^ hv ho frqmxqwr uhvwulfflöq gho frqmxqwr i{ 5 U = s{ s{ 4 4j 51 Yhugdghur1 Uhvroydprv u { 4 { 4 +414;, U @ i{ 5 U = { 4 { 3 a 4 { 9@ 3j @ ^3> 4^ Frqvlghudqgr { 5 U | hohydqgr do fxdgudgr rewhqhprv od lqhfxdflöq htxlydohqwh { 4 { 4 Frpr 4 { A 3 sdud { 5 U @ ^3> 4^ hqwrqfhv { 4 { 4 +, { 4 { gh grqgh { 4 5 Dvð/ { 5` 4> 4@5` _ ^3> 4^@ ^3> 4@5` Sru or wdqwr/ ho frqmxqwr vroxflöq gh od lqhfxdflöq 414; hv V @ ^3> 4@5` Dkrud uhvroydprv s{ s4 { +414<, Odv h{suhvlrqhv s{ | s4 { hvwäq ghfiqlgdv hq U vl | vöor vl { 3 a 4 { 3 oxhjr U @ ^3> 4` | frpr odv lqhfxdflrqhv 414; | 414< vrq htxlydohqwhv hq ^3> 4^ hqwrqfhv vxv frqmxqwrv vroxflrqhv vrq ljxdohv1 61 Idovr1 H{lvwh { @ 4 7 5 U. wdo txh s 4 { @ 5 9 41 Revhuydflöq= Qr frqixqglu or txh vljqlfifdq odv vljxlhqwhv h{suhvlrqhv= l, Od h{suhvlöq +;{ 5 U.,+ 4 s{ 4, hv xqd sursrvlflöq txh dfiupd txh sdud wrgr { hq orv uhdohv srvlwlyrv vh fxpsoh txh s 4 { 4/ or fxdo khprv ylvwr txh hv idovr1 ll, Hq fdpelr od h{suhvlöq i{ 5 U. = 4 s{ 4j lqglfd ho frqmxqwr gh orv uhdohv srvlwlyrv txh fxpsoh frq od lqhfxdflöq s 4 { 4/ hv ghflu/ ho frqmxqwr ^4>4^1
77 FKDSWHU 41 ORV QÝPHURV UHDOHV Qrwhprv txh od ljxdogdg s d5 @ d qr vh fxpsoh sdud wrgr d 5 U > sxhvwr txh d5 hv srvlwlyr | od udð} sdu gh xq qýphur srvlwlyr hv xq qýphur srvlwlyr1 Sru hmhpsor/ vl d @ 5 s d5 @ s+5,5 @ s 7@5 9@ 5 Vhd d 3 sru BB> s d5 @ d 5 5 @ d Vhd d ? 3> frpr d A 3 | sru BB vh wlhqh txh= s d5 @ s+d,5 @ +d, 5 5 @ d Hq frqvhfxhqfld s d5 @ d ;d 5 U. s 3 d5 @ d ;d 5 U Or txh hvfuleluhprv s d5 @ ffl d vl d 3 d vl d ? 3 +4153, Hmhuflflr 6; Hq edvh d or dqwhulru ghwhuplqh vl odv vljxlhqwhv dfiupdflrqhv vrq yhugdghudv r idovdv 41 +;d 5 U,+s7 d9 @ s d6,1 51 Vhd d 5 U./ hqwrqfhv i{ 5 U = {5 @ dj @ isd> sdj1 61 i{ 5 U = {5 @ 7j @ i{ 5 U = { @ s 7j1 71 Vhdq {> | 5 U wdo txh {5 . |5 5{| @ 7 hqwrqfhv { | @ 51 Vroxflöq= 41 Idovr1 Vöor sdud d 3/ vh wlhqh txh= s7 d9 @ d 9 7 @ d 6 5 @ s d61 Sdud d ? 3/ s d6 95 U | s7 d9 5 U/ | od ljxdogdg qr vh gd sdud wrgr d 5 U 51 Yhugdghur1 {5 @ d / {5 d @ 3 / +{ sd,+{ . sd,@3 / { sd @ 3 b { . sd @ 3 / { @ sd b { @ sd Oxhjr i{ 5 U = {5 @ dj @ isd> sdj Qrwh txh qr hv or plvpr uhvroyhu od hfxdflöq {5 @ d txh uhvroyhu od hfxdflöq { @ sd1 61 Idovr1 Sru or dqwhulru hv fodur txh i{ 5 U = {5 @ 7j @ i5> 5j 9@ i{ 5 U = { @ s 7j @ i5j 71 Idovr1 {5 . |5 5{| @ 7 +, +{ |,5 @ 7 +, +{ |,5 7@3 +, ++{ |, 5,++{ |, . 5, @ 3 +, { | @ 5 b { | @ 5 Hq sduwlfxodu/ vh frqvlghudprv { @ 5> | @ 7 hqwrqfhv {5 . |5 5{| @ 7
4161 D[LRPDV GH RUGHQ 78 shur { | @ 5 9@ 5 Hq jhqhudo vl d 5 U. +{ |, 5 @ d hqwrqfhv { | @ sd b { | @ sd Yhuhprv d frqwlqxdflöq or ýwlo txh vrq hvwdv surslhgdghv sdud od uhvroxflöq gh vlvwhpdv gh hfxdflrqhv 41615 Vlvwhpdv gh Hfxdflrqhv qr Olqhdohv Uhvroyhu xq vlvwhpd gh hfxdflrqhv hq q yduldeohv {4> {5> {q 5 U hv hqfrqwudu odv q0xsod +{4> {5> > {q, txh vdwlvidfhq fdgd xqd gh odv hfxdflrqhv vlpxowäqhdphqwh1 Or hvhqfldo hq od uhvroxflöq gh vlvwhpdv qr olqhdohv frpr orv vljxlhqwhv/ hv ghwhuplqdu xqd idfwrul}dflöq gh od irupd de @ 3 | dqdol}du orv fdvrv d @ 3 | e @ 31 Hmhpsor 6< Uhvroydprv {5 . |5 @ 5+{| . 5, { . | @ 9 Vroxflöq= Wudqvirupdqgr ho vlvwhpd gdgr hq od irupd htxlydohqwh +4, +{ |,5 @ 7 +5, { . | @ 9 Vh rewlhqh gh +4, orv vljxlhqwhv fdvrv sru dqdol}du { | @ 5 b { | @ 5 l, Vl { | @ 5/ hqwrqfhv vh wlhqh { | @ 5 { . | @ 9 gh grqgh { @ 7 | | @ 5 ll, Vl { | @ 5 vh wlhqh { | @ 5 { . | @ 9 gh grqgh { @ 5 | | @ 7 Sru or wdqwr h{lvwhq grv vroxflrqhv +{> |, @ +7> 5,> +{> |, @ +5> 7, Hmhuflflr 73 Uhvroyhu +4, 5+{5 . |5,@8{| +5, 7+{ |, @ {| Vroxflöq= Vh wlhqh= +4, +, {5 . |5 @ 8 5 {| +, {5 5{| . |5 @ 8 5 {| 5{| +, +{ |, 5 @ 4 5 {| +4154,
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