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4 VNXSRYL1 IXQNFLMH1 UHDOQL EURMHYL1 6414 RVQRYH PDWHPDWLFNH ORJLNH 1 1 1 1 1 1 1 1 1 1 1 1 1 6

41414 Orjlfnl vxg 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 641415 Rshuludqmh vxgrylpd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 741416 Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9

415 VNXSRYL L UHODFLMH 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 941514 Vnxs 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 941515 Rshuludqmh vnxsrylpd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :41516 Uhodflmh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;41517 Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 43

416 IXQNFLMH 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4441614 Ixqnflmd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4441615 Ixqnflmvnd nrpsr}lflmd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4541616 Lqyhu}qd ixqnflmd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4641617 Nduglqdoql eurm 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4741618 Ymh}eh 1 1 1 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

417 UHDOQL EURMHYL 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4941714 Sulurgql l flmhol eurmhyl 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4941715 Udflrqdoql eurmhyl 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5641716 Dsvroxwqd yulmhgqrvw uhdoqrj eurmd 1 1 1 1 1 1 1 1 1 1 1 5<41717 Srwhqfludqmh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6341718 Nrpelqdwrulfnh rvqryh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6641719 Ymh}eh 1 1 1 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<

418 NRPSOHNVQL EURMHYL 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7441814 Srmdp l rvqryqh rshudflmh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7541815 Jhrphwulmvnl sulnd} nrpsohnvqrj eurmd 1 1 1 1 1 1 1 1 1 7641816 Wuljrqrphwulmvnl }dslv nrpsohnvqrj eurmd 1 1 1 1 1 1 1 7741817 Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 79

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51414 Pdwulfd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7<51415 Ghwhuplqdqwd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8651416 Pdwulflq udqj 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9351417 Umhµdydqmh vxvwdyd olqhduqlk mhgqdg}ded 1 1 1 1 1 1 1 1 9651418 Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9;

515 YHNWRUVND DOJHEUD 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9<51514 Xvpmhuhqd gx}lqd l yhnwru 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9<51515 Yhnwru x suryrnxwqrp nrruglqdwqrp vxvwdyx 1 1 1 1 1 :651516 Yhnwruvnd pqr}hqmd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ::51517 Ymh}eh 1 1 1 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

516 DQDOLWLFND JHRPHWULMD 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;651614 Sudydf x survwrux l x udyqlql 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;651615 Udyqlqd x survwrux 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;851616 Ph¡xrgqrvl wrfdnd/ sudydfd l udyqlqd 1 1 1 1 1 1 1 1 1 ;:51617 Qhnh nulyxomh l sorkh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;<51618 Sroduql/ flolqgulfql l vihuql nrruglqdwql vxvwdyl 1 1 1 1 1 <951619 Ymh}eh 1 1 1 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 NRQYHUJHQFLMD L QHSUHNLGQRVW 434614 HOHPHQWDUQH IXQNFLMH 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 434

61414 ]dgdydqmh ixqnflmd l} U x U 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 43461415 Joredoqd vyrmvwyd uhdoqlk ixqnflmd 1 1 1 1 1 1 1 1 1 1 1 43761416 Rvqryqh hohphqwduqh ixqnflmh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 43961417 Ud}uhged hohphqwduqlk ixqnflmd 1 1 1 1 1 1 1 1 1 1 1 1 1 44561418 Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 447

615 NRQYHUJHQFLMD QL]RYD L UHGRYD 1 1 1 1 1 1 1 1 1 1 1 44861514 Ql} uhdoqlk eurmhyd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 44861515 Vyrmvwyd nrqyhujhqwqlk +srg,ql}ryd 1 1 1 1 1 1 1 1 1 1 44;61516 Uhg uhdoqlk eurmhyd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 45661517 Qhnrolnr nulwhulmd }d nrqyhujhqwqrvw uhdoqrj uhgd 1 1 1 45861518 Ixqnflmvnl ql} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 45<61519 Ixqnflmvnl uhg 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4646151: Ymh}eh 1 1 1 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

616 QHSUHNLGQH IXQNFLMH 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 46861614 Judqlfqd yulmhgqrvw x ehvnrqdfqrvwl 1 1 1 1 1 1 1 1 1 1 46861615 Judqlfqd yulmhgqrvw x wrfnl 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 46961616 Qhsuhnlgqrvw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 47361617 Vyrmvwyd qhsuhnlgqh ixqnflmh qd vhjphqwx 1 1 1 1 1 1 1 47861618 Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 483

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7 LQILQLWH]LPDOQL UDFXQ 484714 GHULYDFLMD 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 484

71414 Ghulydeloqrvw l qmh}lqr }qdfhqmh 1 1 1 1 1 1 1 1 1 1 1 1 1 48471415 Ghulydflmh hohphqwduqlk ixqnflmd 1 1 1 1 1 1 1 1 1 1 1 1 48871416 Glihuhqflmdo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 49371417 Rvqryql whruhpl glihuhqflmdoqrjd udfxqd 1 1 1 1 1 1 1 1 49771418 Wd|oruryd irupxod 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 49;71419 Ghulyludqmh ixqnflmvnrj uhgd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:57141: Rguh¡lydqmh ixqnflmvnrj wlmhnd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:77141; Ymh}eh 1 1 1 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

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71514 Srmdp l rvqryqd vyrmvwyd qhrguh¡hqrj lqwhjudod 1 1 1 4<371515 Rvqryqh lqwhjudflmvnh phwrgh1 1 1 1 1 1 1 1 1 1 1 1 1 1 4<771516 Lqwhjuludqmh qhnlk hohphqwduqlk ixqnflmd 1 1 1 1 1 1 1 1 4<:71517 Lqwhjuludqmh ixqnflmvnrj uhgd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 53671518 Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 538

716 RGUHÓHQL LQWHJUDO 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 53871614 Srmdp l rvqryqd vyrmvwyd rguh¡hqrj lqwhjudod 1 1 1 1 1 53971615 Qhnl suleol}ql lqwhjudflmvnl srvwxsfl1 1 1 1 1 1 1 1 1 1 1 54771616 Qhsudyl lqwhjudo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54<71617 Qhnrolnr sulpmhqd rguh¡hqrj lqwhjudod 1 1 1 1 1 1 1 1 1 55671618 Ymh}eh1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 565

8 IXQNFLMD YL�H YDULMDEOD 568814 IXQNFLMH L] U6 X U 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 568

81414 ]dgdydqmh vndoduqlk ixqnflmd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 56881415 Judqlfqd yulmhgqrvw l qhsuhnlgqrvw 1 1 1 1 1 1 1 1 1 1 1 56;81416 Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 576

815 GLIHUHQFLUDQMH VNDODUQLK IXQNFLMD 1 1 1 1 1 1 1 57781514 Sduflmdoqh ghulydflmh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57781515 Glihuhqflmdo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 57981516 Sduflmdoqh ghulydflmh ylµlk uhgryd 1 1 1 1 1 1 1 1 1 1 1 1 58581517 Hj}dnwqd glihuhqflmdoqd irupd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 58:81518 Wd|oruryd irupxod 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 58<81519 Orndoqh hnvwuhpqh yulmhgqrvwl 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5958151: Revwrmqrvw lpsolflwqh ixqnflmh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5998151; Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 59<

816 LQWHJULUDQMH VNDODUQLK IXQNFLMD 1 1 1 1 1 1 1 1 1 5:481614 Ylµhvwuxnl lqwhjudo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5:481615 L}udfxqdydqmh l sulpmhqd ylµhvwuxnrj lqwhjudod 1 1 1 1 1 5:681616 Lqwhjudo rylvdq r sdudphwux 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5:<

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9 XYRG X YHNWRUVNX DQDOL]X 5;:914 YHNWRUVNH IXQNFLMH 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5;:

91414 Qhsuhnlgqrvw yhnwruvnh ixqnflmh 1 1 1 1 1 1 1 1 1 1 1 1 1 5;:91415 Glihuhqflmdeloqrvw yhnwruvnh ixqnflmh 1 1 1 1 1 1 1 1 1 1 5;<91416 Lqwhjudo yhnwruvnh ixqnflmh mhgqh ydulmdeoh 1 1 1 1 1 1 1 5<691417 Ymh}eh 1 1 1 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<7

915 XYRG X WHRULMX R SROMLPD 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5<891514 Vndoduqr l yhnwruvnr sromh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5<891515 Judglmhqw/ glyhujhqflmd l urwdflmd 1 1 1 1 1 1 1 1 1 1 1 1 5<;91516 Xvpmhuhqd ghulydflmd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 63591517 Qhnd srvheqd yhnwruvnd sromd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 63891518 Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 63<

916 NULYXOMQL LQWHJUDO 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 63<91614 Nulyxomd l qmh}lqr xvpmhuhqmh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 63<91615 Nulyxomql lqwhjudo suyh yuvwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 64491616 Nulyxomql lqwhjudo guxjh yuvwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 64691617 Nulyxomql lqwhjudo x srwhqflmdoqrp sromx 1 1 1 1 1 1 1 1 64991618 Juhhqryd irupxod 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 64<91619 Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 655

917 SOR�QL LQWHJUDO 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 65691714 Jodwnd sorkd l qmh}lqd sorµwlqd 1 1 1 1 1 1 1 1 1 1 1 1 1 65691715 Sorµql lqwhjudo suyh yuvwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 65991716 Sorµql lqwhjudo guxjh yuvwh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 65;91717 Rvwurjudgvnl0Jdxvvryd irupxod 1 1 1 1 1 1 1 1 1 1 1 1 1 66591718 Vwrnhvryd irupxod 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 66791719 Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 66:

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:15 GLIHUHQFLMDOQH MHGQDG]EH SUYRJD UHGD 1 1 1 1 67::1514 Glihuhqflmdoqd mhgqdg}ed v rgmhomlylp ydulmdeodpd 1 1 1 67;:1515 Krprjhqd glihuhqflmdoqd mhgqdg}ed 1 1 1 1 1 1 1 1 1 1 1 683:1516 Olqhduqd glihuhqflmdoqd mhgqdg}ed 1 1 1 1 1 1 1 1 1 1 1 1 684:1517 Hj}dnwqd glihuhqflmdoqd mhgqdg}ed 1 1 1 1 1 1 1 1 1 1 1 685:1518 Vxvwdy rg gylmx relfqlk glihuhqflmdoqlk mhgqdg}ded 1 1 687:1519 Ymh}eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 687

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V+EVF, > +D

WE,WF @ D

W+EWF,>

+ll, DVE @ E

VD > D

WE @ E

WD>

+lll, DV+EWF, @ +D

VE,W_+D

VF,>

DW+EVF, @ +D

WE,V+DWF,>

+ly, DV> @ D> D

W> @ >>

+y, DVX @ X > D

WX @ D>

+yl, DVD @ D> D

WD @ D>

+yll, +DS,S @ D>+ylll, +D

VE,S @ DS

WES> +D

WE,S @ DS

VES1

51 Qhnd vx D�> E� � [ l D2> E2 � [1 Grnd}dwl gd mh+l, D� � +D2

VE2, @ +D� �D2,

V+D� �E2,>

+D�

VE�,�D2 @ +D� �D2,

V+E� �D2,>

+ll, D� � +D2

WE2, @ +D� �D2,

W+D� �E2,>

+D�

WE�,�D2 @ +D� �D2,

W+E� �D2,1

4161 IXQNFLMH 44

61 Grnd}dwl gd mh/ }d vydnl vnxs [/ uhodflmd rguh¡hqd rgqrvrp �elwl srg0vnxs� +�, qd sduwlwlyqrp vnxsx 5f mhgqd uhodflmd sduflmdoqrj xuh¡dmd171 Qhnd mh +[>�, +sduflmdoqr, xuh¡hq vnxs1 Mh ol suhvmhn gydmx vhjphqdwd/lqwhuydod/ � � � x [ rshw vhjphqw/ lqwhuydo/ � � � B �wr vh pr}h uh�fl r xqlml lud}olfl gydmx vhjphqdwd/ lqwhuydod/ � � � B

�%1 &��'�(�

�%1%� &,/� ��

Ixqnflmd mhvw mhgdq rg qdmyd}qlmlk pdwhpdwlfnlk srmpryd1 Gh�qludw �fhprjd vdvylp rs�fhqlwr/ wm1 qd vnxsryqrm ud}lql1

Gh�qlflmd 41614 Srg ixqnflmrp srgud}xplmhydpr vydnl xuh¡hql wurvorj

+[> i> \ ,/ sul fhpx vx [ l \ vnxsryl/ d i mh elor nrmh sudylor sr nrmhpx

vh vydnrp hohphqwx { 5 [ sulgux}xmh wrfqr mhgdq hohphqw | 5 \ 1

Xrelfdmhql }dslv }d ixqnflmx +[>i> \ , mhvw i = [ $ \ lol { :$ | @ i+{,l sulwrp fhvwr nd}hpr gd mh i ixqnflmd l} vnxsd [ x vnxs \ 1 +Ndg vx xqhnrp ud}pdwudqmx [ l \ }dgdql l qhsurpmhqmlyl/ fhvwr vh/ mhgqrvwdyqrvwludgl/ l vdpr sudylor i qd}lyd ixqnflmrp1,

Sulplmhwlpr gd l} gh�qlflmh l}udyqr surl}od}l \ 9@ >/ grn mh [ @ >prjx�fh1 +Grgdwqr pr}hpr srvwxoludwl l �sud}qx ixqnflmx� [ $ >/ suhpgdqdp wdm srmdp rygmh qh �fh wuhedwl1,

]d gdqx ixqnflmx i = [ $ \ / vnxs [ qd}lydpr gh�qlflmvnlp srgux0fmhp +lol grphqrp/ r}qdnd= Gs ,/ vnxs \ yulmhgqrvqlp srguxfmhp +lolnrgrphqrp/ r}qdnd= Us ,/ d hohphqw | @ i+{, yulmhgqrµ�fx +qd hohphqwx{, ixqnflmh i 1 Srqhndg vh nd}h gd mh { qh}dylvqd ydulmdeod +lol dujxphqw,/d | }dylvqd ydulmdeod ixqnflmh i 1 Volnrylwr vh ixqnflmx qdmfhµ�fh sulnd}xmhndr qd fuwh}x=

;

[ \ I�[�

<I

Ixqnflmlq judi mh vnxs Js @ i+{> i+{,, m { 5 [j � [ � \ 1 Sulpmhulfh+volnrylwr,=


\ I�[��[�I�[��

<

;

*I

[

; <

Uh�fl �fhpr gd mh ixqnflmd i� = [� $ \� mhgqdnd ixqnflml i2 = [2 $ \2 lslvdwl i� @ i2/ dnr mh [� @ [2/ \� @ \2 l i�+{, @ i2+{, }d vydnl { 5 [� @[21 Rflwr/ dnr mh i� @ i2 rqgd mh l i2 @ i�1

Lvwdnqlpr vdgd qhnrolnr yd}qlk sulpmhud=

41 Qhnd mh [ � \ / d l = [ $ \ qhnd mh ixqnflmd rguh¡hqd sudylorp+;{ 5 [, l+{, @ {1 Wr mh w}y1 lqnox}lmd +lol xodjdqmh, nrmx �fhpr fhvwrr}qdfdydwl v l = [ /$ \ 1 Srvhelfh/ dnr mh [ @ \ grelydpr w}y1 lghq0

wlfnx ixqnflmx +lol lghqwlwhwx, 4t = \ $ \ / 4t +|, @ |1 +Sulplmhwlpr gdvx/ rs�fhqlwr/ lqnox}lmd l lghqwlwhwd ph¡xvreqr ud}olflwh/ mhu vx lp grphqhud}olflwh1,

51 Qhnd vx ixqnflmh sf = [ � \ $ [ l st = [ � \ $ \ gh�qludqhuhgrp sudylolpd sf+{> |, @ { l st +{> |, @ |/ +{> |, 5 [ � \ 1 Wr vx w}y1surmhnflmh l} gluhnwqrjd surgxnwd [ � \ qd idnwruh [ l \ uhgrp1

61 Qhnd mh X qhsud}dq vnxs/ d [ @ 5L sduwlwlyql vnxs rg X 1 Wdgdvx elqduqh rshudflmh

V/W

l " ixqnflmh l} [ �[ x [1 +Vdvylp rs�fhqlwr/elqduqd rshudflmd qd vnxsx [ mh vydnd ixqnflmh l} [ �[ x [1,

�%1%- &,/� ��+�� #0.#��

Gh�qlflmd 41615 Qhnd vx i = [ $ \ l j = \ $ ] ixqnflmh1 Srg nrp0

sr}lflmrp ixqnflmh i v ixqnflmrp j srgud}xplmhydpr ixqnflmx k = [ $ ]

}dgdqx sudylorp k+{, @ j+i+{,, }d vydnl { 5 [1 ]dslvxmhpr wr ndr k @ j�ilol nud�fh k @ ji 1

Ixqnflmvnx nrpsr}lflmx loxvwulud vomhgh�fl fuwh}=

I J

K JI

[ \ I�[�] J�I�[��

; < =

Wyugqmd 41614 Ixqnflmvnd nrpsr}lflmd mh dvrflmdwlyqd/ wm1 dnr vx i = [ $\ / j = \ $ ] l k = ] $Z ixqnflmh/ rqgd mh k+ji, @ +kj,i 1

4161 IXQNFLMH 46

Grnd}1 ]d vydnl { 5 [ mh +k+ji,,+{, @ k++ji,+{,, @ k+j+i+{,,, @+kj,+i+{,, @ ++kj,i,+{,/ gdnoh/ k+ji, @ +kj,i 1

Wyugqmd 41615 ]d vydnx ixqnflmx i = [ $ \ mh i4f @ i l 4t i @ i / jgmh

vx 4f l 4t sulsdgqh lghqwlwhwh1 +Grnd} suhsxµwdpr flwdwhomx1,

Qhnd vx [ � � [/ \ � � \ +srg,vnxsryl l qhnd }d ixqnflmh i � = [ � $\ � l i = [ $ \ yulmhgl +;{� 5 [ �, i �+{�, @ i+{�,1 Wdgd nd}hpr gd mhi � vx}hqmh +lol uhvwulnflmd, ixqnflmh i / rgqrvqr/ gd mh i surµluhqmh +lolhnvwhq}lmd, ixqnflmh i �1 Dnr mh/ srvhelfh/ \ � @ \ rqgd slµhpr i � @ i mf� 1Sulplmhwlpr gd mh/ }d vydnx ixqnflmx i = [ $ \ / vx}hqmh qd vydnl srgvnxs[ � � [ mhgqr}qdfqr rguh¡hqd wlp srgvnxsrp1 Ph¡xwlp/ dqdorjqd wyugqmd}d surµluhqmh qlmh lvwlqlwd1

�%1%1 �/$��/� 4,/� ��

Gh�qlflmd 41616 Qhnd vx gdql ixqnflmd i = [ $ \ l srgvnxsryl D �[>E � \ 1 Volnrp vnxsd D sr ixqnflml i vpdwudpr vnxs i ^D` @ i| @i+{, m { 5 Dj � \ 1 X voxfdmx D @ [ l i ^[` @ i|j +mhgqrfodq vnxs,/ixqnflmx i qd}lydpr nrqvwdqwqrp ixqnflmrp +lol nrqvwdqwrp, x hoh0

phqw | 5 \ l r}qdfxmhpr i @ f+/ d x voxfdmx i ^[` @ \ 0 vxumhnwlyqrp

ixqnflmrp +lol vxumhnflmrp lol suhvolndydqmhp �qd�,1Reudwqrp volnrp +lol ruljlqdorp, vnxsd E sr ixqnflml i vpdwudpr vnxs

i3�Ê` @ i{ m i+{, 5 Ej � [1 +Rsuh}$ X r}qdfl i3�Ê` glr i3� qh

r}qdfxmh ixqnflmx +l} \ x [,1 Rydm i3� vh lsdn pr}h lqwhusuhwludwl ndrixqnflmd/ dol l} \ x 5f / vwdyomdmx�fl | :$ i3�+|, @ i3�î|j`1 Qdpd �fh/ lqdfh/x rylp suhgdydqmlpd i3� vox}lwl }d r}qdfdydqmh w}y1 lqyhu}qh ixqnflmh 0y1 Whruhp 416141, Dnr mh +;| 5 \ , i3�î|j` @ i{j mhgqrfodq srgvnxs

rg [/ rqgd nd}hpr gd mh i lqmhnwlyqd ixqnflmd +lol lqmhnflmd lol �404�0

suhvolndydqmh,1

Sulpmhu 41614 Mhgqrvwdyqr mh surymhulwl gd mh vydnd lqnox}lmd l = [ /$ \

lqmhnflmd/ d vydnd surmhnflmd sf+st , = [ � \ $ [+\ , vxumhnflmd1

Gh�qlflmd 41617 Uh�fl fhpr gd mh ixqnflmd i = [ $ \ elmhnflmd +lol rer0vwudqr mhgqr}qdfqd ixqnflmd,/ dnr mh i lqmhnflmd l vxumhnflmd1

L} Gh�qlflmh 41616 l}udyqr volmhgl=i = [ $ \ mh lqmhnflmd / ++;{> {� 5 [, i+{, @ i+{�,, { @ {�,>i = [ $ \ mh vxumhnflmd / ++;| 5 \ ,+<{ 5 [, i+{, @ |,1

Ryr vnxsd v Gh�qlflmrp 41617 rgpdk sryodfl=i = [ $ \ mh elmhnflmd / ++;| 5 \ ,+<${ 5 [, i+{, @ |,1

Whruhp 41614 Ixqnflmd i = [ $ \ mh elmhnflmd rqgd l vdpr rqgd/ dnr

srvwrml ixqnflmd j = \ $ [ wdnyd gd mh ji @ 4f l ij @ 4t 1 Ixqnflmd j mh/

wdnr¡hu/ elmhnflmd/ mhglqvwyhqd mh/ d qd}lydpr mx lqyhu}qrp ixqnflmrp

ixqnflmh i l r}qdfxmhpr v i3�1


Grnd}1 Qhnd mh i = [ $ \ elmhnflmd1 Exgx�fl gd wdgd }d vydnl { 5 [

srvwrml wrfqr mhgdq | 5 \ }d nrml mh i+{, @ |/ prjx�fh mh gh�qludwl ixqnflmxj = \ $ [ sudylorp j+|, @ { flp mh | @ i+{,1 ]d nrpsr}lflmh grelydpr=ji+{, @ j+|, @ { @ 4f+{,/ }d vydnl { 5 [/ l ij+|, @ i+{, @ | @ 4t +|,/}d vydnl | 5 \ 1 Reudwqr/ qhnd }d i = [ $ \ srvwrml j = \ $ [ wdnrgd mh ji @ 4f l ij @ 4t 1 Dnr mh | 5 \ / x}plpr { @ j+|, 5 [/ sd

�fh elwl i+{, @ ij+|, @ 4t +|, @ |1 Suhpd wrpx/ i mh vxumhnflmd1 Dnr vx{> {� 5 [ wdnyl gd mh i+{, @ i+{�, 5 \ / rqgd mh { @ 4f+{, @ ji+{, @ji+{�, @ 4f+{�, @ {�1 Volmhgl gd mh i mh lqmhnflmd/ gdnoh l elmhnflmd1 ]dgrnd} mhglqvwyhqrvwl/ suhwsrvwdylpr gd srvwrmh gylmh ixqnflmh j> j� = \ $ [

wdnyh gd mh ji @ 4f @ j�i l ij @ 4t @ ij�1 Wdgd mh/ }d vydnl | 5 \ /j+|, @ 4f+j+|,, @ +j�i,+j+|,, @ j�++ij,+|,, @ j�+4t +|,, @ j�+|,/ gdnoh/j @ j�1

�%1%3 ��/��/� 2�#�

Gh�qlflmd 41618 Qhnd vx [>\ vnxsryl1 Uh�fl �fhpr gd mh [ hnylsrwhqwdq

+lol gd lpd mhgqdnr pqrjr hohphqdwd ndr, \ / dnr srvwrml qhnd elmhnflmd

i = [ $ \ 1

Hnylsrwhqwqrvw vnxsryd mh/ rflwr/ ud}uhgehqd uhodflmd qd nodvl4 vylk vnx0sryd/ sd vh vnxsryl vyuvwdydmx x glvmxqnwqh hnylsrwhqflmvnh ud}uhgh1 Ud}uhgnrmhpx sulsdgd vnxs [ qd}lydpr nduglqdoqlp eurmhp +lol jodyqlp eur0

mhp lol srwhqflmrp, vnxsd [ l r}qdfxmhpr v m[m lol v fdug[1

Gh�qlflmd 41619 Uh�fl �fhpr gd mh vnxs [ ehvnrqdfdq/ dnr srvwrmh sudyl

srgvnxs [ � � [ l elmhnflmd i = [ $ [ �1 Dnr vnxs [ qlmh ehvnrqdfdq/ uh�fl

�fhpr gd mh nrqdfdq1

Sulplmhwlpr gd Gh�qlflmd 41619 l}udyqr sryodfl nrqdfqrvw sud}qrjd vnxsd >1

Whruhp 41615 Qhnd vx [ l \ hnylsrwhqwql vnxsryl/ wm1 m[m @ m\ m1 Wdgd mh

[ +ehv,nrqdfdq wrfqr rqgd ndg mh \ +ehv,nrqdfdq1

Grnd}1 Sr Gh�qlflmdpd 41619 l 41618/ }d grnd} relmx hnylydohqflmd/ gr0yromqr mh grnd}dwl vdpr mhgqx lpsolndflmx1 Qhnd mh [ ehvnrqdfdq1 Wdgdsrvwrmh [ � � [ l elmhnflmd i = [ $ [ �1 Exgx�fl gd m[m @ m\ m/ srvwrml l qhndelmhnflmd j = [ $ \ 1 Qhnd mh \ � @ j^[ �` sd mh vljxuqr \ � � \ sudyl srg0vnxs1 Gh�qludmpr ixqnflmx j� = [ � $ \ � ndr vx}hqmh j mf� 1 Wdgd mh j� greurgh�qludqd l rflwr mh elmhnwlyqd1 Qdsrnrq/ nrpsr}lflmd k @ j�ij3� = \ $ \ �

mh elmhnflmd +ndr nrpsr}lflmd wulmx elmhnflmd/ y1 ¢41618 Ymh}eh/ ]dgdwdn 71,/sd mh l \ ehvnrqdfdq vnxs1

4Srmdp �vnxs vylk vnxsryd� yrgl n surwxvoryomx1 Pqr}lqd vylk vnxsryd qh wyrul vnxs

qhjr rs�fhqlwlml remhnw/ w}y1 nodvx1

4161 IXQNFLMH 48

Gh�qlflmd 4161: Uh�fl �fhpr gd mh nduglqdoql eurm vnxsd [ pdqml lol mhgqdn

rg nduglqdoqrj eurmd vnxsd \ l slvdwl m[m � m\ m +lol fdug[ � fdug\ ,/dnr srvwrml qhnd lqmhnflmd i = [ $ \ 1 Dnr mh m[m � m\ m/ d [ l \ qlvx

hnylsrwhqwql +wm1 m[m 9@ m\ m,/ uh�fl �fhpr gd mh nduglqdoql eurm rg [ pdqml rg

nduglqdoqrj eurmd rg \ l slvdwl m[m ? m\ m1

Sulplmhwlpr gd [ � \ sryodfl m[m � m\ m/ mhu mh lqnox}lmd l = [ /$ \

lqmhnwlyqd ixqnflmd1 Odnr mh surymhulwl gd mh ryd uhodflmd � uh hnvlyqd l wudq0}lwlyqd/ grn qmhqx dqwlvlphwulfqrvw mdpfl w}y1 Fdqwru0Ehuqvwhlqry whruhp/grnd} nrmhjd mh yuor qhwulylmdodq/ d gmhorplfh vh }dvqlyd qd ¢41719/ ]dgdwdn41 Suhpd wrpx/ � mh xuh¡dmqd uhodflmd qd nodvl vylk nduglqdoqlk eurmhyd1

Whruhp 41616 Dnr mh [ nrqdfdq/ d \ ehvnrqdfdq vnxs/ rqgd mh m[m ? m\ m1

Ohpd 41614 Qhnd mh [ � \ 1 Dnr mh [ ehvnrqdfdq vnxs rqgd mh wdndy l \ /

d dnr mh \ nrqdfdq vnxs rqgd mh wdndy l [1

Grnd}1 Dnr mh [ ehvnrqdfdq/ rqgd srvwrmh sudyl srgvnxs [ � � [

l elmhnflmd i = [ $ [ �1 Qhnd mh \ � glvmxqnwqd xqlmd [ �W+\ q [,/ sd mh

\ � � \ sudyl srgvnxs1 Gh�qludmpr ixqnflmx j = \ $ \ � sudylorp i qd [ �

l lqnox}lmrp qd \ q[/ wm1 +;| � { 5 [ �, j+|, @ i+{, l +;| 5 \ q[, j+|, @|1 Wulylmdoqr mh surymhulwl gd mh j elmhnflmd/ µwr }qdfl gd mh \ ehvnrqdfdqvnxs1 V guxjh vwudqh/ dnr mh \ nrqdfdq vnxs rqgd wdndy prud elwl l vydnlqmhjry srgvnxs [/ mhu el surwlyqd suhwsrvwdynd gryhod/ qd rslvdql qdflq/gr }dnomxfnd gd mh \ ehvnrqdfdq 0 surwxvoryomh1

Grnd} +Whruhpd 41616,1 Exgx�fl gd mh � xuh¡dmqd uhodflmd/ wr mh lolm[m � m\ m lol m\ m � m[m1 Suhwsrvwdylpr gd mh m\ m � m[m/ wm1 gd srvwrmlqhnd lqmhnflmd j = \ $ [1 ]d volnx j^\ ` � [ wdgd yulmhgl mj^\ `m @ m\ m1Sr Whruhpx 41615/ vnxs j^\ ` mh ehvnrqdfdq sd mh/ sr Ohpl 41614/ l vnxs [

ehvnrqdfdq 0 surwxvoryomh1 Suhpd wrpx/ prud yulmhglwl m[m � m\ m1 Qdsrnrq/exgx�fl gd mh/ sr Whruhpx 41615/ m[m 9@ m\ m/ wr prud elwl m[m ? m\ m1

�%1%5 ��2�

41 Qhnd mh i = [ $ \ ixqnflmd l qhnd vx D>E � [ l F>G � \ 1 Grnd}dwlryh irupxoh=

+l, i ^DVE` @ i ^D`

Vi Ê`> i3�^F

VG` @ i3�^F`

Vi3�^G`>

+ll, i ^DWE` � i ^D`

Wi Ê`> i3�^F

WG` @ i3�^F`

Wi3�^G`>

+lll, i ^D qE` � i ^D` q i Ê`> i3�^F qG` @ i3�^F` q i3�^G`>+ly, i î3�^F`` @ F

Wi ^[` � F> i3�î ^D`` � D1

Grnd}lpr qsu1 +ll,= Dnr mh x suyrm irupxol l} +ll, volnd i ^DWE` @ > qhpdpr

µwr grnd}lydwl1 Vwrjd/ qhnd mh | 5 i ^DWE` 9@ >1 Wdgd srvwrml { 5 D

WE }d

nrml mh i+{, @ |1 Qx/ wdgd mh { 5 D l { 5 E/ sd mh i+{, 5 i ^D` l i+{, 5 i Ê`/gdnoh/ | @ i+{, 5 i ^D`

Wi Ê`/ µwr mh l wuhedor grnd}dwl1 Gd mh prjx�fh

i ^DWE` 9@ i ^D`

Wi Ê` srnd}xmh voxfdm D

WE @ >/ d i ^D`

Wi Ê` 9@ >1


Sulpmhulfh/ dnd vx D @ id> d�j>E @ ie> e�j � [ glvmxqnwql srgvnxsryl l\ @ i|j/ rqgd mh/ }d vydnx ixqnflmx i = [ $ \ / i ^D

WE` @ i ^>` @ > l

i ^D`Wi Ê` @ \ 9@ >1

]d grnd} guxjh irupxoh l} +ll,/ qhnd mh/ qdmsulmh/ { 5 i3�^FWG`1 Wdgd

srvwrml | 5 FWG }d nrml mh i+{, @ |1 Gdnoh/ | 5 F l | 5 G l i+{, @ |/ µwr

sryodfl { 5 i3�^F` l { 5 i3�^G`/ wm1 { 5 i3�^F`Wi3�^G`1 Reudw vh pr}h

grnd}dwl flwdmx�fl suhwkrgqr }dnomxflydqmh qdwudµnh151 Qhnd vx i = [ $ \ l j = \ $ ] ixqnflmh1 Wdgd yulmhgl=

+d, Dnr mh ji = [ $ ] lqmhnflmd rqgd mh l i lqmhnflmd=+e, dnr mh ji =[ $ ] vxumhnflmd rqgd mh l j vxumhnflmd1

61 Grnd}dwl gd mh/ }d vydnx elmhnflmx i = [ $ \ / +i3�,3� @ i 171 Grnd}dwl gd mh nrpsr}lflmd ji rg elmhnflmd j l i rshw elmhnflmd181 Qhnd mh [ vnxs1 ]d vydnl srgvnxs D � [ gh�qludmpr w}y1 ndudnwhulv0wlfqx ixqnflmx "� = [ $ i3> 4j srgvnxsd D vwdyomdmx�fl

"�+{, @

�4> { 5 D

3> { 5 [ qD1

Grnd}dwl gd mh ixqnflmd I = 5f $ ii m i = [ $ i3> 4jj � i3> 4jf / rguh¡hqdsudylorp I +D, @ "� }d vydnl D 5 5f / elmhnflmd1 +Suhpd wrpx/ sduwlwlyqlvnxs 5f l vnxs i3> 4jf vylk ixqnflmd l} [ x gyrfodql vnxs vx hnylsrwhqwql1,91 Grnd}dwl gd mh m\ m � m[m flp srvwrml qhnd vxumhnflmd i = [ $ \ 1:1 Grnd}dwl gd mh/ }d vydnl vnxs [/ m[m ?

��5f��1 +Wr sryodfl gd qhpdqdmyh�fhjd nduglqdoqrj eurmd1,

�%3 ��)� *�"(��

Vnxs uhdoqlk eurmhyd lpd whphomqx xorjx x flmhorm pdwhpdwlfl/ d x pdwhp0dwlfnrm dqdol}l srvhelfh1 Xrelfdmlod vx vh gyd sulvwxsd ryrpx vnxsx= lqgxn0wlyql +nrqvwuxnwlyql,/ nrml srod}l rg Shdqrylk dnvlrpd }d vnxs sulurgqlkeurmhyd/ l dnvlrpdwvnl +ghgxnwlyql,/ nrml suhwsrvwdyomd greur sr}qdydqmhqhnlk dojheduvnlk vwuxnwxud1 Pl �fhpr vh gu}dwl +vwdulmhjd, lqgxnwlyqrjdsulvwxsd1

�%3%� ��#�/� � �� 2�#��$�

Sulkyd�fdpr/ ndr �lvnxvwyhqx flqmhqlfx�/ revwrmqrvw eduhp mhgqrj vnxsd Qnrml xgryromdyd rylp wulpd xymhwd=

+S4, Q vdgu}l eduhp mhgdq hohphqw 0 r}qdflw �fhpr jd }qdnrp4 l qd}ydwl eurmhp mhgdq +lol mhglqlfrp,>

+S5, Srvwrml lqmhnwlyqd ixqnflmd v = Q $ Q / wdnyd gd mh/ }dvydnl q 5 Q / v+q, 9@ 4>

+S6, Dnr }d srgvnxs P � Q yulmhgl=

+l, 4 5P>

+ll, +;q 5 Q, q 5P , v+q, 5P /

4171 UHDOQL EURMHYL 4:

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4171 UHDOQL EURMHYL 4<

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4171 UHDOQL EURMHYL 54

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Sulpmhu 41714 Srslµlpr vyh shupxwdflmh vnxsd [ @ id> e> fj1 Wr vx=+d> e> f,/ +d> f> e,/ +e> d> f,/ +e> f> d,/ +f> d> e,/ +f> e> d,1

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+{� {f � h,2 . +| � |f,2 m@ 5d/ µwr gdmh

+{� {f,2

d2� +| � |f,

2

e2@ 4> e2 � h2 � d2= +54,

7

2 [�

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W

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;

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@+{ � {f, qd}lydpr klshuerolqlp dvlpswrwdpd1 Qdod}h ol vh klshu0

erolqd }dulµwd qd sudyfx { @ {f xvsruhgqrp v \ 0rvl/ x sulsdgqrm mhgqdg}elreolnd +54, �fh {� {f l | � |f l}plmhqlwl pmhvwd wm1

+| � |f,2

d2� +{� {f,

2

e2@ 4= +55,

Qhnd vh klshuerolqd }dulµwd I� l I2 qdod}h qd sudyfx | � |f @ { � {f+vlphwulfqr v re}lurp qd qmh}lqr vuhglµwh V @ +{f> |f,, l qhnd mh e @ d1Wdgd mh I� @ +{f�d> |f�d,/ I2 @ +{f.d> |f.d, l dvlpswrwh vx xvsruhgqhv nrruglqdwqlp rvlpdp +y1 fuwh},/ d klshuerolqd mhgqdg}ed srsulpd reoln

+{� {f,+| � |f, @d2

5= +56,

5161 DQDOLWL FND JHRPHWULMD <4

2 ;

<

6

[�

\�

Volfqr vh/ ndg mh e @ d l }dulµwd qd sudyfx |�|f @ �+{�{f,/ }d klshuerolqxmhgqdg}ex grelyd

+{� {f,+| � |f, @ �d2

5= +57,

Gh�qlflmd 51616 Qhnd mh s sudydf x udyqlql � l qhnd mh I wrfnd x � l}ydqs1 Vnxs S+I > s, @ iW m g+I>W , @ g+I> s,j vylk wrfdnd W x udyqlql � mhgqdnrxgdomhqlk rg wrfnh I l rg sudyfd s qd}lydpr sduderorp +lol klwqlfrp,1 ]dI nd}hpr gd mh }dulµwh +lol irnxv,/ d }d s gd mh udyqdolfd +lol gluhnwulvd,grwlfqh sduderoh1

R}qdflpr g+I> s, � d/ sd qhnd mh x vxvwdyx +R> l> m, sudydf s xvsruhgdqv \ 0rvl l }dgdq mhgqdg}erp { @ {f � @

2 / d I @ +{f .@

2 > |f, +y1 fuwh},1

<

;

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WW

)

S

[�

[R�BD� [��

BD�

L} gh�qlflmvnrjd xymhwd vh wdgd odnr l}yhgh sulsdgqd sduderolqd mhgqdg}ed+| � |f,

2 @ 5d+{� {f,= +58,Dnr mh udyqdolfd s }dgdqd mhgqdg}erp { @ {f.

@

2 / d }dulµwh I @ +{f� @

2 > |f,/mhgqdg}ed srsulpd reoln

+| � |f,2 @ �5d+{� {f,= +58�,

Ndg mh udyqdolfd s xvsruhgqd v [0rvl l }dgdqd mhgqdg}erp | @ |f � @

2 /rgqrvqr | @ |f .

@

2 / d }dulµwh I @ +{f> |f .@

2 ,/ rgqrvqr I @ +{f> |f � @

2 ,/grelydpr vlphwulfqh mhgqdg}eh

+{� {f,2 @ 5d+| � |f,> +59,

rgqrvqr/+{� {f,

2 @ �5d+| � |f,= +59�,Ndr µwr vpr sudydf x udyqlql prjol }dgdwl gymhpd mhgqdg}edpd vd }d0

mhgqlfnlp sdudphwurp +y1 +6�,,/ d sudydf x survwrux wulpd mhgqdg}edpdvd }dmhgqlfnlp sdudphwurp +y1 +6,,/ wdnr 0 sdudphwduvnl 0 pr}hpr }d0gdwl l nulyxomx x udyqlql lol survwrux1 Sulpmhulfh/ sdudphwduvnh mhgqdg}eh }dnux}qlfx N+R> u, +wm1 {2.|2 @ u2, x udyqlql � v sudyrnxwqlp nrruglqdwqlp


vxvwdyrp +R> l> m, mhvx= { @ u frv w/ | @ u vlq w/ w 5 ^3> 5�l1 Surpwudpr ollvwx nux}qlfx x survwrux H v sudyrnxwqlp nrruglqdwqlp vxvwdyrp+R> l> m>n,+udyqlqd � vh srgxgdud v nrruglqdwqrp [\ 0udyqlqrp,/ sdudphwduvnl }dslvsrsulpd reoln { @ u frv w/ | @ u vlq w/ } @ 3/ ^3> 5�l1 Gdndnr gd nulyxomd/rs�fhqlwr/ pr}h lpdwl ylµh ud}olflwlk sdudphwduvnlk }dslvd 0 mhgqdg}ded1

Sulpmhu 51618 Sdudphwduvnh mhgqdg}eh

{ @ u+w� vlq w,> | @ u+4� frv w,> } @ 3> w 5 U> +5:,

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<

2

;

� Uπ

U

$

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<

;2 UB

�U

Dvwurlgd1

Sulpmhu 51619 Mhgqdg}eh

{ @ u frv� w> | @ u vlq� w> } @ 3> w 5 ^3> 5�l > +5;,

suhgvwdyomdmx sdudphwduvnl }dslv udyqlqvnh nulyxomh dvwurlgh/ µwr mx rslvxmhwrfnd D @ +u> 3> 3, qd nux}qlfl N� � � � +{� �o

e,2 . |2 @ + o

e,2/ } @ 3/ grn vh

N� �nrwuomd� l}qxwud sr nux}qlfl N � � � {2 . |2 @ u2/ } @ 3/ qh l}od}h�fl l}udyqlqh } @ 31

Holplqludqmhp sdudphwud w grelydpr mhgqdg}ex +} @ 3,

{2

� . |2

� @ u2

� = +5;�,

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{ @6dw

4 . w�> | @

6dw2

4 . w�> } @ 3> w 5 U> +5<,

rguh¡xmx w}y1 Ghvfduwhvry olvw

;

<

2

�D�B

�D

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=

; <U

Nux }qd x}yrmqlfd1

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{� . |� � 6d{| @ 3= +5<�,


Sulpmhu 5161; ]d gdqh nrqvwdqwh d l u A 3/ sdudphwduvnh mhgqdg}eh{ @ u frv w> | @ u vlq w> } @ dw> w 5 U> +63,

suhgvwdyomdmx nux}qx x}yrmqlfx +lol flolqgulfqx vsludox, 0 survwruqx nul0yxomx nrmx rslvxmh wrfnd µwr vh qd xgdomhqrvwl u yuwl rnr sudyfd s � � � { @ 3/| @ 3/ l lvwrgreqr vh vwdoqrp eu}lqrp d jled xvsruhgqr v wlp sudyfhp1

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D{2 .E|2 .F}2 .G{| .H{} . I|} .J{.K| . M} .N @ 3>v uhdoqlp nrh�flmhqwlpd D/ E/ F/ G/ H/ I / J/ K/ M l N/ srg xymhwrp gdmh eduhp mhgdq rg D/ E/ F/ G/ H lol I ud}olflw rg qxoh1 Srvheqr �fh qdv}dqlpdwl vdpr qhnh nydgulnh1

Mhgqdg}ed+{� {f,

2 . +| � |f,2 . +} � }f,

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]d gdqh uhdoqh nrqvwdqwh d/ e l f/ mhgqdg}ed+{� {f,

2

d2.

+| � |f,2

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+} � }f,2

f2@ 4 +65,

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Qdgdomh/ mhgqdg}ed{2

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f2@ 4 +66,

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}2

f2.{2

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d mrµ mhgqrp wdnyrp }dpmhqrp grelydpr mhgqdg}ex +�sryodµwhqd� mh [0rv,=

|2

e2.}2

f2�{2

d2@ 4= +66

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,

Mhgqdg}ed

�{2

d2�|2

e2.}2

f2@ 4 +67,

rslvxmh gyrnuloql holswlfql klshuerorlg1

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�}2

f2�{2

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,

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d2.|2

e2@ 5} +68,

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Holswl fql l klshuerorfql sduderorlg1

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d2@ 5} +69,

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d2.|2

e2@}2

f2+6:,

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d2.|2

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{2

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| @ 5d}2 +73,

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Ydomfdvwh sorkh1

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nrml vlmhnx nulyxomx N l xvsruhgql vx v sudyfhp s1 Wuhwludmx�fl vydnl sudydf

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Sulpmhulfh/ holswlfqrm ydomfdvwrm sorkl %2

@2. +2

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]0rv/ d udyqdolfd mrm mh holsvd %2

@2. +2

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Pl �fhpr/ qdmfhµ�fh/ surpdwudwl rqh ydomfdvwh sorkh l}yrgqlfh nrmlk vxnrruglqdwqh rvl/ d udyqdolfh vx lp qhnh rg sr}qdwlk nulyxomd1 +Udyqdolfd/qdudyqr/ qh �fh qx}qr oh}dwl x qhnrm rg nrruglqdwqlk udyqlqd1,


Qdsrphqlpr l wr gd vh survwruqd nulyxomd fhvwr }dgdmx suhvmhnrp gylmxsorkd1 X wdnyrp vh voxfdmx olqhduqlp nrpelqdflmdpd sulsdgqlk mhgqdg}dedgrelydmx mhgqdg}eh l guxjlk sorkd µwr vdgu}h wx nulyxomx1

Sulpmhu 5161< Nux}qlfx +}dgdqx suhvmhnrp vihuh l udyqlqh,

{2 . |2 . }2 @ 7> {. | � 5 @ 3

pr}hpr }dgdwl l suhvmhnrp gylmx ydomfdvwlk sorkd1 Holplqludpr ol/ qdlph/ydulmdeox | l} suyh mhgqdg}eh xyuµwhqmhp +l} rqh guxjh, | @ �{.5/ grelydprmhgqdg}ex ydomfdvwh sorkh

+{� 4,2 . 52

2@ 4>

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+| � 4,2 . 52

2@ 4>

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�fhpr jd vdgd gh�qludwl1

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�$RW

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2 S� � �Dπ S

Duklphgryd vsludod1

2 �D S

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%1

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2,2. |2 @ �

e1 Sroduqd mhgqdg}ed � @ frv 5*

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2 D

S

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2

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vxvwdy +R>sf>!>n, � +R>�f>*f>n, x nrmhpx vh vydnrm wrfnl W pr}h sulgl0mholwl xuh¡hqd wurmnd +�>*> �,/ jgmh vx � l * sroduqh nrruglqdwh/ x vxvwdyx

<; SRJODYOMH 51 OLQHDUQD DOJHEUD

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α

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Τ=(ρ,ϕ,]�

ϕρ

S

=

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N 7

7

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%eol}x 4 ndg mh m{m grvwdwqr yholn1 Ud}xpqr mh/ vwrjd/

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w � � � | � i+{f, @ i �+{f,+{� {f,= +5,

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i � @ .4 +�4,/ r fhpx vdg

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q � � � | � i+{f, @ � �s �E%f�

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ai = [ q i{fj $ U> ai+{, @i+{,� i+{f,

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ixqnflmh i x wrfnl {f1 Rflwr mh gd mh ixqnflmd i / nrmd mh ghulydeloqd vol0mhyd l }ghvqd x wrfnl {f/ ghulydeloqd x {f dnr l vdpr dnr vh wh ghulydflmhvolmhyd l }ghvqd srgxgdudmx1 +Wdm mh eurm rqgd i �+{f,1,

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+ji,�+{f, @ j�+i+{f,,i�+{f,= +4;,

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i �+{f,= +56,

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Qdsrphqd 71415 Dnr mh ixqnflmd i = [ $ U/ [ � U/ }dgdqd lpsolflwqrmhgqdg}erp I +{> |, @ 3 +y1 ¢61414, l dnr mh ghulydeloqd x wrfnl {f/ rqgdghulydflmx i �+{f, rguh¡xmhpr irupdoqlp ghulyludqmhp mhgqdg}eh I +{> |, @ 3+ghulyludmx�fl | slµhpr |� � i �+{, l xyuµwdydpr { @ {f, l vuh¡lydqmhp1

Sulpmhu 71419 Rguhglpr ghulydflmx x wrfnl { @ 3 ixqnflmh { :$ i+{, @ |

lpsolflwqr }dgdqh mhgqdg}erp 5+ � {| . {2 � 5 @ 31 Irupdoqr ghulyludmx�flmhgqdg}ex grelydpr= +5+�{|.{2�5,� @ 3� , 5+ � oq 5 �|��+|.{|�,.5{ @3 , |� @ +32%

2+ *? 23% / sd mh i �+{f, � |�f @ +f32%f2+f *? 23%f

1 Yulmhgqrvw i+3, � |f

grelydpr xyuµwhqmhp x mhgqdg}ex1 Gdnoh 5+f � 3 � |f . 32 � 5 @ 3, 5+f @5, |f @ 41 Qdsrnrq/ i �+3, � |�f @

�32uf2� *? 23f @ �

2 *? 2 @ �*? e 1


3%�%1 �4��/ ��

X srgrgmhomnx ¢71415 vpr srnd}dol gd }d suludvw i+{,�i+{f, @ +�%fi,+{, ��i+{, ghulydeloqh +x wrfnl {f, ixqnflmh i = [ $ U/ [ � U/ { @ {f .�{/yulmhgl irupxod +4:,=

�i+{, @ i �+{f,�{. �+�{,�{/ sul fhpx mh olp{%<f

�+�{, @ 31

Wr qdyrgl gd vh ixqnflml ghulydeloqrm x wrfnl {f sulglmhol +orndoqr, olqhduqdixqnflmd µwr mx rguh¡xmh ghulydflmd i �+{f,1 Wdnr vh grod}l gr w}y1 glihuhqflmd0eloqrvwl/ rgqrvqr/ glihuhqflmdod surpdwudqh ixqnflmh x wrfnl {f= ]d ixqnflmhmhgqh ydulmdeoh/ µwr lk vdgd surxfdydpr/ elw �fh ud}ylgqr gd vx glihuhqflmd0eloqrvw l ghulydeloqrvw hnylydohqwqd vyrmvwyd1 Gd mh glihuhqflmdeloqrvw/ rs�fh0qlwr/ mdfh vyrmvwyr rg ghulydeloqrvwl xrflw �fhpr xvnrur surxfdydmx�fl uhdoqhixqnflmh ylµh ydulmdeod1

Gh�qlflmd 71415 Uh�fl �fhpr gd mh ixqnflmd i = [ $ U/ [ � U/ glihuhqfl0mdeloqd x wrfnl {f/ dnr srvwrml eurm d 5 U wdndy gd mh

i+{,� i+{f, @ d � +{� {f, . u+{� {f,>sul fhpx }d ixqnflmx {� {f :$ u+{� {f,/ { 9@ {f/ prud yulmhglwl

olp%<%f

u+{� {f,

{� {f@ 3 +�rvwdwdn� u+{�{f, wh}l n 3 elwqr eu}h rg {�{f,1

Uh�fl �fhpr gd mh ixqnflmd i glihuhqflmdeloqd/ dnr mh glihuhqflmdeloqd x

vydnrm wrfnl { 5 [1

Ud}ylgqr mh +y1+4:,, gd mh ixqnflmd i glihuhqflmdeloqd x {f flp mh ghuly0deloqd x {f1 Qdlph/ }d wud}hql eurm d vplmhpr x}hwl i �+{f,/ sd mh u+{�{f, @�+{� {f, � +{� {f, l xymhwx mh xgryromhqr1 V guxjh vwudqh/ odnr mh grnd}dwlgd mh eurm d mhglqvwyhq flp mh i glihuhqflmdeloqd x {f/ d rqgd mh

d @ olp%<%f

i+{,� i+{f,

{� {f. olp

%<%f

u+{� {f,

{� {f@ i �+{f,=

Suhpd wrpx/ ixqnflmd i mh glihuhqflmdeloqd +x wrfnl {f, wrfqr rqgd ndg mhghulydeloqd +x {f,1 Sulsdgqx olqhduqx ixqnflmx qd}lydpr glihuhqflmdorp

ixqnflmh i x wrfnl {f l r}qdfxmhpr vgi+{f, = U$ U> gi+{f,+{, @ i �+{f, � {= +69,

Sulplmhwlpr gd mh/ sr +4:,/

olp{%<f

{sE%�3_sE%f�E{%�{%

@ olp{%<f

oE{%�{%

@ olp{%<f

�+�{, @ 3=

Sulpmhu 7141: Rguhglpr glihuhqflmdoh gi+{f,/ x vydnrm wrfnl {f 5 U/ rv0qryqlk hohphqwduqlk ixqnflmd vlq/ srwhqflmh v hnvsrqhqwlqd 5 l olqhduqh ixqnflmhv nrh�flmhqwrp n/ wm1 ixqnflmd { :$ vlq{/ { :$ {2 +nydguludqmh, l { :$ n{

+n 5 U nrqvwdqwd,1Qhnd mh {f 5 U elor nrmd wrfnd +surpdwudqh ixqflmh vx ghulydeloqh,1 Yulmhg0qrvwl gi+{f,+{, wud}hqlk glihuhqflmdod gi+{f, mhvx uhgrp=

g+vlq{f,+{, @ vlq� { m%'%f �{ @ frv{f � {>g+{2f,+{, @ +{2,� m%'%f �{ @ 5{f{>g+n{f,+{, @ +n{,� m%'%f �{ @ n{1

7141 GHULYDFLMD 494

Sulplmhwlpr gd mh glihuhqflmdo olqhduqh ixqnflmh/ x vydnrm wrfnl {f 5 U/ mhgqdnwrm ixqnflml1

Gd elvpr µwr mhgqrvwdyqlmh rshuludol glihuhqflmdorp/ lvnrulvwlpr flqmhqlfx+y1 Olqhduqx dojheux,

gi+{f, 5 Krp+U>U, �@ U/wm1 gd mh gi+{f, yhnwru x gxdoqrpx survwrux +vylk olqhduqlk ixqnflmd, rg U1Dnr v g 5 krp+U>U, r}qdflpr ed}ql yhnwru gxdodq yhnwrux h @ 4 5 U/vplmhpr slvdwl

gi+{f, @ i �+{f,g/d }d qmhjryh yulmhgqrvwl

gi+{f,+{, @ i �+{f,g{/ { 5 U1Srmhgqrvwdyqmxmx�fl }dslvlydqmh/ ndg jrg qh pr}h gr�fl gr }dexqh/ xrelfdmlorvh glihuhqflmdo gi+{f, wuhwludwl ndr ixqnflmx rg {� {f @ �{/ { 5 U/ l slvdwl

gi+{f,+�{, � gi+{, @ i �+{,g{/sd �fhpr vh l pl wrjd gu}dwl1 Qd wdm vh qdflq jruqml sulpmhul prjx }dslvdwlndnr volmhgl=

g vlq{ @ frv{g{/ g{2 @ 5{g{/ g+n{, @ ng{1Sulplmhwlpr gd wdnyr vnud�fhqr }dslvlydqmh grsxµwd }dslvdwl ghulydflmx ndrnrolfqln

i �+{, @ _sE%�_%

� _+_%=

Sulwrp qd yholflqx g{ vplmhpr johgdwl ndr qd �qhl}pmhuqr pdol� �{ mhu mhi �+{f,�{ @ gi+{f,+�{, � gi+{, @ i �+{,g{1

Udgl eromhjd srlpdqmd/ nrulvqr mh srmdvqlwl glihuhqflmdoryr jhrphwulmvnr}qdfhqmh +y1 fuwh} gromh,1 Exgx�fl gd mh i �+{f, @

_sE%�_%

/ { @ {f.�{/ g{ @ �{/wdqjhqwlq vpmhuryql nrh�flmhqw +w x wrfnl +{f> i+{f,, 5 Js / y1 +5,,> volmhglgd mh yulmhgqrvw sulsdgqrjd glihuhqflmdod gi+{f,+g{, � gi+{, @ i �+{f,g{�suludvw gr wdqjhqwh� w/ grn mh �i+{, sulsdgql ixqnflmvnl suludvw/ wm1 �suludvwgr judid� Js $,

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Whruhp 71415 lpd vyrm orndoql glihuhqflmdoql dqdorjrq1

Whruhp 71418 +Glihuhqflmdo l rvqryqh udfxqvnh rshudflmh, Qhnd vx ixqnflmh

i> j = [ $ U/ [ � U/ glihuhqflmdeloqh x wrfnl {f1 Wdgd }d sulsdgqh glihuhq0

flmdoh x {f yulmhgl=

gfo+{f, @ ff +i � fo> ff � nrqvwdqwqh ixqnflmh,> +6<,

g+i . j,+{f, @ gi+{f, . gj+{f,> +73,


g+i � j,+{f, @ gi+{f,� gj+{f,> +74,

g+i � j,+{f, @ j+{f,gi+{f, . i+{f,gj+{f,> +75,

g+fo � j,+{f, @ fogj+{f, +i @ fo,> +76,

g+i

j,+{f, @

j+{f,gi+{f,� i+{f,gj+{f,

j+{f,2+j+{f, 9@ 3,= +77,

Grnd}1 Grnd}lpr/ sulpmhud udgl/ irupxox +75,$ X}plpr elor nrml { 5 U1Wdgd mh x Krp+U>U, g+i � j,+{f,+{, @ +i � j,�+{f,g{ +45,

@ +i �+{f,j+{f, .i+{f,j

�+{f,,g{ @ j+{f,i�+{f,g{ . i+{f,j

�+{f,g{ @ j+{f,gi+{f,+{,.i+{f,gj+{f,+{, @ +j+{f,gi+{f, . i+{f,gj+{f,,+{,1

Sulpmhu 7141; Rguhglpr ixqnflml { :$ i+{, @ h% vlq{ glihuhqflmdo x elor nr0

mrm wrfnl {f 5 U1 gi+{f,+{, +75,@ +vlq{fg+h

%f,.h%fg+vlq{f,,+{, @ vlq{fh%fg{.

h%f frv{fg{ @ h%f+vlq{f.frv{f,g{1 Lol vnud�fhqr/ g+h% vlq{, @ +h% vlq{,�g{ @

h%+vlq{. frv{,g{

Whruhp 71419 Qhnd mh ixqnflmd i = [ $ U/ [ � U/ ghulydeloqd x wrfnl {fl qhnd mh i �+{f, 9@ 31 Wdgd mh

olp{%<f

�i+{,

gi+{,@ 4=

Grnd}1 Sr +4:, l +69, mh {sE%�_sE%� @ s �E%f�_%nkE_%�_%

s �E%f�_%@ 4 . kE_%�

s �E%f�l

olp_%<f

�+g{, @ 3= Suhpd wrpx/ olp{%<f

{sE%�_sE%� @ 4. �

s �E%f�olp_%<f

�+g{, @ 4=

Whruhp 71419 srwyu¡xmh gd mh ixqnflmlq suludvw �i+{f,+�{, � �i+{, ��| suleol}qr mhgqdn glihuhqflmdoryrm yulmhgqrvwl gi+{f,+�{, � gi+{, � g|

flp mh ydulmdeolq suludvw �{ @ g{ grvwdwqr pdohq1 Slµhpr �| � g|1 Ryd vh

flqmhqlfd fhvwr sulplmhqmxmh x suleol}qrp l}udfxqdydqmx ixqnflmvnlk yulmhg0qrvwl/ ndr l x surfmhqmlydqmx srjumhµdnd nrmh vx srvomhglfd qhwrfqlk pmhuhqmd1Dnr mh qsu1 sr}qdwd ixqnflmlqd yulmhgqrvw |f @ i+{f,/ qmh}lqd vh yulmhg0

qrvw x wrfnl { @ {f . g{ +i ghulydeloqd x {f l g{ grvwdwqr pdohq, vplmhdsurnvlpludwl sr irupxol

i+{f . g{, @ |f .�| � |f . g|= +78,Qdgdomh/ dnr mh ydulmdeod { l}pmhuhqd v srjumhµnrp nrmd dsvroxwqr qh suh0pdµxmh mg{m/ rqgd vh qdmyh�fd dsvroxwqd srjumhµnd ixqnflmlqh yulmhgqrvwl| @ i+{, vplmh surflmhqlwl ndr

m�|m � mg|m @ mi �+{,mg{> +79,uhodwlyqd srjumhµnd ndr

�|

|� g|

|> +7:,

d srvwrwqd srjumhµnd ndr�|

|� 433( � g|

|� 433(= +7;,

7141 GHULYDFLMD 496

Sulpmhu 7141< L}udfxqdmpr qhnx suleol}qx +udflrqdoqx, yulmhgqrvw ludflr0qdoqrjd eurmd e

s;7 udeh�fl dsurnvlpdflmx glihuhqflmdorp1

X umhµdydqmx rydnyrjd }dgdwnd wuhed yrglwl udfxqd r wulpd vwydulpd= Suyr/gd vh rgdehuh µwr mh prjx�fh mhgqrvwdyqlmd hohphqwduqd ixqnflmd i +sul0pmhuhqd gdqrp }dgdwnx,> gd vh rgdehuh rgjrydudmx�fd wrfnd {f }d nrmxvh +wrfqd, yulmhgqrvw i+{f, pr}h odnr l}udfxqdwl> l wuh�fh/ gd {f . g{ exghgdqd ydulmdeolqd yulmhgqrvw v uhodwlyqr pdolp g{1Sulplmhwlpr/ qdmsulmh/ gd mh e

s;7 @ +;4 . 6,

�

e @ 6+4 . �2.,

�

e 1Vwrjd vh +qh }dkwlmhydpr ol eromx dsurnvlpdflmx,/ ndr sulnodgdq l}eru qdph�fx

ixqnflmd { :$ i+{, @ 6+4.{,�

e l wrfnd {f � 3/ sul fhpx mh g{ @ �2. @ {f.g{1

Wdgd/ sulplmhqmxmx�fl irupxox +78, grelydpr=es;7 @ 6+4 . �

2.,�

e @ i+3 . �2., � i+3, . i �+3, � �

2. @

6+4 . 3,�

e . �

eE�nf��

e

� �2. @ 6 . �

�S @ �fb�S � 6> 35;1

Qhnd mh i = [ $ U/ [ � U/ ghulydeloqd +hnylydohqwqr/ glihuhqflmd0eloqd, ixqnflmd1 Wdgd x vydnrm wrfnl { srvwrml glihuhqflmdo gi+{, = U $ U/gi+{,+{, @ i �+{,g{/ sd mh greur gh�qludqd ixqnflmd gi = [ $ Krp+U>U,/{ :$ gi+{,1 Exgx�fl gd mh Krp+U>U, l}rpruidq yhnwruvnrpx survwrux U/Krp+U>U, �@ U/ wr qd gi vplmhpr johgdwl l ndr qd ixqnflmx l} [ x U1

Qhnd mh ixqnflmd i = [ $ U/ [ � U/ ghulydeloqd qd vnxsx D � [/wm1 qhnd srvwrml ixqnflmd +i m�,� = D $ U/ { :$ i �+{,1 Dnr mh ixqnflmd +i m�,�ghulydeloqd x wrfnl {f/ rqgd nd}hpr gd mh ixqnflmd i gydsxw ghulydeloqd x

wrfnl {f/ d yulmhgqrvw ++i m�,�,�+{f, � i ��+{f, qd}lydpr guxjrp ghulydfl0

mrp ixqnflmh i x wrfnl {f1 Uh�fl �fhpr gd mh ixqnflmd i gydsxw ghulydeloqdqd D dnr mh i gydsxw ghulydeloqd x vydnrm wrfnl { 5 D1 Sulsdgqx ixqnflmxqd}lydpr guxjrp ghulydflmrp ixqnflmh i qd vnxsx D l r}qdfxmhpr v+i m�,�� = D $ U1 X voxfdmx D @ [ jryrulpr r gydsxw ghulydeloqrm

ixqnflml i l r qmh}lqrm guxjrm ghulydflml i �� +i �,� = [ $ U1 Qdvwdy0omdmx�fl lqgxnwlyqr/ lpd vplvod jryrulwl r wuh�frm/ fhwyuwrm/ � � � > q0wrm/ � � �ghulydflml ixqnflmh i +qd vnxsx D> x wrfnl {f,1 Sulsdgqh r}qdnh vx i ��/i +7,/ � � � / i E?�/ � � � 1 Gdnoh +y1 Whruhp 41715,/

i E?�+{f, @ +i E?3��,�+{f,> q 5 Q +i Ef� � i,= +83,

Sulpmhu 714143 Rguhglpr +dnr srvwrmh, vyh ghulydflmh i E?� = U $ U/ q 5Q/ ixqnflmh i = U$ U/ i+{, @ ho%1

Exgx�fl gd mh i ixqnflmvnd nrpsr}lflmd pqr}hqmd nrqvwdqwrp +u, v +sulurg0qrp, hnvsrqhqflmdoqrp ixqnflmrp +h{se,/ wr mh i ghulydeloqd ixqnflmd1 Volfqryulmhgl l }d qmh}lqh ghulydflmh1 Sulwrp grelydpr=

i �+{, @ uho%/ i ��+{, @ +uho%,� @ u2ho%/ � � � /i E?�+{, @ +u?3�ho%,� @ u?ho%/ � � � 1

Srvyh volfqr vh }d ixqnflmx i = [ $ U/ [ � U/ gh�qludmx glihuhq0flmdol ylµlk uhgryd +x wrfnl {f,1 Suhwsrvwdylpr gd mh i glihuhqflmdeloqd


ixqnflmd1 Guxjlp glihuhqflmdorp ixqnflmh i x wrfnl {f qd}lydpr glihu0hqflmdo g+gi,+{f, ixqnflmh0glihuhqflmdod gi = [ $ Krp+U>U, �@ U x wrfnl {f1Udgl vh/ gdnoh/ r olqhduqrm ixqnflml/ nrmx r}qdfxmhpr v g2i+{f, = U $ U/rguh¡hqrm �nrh�flmhqwrp� i ��+{f,g{/ wm1

g2i+{f, @ g+gi+{,,m%'%f @ g+i �+{,g{,m%'%f @ i ��+{f,g{/

+rgqrvqr/ g2i+{f, @ +i ��+{f,g{,g 5 Krp+U>U,, v yulmhgqrvwlpd

g2i+{f,+{, � g2i+{, @ +i ��+{f,g{,g{ � i ��+{f,g{2> +84,

Yholflqd g{2 @ g{�g{ qrvl redylmhvw r mrµ mhgqrm }dpmhql survwrud Krp+U>U,survwrurp U1 Sulwrp nd}hpr gd mh ixqnflmd i gydsxw glihuhqflmdeloqd x

wrfnl {f1 Dnr mh i gydsxw glihuhqflmdeloqd x vydnrm wrfnl { 5 [ rqgd }dqmx nd}hpr gd mh gydsxw glihuhqflmdeloqd ixqnflmd1 X wrp voxfdmx srvwrmlixqnflmd +guxjl glihuhqflmdo rg i,

g2i = [ $ Krp+U>Krp+U>U,, �@ Krp+U>U, �@ U> { :$ g2i+{,=

Rs�fhqlwr/ q0wl glihuhqflmdo g?i+{f, = U$ U ixqnflmh i x wrfnl {f/ q 5 Q/q � 5/ gh�qludpr ndr glihuhqflmdo +q � 4,0yrjd glihuhqflmdod rg i / g?3�i =[ $ U/ x {f/ wm1 g?i+{f, @ g+g?3�i,+{f,1 Vwrjd yulmhgl irupxod +g{? �+g{,?,

g?i+{f,+{, � g?i+{, @ i E?�+{f,g{?> +85,

sul fhpx yholflqd g{? @ g{?3� � g{ qrvl redylmhvw r q }dpmhqd survwrudKrp+U>U, survwrurpU1 Revwrmqrµ�fx q0wrj glihuhqflmdod rg i +x {f, gh�qludvh q sxwd glihuhqflmdeloqrvw ixqnflmh i +x wrfnl {f,1 Ud}ylgqr mh gd vxwd vyrmvwyd hnylydohqwqd rgjrydudmx�frm ghulydeloqrvwl1

Sr +85,/ irupxod +6;, lpd vomhgh�fh srrs�fhqmh qd vydnl q 5 Q=i E?�+{, @

g?i+{,

g{?= +86,

Qhnd mh mhgqdg}edpd { @ *+w, l | @ #+w, sdudphwduvnl }dgdqd ixqnflmd{ :$ i+{, @ | +y1 ¢61414,1 Exgx�fl gd vh/ rs�fhqlwr/ sdudphwdu w qh pr}hholplqludwl/ rguhglw �fhpr ghulydflmx rg i x wrfnl {f @ *+wf, +flp srvwrml,srpr�fx glihuhqflmdod rg * l # x wrfnl wf1

i �+{f, @_+_%� _�E|�

_)E|� @��E|f�_|)�E|f�_|

@ ��E|f�)�E|f�

> *�+wf, 9@ 3>

µwr vh nud�fh l rs�fhqlwr }dslvxmh ndr

|� � i �+{, @b|

b{> b{ @ *�+w,> b| @ #�+w,= +87,

Qd volfdq qdflq grelydpr +dnr srvwrmh, l ghulydflmh ylµlk uhgryd/ sulpmhulfh+g{ l gw wuhwludpr ndr nrqvwdqwh,/

|�� @ _2+_%2

@ _E+�_%�_%2

@_E �+

�%_%�

_%u_% @_E �+

�%�

_% @_|E

�+

�%_|�

�%_| @

��+ �%3 �+

��%

�%2

�% > wm1

|�� @�| b{� b|�{

b{�> b{ 9@ 3> �{ @ *��+w,> �| @ #��+w,= +88,

3%�%3 "+/#$/� 6�#��0� ��4��/ ��/#�� ,/�

Rygmh �fhpr grnd}dwl qhnrolnr whphomqlk whruhpd r yd}qlp vyrmvwylpd ghul0ydeloqlk ixqnflmd1

7141 GHULYDFLMD 498

Whruhp 7141: +Ihupdwry whruhp, Qhnd vx}hqmh i m'@cK� ixqnflmh i = [ $U/ [ � U/ srsulpd x wrfnl {f 5 kd> el � [ vyrmx qdmpdqmx lol qdmyh�fx

yulmhgqrvw1 Dnr mh i ghulydeloqd x {f rqgd mh i �+{f, @ 31

Grnd}1 Suhwsrvwdylpr gd mh i+{f, plqlpxp rg i m'@cK�/ wm1 i+{f, � i+{,}d vydnl { 5 kd> el1 ]dslvxmx�fl { @ {f.�{ grelydpr i+{f.�{,�i+{f, � 31Exgx�fl gd mh i ghulydeloqd x {f 5 kd> el/ wr mh i ghulydeloqd volmhyd l }ghvqdx {f l sulsdgqh vh ghulydflmh srgxgdudmx v i �+{f,/ wm1

olp{%<f3f

sE%fn{%�3sE%f�{% @ i �+{f, @ olp

{%<fnf

sE%fn{%�3sE%f�{% =

Sulplmhwlpr gd mh qd}lyqln �{ x ghulydflml volmhyd vwdoqr qhjdwlydq +{ ? {f,/grn mh x ghulydflml }ghvqd vwdoqr sr}lwlydq +{ A {f,1 Exgx�fl gd vx eurmqlflvwdoqr sr}lwlyql/ wr mh ghulydflmd volmhyd rg i x {f pdqmd lol mhgqdnd qxod/ drqd }ghvqd yh�fd lol mhgqdnd qxod1 Volmhgl }dnomxfdn i �+{f, @ 31 X voxfdmxpdnvlpxpd i+{f, grnd}xmh vh qd lvwl qdflq1

Whruhp 7141; +Uroohry whruhp, Qhnd mh ixqnflmd i = [ $ U/ [ � U/

qhsuhnlgqd qd vhjphqwx ^d> e` � [ l ghulydeloqd qd lqwhuydox kd> el = Dnr mh

i+d, @ i+e, rqgd srvwrml wrfnd {f 5 kd> el wdnyd gd mh i �+{f, @ 31

Grnd}1 Sr Whruhpx 616145/ vx}hqmh i md@cKo srvwl}h qd ^d> e` vyrmx qdm0pdqmx l vyrmx qdmyh�fx yulmhgqrvw/ wm1 srvwrmh {�> {2 5 ^d> e` wdnyl gd mh/ }dvydnl { 5 ^d> e`/ i+{�, � i+{, � i+{2,1 Dnr mh i+{2, @ i+{�, rqgd mh i md@cKonrqvwdqwqd ixqnflmd fsE%��/ sd mh +i m'@cK�,� @ ffm'@cK� 0 grnd} jrwry1 Qhndmh i+{2, 9@ i+{�,/ wm1 i+{�, ? i+{2,1 Wdgd qh prjx remh yulmhgqrvwl elwlmhgqdnh i+d, +@ i+e,,1 Suhwsrvwdynd i+{�, 9@ i+d, sryodfl d 9@ {� 9@ e/ wm1{� 5 kd> el/ d suhwsrvwdynd i+{2, 9@ i+d, sryodfl d 9@ {2 9@ e/ wm1 {2 5 kd> el1Sr Whruhpx 7141: }dnomxfxmhpr gd mh {� lol {2 wud}hqd wrfnd {f1

Whruhp 7141< +Odjudqjhry whruhp r vuhgqmrm yulmhgqrvwl, Qhnd mh ixqnflmd

i = [ $ U/ [ � U/ qhsuhnlgqd qd vhjphqwx ^d> e` � [ l ghulydeloqd qd

lqwhuydox kd> el = Wdgd srvwrml wrfnd {f 5 kd> el wdnyd gd mh

i �+{f, @i+e,� i+d,

e� d= +89,

Grnd}1 Surpdwudmpr ixqnflmx j = [ $ U> j+{, @ i+{,� sEK�3sE@�K3@ +{�d,

+y1 fuwh} gromh,1 Ixqnflmd j mh qhsuhnlgqd qd ^d> e` l ghulydeloqd qd kd> el l nwrpx mh j+d, @ i+d, @ j+e,1

<

2

;D EF� F�

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Sr Whruhpx 7151; srvwrml qhnd wrfnd {f 5 kd> el x nrmrm mh j�+{f, @ 3 @

i �+{f,� sEK�3sE@�K3@ > wm1 i �+{f, @

sEK�3sE@�K3@ =

Odjudqjhryd irupxod +89, vh pr}h qdslvdwl qd qhnrolnr qdflqd1 Wdnrqsu1 pqr}hqmhp irupxoh +89, idnwrurp e� d grelydpr

i+e,� i+d, @ i �+{f,+e� d,= +89�

,Qdgdomh/ exgx�fl gd mh {f @ d . %f3@

K3@ +e � d, l 3 ? %f3@K3@ � & ? 4> vplmhpr

slvdwli+e,� i+d, @ i �+d. &+e� d,,+e� d,> 3 ? & ? 4= +89

��

,Qdsrnrq/ r}qdflpr ol d @ { l e @ {.�{/ grelydpr

�i+{, @ i+{.�{,� i+{, @ i �+{. &�{,�{> 3 ? & ? 4= +89��

,


phqwx ^d> e` � [ l ghulydeloqd qd lqwhuydox kd> el1 Dnr mh i �+{, @ 3 }d vydnl

{ 5 kd> el/ rqgd mh vx}hqmh i md@cKo nrqvwdqwqd ixqnflmd1

Grnd}1 Rgdehulpr elor nrml { 5 kd> e`1 Wdgd vx qd vhjphqwx ^d> {`lvsxqmhql xymhwl Whruhpd 7141< sd srvwrml wrfnd {f 5 kd> {l � kd> el }d nrmxmh i+{, � i+d, @ i �+{f,+{ � d,1 Sr suhwsrvwdyfl mh i �+{f, @ 3/ µwr vdgdsryodfl i+{, @ i+d, }d vydnl { 5 kd> e`1Nrurodu 71414 Qhnd vx ixqnflmh i> j = [ $ U/ [ � U/ qhsuhnlgqh qd

vhjphqwx ^d> e` � [ l ghulydeloqh qd lqwhuydox kd> el1 Dnr mh i �+{, @ j�+{,}d vydnl { 5 kd> el rqgd mh vx}hqmh +i � j,md@cKo nrqvwdqwqd ixqnflmd1

Grnd}1 Ixqnflmd k � i�j = [ $ U xgryromdyd xymhwlpd Whruhpd 714143mhu mh/ qd kd> el/ k� @ i � � j� @ 31 Volmhgl gd mh ixqnflmd i � j nrqvwdqwqd qd^d> e`1

Odjudqjhryx irupxox �fhpr vdgd xsrwulmhelwl x grnd}x w}y1 O*Krvsl0wdoryd sudylod/ µwr jd udelpr }d rguh¡lydqmh judqlfqlk yulmhgqrvwl x pqrjlpvoxfdmhylpd w}y1 qhrguh¡hqlk reolnd=

3

3>44 > 3 � 4> 4�4> 3f> 4"> 4f=

+4 rygmh r}qdfxmh elor .4 elor �4$, Sulpmhulfh/ qhrguh¡hql reoln ff vh

mdyomd sul rguh¡lydqmx judqlfqh yulmhgqrvwl olp%f

s} flp mh olp

%fi @ 3 @ olp

%fj/

d 4f sul rguh¡lydqmx olp%f

i} flp mh olp%f

i @ 4 l olp%f

j @ 31 Vomhgh�fl whruhp

ud}pdwud vdpr reoln ff / grn vh suhrvwdol grsxvwlylp suhlqdndpd vyrgh qd

qmhjd1

Whruhp 714144 +O*Krvslwdoryr sudylor, Qhnd }d ixqnflmh i> j = [ $ U/

[ � U/ yulmhgl olp%f

i @ 3 @ olp%f

j1 Dnr srvwrml lqwhuydo L wdndy gd mh {f 5 L �[ l gd vx i l j qhsuhnlgqr ghulydeloqh qd L/ sul fhpx mh j�+{, 9@ 3 }d vydnl

{ 5 L/ rqgd mh

olp%f

i

j@ 3 @ olp

%f

i �

j�= +8:,

7141 GHULYDFLMD 49:

X grnd}x �fhpr wuhedwl ryx ohpx=

Ohpd 71414 Qhnd mh ixqnflmd j = [ $ U/ [ � U/ ghulydeloqd x wrfnl {fl qhnd mh j+{f, @ 3/ d j�+{f, 9@ 31 Wdgd srvwrml qhnl � A 3 wdndy gd vx qd

vnxsx D" @ +[ q i{fj,W k{f � �> {f . �l vyh yulmhgqrvwl j+{, 9@ 31

Grnd}1 Suhwsrvwdylpr surwlyqr/ wm1 gd }d vydnl � A 3 srvwrml qhnl{" 5 D" }d nrml mh j+{", @ 31 Eludmx�fl � @ �

?/ q 5 Q/ grelydpr wdnr

ql} +{?,/ {? 5 D �?

� [ q i{fj/ nrml nrqyhujlud suhpd {f/ grn mh +j+{?,,

nrqvwdqwdq qxoql} +3,1 Wdgd mh l +}E%?�3}E%f�%?3%f, @ +3,1 Sr suhwsrvwdyfl/ srvwrml

j�+{f, @ olp%<%f

}E%�3}E%f�%3%f l j�+{f, 9@ 3= ]erj olp+{?, @ {f prud elwl +y1

Whruhp 61617, l j�+{f, @ olp+}E%?�3}E%f�%?3%f , @ olp+3, @ 3> µwr mh surwxvoryomh1

Grnd}1 +Whruhpd 7141441, Sr Whruhpx 71414 volmhgl vx ixqnflmh i l jqhsuhnlgqh qd lqwhuydox L sd mh i+{f, @ olp

%fi @ 3 l j+{f, @ olp

%fj @ 3

+y1 Whruhp 61619,1 Sr Ohpl 71414 }dnomxfxmhpr gd srvwrml srgvnxs D" @LW k{f � �> {f . �l � [/ � A 3/ wdndy gd mh j+{, 9@ 3 }d vydnl { 5 D" q i{fj=

Gdnoh/ nyrflmhqw sE%�}E%� mh gh�qludq flp mh { 5 D" q i{fj1 �wrylµh/

olp%f

s} @ olp

%<%f

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%f

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uhp 7141<, @ olp%f

s �E%��}�E%2�

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@ +i �> j� qhsuhnlgqh qd L, @*�4%f

s �

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61615+lll,> j�+{, 9@ 3> { 5 D", @ olp%f

s �

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Qdsrphqd 71416 Srvwrmh ud}qh ydulmdqwh l srrs�fhqmd xsudyr grnd}dqrjdwhruhpd1 Wdnr qsu1 O*Krvslwdoryr sudylor yulmhgl l }d qhrguh¡hqh reolnh �"

�" /}d judqlfqh yulmhgqrvwl ndg { $ 4 ndr l x voxfdmhylpd ndg vh judqlfqhyulmhgqrvwl +ghulydflmh, }dplmhqh judqlfqlp yulmhgqrvwlpd volmhyd lol }ghvqd+ghulydflmdpd volmhyd lol }ghvqd,1

Qdsrphqd 71417 O*Krvslwdoryr sudylor vh vplmh sulplmhqlwl ylµh sxwd x}d0vwrsfh/ flp suhwkrgqd sulpmhqd rshw gdmh reoln f

f +"", l dnr qryh ixqnflmhxgryromdydmx srvwdyomhqlp xymhwlpd1

Qdsrphqd 71418 Dnr vh x rguh¡lydqmx judqlfqh yulmhgqrvwl srmdyl qhnlrg suhrvwdolk +sulmh qdyhghqlk, qhrguh¡hqlk reolnd l dnr vh }hol sulplmhqlwlO*Krvslwdoryr suylor/ wuhed wdm qhrguh¡hql reoln srjrgqrp wudqvirupdflmrpvyhvwl qd reoln f

f lol qd reoln "" =

Sulpmhu 714144 olp%<f

�3ULt e%%2

ED.�@Eff�olp%<f

e t�? e%2%

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i ixqnflmh { :$ i+{, @

{%1 Exgx�fl gd mh gh�qlflmvnr srguxfmh rg i vnxs Un/ wm1 vnxs vylk { A 3/ wrmh wud}hqd judqlfqd yulmhgqrvw lvwr µwr l sulsdgqd judqlfqd yulmhgqrvw }ghvqd/olp%<f

i @ olp%<nf

i 1 +Judqlfqd yulmhgqrvw x 3 volmhyd qhpd vplvod$, Suhpd wrpx/

olp%<f

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+ghulydeloqh qd lqwhuydox L, x wrfnl { sr irupxoli+{, @ i+{f, .Uf+{,> Uf+{, @ +{� {f,i

�+{f . &+{� {f,,> 3 ? & ? 4=Guxjlp ulmhflpd/ vx}hqmh i mU vh dsurnvlplud srolqrprp s qxowrjd vwxsqmd+nrqvwdqwqrp ixqnflmrp s @ fsE%f�,/ sul fhpx ixqnflmd Uf/ w}y1 rvwdwdn/suhgvwdyomd dsurnvlpdflmvnx srjumhµnx1 Dnr mh ixqnflmd i qd L ghulydeloqdq . 4 sxwd/ sxqr eromx dsurnvlpdflmx gdmh w}y1 Wd|oruryd irupxod/ nrmdsrrs�fxmh Odjudqjhryx xnomxfxmx�fl srvwrmh�fh ghulydflmh ylµlk uhgryd1 Srv0wrmh ol ghulydflmh rg i sr yroml ylvrnrj uhgd/ dsurnvlpludqmh suhx}lpd w}y1Wd|orury uhg1

Whruhp 714145 Qhnd mh ixqnflmd i = [ $ U/ [ � U/ qd lqwhuydox kd> el �[ ghulydeloqd q . 4 sxwd/ wh qhnd mh {f 5 kd> el elor nrmd wrfnd1 Wdgd }dvydnl { 5 kd> el yulmhgl Wd|oruryd irupxod=+

i+{, @ i+{f, .s �E%f��- +{� {f, . � � �. sE?�E%f�

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U?+{, @sE?n��E%fniE%3%f��

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7141 GHULYDFLMD 49<

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i+{, @ i+{f, .s �E%f��- +{� {f, . � � �. sE?�E%f�

?- +{� {f,? .U?+{,>

sul fhpx mh U?+{, @ +{ � {f,Rk+{,/ }d qhnl s 5 Q l yulmhgqrvw k+{, qhnh

ixqnflmx k = k{f> el $ U1 Wuhed/ qdudyqr/ }d gdql s/ rguhglwl ixqnflmx k= X wxvyukx/ }d elor nrml fyuvwl { 5 k{f> el/ gh�qludmpr ixqnflmx j% � j = ^{f> el $ U

vwdyomdmx�fl

j+w, @ i+w, . s �E|��- +{� w, . � � �. sE?�E|�

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Sulplmhwlpr gd mh j+{f, @ j+{, +@ i+{,, l gd mh j ghulydeloqd qd k{f> {l/j�+w, @ i �+w,� s �E|�

�- . s ��E|��- +{� w,� � � � � s E?�E|�

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Suhpd wrpx/ ixqnflmd j qd vhjphqwx ^{f> {` lvsxqmd xymhwh Uroohryd whruhpd/sd srvwrml qhnd wrfnd wf � {f . &+{� {f, 5 k{f> {l/ 3 ? & ? 4/ wdnyd gd mhj�+wf, @ 31 Volmhgl=

3 @ s E?n��E%fniE%3%f��?- +{�{f�&+{�{f,,?�s+{�{f�&+{�{f,,R3�k+{,> wm1

k+{, @ sE?n��E%fniE%3%f��Ru?- +{� {f,

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Qdyhghql }dslv }d U?+{, vh qd}lyd rvwdwnrp x Vfkoùplofkryx reolnx1 Dnrmh s @ 4/ rvwdwdn U?+{, srsulpd w}y1 Fdxfk|mhy reoln =

U?+{, @i E?n��+{f . &+{� {f,,

q$+{� {f,

?n�+4� &,?> +8<,

d dnr mh s @ q. 4 0 w}y1 Odjudqjhry reoln =

U?+{, @i E?n��+{f . &+{� {f,,

+q. 4,$+{� {f,

?n�= +93,

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i+{,�?S

&'f

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&=

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Whruhp 714146 Qhnd ixqnflmd i = [ $ U/ [ � U/ lpd qd lqwhuydox L � [

vyh ghulydflmh +i mU,E?�/ q 5 Q/ l qhnd mh {f 5 L elor nrmd wrfnd1 Wdgd mh/ }dvydnl { 5 L>

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?- +{� {f,? . � � �

�"S?'f

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? +94,

rqgd l vdpr rqgd/ dnr ql} +U?+{,, nrqyhujlud suhpd qxol/ olp+U?+{,, @ 31

Srwhqflmvnl uhg x +94, qd}lydprWd|orurylp uhgrp +lol ud}yrmhp, ixqnflmhi x wrfnl {f1 Dnr mh {f @ 3/ jryrulpr r Pdfodxulqryx uhgx +lol ud}yrmx,ixqnflmh i / wm1 r

4:3 SRJODYOMH 71 LQILQLWH]LPDOQL UD FXQ

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?- {? . � � � �"S?'f

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Qdsrphqd 71419 ]hol ol vh x Wd|orury uhg ud}ylwl elor nrmx hohphqwduqxixqnflmx/ grvwdwqr mh xyuvwlwl rgjrydudmx�fh ghulydflmh x ghvqx vwudqx iru0pxoh +94, wh rguhglwl nrqyhujhqflmvnl lqwhuydo grelyhqrjd uhgd1 Irupxod+94, �fh yulmhglwl x vydnrm wrfnl { wrjd lqwhuydod/ wm1 elw �fh olp+U?+{,, @31 Ryr/ ph¡xwlp/ qh yulmhgl }d vydnx sr yroml pqrjr sxwd ghulydeloqxixqnflmx$ Qdlph/ dnr ixqnflmd i qlmh hohphqwduqd/ prjx�fh mh gd/ x qhnrmwrfnl { 5 [/ qmh}lq Wd|orury uhg nrqyhujlud dol qh suhpd i+{,/ wm1 prjx�fhmh l olp+U?+{,, 9@ 31 Sulpmhu wdnyh ixqnflmh mhvw

i = U$ U> i+{, @

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>

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[ � U6/ x wrfnl {f/ olp

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8141 IXQNFLMH L] U� X U 574

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>U, �@ U6o

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�fhpr surpdwudwl vdpr rqh nrmlpd mh gh�qlflmvnr srguxfmh [ � U6 l yulmhg0qrvqr srguxfmh U?1

Gh�qlflmd 91414 ]d elor nrmx ixqnflmx z = [ $ U?/ sul fhpx vx [ � U

6

l q � 5/ nd}hpr gd mh yhnwruvnd ixqnflmd1

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5;:

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Gh�qlflmd 91415 Uh�fl �fhpr gd mh yhnwru |f 5 U? judqlfqd yulmhgqrvw +lol

olphv, yhnwruvnh ixqnflmh z = [ $ U?/ [ � U6/ x wrfnl {f 5 U

6/ dnr

+;� A 3,+<� A 3,+;{ 5 [ q i{fj, g+{> {f, ? � , g+z+{,> |f, ? �1

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z @ |f lol olp%<%f

z+{, @ |f1

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Gh�qlflmd 91416 Uh�fl �fhpr gd mh yhnwruvnd ixqnflmd z = [ $ U?/ [ � U6/

qhsuhnlgqd x wrfnl {f 5 [ dnr

+;� A 3,+<� A 3,+;{ 5 [, g+{> {f, ? � , g+z+{,> z+{f,, ? �1

Dnr mh yhnwruvnd ixqnflmd z qhsuhnlgqd x vydnrm wrfnl { 5 D � [/ rqgd

nd}hpr gd mh z qhsuhnlgqd qd vnxsx D1 X voxfdmx D @ [ jryrulpr r

qhsuhnlgqrm yhnwruvnrm ixqnflml z1

X sudnvl mh yuor nrulvqr gd vh qhsuhnlgqrvw yhnwruvnh ixqnflmh vyrgl qdqhsuhnlgqrvw qmh}lqlk nrruglqdwqlk ixqnflmd1 R wrpx jryrul vomhgh�fl whruhp=

Whruhp 91414 Yhnwruvnd ixqnflmd z = [ $ U?/ [ � U

6/ mh qhsuhnlgqd

+x wrfnl {f, rqgd l vdpr rqgd/ dnr vx vyh qmh}lqh nrruglqdwqh ixqnflmh

z�> � � � > z? = [ $ U qhsuhnlgqh +x wrfnl {f,1

Grnd}1 Volmhgl l}udyqr l} Gh�qlflmh 91416 l Gh�qlflmh 814181

Ndr l }d vndoduqx ixqnflmx/ qhsuhnlgqrvw yhnwruvnh ixqnflmh x qhl}roludqrmwrfnl prjx�fh mh rslvdwl srpr�fx sulsdgqh judqlfqh yulmhgqrvwl1

Whruhp 91415 Yhnwruvnd ixqnflmd z = [ $ U?/ [ � U

6/ mh qhsuhnlgqd x

qhl}roludqrm wrfnl {f 5 [ rqgd l vdpr rqgd/ dnr mh olp%f

z @ z+{f,1

Grnd}1 Rflwr volmhgl l} Gh�qlflmd 91415 l 914161

9141 YHNWRUVNH IXQNFLMH 5;<

Sulpmhu 91414 Lvwud}lpr +qh,suhnlgqrvw yhnwruvnh ixqnflmh z = U$ U�/

z+{, @

�+{� 4,l. {2n> { ? 4

+oq{,m . +4� {2,n/ { � 41

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z�+{, @

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3/ { � 4/ z2+{, @

�3/ { ? 4

oq{/ { � 4/

z�+{, @

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+z�+4,> z2+4,> z�+4,, @ +3> 3> 3, @ z+4,1

Exgx�fl gd mh

olp%<�3f

z�+{, @ 3 @ olp%<�nf

z�+{, @ z�+4,/ wm1 olp�z� @ z�+4,/

wr mh ixqnflmd z� qhsuhnlgqd x wrfnl { @ 41 Volfqr vh srnd}xmh gd mh olp�z2 @

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olp%<�3f

z�+{, @ 4 9@ 3 @ olp%<�nf

z�+{, @ z�+4,/

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suhnlgqd x wrm wrfnl1 Sr Whruhpx 91414 mh l yhnwruvnd ixqnflmd z suhnlgqd+vdpr, x wrfnl { @ 41

7%�%- �4��/ ��2��/#+6 $��6#�+�� 4,/� ��

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Gh�qlflmd 91417 Uh�fl �fhpr gd mh yhnwruvnd ixqnflmd z = [ $ U?/ [ � U6/

glihuhqflmdeloqd x wrfnl {f 5 [/ dnr srvwrml olqhduql rshudwru D = U6 $U? wdndy gd mh

z+{,�z+{f, @ D+{� {f, . u+{� {f,>

sul fhpx }d ixqnflmx {� {f :$ u+{� {f, 5 U? prud yulmhglwl

olp%<%f

oE%3%f�8%3%f8 @ 3

+�rvwdwdn� u+{� {f, wh}l n qxoyhnwrux elwqr eu}h rg {� {f,1

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vydnrm wrfnl { 5 [1

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olp%<%f

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olp%<%f

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5<3 SRJODYOMH 91 XYRG X YHNWRUVNX DQDOL]X

Sulpmhu 91415 +d, Vydnd nrqvwdqwqd yhnwruvnd ixqnflmd z � f� = [ $ U?/

[ � U6 rwyruhq/ x elor nrml yhnwru y 5 U

?/ { :$ f�+{, @ y/ mh glihuhq0flmdeloqd l glihuhqflmdo x vydnrm wrfnl mrm mh qxorshudwru/ wm1 gi+{, @ R =U6 $ U

?/ }d vydnl { 5 [1 +Sulwrp mh l u nrqvwdqwqd qxoixqnflmd$,1 Qdlph/gh�qlflmvnd mhgqdnrvw rygmh srvwdmh y � y @ D+{� {f, . u+{� {f, sd xymhwr judqlfqrm yulmhgqrvwl sryodfl uhfhql }dnomxfdn1

+e, Qhnd mh z = [ $ U? vx}hqmh Z mf olqhduqrj rshudwrud Z = U6 $

U6 qd rwyruhql srgvnxs [ � U

61 Wdgd mh ixqnflmd z glihuhqflmdeloqd lgz+{, @Z x vydnrm wrfnl { 5 [1

Rguhglpr }dslv +pdwulfx, +d��, olqhduqrj rshudwrud D � gz+{f, = U6 $

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z @ +z�> � � � > z?,/ wr mh qmh}lqd glihuhqflmdeloqrvw x {f/ wm1z+{,�z+{f, @ gz+{f,+g{, . u+g{, l

olp_%<f

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hnylydohqwqd glihuhqflmdeloqrvwl vylk sulsdgqlk nrruglqdwqlk ixqnflmd z� x{f/ wm1

z�+{,�z�+{f, @ gz�+{f,+g{, . u�+g{, l

olp_%<f

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Exgx�fl gd x wud}hqrpx }dslvx prud m0wl uhgdn rg +d��, rgjrydudwl }dslvx

olqhduqrjd ixqnflrqod +y1¢81515/ +5,, gz�+{f, @ ^ Y��E%f�Y%�

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vyh sduflmdoqh ghulydflmhY��Y%�

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flp lpdmx vplvod1 Sulpmhulfh/g+z . x,+{f, @ gz+{f, . gx+{f, +glihuhqfludqmh }eurmd,>g+zmx,+{f, @ +x+{f,mgz+{f,, . +z+{f,mgx+{f,, +glihuhqfludqmh vndoduqrj

xpqrµnd,>g+x � z,+{f, @ gx+z+{f,, � gx+{f, +glihuhqfludqmh nrpsr}lflmh,1

Rygmh vh qh �fhpr xsxµwdwl x ud}pdwudqmh r glihuhqflmdeloqrvwl l glihuhq0flmdolpd ylµlk uhgryd yhnwruvnh ixqnflmh/ qhjr �fhpr vh mrµ pdor }dgu}dwl qd

9141 YHNWRUVNH IXQNFLMH 5<4

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Nrurodu 91414 Yhnwruvnd ixqnflmd z = [ $ U?/ [ � U/ mh ghulydeloqd x

wrfnl {f 5 [ rqgd l vdpr rqgd/ dnr yhnwruvnd ixqnflmd2z = [ q i{fj $ U

?>2z+{, @ �E%�3�E%f�

%3%f >

lpd judqlfqx yulmhgqrvw olp%f

2z x wrfnl {f1

+Sulsdgql yhnwru r}qdfxmhpr v z�+{f,/ wm1 olp%f

z @ olp%<%f

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l qd}lydpr ghulydflmrp yhnwruvnh ixqnflmh z x wrfnl {f1,

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�?+{f,,1

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z�+{f,g{1 Dnr mh yhnwruvnd ixqnflmd z = [ $ U?/ [ � U/ ghulydeloqd x

vydnrm wrfnl { 5 [/ rqgd mh greur gh�qludqd ixqnflmd +ghulydflmd rg z,z� = [ $ U

?/ { :$ z�+{,1 Mdvqr mh l ndnr wuhed gh�qludwl ghulydflmh ylµlk

uhgryd l ylµh glihuhqflmdoh1 Qdlph +xvs1 ¢71416/ +84, l +85,,/

z�� ghi1@ +z�,�> ===> zE?n�� ghi1

@ +zE?�,� whg2z+{f,g{ � g2z+{, @ z��+{,g{2> ===> goz+{f,g{ � goz+{, @ zEo�+{,g{o1

Sulpmhu 91416 Jledqmh pdwhulmdoqh wrfnh rslvxmh mhgqdg}edv+w, @ 5 frv wl. 5 vlq wm . 6wn/ w 5 ^3> �l +yulmhph,1

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2 1

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µwr vh redylmdmx�fl ydomdn {2 . |2 @ 7 �shqmh eu}lqrp� } @ 6w +y1 fuwh},1

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y+w, @ v�+w, @ �5 vlq wl. 5frv wm . 6n/ w 5 ^3> �l1Qdgdomh/ ghulyludmx�fl eu}lqx grelydpr xeu}dqmh/ wm1

d+w, @ y�+w, @ �5 frv wl�5 vlq wm/ w 5 ^3> �l1Qdsrnrq/ dnr mh w @ 3 rqgd mh y+3, @ 5m . 6n l d+3, @ �5l/ d dnr mh w @ Z

2rqgd mh y+Z2 , @ �5l. 6n l d+Z2 , @ �5m1

Qduhgql whruhp grqrvl ghulydflmvnd sudylod }d rvqryqh rshudflmh qdg yhn0wruvnlp ixqnflmdpd vndoduqh ydulmdeoh1

Whruhp 91416 Qhnd vx z> x = [ $ U?/ [ � U/ ghulydeloqh yhnwruvnh

ixqnflmh/ i = [ $ U ghulydeloqd +vndoduqd, ixqnflmd l �>� 5 U1 Wdgdyulmhgh ryd sudylod=+l, +�z . �x,� @ �z� . �x�>+ll, +iz,� @ i �z. iz�/ sul fhpx mh +iz,+{, @ i+{,z+{,>

+lll, +z

i,� @

iz� � i �z

i2/ sul fhpx mh +

z

i,+{, @

z+{,

i+{,l i+{, 9@ 3>

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�'�z�+{,x�+{, vndoduql

xpqr}dn1X voxfdmx q @ 6/ }d yhnwruvnl xpqr}dn z � x +y1 ¢51516,/ wm1 }d

+z� x,+{, @ z+{,� x+{, @

��

l m n

z�+{, z2+{, z�+{,x�+{, x2+{, x�+{,

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+y, +z � x,� @ z� � x.z� x�1

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Sulpmhu 91417 Rguhglpr ghulydflmx yhnwruvnh ixqnflmh $ = [ $ U�/ [ �

U/ dnr mh $ @ x�+y�z,/ sul fhpx vx x> y>z = [ $ U� ghulydeloqh yhnwruvnh

ixqnflmh1 Grelyhql lvkrg surymhulpr qd ixqnflmdpd x+{, @ +5{> 3>�6{2,/y+{, @ +{�> 3> 5 . {2, l z+{, @ +{> {2> {�,1Sulplmhqlpr +gydsxw, Whruhp 91416+y,=

$�+{, @ +x� +y �z,,�+{, @ x�+{,� +y �z,+{, . x+{,� +x� y,�+{, @x�+{,� +y+{,�z+{,, . x+{,� +y�+{,�z+{,, . x+{,� +y+{,�z�+{,,1

Exgx�fl gd mh x�+{, @ +5> 3>�9{,/ y�+{, @ +6{2> 3> 5{, l z�+{, @ +4> 5{> 6{2,wh

y+{,�z+{, @ +{S � {e � 5{2> {� . 5{>�{e,/

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Vdgd xyuµwdydqmhp l vuh¡lydqmhp grelydpr=+x+{,�+y+{,�z+{,,,� @ +6{D.9{�>�6{H.6{S.5{D.9{e> 5{e.7{2,� @+48{e . 4;{2>�57{. . 4;{D . 43{e . 57{�> ;{� . ;{,/ lx�+{,� +y+{,�z+{,, . x+{,� +y�+{,�z+{,, . x+{,� +y+{,�z�+{,, @+9{e.45{2>�9{..9{D.5{e.45{�> 5{�.7{,.+9{e>�<{..9{D.9{e> 7{�,.+6{e . 9{2>�<{. . 9{D . 5{e . 45{�> 5{� . 7{, @+48{e . 4;{2>�57{. . 4;{D . 43{e . 57{�> ;{� . ;{,1

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