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dvvxpswlrq1 Wklv jlyhv ulvh wr dq huuru lq wkh irupxod ghulyhg iurp GQD1
Iljxuh : vkrzv wkdw irupxod +48, lv pxfk pruh dffxudwh wkdq wkh irupxodh
rewdlqhg zkhq zh xvh GQD +iru yroxph iudfwlrqv yL A 3=7> l1h1 uhodwlyh wklfnqhvv
w@o ? 3=6:/ wkh dffxudf| ri +48, lv hyhq ehwwhu wkdq 4=;(,1 Lq dgglwlrq zh qrwh
wkdw wklv lpsuryhg irupxod +48, lv mxvw dv vlpsoh dv +<, dqg +43,1
43
Wdeoh 5/ Ydoxhv ri htxdwlrqv +<,/+43,/ dqg +48,
Fodvvlfdo
sodwh wkhru|
+ehqglqj.vkhdu,
yL
3=<
3=;
3=:
3=9
3=8
3=7
3=6
3=5
3=4
Htxdwlrq +<,
@ 4> n @ 4
8=8
+plq= 7= 9,
47= 4
+plq= 43= 9,
58=3
+plq= 4;= 3,
69=<
+plq= 59=:,
7;=9
+plq= 69=6,
8<=4
+plq= 79=8,
9:= 8
+plq= 89= <,
:6= 8
+plq= 9:=3,
:9=9
+plq= :9=3,
pg J
W>78 ^(`
Fodvvlfdo
sodwh wkhru|
+sxuh ehqglqj,
Htxdwlrq +43,
@ 4> n @ 3
7=53
+plq= 7=4:,
<= 3;
+plq= ;= <5,
47= 99
+plq= 47= 3;,
53= <6
+plq= 4<= 75,
5:= ;8
+plq= 57= 89,
68= 6:
+plq= 5;= ;:,
76= 68
+plq= 64= 5:,
84= 78
+plq= 5<= 73,
8;= :5
+plq= 49= <<,
ph J W>78
Lpsuryhg
irupxod
Htxdwlrq +48,
@ 4> @ 4
3=<:<
+plq= 3= <;7,
4= 8:<
+plq= 4= 8;8,
4= :59
+plq= 4=:65,
4= 636
+plq= 4=649,
3= 544
+plq= 3=54<,
4=:93
+plq= 4= 8;;,
7=;46
+plq= 6= <<6,
<=389
+plq= 9= 739,
47=369
+plq= := 83:,
3=347;
3=3537
3=356<
3=3658
3=37;7
3=3745
3=36;7
3=349:
3=34<3
Wdeoh 5
Uhihuhqfhv
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fdofxodwlqj vwlqhvv surshuwlhv ri shulrglf vwuxfwxuhv1 Dssolfdwlrq ri Pdwk1 +wr
dsshdu,1
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ri vtxduh krqh|frpev dqg rwkhu xqlgluhfwlrqdo frpsrvlwhv1 Frpsrvlwhv Sduw E1
+wr dsshdu,1
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wlrq ri qhwzrunv dqg mxqfwlrqv/ Wr dsshdu lq= Dv|pswrwlf Dqdo1
44
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wxuhv/ Dssolhg Pdwkhpdwlfdo Vflhqfh +hg1 Pduvghq M1 H1 dqg Vlurylfk O1,/ 469/
Vsulqjhu/ Qhz \run1
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4774047951
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Phfkdqlfv/ 83/ 7;408381
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ri Vrolg Phfkdqlfv/ NWK Uhvhdufk Uhsruw WULWD0KOI035831
^43` Kroperp D1/ Shuvvrq/ O1 H1 dqg Vydqvwhgw Q1/ +4<<5,/ D krprjhql}dwlrq surfh0
gxuh iru frpsxwlqj hhfwlyh prgxol dqg plfurvwuhvvhv lq frpsrvlwh pdwhuldov1 M1
Frpsrvlwh Hqjlqhhulqj 5/ Qr 7/ 57<058<1
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rshudwruv dqg lqwhjudo ixqfwlrqdov/ Vsulqjhu0Yhuodj/ Ehuolq1
^45` Oxnndvvhq G1/ +4<<<,/ D qhz uhlwhudwhg vwuxfwxuh zlwk rswlpdo pdfurvfrslf eh0
kdylru/ VLDP M1 Dsso1 Pdwk1 8</ 8/ 4;5804;751
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hpdwlfdo dvshfwv rq wkh krprjhql}dwlrq phwkrg1 Frpsrvlwhv Hqjlqhhulqj 8/ 8/
84<08641
^47` Phlghoo/ D1 dqg Zdoo/ S1/ +4<<:,/ Krprjhql}dwlrq dqg ghvljq ri vwuxfwxuhv zlwk rs0
wlpdo pdfurvfrslf ehkdylru/ lq Khuqäqgh} V1 dqg Euhggld F1D1 +hgv1,/ Frpsxwhu
dlghg rswlpxp ghvljq ri vwuxfwxuhv Y/ Frpsxwdwlrqdo Phfkdqlfv Sxeolfdwlrqv/
Vrxwkdpswrq/ 6<607351
^48` Phlghoo/ D1/ +4<<;,/ Wkh rxw0ri0sodqh vkhdu prgxoxv ri wzr0frpsrqhqw uhjxodu
krqh|frpev zlwk duelwudu| wklfnqhvv/ lq Phfkdqlfv ri Frpsrvlwh Pdwhuldov dqg
Vwuxfwxuhv +hgv1 F1D1 Prwd Vrduhv/ F1P1 Prwd Vrduhv dqg P1M1P1 Iuhlwdv,/ QDWR
DVL/ Wurld/ Sruwxjdo/ Yro1 LLL/ 69:06:<1
^49` Phlghoo D1 dqg Shuvvrq O1 H1/ +4<<<,/ Rswlpdo ghvljq ri uhfwdqjxodu krqh|frpev lq
uhfwdqjxodu vdqgzlfk sodwhv/ Surfhhglqjv ri 9 Lqwhuqdwlrqdo Frqihuhqfh rq Frp0
srvlwhv Hqjlqhhulqj/ 89:089;/ Ruodqgr1
^4:` Sdqdvhqnr J1S1/ +4<<5,/ Dv|pswrwlf vroxwlrqv ri wkh hodvwlvlw| wkhru| v|vwhp ri
htxdwlrqv iru odwwlfh dqg vnhohwdo vwuxfwxuhv/ Pdwk1 Veruqln/ 4;6/ 4/ ;<0446 +lq
Uxvvldq,1 Hqjolvk wudqvodwlrq= +4<<6,/ Dfdg1 Vfl1 Veruqln Pdwk1/ :8/ 4/ ;804431
^4;` Shuvvrq O1H1/ Shuvvrq O1/ Vydqvwhgw Q1 dqg Z|oohu M1/ +4<<6,/ Wkh krprjhql}dwlrq
phwkrg= Dq lqwurgxfwlrq/ Vwxghqwolwwhudwxu/ Oxqg1
45
^4<` Wlprvkhqnr V1 S1 dqg Jrrglhu M1 Q1/ +4<84,/ Wkhru| ri hodvwlvlw|/ Wklug Hglwlrq
+4<;:,/ PfJudz0Kloo/ Lqf1/ Qhz \run1
^53` Wrutxdwr V1/ Jleldqvnl O1Y1/ dqg Jlevrq O1 M1/ +4<<;,/ Hhfwlyh Phfkdqlfdo dqg
Wudqvsruw Surshuwlhv ri fhooxodu Vrolgv/ Phfkdqlfdo Vflhqfhv/ 73/ :40;51
^54` ]lhqnlhzlf} R1 F1 dqg ]kx M1 ]1/ +4<;:,/ D vlpsoh huuru hvwlpdwru dqg dgdswlyh
surfhgxuh iru sudfwlfdo hqjlqhhulqj dqdo|vlv/ Lqw1 M1 iru Qxphulfdo Phwkrgv lq
Hqjlqhhulqj/ Yro1 57/ 66:068:1
46
Dsshqgl{ D
Wkh lqhtxdolwlhv +46, duh suryhg dv iroorzv= E| xvlqj wkh lqhtxdolw| md . em mdm mem zh jhw
. J J J
J
J
J
W>78
Irup
W>78
W>78
Irup
W>78
@ 433
pg J
W>78 @ 433
J
J
W>78
W>78
433
J
W>78 JIrup
J
W>78
433
JW>78 JW>78 J
W>78
JW>78 JW>78 W>78 JIrup
433
433
JW>78
J
W>78
J
W>78 JIrup
ph JW>78 >
433
JW>78
J
zklfk jlyhv wkh uljkw vlgh ri +46,1 Vlploduo|/ e| xvlqj wkh lqhtxdolw| md . em mdm . mem zh rewdlq
J
JW>78 JIrup . J
W>78
W>78
pg J
W>78 @ 433
JW>78
Irup J
JW>78 J
J
W>
78
W>78
. 433
433
J
J
W>78
W>78
J
W>78 JIrup
. ph JW>78 >
433
JW>78
zklfk jlyhv wkh ohiw vlgh ri +46,1
Dsshqgl{ E
Wkh qxphulfdo ydoxhv duh suhvhqwhg khuh1
47
3=<8
3=<
3=;
3=:
3=9
3=8
3=7
3=6
3=5
3=4
3
3=4
3=5
3=6
3=7
3=78
3=<8
3=<
3=;
3=:
3=9
3=8
3=7
3=6
3=5
3=4
3
3=4
3=5
3=6
3=7
3=78
yL @ 3=4
J
W>78 @H
6=<:4:984343
5=34:3947963
4=37454<:563
3=:4:73388:3
3=889:353796
3=79468633:4
3=6<;::<:464
3=688367;866
3=656488::5<
3=5<<5<854;7
3=5;44869698
3=59:5<6639;
3=589::37393
3=57;<943646
3=57677649:
3=574786<653
ph JW>78 ^(
3=34<3
3=34;;
3=34;6
3=34:;
3=34:5
3=349:
3=3494
3=3488
3=347<
3=3476
3=3469
3=3463
3=3456
3=3449
3=3443
3=343;
yL @ 3=5
J
@H
W>78
4=:;75787693
<=4465376;33 434
7=:9484;4533 434
6=656;547853 434
5=9493953833 434
5=5349867583 434
4=<686473873 434
4=:883;76<:3 434
4=9635;:9783 434
4=87758;9643 434
4=7;:797<;:3 434
4=78773<9833 434
4=7754:43:<3 434
4=77<:565<63 434
4=7::9<88;73 434
4=8333556683 434
yL @ 3=6
J
@H
ph JW>78 ^(`
W>78
;=6:7896833 434
7=5;;333683 434
5=585699843 434
4=8;48535:7 434
4=585;69468 434
4=394<;7<5: 434
<=74389;883 435
;=944;43963 435
;=3;5363443 435
:=:78<<8<33 435
:=896356653 435
:=8463:<543 435
:=8<3<968:3 435
:=;37;5;893 435
;=4:;8:55;3 435
;=76;:57983 435
3=36;7
3=36;5
3=36:<
3=36:8
3=36:5
3=369;
3=3698
3=3694
3=368;
3=3687
3=3684
3=367<
3=367:
3=367:
3=3684
3=3688
48
ph JW>78 ^(`
3=349:
3=3499
3=3497
3=3495
3=3493
3=348:
3=3488
3=3485
3=3483
3=347:
3=3478
3=3475
3=346<
3=346:
3=3469
3=3469
yL @ 3=7
J
@H
ph JW>78 ^(`
W>78
6=<6<75;783 434
5=34<76:9:5 434
4=396759;<: 434
:=7;:::5443 435
8=<83393;43 435
8=3945;<853 435
7=8359<8:<3 435
7=46<43<5:3 435
6=<37966;;3 435
6=:97;;4963 435
6=:3544:<83 435
6=:3;<76883 435
6=:;94:76;3 435
6=<75<8<5:3 435
7=4<<9<3:<3 435
7=6:9988333 435
3=369;
3=369;
3=3699
3=3697
3=3696
3=3695
3=3694
3=3693
3=3693
3=3694
3=3696
3=3699
3=36:4
3=36;4
3=36<:
3=3745
3=<8
3=<
3=;
3=:
3=9
3=8
3=7
3=6
3=5
3=4
3
3=4
3=5
3=6
3=7
3=78
3=<8
3=<
3=;
3=:
3=9
3=8
3=7
3=6
3=5
3=4
3
3=4
3=5
3=6
3=7
3=78
yL @ 3=8
J
@H
W>78
4=:;8;97733 434
<=48<<:;963 435
7=;5<7;;483 435
6=738489983 435
5=:43336:43 435
5=63<4:5<:3 435
5=38;684<<6 435
4=;<96<:67: 435
4=:<6973;54 435
4=:67:7673: 435
4=:4535;676 435
4=:55:7756< 435
4=:9;4:83<3 435
4=;87385<98 435
4=<<5835688 435
5=3;;3:3743 435
ph JW>78 ^(`
3=379:
3=3798
3=3795
3=378<
3=3789
3=3786
3=3784
3=377<
3=377:
3=3778
3=3778
3=3779
3=377;
3=3788
3=379<
3=37;7
yL @ 3=:
J
@H
ph JW>78 ^(`
W>78
5=94;9:3;9; 435
4=676:33778 435
:=3<3989683 436
8=33765:7;3 436
6=<;:38:7;7 436
6=7348833:; 436
6=369685<3; 436
5=;34<;93;; 436
5=98848<5:5 436
5=8:69<:5;< 436
5=879:95596 436
5=8:43476;8 436
5=97<7:4;;4 436
5=:<57<3376 436
6=35484:695 436
6=4;33<;76< 436
3=3558
3=3557
3=3556
3=3555
3=3554
3=3553
3=3553
3=354<
3=354<
3=354<
3=354<
3=3553
3=3555
3=3559
3=3566
3=356<
49
yL @ 3=9
J
@H
W>78
:=759<:6763 435
6=;437834;3 435
5=3435395<4 435
4=74;5<:663 435
4=45<8<;:86 435
<=9666;6:33 436
;=8<8599<33 436
:=<5::88673 436
:=83:;79993 436
:=5:566<;43 436
:=4<34766;3 436
:=58444;8:3 436
:=795:8<493 436
:=;8593:7:3 436
;=7::<<9;53 436
;=<437;;683 436
ph JW>78 ^(`
3=363;
3=363:
3=3638
3=3637
3=3635
3=3634
3=35<<
3=35<;
3=35<:
3=35<:
3=35<:
3=35<;
3=3633
3=3638
3=3647
3=3658
yL @ 3=;
J
@H
ph JW>78 ^(`
W>78
9=96:44<363 436
6=738;687<7 436
4=:<:77;794 436
4=59;:5<7:8 436
4=343<9::<5 436
;=959765483 437
:=:3499<843 437
:=43;9;;:73 437
9=:6:;7;583 437
9=865<<9<;3 437
9=799;:;733 437
9=864544:;3 437
9=:67384:53 437
:=435678343 437
:=9<4:8;9;3 437
;=433337363 437
3=34:5
3=34:5
3=34:4
3=34:4
3=34:3
3=34:3
3=34:3
3=34:3
3=34:4
3=34:5
3=34:6
3=34:9
3=34:<
3=34;8
3=34<8
3=3537
3=<8
3=<
3=;
3=:
3=9
3=8
3=7
3=6
3=5
3=4
3
3=4
3=5
3=6
3=7
3=78
yL @ 3=<
J
@H
W>78
:= 5669;776< 437
6= :45344866 437
4= <8<439438 437
4= 6;5;;<<83 437
4= 434<;4<83 437
<=7367<7743 438
;=6<8<47;93 438
:=:7<<<3859 438
:=67958879; 438
:=456869739 438
:=3854:8:;9 438
:=4565:59:9 438
:=678:35558 438
:=:7<385<43 438
;=6<77898:3 438
;=;748;:753 438
ph JW>78 ^(`
3=3454
3=3454
3=3453
3=3453
3=3453
3=3453
3=3453
3=3453
3=3453
3=3454
3=3456
3=3458
3=345;
3=3466
3=3474
3=347;
Zh kdyh xvhg txdgulodwhudo hohphqwv zlwk hljkw qrghv wkdw ghqh wkh hoh0
phqw1 Hdfk qrgh kdv wzr ghjuhhv ri iuhhgrp> wudqvodwlrqv lq wkh qrgdo {4 dqg
{5 gluhfwlrqv1 Lq wkh wdeoh ehorz zh kdyh olvwhg wkh vshflf qxpehu ri hoh0
phqwv dqg qrghv xvhg iru hdfk yroxphiudfwlrq zkhq vroylqj wkh qlwh hohphqw
sureohpv1 Wkh vdph hohphqwphvk lv xvhg iru hdfk Srlvvrq*v udwlr zlwklq hdfk
yroxphiudfwlrq1
yL
3=4
3=5
3=6
3=7
3=8
3=9
3=:
3=;
3=<
Qxpehu ri
hohphqwv qrghv
;798
58<5;
;;44
5<<8;
:<<8
5778:
;;<:
5:495
<975
5<754
94;5
4<344
:696
55975
:836
56463
97;3
5344<
4: