lial_22816_IRG/SM_TTL_CPY.qxd
5/18/04
8:07 AM
Page 1
INSTRUCTOR’S RESOURCE GUIDE AND SOLUTIONS MANUAL to accompany
CALCULUS WITH APPLICATIONS EIGHTH EDITION
AND
CALCULUS WITH APPLICATIONS Brief Version EIGHTH EDITION
Lial
•
Greenwell
•
Ritchey
lial_22816_IRG/SM_TTL_CPY.qxd
5/18/04
8:07 AM
Page 2
This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes. All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials.
Reproduced by Pearson Addison-Wesley from supplied electronic files. Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. ISBN
0-321-22816-2
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FKDSWHU 4 OLQHDU IXQFWLRQV 414 Vorshv dqg Htxdwlrqv ri Olqhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 74 415 Olqhdu Ixqfwlrqv dqg Dssolfdwlrqv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 7: 416 Wkh Ohdvw Vtxduhv Olqh = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 7< Fkdswhu 4 Uhylhz H{huflvhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 87 H{whqghg Dssolfdwlrq= Xvlqj H{wudsrodwlrq wr Suhglfw Olih H{shfwdqf| = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 8:
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FKDSWHU 6 GHULYDWLYH 614 Olplwv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = <: 615 Frqwlqxlw| = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 435 616 Udwhv ri Fkdqjh = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 437 617 Gh qlwlrq ri wkh Ghulydwlyh = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 43< 618 Judsklfdo Gl huhqwldwlrq = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 448 Fkdswhu 6 Uhylhz H{huflvhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 449
FKDSWHU 7 FDOFXODWLQJ WKH GHULYDWLYH 714 Whfkqltxhv iru Ilqglqj Ghulydwlyhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 454 715 Ghulydwlyhv ri Surgxfwv dqg Txrwlhqwv= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =458 716 Wkh Fkdlq Uxoh= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =45; 717 Ghulydwlyhv ri H{srqhqwldo Ixqfwlrqv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 465 718 Ghulydwlyhv ri Orjdulwkplf Ixqfwlrqv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 469 Fkdswhu 7 Uhylhz H{huflvhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 46<
FKDSWHU 8 JUDSKV DQG WKH GHULYDWLYH 814 Lqfuhdvlqj dqg Ghfuhdvlqj Ixqfwlrqv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 478 815 Uhodwlyh H{wuhpd = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 47< 816 Kljkhu Ghulydwlyhv/ Frqfdylw|/ dqg wkh Vhfrqg Ghulydwlyh Whvw = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 487 ly
817 Fxuyh Vnhwfklqj = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 493 Fkdswhu 8 Uhylhz H{huflvhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 49; FKDSWHU 9 DSSOLFDWLRQV RI WKH GHULYDWLYH 914 Devroxwh H{wuhpd = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 4:7 915 Dssolfdwlrqv ri H{wuhpd = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 4:: 916 Ixuwkhu Exvlqhvv Dssolfdwlrqv= Hfrqrplf Orw Vl}h> Hfrqrplf Rughu Txdqwlw|> Hodvwlflw| ri Ghpdqg = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 4;7 917 Lpsolflw Gl huhqwldwlrq = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =4;9 918 Uhodwhg Udwhv= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =4<3 919 Gl huhqwldov= Olqhdu Dssur{lpdwlrq = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 4<7 Fkdswhu 9 Uhylhz H{huflvhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 4<8 H{whqghg Dssolfdwlrq= D Wrwdo Frvw Prgho iru d Wudlqlqj Surjudp = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 533 FKDSWHU : LQWHJUDWLRQ :14 Dqwlghulydwlyhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 534 :15 Vxevwlwxwlrq = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 538 :16 Duhd dqg wkh Gh qlwh Lqwhjudo= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =53; :17 Wkh Ixqgdphqwdo Wkhruhp ri Fdofxoxv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =547 :18 Wkh Duhd Ehwzhhq Wzr Fxuyhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 553 :19 Qxphulfdo Lqwhjudwlrq = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =559 Fkdswhu : Uhylhz H{huflvhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 564 H{whqghg Dssolfdwlrq= Hvwlpdwlqj Ghsohwlrq Gdwhv iru Plqhudov = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 56: FKDSWHU ;
IXUWKHU WHFKQLTXHV DQG DSSOLFDWLRQV RI LQWHJUDWLRQ
;14 Lqwhjudwlrq e| Sduwv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 56< ;15 Yroxph dqg Dyhudjh Ydoxh = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =574 ;16 Frqwlqxrxv Prqh| Iorz = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 578 ;17 Lpsurshu Lqwhjudov = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 57: Fkdswhu ; Uhylhz H{huflvhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 584 H{whqghg Dssolfdwlrq= Hvwlpdwlqj Ohduqlqj Fxuyhv lq Pdqxidfwxulqj zlwk Lqwhjudov = = = = = = = = = = = = = = = = = = = = = 588 y
FKDSWHU < PXOWLYDULDEOH FDOFXOXV <14 Ixqfwlrqv ri Vhyhudo Yduldeohv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 589 <15 Sduwldo Ghulydwlyhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 593 <16 Pd{lpd dqg Plqlpd = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =599 <17 Odjudqjh Pxowlsolhuv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 5:4 <18 Wrwdo Gl huhqwldov dqg Dssur{lpdwlrqv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 5:9 <19 Grxeoh Lqwhjudov = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 5:< Fkdswhu < Uhylhz H{huflvhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 5;7 H{whqghg Dssolfdwlrq= Xvlqj Pxowlyduldeoh Ilwwlqj wr Fuhdwh d Uhvsrqvh Vxuidfh Ghvljq = = = = = = = = = = = = = = = = 5<5
FKDSWHU 43 GLIIHUHQWLDO HTXDWLRQV 4314 Vroxwlrqv ri Hohphqwdu| dqg Vhsdudeoh Gl huhqwldo Htxdwlrqv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 5<7 4315 Olqhdu IluvwRughu Gl huhqwldo Htxdwlrqv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 633 4316 Hxohu*v Phwkrg = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 637 4317 Dssolfdwlrqv ri Gl huhqwldo Htxdwlrqv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 63: Fkdswhu 43 Uhylhz H{huflvhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =645 H{whqghg Dssolfdwlrq= Srooxwlrq ri wkh Juhdw Odnhv = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 64:
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Udwhv ri Fkdqjh
51 | @ 7{5 9 @ i+{, ehwzhhq { @ 5 dqg { @ 8 Dyhudjh udwh ri fkdqjh @
i+8, i+5, 85
431 v+w, @ w5 . 8w . 5 v+4 . k, v+4, k ^+4 . k,5 . 8+4 . k, . 5`^+4,5 . 8+4, . 5` @ olp k$3 k 5 ^4 . 5k . k . 8 . 8k . 5`^4 . 8 . 5` @ olp k$3 k 5 ; . :k . k ; @ olp k$3 k :k . k5 @ olp k$3 k k+: . k, @ olp k$3 k @ olp +: . k, @ : olp
k$3
k$3
Wkh lqvwdqwdqhrxv yhorflw| dw w @ 4 lv :1
+439, +55, 85 439 . 55 @ 6 ;7 @ 5; @ 6
451 v+w, @ w6 . 5w . <
@
v+7 . k, v+7, k$3 k +7.k,6.5+7.k,.<^76.5+7,.<` @ olp k$3 k 76.6+75 ,k.6+7k5 ,.k6.;.5k.<+76.;.<, @ olp k$3 k 6 5 k . 45k . 7;k . 5k @ olp k$3 k 6 5 k . 45k . 83k @ olp k$3 k 5 k+k . 45k . 83, @ olp k$3 k 5 @ olp +k . 45k . 83, @ 83 olp
71 | @ 6{6 . 5{5 7{ . 4 @ i+{, ehwzhhq { @ 3 dqg { @ 4 i+4, i+3, Dyhudjh udwh ri fkdqjh @ 43 +7, +4, @ 43 8 @ @ 8 4 s 91 | @ 6{ 5 @ i+{, ehwzhhq { @ 4 dqg { @ 5 i+5, i+4, 54 54 @ 54 4 @ @4 4
Dyhudjh udwh ri fkdqjh @
;1 | @
WKH GHULYDWLYH
8 @ i+{, ehwzhhq { @ 5 dqg { @ 7 5{ 6 i+7, i+5, 75 4 +8, @ 75 4 . 8 @ 5 7 @ @5 5
Dyhudjh udwh ri fkdqjh @
k$3
Wkh lqvwdqwdqhrxv yhorflw| dw w @ 7 lv 831 471 v+w, @ 7w5 9 dw w @ 5 v+5 . k, v+5, k 7+5 . k,5 9 ^7+5,5 9` @ olp k$3 k 7+7 . 7k . k5 , 9 . 49 . 9 @ olp k$3 k 49k 7k5 @ olp k$3 k k+49 7k, @ olp k$3 k @ olp +49 7k,
olp
k$3
k$3
@ 49 Wkh lqvwdqwdqhrxv udwh ri fkdqjh dw w @ 5 lv 49=
Vhfwlrq 616
Udwhv ri Fkdqjh
491 I +{, @ {5 . 5 dw { @ 3 olp
k$3
I +3 . k, I +3, k
+3 . k,5 . 5 ^35 . 5` @ olp k$3 k
43<
531 i+{, @ {oq
{
dw { @ 6
k =34
k5 @ olp k$3 k
i+6 . =34, i+6, =34 @
6=34oq
6=34
6oq
6
=34
@ 5=78:6
@ olp k k$3
@3
=334
Wkh lqvwdqwdqhrxv udwh ri fkdqjh dw { @ 3 lv 31
i+6 . =334, i+6, =334 @
{
4;1 i+{, @ { dw { @ 6
6=334oq
6=334
6oq
6
=334
@ 5=77<8
k =34
i+6 . =34, i+6, =34 @
6=346=34 66 =34
@ 8:=63:5 =334
i+6 . =334, i+6, =334
6=33334oq 6=33334 6oq =33334
6
@ 5=77;9 Wkh lqvwdqwdqhrxv udwh ri fkdqjh dw { @ 6 lv 5177;91
i+6 . =33334, i+6, =33334
571 +d, s+4<<8, @ 4:=7 s+4<<:, @ 49=7
6=333346=33334 66 =33334
@ 89=9965 i+6 . =333334, i+6, =333334 @
6=3333346=333334 66 =333334
@ 89=9959 =3333334
@
@ 89=:598
@
=333334
i+6 . =33334, i+6, =33334
551 Li wkh lqvwdqwdqhrxv udwh ri fkdqjh ri i+{, zlwk uhvshfw wr { lv srvlwlyh zkhq { @ 4> wkh ixqfwlrq zrxog eh lqfuhdvlqj1
@
=33334
6=3346=334 66 =334
=33334
i+6 . =3333334, i+6, =3333334 6=33333346=3333334 66 @ =3333334 @ 89=9958
Wkh lqvwdqwdqhrxv udwh ri fkdqjh dw { @ 6 lv 89199581
s+4<<:, s+4<<8, 4<<: 4<<8 49=7 4:=7 @ 5 4 @ =8 @ 5
Vorsh @
Wkh udwh ri fkdqjh lv =8 zklfk phdqv wkdw wkh shufhqwdjh ri vdohv frqvlvwlqj ri lpsruwv ghfuhdvhg dq dyhudjh ri 18( shu |hdu iurp 4<<8 wr 4<<:1 +e, s+4<<<, @ 54=4 s+5334, @ 58=3 s+5334, s+4<<<, 5334 4<<< 58=3 54=4 @ 5 6=< @ 4=<8 @ 5
Vorsh @
443
Fkdswhu 6
WKH GHULYDWLYH
Wkh udwh ri fkdqjh lv 4=<8 zklfk phdqv wkdw wkh shufhqwdjh ri vdohv frqvlvwlqj r psruwv lqfuhdvhg dq dyhudjh ri 41<8( shu |hdu iurp 4<<< wr 53341 +f, Qr 591 +d, E| orrnlqj dw wkh judskv/ zh vhh wkdw iurp 4<<4 wr 4<<: wkh fxuyh uhsuhvhqwlqj wkh udwlr ri Lqwhuqhw phvvdjhv wr whohskrqh fdoov lv vwhhshu wkdq wkh fxuyh uhsuhvhqwlqj wkh udwlr ri krxvhkrogv zlwk SFv wr krxvhkrogv zlwkrxw SFv> wkxv wkh fxuyh uhsuhvhqwlqj wkh udwlr ri Lqwhuqhw phvvdjhv wr whohskrqh fdoov kdv d juhdwhu udwh ri fkdqjh ryhu wklv wlph lqwhuydo1 +e, Wkh qrqolqhdu fxuyh/ wkdw lv/ wkh fxuyh uhsuhvhqwlqj wkh udwlr ri Lqwhuqhw phvvdjhv wr whohskrqh fdoov/ kdv dq lqfuhdvlqj udwh ri fkdqjh1 5;1 U @ 43{ =335{5 +d, Dyhudjh udwh ri fkdqjh @
;338=<<; ;333 U+4334, U+4333, @ @ 8=<<; 4334 4333 4
Wkh dyhudjh udwh ri fkdqjh lv '8<<;1 +e, Pdujlqdo uhyhqxh @ olp
k$3
@ olp
k$3
@ olp
k$3
@ olp
k$3
@ olp
k$3
U+4333 . k, U+4333, k ^43+4333 . k, =335+4333 . k,5 ` ^43+4333, =335+4333,5 ` k ^43/333 . 43k =335+4/333/333 . 5333k . k5 ` ;333 k 43/333 . 43k 5333 7k =335k5 ;333 k 5 9k =335k k+9 =335k, @ olp @ olp +9 =335k, @ 9 k$3 k$3 k k
Wkh pdujlqdo uhyhqxh lv '93331 +f, Dgglwlrqdo uhyhqxh @ U+4334, U+4333, @ ^43+4334, =335+4334,5 ` ^43+4333, =335+4333,5 ` @ ;338=<<; ;333 Wkh dgglwlrqdo uhyhqxh lv '8<<;1 +g, Wkh dqvzhuv wr sduwv +d, dqg +f, duh wkh vdph1 631 s+w, @ w5 . w +d, s+4, @ 45 . 4 @ 5 s+7, @ 75 . 7 @ 53 Dyhudjh udwh ri fkdqjh @
53 5 s+7, s+4, @ @9 74 6
Wkh dyhudjh udwh ri fkdqjh lv 9( shu gd|1 s+6 . k, s+6, +6 . k,5 . +6 . k, +65 . 6, < . 9k . k5 . 6 . k 45 @ olp @ olp k$3 k$3 k$3 k k k k5 . :k k+k . :, @ olp @ olp +k . :, @ : @ olp k$3 k$3 k$3 k k
+e, olp
Wkh lqvwdqwdqhrxv udwh ri fkdqjh lv :( shu gd|1
Vhfwlrq 616
Udwhv ri Fkdqjh
444
651 +d, S +5, @ 8> S +4, @ 6 86 5 S +5, S +4, @ @ @5 54 54 4 Iurp 4 plq wr 5 plq/ wkh srsxodwlrq ri edfwhuld lqfuhdvhv/ rq wkh dyhudjh/ 5 ploolrq shu plq1 Dyhudjh udwh ri fkdqjh @
+e, S +6, @ 7=5> S +5, @ 8 7=5 8 S +6, S +5, =; @ @ @ =; 65 65 4 Iurp 5 plq wr 6 plq/ wkh srsxodwlrq ri edfwhuld ghfuhdvhv/ rq wkh dyhudjh/ =; ploolrq ru ;33/333 shu plq1 Dyhudjh udwh ri fkdqjh @
+f, S +6, @ 7=5> S +7, @ 5 Dyhudjh udwh ri fkdqjh 5 7=5 S +7, S +6, 5=5 @ @ @ 5=5 76 76 4 Iurp 6 plq wr 7 plq/ wkh srsxodwlrq ri edfwhuld ghfuhdvhv/ rq wkh dyhudjh/ 5=5 ploolrq shu plq1 @
+g, S +7, @ 5> S +8, @ 4 S +8, S +7, 45 4 @ @ @ 4 87 87 4 Iurp 7 plq wr 8 plq/ wkh srsxodwlrq ghfuhdvhv/ rq wkh dyhudjh/ 4 ploolrq shu plq1 Dyhudjh udwh ri fkdqjh@
+h, Wkh srsxodwlrq lqfuhdvhg xs wr 5 plq diwhu wkh edfwhulflgh zdv lqwurgxfhg/ exw ghfuhdvhg diwhu 5 plq1 +i , Wkh udwh ri ghfuhdvh ri wkh srsxodwlrq vorzv grzq dw derxw 6 plq1 Wkh judsk ehfrphv ohvv dqg ohvv vwhhs diwhu wkdw srlqw1 671 +d,
+e, Wkh dyhudjh udwh ri fkdqjh gxulqj wkh uvw krxu lv I +4, I +3, ;4=84 43 nlormrxohv shu krxu shu krxu1 +f, Vwruh I +w, lq d ixqfwlrq phqxv ri d judsklqj fdofxodwru1 4 +4, Vwruh \4 +4.[,\ dv \5 lq wkh ixqfwlrq phqx/ zkhuh \4 uhsuhvhqwv I +w,1 Vxevwlwxwh vpdoo ydoxhv iru [ lq \5 [ shukdsv zlwk xvh ri d wdeoh ihdwxuh ri wkh judsklqj fdofxodwru1 Dv [ lv doorzhg wr jhw vpdoohu/ \5 dssurdfkhv 4;1;4 nlormrxohv shu krxu shu krxu1 +g, Wkurxjk xvh ri d PD[2ihdwxuh surjudp ri d judsklqj fdofxodwru/ wkh pd{lpxp srlqw vhhq lq sduw +d, lv hvwlpdwhg wr rffxu dw dssur{lpdwho| w @ 4=6 krxuv1
445
Fkdswhu 6
691 Ohw F+w, eh fdqqdelv surgxfwlrq +lq phwulf wrqv, iru |hdu w dqg R+w, eh rslxp surgxfwlrq +lq phwulf wrqv, iru |hdu w1 +d, F+4<<;, @ 48/;33 F+5333, @ 47/833 F+5334, @ 47/<33 Dyhudjh udwh ri fkdqjh iurp 4<<; wr 5333 F+5333, F+4<<;, 5333 4<<; 47/833 48/;33 @ 5 @ 983
@
Rq dyhudjh/ wkh surgxfwlrq ghfuhdvhg derxw 983 phwulf wrqv shu |hdu1 Dyhudjh udwh ri fkdqjh iurp 5333 wr 5334 F+5334, F+5333, 5334 5333 47/<33 47/833 @ 4 @ 733 @
Rq dyhudjh/ wkh surgxfwlrq lqfuhdvhg derxw 733 phwulf wrqv shu |hdu1 R+4<<;, @ 77;9 R+5333, @ 8343 R+5334, @ 4569 Dyhudjh udwh ri fkdqjh iurp 4<<; wr 5333 R+5333, R+4<<;, 5333 4<<; 8343 77;9 @ 5 @ 595 @
Rq dyhudjh/ wkh surgxfwlrq lqfuhdvhg derxw 593 phwulf wrqv shu |hdu1 Dyhudjh udwh ri fkdqjh iurp 5333 wr 5334 R+5334, R+5333, 5334 5333 4569 8343 @ 4 @ 6::7 @
Rq dyhudjh/ wkh surgxfwlrq ghfuhdvhg derxw 6;33 phwulf wrqv shu |hdu1 +e, Rslxp kdg wkh juhdwhvw fkdqjh lq qhw surgxf0 wlrq1
WKH GHULYDWLYH
W +6333, W +4333, 56 48 @ 6333 4333 5333 ; @ 5333 7 @ 4333 Iurp 4333 wr 6333 iw/ wkh whpshudwxuh fkdqjhv derxw 7 shu 4333 iw> wkh whpshudwxuh ulvhv +rq wkh dyhudjh,1
6;1 +d,
W +8333, W +4333, 55 48 @ 8333 4333 7333 : @ 7333 4=:8 @ 4333 Iurp 4333 wr 8333 iw/ wkh whpshudwxuh fkdqjhv derxw 41:8 shu 4333 iw> wkh whpshudwxuh ulvhv +rq wkh dyhudjh,1 +e,
48 56 W +<333, W +6333, @ <333 6333 9333 ; @ 9333 76 @ 4333 Iurp 6333 wr <333 iw/ wkh whpshudwxuh fkdqjhv derxw 76 shu 4333 iw> wkh whpshudwxuh idoov +rq wkh dyhudjh,1
+f,
W +<333, W +4333, 48 48 @ @3 <333 4333 ;333 Iurp 4333 wr <333 iw/ wkh whpshudwxuh fkdqjhv derxw 3 shu 4333 iw> wkh whpshudwxuh vwd|v frq0 vwdqw +rq wkh dyhudjh,1 +g,
+h, Wkh whpshudwxuh lv kljkhvw dw 6333 iw dqg orz0 hvw dw 4333 iw1 Li :333 iw lv fkdqjhg wr 43/333 iw/ wkh orzhvw whpshudwxuh zrxog eh dw 43/333 iw1 +i , Wkh whpshudwxuh dw <333 iw lv wkh vdph dv 4333 iw1 731 +d, Dyhudjh udwh ri fkdqjh iurp 18 wr 4= 7; 53 i+4, i+=8, @ @ 89 psk 4 =8 =8 Dyhudjh udwh ri fkdqjh iurp 4 wr 418= i+4=8, i+4, ;3 7; @ @ 97 psk 4=8 4 =8 Hvwlpdwh ri lqvwdqwdqhrxv yhorflw| lv 89 . 97 @ 93 psk1 5
Vhfwlrq 617
Gh qlwlrq ri wkh Ghulydwlyh
+e, Dyhudjh udwh ri fkdqjh iurp 418 wr 5= i+5, i+4=8, 437 ;3 @ @ 7; psk 5 4=8 =8 Dyhudjh udwh ri fkdqjh iurp 5 wr 518 i+5=8, i+5, 459 437 @ @ 77 psk 5=8 5 =8
446
Vwhs 5 i+{ . k, i+{, @ 9{5 . 45{k . 9k5 7{ 7k 9{5 . 7{ @ 9k5 . 45{k 7k @ k+9k . 45{ 7, i+{ . k, i+{, k+9k . 45{ 7, @ k k
Vwhs 6
@ 9k . 45{ 7
Hvwlpdwh ri lqvwdqwdqhrxv yhorflw| lv 7; . 77 @ 79 psk1 5
Vwhs 7 i 3 +{, @ olp
k$3
i+{ . k, i+{, k
@ olp +9k . 45{ 7, k$3
617
@ 45{ 7
Gh qlwlrq ri wkh Ghulydwlyh
i 3 +5, @ 45+5, 7 @ 5; i 3 +3, @ 45+3, 7 @ 7 i 3 +6, @ 45+6, 7 @ 65
51 +d,
471 i+{, @
9 {
i+{ . k, @
9 {.k 9 9 {.k { 9{ 9+{ . k, @ {+{ . k,
s Wkh olqh wdqjhqw wr j+{, @ 6 { dw { @ 3 lv d yhuwlfdo olqh1 Vlqfh wkh vorsh ri d yhuwlfdo olqh lv xqgh qhg/ j3 +3, grhv qrw h{lvw1
i+{ . k, i+{, @
71 Li wkh udwh ri fkdqjh ri i+{, lv }hur zkhq { @ d/ wkh wdqjhqw olqh dw wkdw srlqw pxvw kdyh d vorsh ri }hur1 Wkxv/ wkh wdqjhqw olqh lv krul}rqwdo dw wkdw srlqw1
@
91 Xvlqj wkh srlqwv +5> 5, dqg +5> 9,> zh kdyh p@
95 7 @ @ 4= 5 5 7
;1 Xvlqj wkh srlqwv +6> 4, dqg +5> 6,> zh kdyh p@
7 7 6 +4, @ @ = 8 5 6 8
431 Wkh olqh wdqjhqw wr wkh fxuyh dw +7> 5, lv d yhuwlfdo olqh1 D yhuwlfdo olqh kdv xqgh qhg vorsh1 451 i+{, @ 9{5 7{ Vwhs 4 i+{ . k, @ 9+{ . k,5 7+{ . k, @ 9+{5 . 5{k . k5 , 7{ 7k @ 9{5 . 45{k . 9k5 7{ 7k
9k {+{ . k,
i+{ . k, i+{, 9k @ k k{+{ . k, 9 @ {+{ . k, i+{ . k, i+{, k 9 @ olp k$3 {+{ . k, 9 @ 5 {
i 3 +{, @ olp
k$3
i 3 +5, @ i 3 +3, @
6 9 9 @ @ 5 +5, 7 5
9 zklfk lv xqgh qhg vr i 3 +3, 35
grhv qrw h{lvw1 i 3 +6, @
5 9 9 @ @ 65 < 6
447
Fkdswhu 6
s 491 i+{, @ 6 { Vwhsv 406 duh frpelqhg1 i+{ . k, i+{, k s s 6 { . k . 6 { @ k Udwlrqdol}h wkh qxphudwru1 s s s s 6 { . k . 6 { 6 { . k 6 { s @ s k 6 { . k 6 { @
<+{ . k, <{ s s k+6 { . k 6 {,
@
<{ .
@
< s s 6 { . k 6 {
@
6 s s {.k { 6 s s {.k { 6 6 s s @ @ s { { 5 {
i 3 +{, @ olp
k$3
6 s i +5, @ zklfk lv xqgh qhg vr i 3 +5, 5 5
i 3 +4, @ 5+4, @ 5 lv wkh vorsh ri wkh wdqjhqw olqh dw { @ 4= Xvh p @ 5 dqg +4> 8, lq wkh srlqw0vorsh irup1 | 8 @ 5+{ . 4, | 8 @ 5{ . 5 | @ 5{ . : 531 i+{, @
i 3 +3, @
6 6 s @ zklfk lv xqgh qhg vr 3 5 3
i 3 +3, grhv qrw h{lvw1 i 3 +6, @
6 6 s @ s 5 6 5 6
4;1 i+{, @ 9 {5 > { @ 4 i+{ . k, i+{, k ^9 +{ . k,5 ` ^9 +{,5 ` @ k @
^9 +{5 . 5{k . k5 ,` ^9 {5 ` k
@
9 {5 5{k k5 9 . {5 k
k+5{ k, 5{k k5 @ @ @ 5{ k k k i 3 +{, @ olp +5{ k, @ 5{ k$3
6 > {@4 {.4
i+{ . k, i+{, k 6 6 +{ . k, . 4 { . 4 @ k 6+{ . 4, . 6+{ . k . 4, +{ . 4,+{ . k . 4, @ k 6{ 6 . 6{ . 6k . 6 +{ . 4,+{ . k . 4, @ k 6k @ k+{ . 4,+{ . k . 4, i 3 +{, @ olp
k$3
@
3
grhv qrw h{lvw1
WKH GHULYDWLYH
6 +{ . 4,+{ . k . 4,
6 +{ . 4,5
6 6 i 3 +4, @ +4.4, 5 @ 7 lv wkh vorsh ri wkh wdqjhqw olqh dw { @ 4= Xvh p @ 67 dqg 4> 65 lq wkh srlqw0 vorsh irup1 6 6 | @ +{ 4, 5 7 6 6 6 |. @ { 5 7 7 < 6 | @ { 7 7 s 551 i+{, @ {> { @ 58
i+{ . k, i+{, k s s {.k { @ k s s s s {.k { {.k. { @ s s k {.k. { @
{.k{ s s k+ { . k . {,
@
k 4 s s @s s k+ { . k . {, {.k. {
Vhfwlrq 617
Gh qlwlrq ri wkh Ghulydwlyh
4 4 i 3 +{, @ olp s s @ s k$3 5 { {.k. { 4 4 4 i 3 +58, @ s @ @ 58 43 5 58 Xvh p @
4 43
448
5;1 i+{, @
9 { {9 k 9{ 9+{ . k, @ k{+{ . k, 9 @ {+{ . k,
i+{ . k, i+{, @ k
dqg +58> 8, lq wkh srlqw0vorsh irup1 4 +{ 58, 43 4 58 |8 @ { 43 43 8 4 | @ {. 43 5 |8 @
571 i+{, @ 9{5 7{ i+{ . k, i+{, k 9+{ . k,5 7+{ . k, +9{5 7{, @ k 45{k . 9k5 7k @ k @ 45{ . 9k 7 i 3 +{, @ olp +45{ . 9k 7, k$3
@ 45{ 7 i 3 +5, @ 45+5, 7 @ 57 7 @ 53 i 3 +49, @ 45+49, 7 @ 4<5 7 @ 4;; i 3 +6, @ 45+6, 7 @ 69 7 @ 73 591 i+{, @ <{ 8 i+{ . k, i+{, k <+{ . k, 8 ^<{ 8` @ k <{
3
i +5, @ < i 3 +49, @ < i 3 +6, @ <
i 3 +{, @ olp
k$3
9 {.k
9 {+{ . k,
9 {5 6 9 9 @ i 3 +5, @ @ +5,5 7 5 6 9 9 @ i 3 +49, @ 5 @ 49 589 45; 5 9 9 @ i 3 +6, @ @ 5 +6, < 6 @
s 631 i+{, @ 6 { i+{ . k, i+{, k s s 6 { . k . 6 { @ k s s s s 6 { . k . 6 { 6 { . k 6 { s @ s k 6 { . k 6 { <+{ . k, <{ s s k+6 { . k 6 {, < s @ s 6 { . k 6 { 6 @ s s {.k { @
6 s s k$3 { . k { 6 6 s @ s @ s 5 { { { 6 i 3 +5, @ s 5 5 6 6 i 3 +49, @ s @ ; 5 49 i 3 +{, @ olp
i 3 +6, @
6 s > 5 6
zklfk lv qrw d uhdo qxpehu/ vr
i 3 +6, grhv qrw h{lvw1
449
Fkdswhu 6
651 Qr ghulydwlyh h{lvwv dw { @ 9 ehfdxvh wkh ixqf0 wlrq lv qrw gh qhg dw { @ 9= 671 Iru { @ 8 dqg { @ 3/ wkh ixqfwlrq i+{, lv qrw gh qhg1 Iru { @ 6 dqg { @ 5 wkh judsk ri i+{, kdv vkdus srlqwv1 Iru { @ 7> wkh wdqjhqw wr wkh judsk lv yhuwlfdo1 Wkhuhiruh/ qr ghulydwlyh h{lvwv iru { @ 8> { @ 6> { @ 3> { @ 5> ru { @ 7= 691 Wkh }hurv ri judsk +e, fruuhvsrqg wr wkh wxuqlqj srlqwv ri judsk +d,/ wkh srlqwv zkhuh wkh ghulydwlyh lv }hur1 Judsk +d, jlyhv wkh glvwdqfh/ zkloh judsk +e, jlyhv wkh yhorflw|1
k
+d,
k =34
+e, Judsk wkh ixqfwlrq rq d judsklqj fdofxodwru dqg pryh wkh fxuvru wr dq {0ydoxh qhdu { @ 5= D jrrg fkrlfh iru wkh lqlwldo ylhzlqj zlqgrz lv ^3> 6` e| ^3> 43`= Qrz }rrp lq rq wkh ixqfwlrq vhyhudo wlphv1 Hdfk wlph |rx }rrp lq/ wkh judsk zloo orrn ohvv olnh d fxuyh dqg pruh olnh d vwudljkw olqh1 Zkhq wkh judsk dsshduv wr eh d vwudljkw olqh/ xvh wkh WUDFH ihdwxuh wr vhohfw wzr srlqwv rq wkh judsk/ dqg uhfrug wkhlu frruglqdwhv1 Xvh wkhvh wzr srlqwv wr frpsxwh wkh vorsh1 Wkh uhvxow zloo eh yhu| forvh wr wkh prvw dffxudwh ydoxh irxqg lq sduw +d,/ zklfk lv 91::591 731 i+{, @ {4@{ > d @ 5
6;1 i+{, @ {{ > d @ 5 +d,
i+5 . =34, i+5, =34
134
@ 9=;7
@ =43:4 i+5 . =334, i+5, =334
5=3345=334 55 @ =334
1334
@
5=33345=3334 55 =3334
13334
5=3333345=333334 55 =333334
5=33344@5=3334 54@5 =3334
i+5 . =33334, i+5, =33334 133334
@
5=333344@5=33334 54@5 =33334
@ =43;8
i+5 . =333334, i+5, =333334 @
@
@ =43;8
@ 9=::5:
=333334
5=3344@5=334 54@5 =334
i+5 . =3334, i+5, =3334
@ 9=::6 i+5 . =33334, i+5, =33334 5=333345=33334 55 @ =33334
@
@ =43;7
i+5 . =3334, i+5, =3334
=33334
5=344@5=34 54@5 =34
@
@ 9=::<
=3334
i+5 . =34, i+5, =34
5=345=34 55 @ =34 i+5 . =334, i+5, =334 =334
WKH GHULYDWLYH
i+5 . =333334, i+5, =333334 1333334
@
5=3333344@5=333334 55 =333334
@ 9=::59
@ =43;8
Lw dsshduv wkdw i 3 +5, @ 9=::59=
Lw dsshduv wkdw i 3 +5, @ =43;8=
Vhfwlrq 617
Gh qlwlrq ri wkh Ghulydwlyh
+e, Judsk wklv ixqfwlrq rq d judsklqj fdofxodwru dqg pryh wkh fxuvru wr dq {0ydoxh qhdu { @ 5= D jrrg fkrlfh iru wkh lqlwldo ylhzlqj zlqgrz lv ^3> 8` e| ^3> 6`= Iroorz wkh surfhgxuh rxwolqhg lq wkh vroxwlrq iru H{huflvh 6;/ sduw +e,1 Wkh qdo uhvxow zloo eh forvh wr wkh ydoxh irxqg lq sduw +d, ri wklv h{huflvh/ zklfk lv 143;81
44:
Froxpq E Judsk | @ h{
751 Iru Froxpq D/ ohw k @ =34 i+{, @ oq m{m Judsk | @
oq m{ . =34m oq m{m 1 =34
i+{, @ h{ Judsk | @
h{.=34 h{ 1 =34
Judsk | @ 6{5
Judsk | @
4 {
Zh revhuyh wkdw wkh judsk ri i+{, @ {6 Judsk | @
+{ . =34,6 {6 1 =34
|@
oq m{ . =34m oq m{m =34
lv yhu| vlplodu wr wkh judsk ri |@
4 / {
wkh judsk ri |@
h{.=34 h{ =34
lv yhu| vlplodu wr wkh judsk ri | @ h{ / dqg
44;
Fkdswhu 6
+e, i 3 +6, @ ;+6, . 44 @ 46
wkh judsk ri |@
+{ . =34,6 {6 =34
lv yhu| vlplodu wr wkh judsk ri | @ 6{5 1 Wkxv wkh ghulydwlyh ri oq { lv {4 / wkh ghulydwlyh ri h{ lv h{ / dqg wkh ghulydwlyh ri {6 lv 6{5 1 771 +d, i+{, @ 7{5 . 44{ i+{ . k, @ 7+{ . k,5 . 44+{ . k, @ 7+{5 . 5{k . k5 , . 44+{ . k, @ 7{5 ;{k 7k5 . 44{ . 44k i+{ . k, i+{, @ 7{5 ;{k 7k5 . 44{ . 44k . 7{5 44{ @ ;{k 7k5 . 44k i+{ . k, i+{, ;{k 7k5 . 44k @ k k @ ;{ . 7k . 44 i 3 +{, @ olp
k$3
i+{ . k, i+{, k
@ olp +;{ . 7k . 44, k$3
@ ;{ . 44 i 3 +6, @ ;+6, . 44 @ 46 i+6 . =4, i+6, =4 i+6=4, i+6, @ =4 7+6=4,5 . 44+6=4, +7+6,5 . 44+6,, =4 7+<=94, . 44+6=4, . 7+<, 44+6, @ =4 6;=77 . 67=4 . 69 66 @ =4 4=67 @ @ 46=7 =4 i+6 . =4, i+6 =4, 5+=4, @
i+6=4, i+5=<, @ =5 7+6=4,5 . 44+6=4, +7+5=<,5 . 44+5=<,, =5 7+<=94, . 44+6=4, . 7+;=74, 44+5=<, @ =5 5=9 6;=77 . 67=4 . 66=97 64=< @ @ 46 @ =5 =5 @
WKH GHULYDWLYH
i+6 . =34, i+6, =34 i+6=34, i+6, @ =34 7+6=34,5 . 44+6=34, +7+6,5 . 44+6,, =34 7+<=3934, . 44+6=34, . 7+<, 44+6, @ =34 69=5737 . 66=44 . 69 66 @ =34 =4637 @ 46=37 @ =34 i+6 . =34, i+6 =34, 5+=34, @
@
i+6=34, i+5=<<, =35
7+6=34,5 . 44+6=34, +7+5=<<,5 . 44+5=<<,, =35 7+<=3934, . 44+6=34, . 7+;=<734, 44+5=<<, @ =35 69=5737 . 66=44 . 68=:937 65=;< @ =35 =59 @ @ 46 =35 5 i+{, @ +f, { 5 i+{ . k, @ {.k 5 5 i+{ . k, i+{, @ {.k { 5 5 . @ {.k { 5{ . 5+{ . k, @ {+{ . k, @
@ i+{ . k, i+{, @ k
5k {+{ . k,
5k {+{.k,
k 5k 4 @ {+{ . k, k 5 @ {+{ . k, i+{ . k, i+{, 3 i +{, @ olp k$3 k 5 5 @ 5 @ olp k$3 {+{ . k, {
Vhfwlrq 617
Gh qlwlrq ri wkh Ghulydwlyh
44<
5 5 @ =555555 65 < i+6=4, i+6, i+6 . =4, i+6, @ =4 =4 5 5 6 @ 6=4 =4 i 3 +6, @
=548387 i+6=4, i+5=<, i+6 . =4, i+6 =4, @ 5+=4, =5 @
5 6=4
=5
5 5=<
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5 6=34
5 5=<< =35
=555558 +h, i+{, @
s {
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i+{ . k, i+{, k s s s s {.k { {.k. { @ s s k {.k. { @
i+6 . =4, i+6 =4, i+6=4, i+5=<, @ 5+=4, =5 s s 6=4 5=< @ =5
+{ . k, { s s k+ { . k . {,
k @ s s k+ { . k . {, 4 @ s s k+ { . k . {, 4 4 i 3 +{, @ olp s s @ s k$3 k+ { . k . {, 5 { 4 i 3 +6, @ s =5;;9:8 5 6 i+6 . =4, i+6, i+6=4, i+6, @ =4 =4 s s 6=4 6 @ =4 =5;963<
S 3 +9, @ 65 7+9, @ 65 57 @ ; \hv/ wkh uvw vkrxog lqfuhdvh surgxfwlrq/ vlqfh wkh pdujlqdo sur w lv srvlwlyh1 +f, '45/333/ { @ 45 S 3 +45, @ 65 7+45, @ 65 7; @ 49 Qr/ ehfdxvh wkh pdujlqdo sur w lv qhjdwlyh1 +g, '53/333/ { @ 53 S 3 +53, @ 65 7+53, @ 65 ;3 @ 7; Qr/ ehfdxvh wkh pdujlqdo sur w lv qhjdwlyh1 7;1 F+{, @ 4333 . =57{5 > 3 { 63/333 +d, Wkh pdujlqdo frvw lv jlyhq e| F 3 +{, @ =7;{> 3 { 63/3331 +e, F 3 +433, @ =7;+433, @ 7;
453
Fkdswhu 6
WKH GHULYDWLYH
+f, Wklv uhsuhvhqwv wkh idfw wkdw wkh frvw ri sur0 gxflqj wkh qh{w +434vw, wdfr lv dssur{lpdwho| 7;1
57 plqxwhv diwhu wkh phdo vwduwv wkh udwh ri irrg frqvxpswlrq lv 31
Wkh h{dfw frvw wr surgxfh wkh 434vw wdfr lv
+f, Diwhu 57 plqxwhv wkh udwh ri irrg frqvxpswlrq lv qhjdwlyh dffruglqj wr wkh ixqfwlrq zkhuh d udwh ri }hur lv pruh dffxudwh1 D orjlfdo udqjh iru wklv ixqfwlrq lv 3 w 57=
F+434, F+433, @ ^4333 . =57+434,5 ` ^4333 . =57+433,5 ` @ 4333 . 577;=57 4333 5733 @ 7;=57= +g, Wkh h{dfw frvw ri surgxflqj wkh 434vw wdfr lv 157 juhdwhu wkdq wkh dssur{lpdwh frvw1 Wkh| duh yhu| forvh1 831 +d, Iurp wkh judsk/ Yps lv mxvw derxw dw wkh wxuq0 lqj srlqw ri wkh fxuyh1 Wkxv/ wkh vorsh ri wkh wdqjhqw olqh lv dssur{lpdwho| }hur1 Wkh srzhu h{0 shqglwxuh lv qrw fkdqjlqj1 +e, Iurp wkh judsk/ wkh vorsh ri wkh wdqjhqw olqh dw Ypu lv dssur{lpdwho| 141 Wkh srzhu h{shqghg lv lqfuhdvlqj 14 xqlw shu xqlw lqfuhdvh lq vshhg1 +f, Wkh vorsh ri wkh wdqjhqw olqh dw Yrsw lv d elw juhdwhu wkdq wkdw dw Ypu > derxw 1451 Wkh srzhu h{shqghg lqfuhdvhv 145 xqlwv iru hdfk xqlw lqfuhdvh lq vshhg1
871 Wkh vorsh ri wkh wdqjhqw olqh wr wkh judsk dw wkh uvw srlqw lv irxqg e| qglqj wzr srlqwv rq wkh wdqjhqw olqh1 +{4 > |4 , @ +4333> 46=8, +{5 > |5 , @ +3> 4;=8, p@
4;=8 46=8 8 @ @ =338 3 4333 4333
Dw wkh vhfrqg srlqw/ zh kdyh +{4 > |4 , @ +4333> 46=8, +{5 > |5 , @ +5333> 54=8,= 54=8 46=8 5333 4333 ; @ 4333
p@
+g, Wkh srzhu ohyho uvw ghfuhdvhv wr Yps > wkhq lqfuhdvhv dw juhdwhu udwhv1 +h, Ypu lv wkh srlqw zklfk surgxfhv wkh vpdoohvw vorsh ri d olqh1 851 L+w, @ 5: . :5w 4=8w5 +d, L+w . k, L+w, k$3 k 5:.:5w.:5k4=8w5 6wk4=8k5 5::5w.4=8w5 @ olp k$3 k :5k 6wk 4=8k5 @ olp k$3 k
@ =33; Dw wkh wklug srlqw/ zh kdyh +{4 > |4 , @ +8333> 53, +{5 > |5 , @ +6333> 55=8,= 55=8 53 6333 8333 5=8 @ 5333
p@
L 3 +w, @ olp
@ olp :5 6w 4=8k k$3
@ :5 6w L 3 +8, @ :5 6+8, @ :5 48 @ 8: Wkh udwh ri fkdqjh ri wkh lqwdnh ri irrg 8 plqxwhv lqwr d phdo lv 8: judpv shu plqxwh1 +e,
B
L 3 +57, @ 3 B
:5 6+57, @ 3 3@3
@ =33458 Dw 833 iw/ wkh whpshudwxuh ghfuhdvhv 1338 shu irrw1 Dw derxw 4833 iw/ wkh whpshudwxuh lqfuhdvhv 133; shu irrw1 Dw 8333 iw/ wkh whpshudwxuh gh0 fuhdvhv 133458 shu irrw1 891 Wkh vorsh ri wkh judsk dw { @ 57 fdq eh hvwlpdwhg xvlqj wkh srlqwv +57> 693, dqg +66> 6<8,1 vorsh @
68 6<8 693 @ @ 6 ;< 66 57 <
Wkxv/ wkh ghulydwlyh iru d 57 rxqfh edw lv derxw 7 iw shu r} zklfk phdqv wkdw wkh glvwdqfh wkh edoo wudyhov lv lqfuhdvlqj 7 ihhw shu rxqfh1
Vhfwlrq 618
Judsklfdo Gl huhqwldwlrq
Wkh vorsh ri wkh judsk dw { @ 84 fdq eh hvwlpdwhg xvlqj wkh srlqwv +93> 6;3, dqg +84> 6<3,1 vorsh @
43 43 6<3 6;3 @ 4=4 @ 84 93 < <
Wkxv/ wkh ghulydwlyh iru d 84 rxqfh edw lv derxw 4=4 iw shu r} zklfk phdqv wkdw wkh glvwdqfh wkh edoo wudyhov lv ghfuhdvlqj 414 ihhw shu rxqfh1
618
Judsklfdo Gl huhqwldwlrq
71 Vlqfh wkh {0lqwhufhswv ri wkh judsk i 3 rffxu zkhq0 hyhu wkh judsk ri i kdv d krul}rqwdo wdqjhqw olqh/ \5 lv wkh ghulydwlyh ri \4 1 Qrwlfh wkdw \5 kdv 5 {0lqwhufhswv> hdfk rffxuv dw dq {0ydoxh zkhuh wkh wdqjhqw olqh wr \4 lv krul}rqwdo1 Qrwh dovr wkdw \5 lv qhjdwlyh zkhqhyhu \4 lv gh0 fuhdvlqj/ dqg \5 lv srvlwlyh zkhqhyhu \4 lv lqfuhdv0 lqj1
454
Rq wkh lqwhuydo +6> 4,> wkh vorsh ri i lv dovr frq0 vwdqw1 Wr qg wklv vorsh/ xvh wkh srlqwv +6> 3, dqg +9> 6,= p@
6 3 6 @ @ 4 96 6
Wkxv/ zh kdyh i 3 +{, @ 4 rq +4> 6, dqg i 3 +{, @ 4 rq +6> 4,= Ehfdxvh wkh judsk ri i kdv vkdus srlqwv dw { @ 6 dqg { @ 6> zh nqrz wkdw i 3 +6, dqg i 3 +6, gr qrw h{lvw1 Zh vkrz wklv rq wkh judsk ri i 3 +{, e| xvlqj rshq flufohv1 Zh dovr revhuyh wkdw wkh vorshv ri wdqjhqw olqhv duh qhjdwlyh rq +6> 3,> wkdw wkh judsk kdv d krul0 }rqwdo wdqjhqw dw { @ 3/ dqg wkdw wkh vorshv ri wdqjhqw olqhv duh srvlwlyh rq +3> 6,= Wkxv/ i 3 lv qhjdwlyh rq +4> 6,> 3 dw { @ 3> dqg srvlwlyh rq +6> 4,= Ixuwkhupruh/ e| gudzlqj wdqjhqwv/ zh vhh wkdw rq +6> 6,> wkh ydoxh ri i 3 lqfuhdvhv iurp 4 wr 41
91 Vlqfh wkh {0lqwhufhswv ri wkh judsk i 3 rffxu zkhq0 hyhu wkh judsk ri i kdv d krul}rqwdo wdqjhqw olqh/ \4 lv wkh ghulydwlyh ri \5 1 Qrwlfh wkdw \4 kdv 7 {0lqwhufhswv> hdfk rffxuv dw dq {0ydoxh zkhuh wkh wdqjhqw olqh wr \5 lv krul}rqwdo1 Qrwh dovr wkdw \4 lv qhjdwlyh zkhqhyhu \5 lv gh0 fuhdvlqj dqg \4 lv srvlwlyh zkhqhyhu \5 lv lqfuhdv0 lqj1 ;1 Wr judsk i 3 / revhuyh wkh lqwhuydov zkhuh wkh vorshv ri olqhv duh srvlwlyh dqg zkhuh wkh| duh qhjdwlyh wr ghwhuplqh zkhuh wkh ghulydwlyh lv srvlwlyh dqg zkhuh lw lv qhjdwlyh1 Dovr/ zkhuhhyhu i kdv d kru0 l}rqwdo wdqjhqw/ i 3 zloo eh 31 Hvwlpdwh wkh pdjqlwxgh ri wkh vorsh dw vhyhudo srlqwv e| gudzlqj wdqjhqwv wr wkh judsk ri i=
431 Rq wkh lqwhuydo +4> 6,> wkh judsk ri i lv d vwudljkw olqh/ vr lwv vorsh lv frqvwdqw1 Wr qg wklv vorsh/ xvh wkh srlqwv +9> 6, dqg +6> 3,= p@
3 +6, 6 @ @4 6 6 +9,
451 Rq wkh lqwhuydo +4> 3,> wkh judsk ri i lv d krul0 }rqwdo olqh/ vr lwv vorsh lv 31 Wkxv/ wkh judsk ri i 3 lv | @ 3 +wkh {0d{lv, rq +4> 3,= Vlqfh i lv glvfrqwlqxrxv dw { @ 3> i 3 +3, grhv qrw h{lvw1 Wkxv/ wkh judsk ri i 3 kdv dq rshq flufoh dw { @ 3= Rq wkh lqwhuydo +3> 4,> wkh vorshv duh srvlwlyh exw ghfuhdvlqj dw d vorzhu udwh dv { jhwv odujhu1 Wkhuh0 iruh/ wkh ydoxh ri i 3 zloo eh srvlwlyh exw ghfuhdv0 lqj rq wklv lqwhuydo1 Wklv ydoxh dssurdfkhv 3/ exw qhyhu ehfrphv 31
455
Fkdswhu 6
471 Wkh judsk ri i lv d vwhs ixqfwlrq1 +Wklv lv wkh juhdwhvw lqwhjhu ixqfwlrq/ i+{, @ ^^{``=, Wkh judsk lv pdgh xs ri dq lq qlwh vhulhv ri krul}rqwdo olqh vhjphqwv1 Wkxv/ wkh ghulydwlyh zloo eh 3 hyhu|0 zkhuh lw lv gh qhg1 Krzhyhu/ vlqfh i lv glvfrqwlq0 xrxv zkhuhyhu { lv dq lqwhjhu/ i 3 +{, grhv qrw h{lvw dw dq| lqwhjhu1 Wkh judsk ri i 3 lv dq lq qlwh vhw ri olqh vhjphqwv rq wkh {0d{lv +zkhuh | @ i 3 +{, @ 3, vhsdudwhg e| rshq flufohv zkhuhyhu { lv dq lqwhjhu1
WKH GHULYDWLYH
Xvh wkh wdqjhqwv gudzq rq wkh judsk dqg dggl0 wlrqdo wdqjhqw dv qhhghg wr hvwlpdwh wkh vorsh dw vhyhudo srlqwv rq wkh judsk ri Y wr lpsuryh wkh dffxudf| ri wkh judsk ri Y 3 =
531
491 Qrwlfh krz wkh vorsh ri wkh judsk fkdqjhv zlwk wlph1 Wkh vorsh lv srvlwlyh xqwlo hduo| Dxjxvw/ wkhq lw lv qhjdwlyh xqwlo Qryhpehu/ wkhq lw lv srvl0 wlyh xqwlo Ghfhpehu/ wkhq lw lv qhjdwlyh xqwlo plg0 Mdqxdu|/ wkhq lw lv srvlwlyh xqwlo Dsulo/ wkhq lw lv qhjdwlyh iru wkh uhpdlqghu ri wkh gdwd1 Xvh wklv lq0 irupdwlrq wr vnhwfk wkh judsk ri wkh udwh ri fkdqjh zlwk uhvshfw wr wlph1
Fkdswhu 6 Uhylhz H{huflvhv 71 Wkh ghulydwlyh fdq eh xvhg wr qg wkh lqvwdqwd0 qhrxv udwh ri fkdqjh dw d srlqw rq d ixqfwlrq/ dqg wkh vorsh ri d wdqjhqw olqh dw d srlqw rq d ixqfwlrq1 91 +d, +e,
4;1 Ohw S +Y , uhsuhvhqw wkh srzhu fruuhvsrqglqj wr d jlyhq ydoxh ri wkh whuq*v vshhg/ Y= Wkh udwh ri fkdqjh ri srzhu dv d ixqfwlrq ri wlph lv jlyhq e| wkh ghulydwlyh ri wklv ixqfwlrq/ S 3 +Y ,= Zh xvh wkh judsk ri S wr vnhwfk wkh judsk ri S 3 = Iluvw/ zh revhuyh wkdw wkh judsk ri S kdv rqh wxuq0 lqj srlqw/ dw Y @ Yps 8=8= Dw wklv ydoxh ri Y> wkh judsk kdv d krul}rqwdo wdqjhqw/ vr wkh judsk ri Y 3 kdv dq {0lqwhufhsw dw wklv ydoxh ri Y= Vlqfh wkh vorshv ri wdqjhqw olqhv duh qhjdwlyh rq wkh lqwhuydo +3> Yps , dqg srvlwlyh zkhq Y A Yps > wkh ydoxh ri Y 3 lv qhjdwlyh rq +3> Yps , dqg srvlwlyh zkhq Y A Yps =
olp j+{, @ 5
{$4
olp j+{, @ 5
{$4.
+f, olp j+{, grhv qrw h{lvw vlqfh sduwv +d, dqg {$4
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+e, olp. k+{, @ 4 {$5 +f, olp k+{, @ 4 {$5
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Fkdswhu 6 Uhylhz H{huflvhv
451 Ohw i+{, @
456
5{ . 8 = {6
{ i+{,
5=< 43;
{ i+{,
6=4 445
591 i+{, @
5=<< 43<; 6=34 4435
5=<<< 43/<<;
Wkh ixqfwlrq lv glvfrqwlqxrxv dw { @ 4 dqg { @ 5 ehfdxvh wkrvh ydoxhv pdnh wkh ghqrplqdwru ri wkh iudfwlrq htxdo wr }hur1 olp i+{, grhv qrw h{lvw vlqfh olp i+{, @ 4
6=334 44/335
{$4
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grhv qrw h{lvw1 471 olp
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Vhfwlrq 815
Uhodwlyh H{wuhpd
771 +d, I +w, @ 43=5; . 4:8=<whw@4=6 I 3 +w, @ +4:8=<,+hw@4=6 ,
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815
Uhodwlyh H{wuhpd
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493
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5 4 471 i+{, @ {6 {5 . 6{ 7 6 5
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Uhodwlyh H{wuhpd
494
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495
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JUDSKV DQG WKH GHULYDWLYH
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Vhfwlrq 815
Uhodwlyh H{wuhpd
j3 +4, @ j3 +=3334, @ j3 +=3334, @ j3 +4, @
496
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9
45 . +4=5 43 ,+=3334, A3 +=3334,46 45 . +4=5 434< ,+=3334,9 ?3 +=3334,46 45 . +4=5 434< ,49 A3 446
j+=334, @ 4369 > j+=334, @ 4369 Plqlpxp ri 4369 dw { @ =334
5: A3 644 44 ?3 S 3 +:, @ 436 S 3 +8, @
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5
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3
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Wkh pd{lpxp sur w lv ':3/9581
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497
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+{ . 5|,5 8 8| 5+{ . 5|, 5 +{ . 5|,7 8{ 43| @ +{ . 5|,6
n{| +{> |, @
+{ . 5|,5 +8, +8{, 5+{ . 5|, +{ . 5|,7 8{ 43| @ +{ . 5|,6
n|{ +{> |, @
5;1 } @ 6|h{
PXOWLYDULDEOH FDOFXOXV
} @ +| . 4, oq {6 | @ +| . 4,+6 oq m{m . oq m|m, 4 }{ +{> |, @ +| . 4, 6 { @
6+| . 4, {
@ 6{4 +| . 4, 4 }| +{> |, @ +| . 4, . +6 oq { . oq |, 4 | |.4 . 6 oq m{m . oq m|m @ | 6+| . 4, {5 | 4 +| . 4, 4 4 }|| +{> |, @ . |5 | 4 4 @ 5 . | |
}{{ +{> |, @ 6{5 +| . 4, @
6 { 4 6 }|{ +{> |, @ 6 @ { { }{| +{> |, @ 6{4 @
671 i+{> |, @ 83 . 7{ 8| . {5 . |5 . {| i{ +{> |, @ 7 . 5{ . |> i| +{> |, @ 8 . 5| . {
}{ +{> |, @ 6|h{ }{{ +{> |, @ 6|h{ }{| +{> |, @ 6h{ }| +{> |, @ 6h{ }|| +{> |, @ 3 }|{ +{> |, @ 6h{
Vroyh wkh v|vwhp 7 . 5{ . | @ 3 8 . { . 5| @ 3= Pxowlso| wkh vhfrqg htxdwlrq e| 5 dqg dgg1
631 n @ oq m8{ :|m 8 @ 8+8{ :|,4 8{ :| : n| +{> |, @ @ :+8{ :|,4 8{ :|
n{ +{> |, @
n{{ +{> |, @ 8+8{ :,5 8 @ 58+8{ :|,5
ru
58 +8{ :|,5
n|| +{> |, @ :+8{ :|,5 +:, @ 7<+8{ :|,5 5
n{| +{> |, @ 8+8{ :,
@ 68+8{ :|,5
ru
7< +8{ :|,5
+:, ru
68 +8{ :|,5
n|{ +{> |, @ :+8{ :|,5 +8, @ 68+8{ :|,5
ru
68 +8{ :|,5
7 . 5{ . | @ 3 43 5{ 7| @ 3 47 6| @ 3 47 |@ 6 Vxevwlwxwh lqwr wkh vhfrqg htxdwlrq wr jhw { @ 46 47 46 6 = Wkh vroxwlrq lv { @ 6 > | @ 6 = 691 i+{> |, @ 5533 . 5:{6 . :5{| . ;| 5 i{ +{> |, @ ;4{5 . :5|> i| +{> |, @ :5{ . 49| Vroyh wkh v|vwhp ;4{5 . :5| @ 3 +4 , :5{ . 49| @ 3= +5 , Glylgh htxdwlrq +4, e| < dqg htxdwlrq +5, e| ;1 <{5 . ;| @ 3 +6 , <{ . 5| @ 3 +7 ,
Vhfwlrq <15
Sduwldo Ghulydwlyhv
5;6
751 i+{> |> }, @ oq ;{| . 8|} {6
Iurp htxdwlrq +7,/ | @ <5 {= Vxevwlwxwh lqwr htxdwlrq +6,1
i{ +{> |> }, @ <{5 69{ @ 3 <{+{ 7, @ 3 { @ 3 ru { @ 7
@ i| +{> |> }, @
Vlqfh | @ <5 {>
@
li { @ 3> | @ 3> li { @ 7> | @
<5 +7,
i} +{> |> }, @ @ 4;= @
Wkh vroxwlrqv duh { @ 3> | @ 3> dqg { @ 7> | @ 4;=
6;1 i+{> |> }, @ 6{8 {5 . |8 i{ +{> |> }, @ 48{7 5{> i| +{> |> }, @ 8|7 > i} +{> |> }, @ 3> i|} +{> |> }, @ 3
4 +7{ . |, |} 5 7{ . | @ |} 5 +|} 5, { +5{5 . {|, } +|} 5,5 5{ 5{5 } +|} 5,5
@
+5{5 | . {|5 , +|} 5,5
+5=33334 . 45 ,5+5=33334,.4 +5 . 45 ,5+5,.4 =33334
+6=33334,8=33335 68 =33334
<6;=< i+5> 4 . k, i+5> 4, k
Djdlq/ ohw k @ =33334= i| +5> 4,
5
+|} 5, +5{ , +5{ 5{ }, 5+|} 5, | +|} 5,7 7{5 . 7{| . 5{5 |} +|} 5,6
i+5=33334> 4, i+5> 4, =33334
k$3
i|} +{> |> }, @
i+5 . k> 4, i+5> 4, k
+e, i| +5> 4, @ olp
i} +{> |> }, @ +5{5 . {|,+|} 5,5 |
5
771 i+{> |, @ +{ . | 5 ,5{.|
i{ +5> 4,
i{ +{> |> }, @
5
8{6 +;{| . 8|} {6 ,5
Zh zloo xvh d vpdoo ydoxh iru k= Ohw k @ =33334=
@ +5{5 . {|,+|} 5,4
@
+;{| . 8|} {6 , 8 +;{ . 8}, 8| +;{| . 8|} {6 ,5
k$3
5{5 . {| |} 5 4 +5{5 . {|, @ |} 5
@
@
;| 6{5 ;{| . 8|} {6 4 +;{ . 8}, ;{| . 8|} {6 ;{ . 8} ;{| . 8|} {6 4 8| ;{| . 8|} {6 8| ;{| . 8|} {6
+d, i{ +5> 4, @ olp
731 i+{> |> }, @
i| +{> |> }, @
i|} +{> |> }, @
4 +;| 6{5 , ;{| . 8|} {6
i+5> 4=33334, i+5> 4, =33334
^5 . +4=33334,5 `5+5,.4=33334 +5 . 45 ,5+5,.4 =33334
^5 . +4=33334,5 `8=33334 68 =33334
43::
5;7
Fkdswhu <
791 U+{> |, @ 8{5 . <|5 7{| +d, U+<> 8, @ 8+<,5 . <+8,5 7+<,+8, @ 783 U{ +{> |, @ 43{ 7| U{ +<> 8, @ 43+<, 7+8, @ :3
zklfk lv wkh dssur{lpdwh fkdqjh lq surgxfwlrq +lq wkrxvdqgv ri xqlwv, iru d 4 xqlw fkdqjh lq fdslwdo1 +e, Lqfuhdvlqj wr 98 xqlwv ri oderu zrxog uhvxow lq dq lqfuhdvh ri dssur{lpdwho| 58 +4333, 4896 edwwhulhv1 49
U zrxog lqfuhdvh e| ':31 +e, U| +{> |, @ 4;| 7{ U| +<> 8, @ 4;+8, 7+<, @ 87 U zrxog lqfuhdvh e| '871 s 7;1 S +{> |, @ 433 {5 . | 5 @ 433+{5 . |5 ,4@5 > zkhuh { lv oderu/ | lv fdslwdo1 +d,
433{ CS @ 83+{5 . |5 ,4@5 5{ @ s C{ {5 . |5
CS 433+7, +7> 6, @ s @ ;3 xqlwv C{ +7,5 . +6,5 CS 433| @ 83+{5 . | 5 ,4@5 5| @ s +e, C| {5 . |5 433+6, CS +7> 6, @ s @ 93 xqlwv C{ +7,5 . +6,5 831 } @ {=: |=6 > zkhuh { lv oderu/ | lv fdslwdo1 Pdujlqdo surgxfwlylw| ri oderu lv C} @ =:{=6 | =6 = C{ Pdujlqdo surgxfwlylw| ri fdslwdo lv C} @ =6{=: | =: = C| 851 i+{> |, @ 6{4@6 | 5@6 > zkhuh { lv oderu/ | lv fdslwdo1 | 5@6 |5@6 +d, i{ +{> |, @ {5@6 |5@6 @ 5@6 @ { { 5@6 458 58 @ 4=8958> i{ +97> 458, @ @ 97 49 zklfk lv wkh dssur{lpdwh fkdqjh lq surgxfwlrq +lq wkrxvdqgv ri xqlwv, iru d 4 xqlw fkdqjh lq oderu1 i| +{> |, @ 5{4@6 |4@6
4@6 {4@6 { @ 5 4@6 @ 5 | | 4@6 97 ; i| +97> 458, @ 5 @ @ 4=9> 458 8
PXOWLYDULDEOH FDOFXOXV
+f, Dq lqfuhdvh ri dssur{lpdwho| 49 +4333, @ 6533 edwwhulhv 8 zrxog eh wkh h hfw ri lqfuhdvlqj fdslwdo wr 459 zkloh kroglqj oderu dw 971 Lqfuhdvlqj fdslwdo lv wkh ehwwhu rswlrq1 871 p+W> I> z, @ 5=8+W I ,z=9: @ 5=8W z=9: 5=8I z=9: +d, Lqfuhdvlqj W iurp 6; F wr 6< F zkloh I uh0 pdlqv dw 45 F dqg z uhpdlqv dw 63 nj uhvxowv lq d fkdqjh lq r{|jhq frqvxpswlrq ri pW +6;> 45> 63, @ 5=8z=9: @ 5=8+63,=9: =589= +e, Lqfuhdvlqj I iurp 47 F wr 48 F zkloh W uh0 pdlqv dw 69 F dqg z uhpdlqv dw 58 nj uhvxowv lq d fkdqjh ri r{|jhq frqvxpswlrq ri pI +69> 47> 58, @ 5=8z=9: @ 5=8+58=9: , =5;<= e @ e+d y,4 dy 533 +d, F+493> 533> 458, @ 493 458 533 @ 68
891 F+d> e> y, @
8=:4 +e, Fd +d> e> y, @ e+d y,5 4 @
e +d y,5 533 +493 458,5 533 @ 5 68
Fd +493> 533> 458, @
=496
Vhfwlrq <15
Sduwldo Ghulydwlyhv
5;8
+f, Fe +d> e> y, @ +d y,4 Fe +493> 533> 458, @ +493 458,4 @
4 68
931 s+o> w, @ 4 .
@ 4 . +4,+4 55 ,
533 +493 458,5 533 @ 5 68
Fy +493> 533> 458, @
=496 +h, Fkdqjlqj d e| 4 xqlw surgxfhv wkh juhdwhvw gh0 fuhdvh lq wkh olwhuv ri eorrg sxpshg/ zkloh fkdqj0 lqj y e| 4 xqlw surgxfhv wkh vdph dprxqw ri lq0 fuhdvh lq wkh olwhuv ri eorrg sxpshg1
8;1 E+z> k, @
:36z k5
+d, E+553> :7, @ +e,
:36+553, 5; :75
CE :36 @ 5 Cz k CE 4739z 5+:36,z @ @ Ck k6 k6
Vlqfh Vlqfh
CE Cz CE Ck
lv srvlwlyh/ dv z lqfuhdvhv/ vr grhv E1 lv qhjdwlyh/ E ghfuhdvhv dv k lqfuhdvhv1
+f, Li zp dqg kp uhsuhvhqw d shuvrq*v zhljkw dqg khljkw/ lq nlorjudpv dqg phwhuv/ uhvshfwlyho|/ wkhq zp @ =7888z> zkhuh z lv zhljkw lq srxqgv/ dqg kp @ =3587k> zkhuh k lv khljkw lq lqfkhv1 Wr wudqvirup wkh irupxod E @ :36z k5 wr kdqgoh phwulf zp lqsxwv/ pdnh wkh vxevwlwxwlrqv z @ =7888 dqg k @ kp = =3587 :36 zp :36+=3587,5 zp Ep @ =7888 @ 5 = kp 5 =7888 kp =3587
:36+=3587,5 zp Vlqfh =<<9 4> xvh Ep @ 5 = =7888 kp
6 7
@4.
+g, Fy +d> e> y, @ e+d y,5 +4, e +d y,5
66 +4 543@8 , 66
+d, s+66> 43, @ 4 .
=35;9
@
o +4 5w@8 , 66
@ 4=:8 Wkh suhvvxuh dw 66 ihhw iru d 430plqxwh glyh lv 41:8 dwprvskhuhv1 +e,
4 +4 5w@8 , 66 4 w@8 o sw +o> w, @ ++oq 5, 5 , 66 8 o @ +oq 5,+5w@8 , 498 4 4 6 so +66> 43, @ +4 543@8 , @ 66 66 7 so +o> w, @
=356 dwp2iw Lqfuhdvlqj wkh ghswk ri wkh glyh e| 4 irrw +wr 67 ihhw,/ zkloh nhhslqj wkh glyh ohqjwk dw 43 plqxwhv/ lqfuhdvhv wkh suhvvxuh e| dssur{lpdwho| 1356 dw0 prvskhuhv1 66 +oq 5,+543@8 , 498 4 @ oq 5 53
sw +66> 43, @
=368 dwp2plq Lqfuhdvlqj wkh glyh wlph e| 4 plqxwh +wr 44 plqxwhv,/ zkloh nhhslqj wkh ghswk ri wkh glyh wr 66 ihhw lq0 fuhdvhv wkh suhvvxuh e| dssur{lpdwho| 1368 dwprvskhuhv1 +f, Vroyh s+99> w, @ 5=48 4.
99 +4 5w@8 , @ 5=48 66 5+4 5w@8 , @ 4=48 4 5w@8 @ =8:8 5w@8 @ =758 w oq 5 @ oq =758 8 8 oq =758 w@ oq 5 w 9=4:
Wkh pd{lpxp glyh ohqjwk lv 914: plqxwhv1
5;9
Fkdswhu <
951 +d, i+<3> 63, <3
{.| 4 . {| f5
991 z+{> |, @
+e, i+<3> :8, 43<
PXOWLYDULDEOH FDOFXOXV
+d, z+83/333/ 483/333,
+f, i+;3> :8, ;9
@
i+<8> 63, i+<3> 63, +g, iW +<3> 63, 8 <8 <3 @4 8
@
{| f5
|5 f5
Wkh lqvwdqwdqhrxv udwh ri fkdqjh lv 156; p2vhf shu p2vhf=
971 s @ i+v> q> d, @ =336d . =4+vq,4@5 +d, i+;> 9> 783, @ =336+783, . =4^+;,+9,`4@5 @ 4=68 . =4+7;,4@5 @ 5=375; s 5=37(
id +v> q> d, @ =336 iru doo rughuhg wulsohv +v> q> d,= Wkhuhiruh/ sd @ =336(= sq @ =38( lv wkh udwh ri fkdqjh ri wkh suredelolw| iru dq dgglwlrqdo vhphvwhu ri kljk vfkrro pdwk1 sd @ =336( lv wkh udwh ri fkdqjh ri wkh suredelolw| shu xqlw ri fkdqjh lq dq VDW vfruh1
{| f5
z{ +83/333/ 483/333, 5 4 483/333 4;9/5;5 @ 5 =56; +83/333,+483/333, 4. +4;9/5;5, 5
i+<3> ;3, i+<3> :8, 8 447 43< @4 8
+j, iK +<3> :8,
sq @ =38(
4.
5 4 . {| f5 5 4 |f @ 5 4 . {| f5
i+<8> :8, i+<3> :8, +i, iW +<3> :8, 8 463 43< @ 7=5 8
4 4@5 +v, +f, iq +v> q> d, @ =336+3, . =4 +vq, 5 4 ^+6,+6,`4@5 +6, iq +6> 6> 653, @ =4 5 4 4 @ +=4, +6, @ =38 5 6
+83/333,+483/333, +4;9/5;5, 5
Wkh urfnhw lv wudyholqj 497/789 p2vhf uhodwlyh wr wkh vwdwlrqdu| revhuyhu1 | 4 . {| f5 f5 +{ . |, +e, z{ @ 5 4 . {| f5
i+<3> 68, i+<3> 63, 8 <5 <3 @ =7 8
4.
497/789
+h, iK +<3> 63,
+e, i+6> 6> 653, @ =336+653, . =4^+6,+6,`4@5 @ =<9 . =4+<,4@5 @ 4=59 s @ 4=59(
83/333 . 483/333
+f, z+f> f, @
f.f 4.
+f,+f, f5
@
5f @f 5
Wkh vshhg lv wkh vshhg ri oljkw/ f=
<16
Pd{lpd dqg Plqlpd
51 i+{> |, @ 7{| . ;{ <| i{ +{> |, @ 7| . ;> i| +{> |, @ 7{ <> i{ +{> |, @ 3 dqg i| +{> |, @ 3 zkhq
dqg
7| . ; @ 3 | @ 5 7{ < @ 3
< {@ = 7 i{{ +{> |, @ 3> i|| +{> |, @ 3> i{| +{> |, @ 7 Vlqfh G @ i{{
< < < > 5 i|| > 5 i{| > 5 7 7 7
@ 3 3 75 @ 49 ? 3> < 7 > 5 lv d vdggoh srlqw1
Vhfwlrq <16
Pd{lpd dqg Plqlpd
71 i+{> |, @ {5 . {| . |5 9{ 6 i{ +{> |, @ 5{ . | 9> i| +{> |, @ { . 5| 5{ . | 9 @ 3 { . 5| @3 5{ . | 9 @ 3 @3 5{ 7| 6| 9 @ 3 | @ 5 { . 5+5, @ 3 {@7 i{{ +{> |, @ 5> i|| +{> |, @ 5> i{| +{> |, @ 4 G @ 5 5 45 @ 6 A 3 dqg i{{ +{> |, A 3 Uhodwlyh plqlpxp dw +7> 5, 5
5
91 i+{> |, @ { . {| . | . 6{ 6| i{ +{> |, @ 5{ . | . 6> i| +{> |, @ { . 5| 6 5{ . | . 6 @ 3 { . 5| 6 @ 3 5{ . | . 6 @ 3 5{ 7| . 9 @ 3 6| . < @ 3 |@6 { . 5+6, 6 @ 3 { @ 6 i{{ +{> |, @ 5> i|| +{> |, @ 5> i{| +{> |, @ 4 G @ 5 5 45 @ 6 A 3 dqg i{{ +{> |, A 3 Uhodwlyh plqlpxp dw +6> 6, ;1 i+{> |, @ 8{| :{5 | 5 . 6{ 9| 7 i{ +{> |, @ 8| 47{ . 6> i| +{> |, @ 8{ 5| 9 8| 47{ . 6 @ 3 5| . 8{ 9 @ 3 43| 5;{ . 9 @ 3 43| . 58{ 63 @ 3 6{ 57 @ 3 { @ ; 5| . 8+;, 9 @ 3 5| @ 79 | @ 56 i{{ +{> |, @ 47> i|| +{> |, @ 5> i{| +{> |, @ 8 G @ +47,+5, 85 @ 6 A 3 dqg i{{ +{> |, ? 3 Uhodwlyh pd{lpxp dw +;> 56,
5;:
431 i+{> |, @ {5 . {| . 6{ . 5| 9 i{ +{> |, @ 5{ . | . 6> i| +{> |, @ { . 5 5{ . | . 6 @ 3 {.5@3 { @ 5 5+5, . | . 6 @ 3 |@4 i{{ +{> |, @ 5> i|| +{> |, @ 3> i{| +{> |, @ 4 G @ 5 3 45 @ 4 ? 3 Vdggoh srlqw dw +5> 4, 451 i+{> |, @ {5 . {| . |5 6{ 8 i{ +{> |, @ 5{ . | 6> i| +{> |, @ { . 5| 5{ . | 6 @ 3 { . 5| @3 5{ . | 6 @ 3 @3 5{ 7| 6| 6 @ 3 | @ 4 { . 5+4, @ 3 {@5 i{{ +{> |, @ 5> i|| +{> |, @ 5> i{| +{> |, @ 4 G @ 5 5 45 @ 6 A 3 dqg i{{ +{> |, A 3 Uhodwlyh plqlpxp dw +5> 4, 471 i+{> |, @ 8{6 . 5|5 93{| 6 i{ +{> |, @ 48{5 93|> i| +{> |, @ 7| 93{ 48{5 93| @ 3 +4 , 7| 93{ @ 3 +5 , Glylgh htxdwlrq +4, e| 48 dqg htxdwlrq +5, e| 71 {5 7| @ 3 +6 , | 48{ @ 3 +7 , Vroyh htxdwlrq +7, iru |= | @ 48{ Vxevwlwxwlqj lq htxdwlrq +6,/ zh kdyh {5 7+48{, @ 3 {5 93{ @ 3 { @ 3 ru { @ 93 | @ 3 ru | @ <33=
5;;
Fkdswhu <
i{{ +{> |, @ 63{> i|| +{> |, @ 7> i{| +{> |, @ 93 Dw +3> 3,> G @ 3 7 +93,5 @ 6933 ? 3= Vdggoh srlqw dw +3> 3, Dw +93> <33,>
551 } @
PXOWLYDULDEOH FDOFXOXV
6 4 4 | |6 {5 | . 5 5 49
C} @ 5{|> C{
C} 6 6 @ |5 {5 C| 5 5
5{| @ 3 { @ 3 ru | @ 3
G @ 4;33 7 +93,5 @ 6933 A 3 dqg i{{ +{> |, A 3=
6 6 5 | {5 @ 3 5 5
Uhodwlyh plqlpxp dw +93> <33, 491 i+{> |, @ 6{5 . :|6 75{| . 8
Li { @ 3> 6 6 5 | @3 5 5
i{ +{> |, @ 9{ 75|> i| +{> |, @ 54|5 75{ 9{ 75{ @ 3 +4 , 54| 5 75{ @ 3 +5 ,
| @ 4= Li | @ 3>
Glylgh htxdwlrq +4, e| 9 dqg htxdwlrq +5, e| 541
6 {5 @ 3 5
{ :| @ 3 +6 , | 5 5{ @ 3 +7 ,
u
{@
Vroyh htxdwlrq +6, iru {= { @ :| Vxevwlwxwlqj lq htxdwlrq +7,/ zh kdyh 5
| 5+:|, @ 3 |5 47| @ 3 | @ 3 ru | @ 47 { @ 3 ru { @ <;=
C5} C5} C5} @ 5|> @ 6|> @ 5{ 5 5 C{ C| C|C{ G @ +5|,+6|, +5{,5 @ 9| 5 7{5 Dw +3> 4,> G @ 9 A 3 dqg
i{{ +{> |, @ 9> i|| +{> |, @ 75|> i{| +{> |, @ 75 }@
Dw +3> 3,>
s 6 9 @ = 5 5
C5} @ 5 ? 3= C{5
4 4 6 4 @4 . 5 5 49 49
G @ 9 3 +75,5 @ 4:97 ? 3=
4 dw +3> 4, Uhodwlyh pd{lpxp ri 4 49
Vdggoh srlqw dw +3> 3, Dw +<;> 47,>
Dw +3> 4,>
G @ 9 8;; +75,5 @ 4:97 A 3 dqg i{{ +{> |, A 3= Uhodwlyh plqlpxp dw +<;> 47, 4;1 i+{> |, @ {5 . h| |
i{ +{> |, @ 5{> i| +{> |, @ h
5{ @ 3 h| @ 3 Wkh odwwhu htxdwlrq kdv qr vroxwlrqv1 Wkhuh duh qr h{wuhpd dqg qr vdggoh srlqwv1
G @ 9 A 3 dqg
C5} @ 5 A 3= C{5
4 48 6 4 @ }@ . . 5 5 49 49 Uhodwlyh plqlpxp ri 48 49 dw +3> 4, s Dw 59 > 3 > G @ 9 ? 3= Vdggoh srlqwv dw
s
9 5 >3
Wkh fruuhfw judsk lv +g,1
s dqg 59 > 3
Vhfwlrq <16
Pd{lpd dqg Plqlpd
571 } @ 5{6 6| 7 . 9{|5 .
5;<
4 49
C} C} @ 9{5 . 9|5 > @ 45| 6 . 45{| C{ C| 9{5 . 9|5 @ 3 +4 , 45| 6 . 45{| @ 3 +5 , Glylgh htxdwlrq +4, e| 9 dqg htxdwlrq +5, e| 451 {5 . |5 @ 3 +6 , | 6 . {| @ 3 +7 , Vroyh htxdwlrq +6, iru |= Vxevwlwxwh | @ { lq htxdwlrq +7,1 Li | @ {> {6 . {5 @ 3 { +{ . 4, @ 3 { @ 3 ru { @ 4=
591 } @ | 7 . 7{| 5{5 .
4 49
C} C} @ 7| 7{> @ 7|6 . 7{ C{ C| 7| 7{ @ 3 +4 , 7| 6 . 7{ @ 3 +5 , Glylgh htxdwlrq +4, e| 7 dqg htxdwlrq +5, e| 7= | { @ 3 +6 , | 6 . { @ 3 +7 ,
5
Vroyh htxdwlrq +6, iru |=
Li { @ 3> | @ 3= Li { @ 4> | @ 4=
|@{
Li | @ {> {6 {5 @ 3 {5 +{ 4, @ 3 { @ 3 ru { @ 4= Li { @ 3> | @ 3 Li { @ 4> | @ 4= C5} C5} @ 45{> @ 69| 5 . 45{> 5 C{ C| 5 C5} @ 45| C|C{
Vxevwlwxwlqj/ zh kdyh {6 . { @ 3 {+{5 . 4, @ 3 { @ 3 ru { @ 4 ru { @ 4= Li { @ 3> | @ 3= Li { @ 4> | @ 4= Li { @ 4> | @ 4=
G @ +45{,+69|5 . 45{, +45| 5 , Dw +3> 3,> G @ 3> zklfk jlyhv qr lqirupdwlrq1 Dw +4> 4,> G @ 45+69 . 45, 477 @ 477 A 3 dqg } @ 5 6 . 9 .
C5} C{5
@ 45 ? 3=
4 4 @4 49 49
4 dw +4> 4,= vr wkhuh lv d uhodwlyh pd{lpxp ri 4 49 Dw +4> 4,>
G @ 45+69 . 45, 477 @ 477 A 3 dqg
C5} C{5
@ 45 ? 3>
4 dw +4> 4,= vr wkhuh lv d uhodwlyh pd{lpxp ri 4 49 Wkh fruuhfw judsk lv +f,1
C5} C5} C5} @ 7> @ 45| 5 > @7 5 5 C{ C| C|C{ G @ 7+45|5 , 75 @ 7;| 5 49 Dw +3> 3,> G @ 49 ? 3= Vdggoh srlqw dw +3> 3, Dw +4> 4, dqg +4> 4,> G @ 7; 49 @ 65 A 3 dqg
C5} C{5
@ 7 ? 3=
4 Uhodwlyh pd{lpxp ri 4 49 dw +4> 4, dqg dw +4> 4,= Wkh fruuhfw judsk lv +i,=
5<3
Fkdswhu <
5;1 i+{> |, @ {6 . +{ |,5 i{ +{> |, @ 6{5 . 5+{ |,> i| +{> |, @ 5+{ |, 6{5 . 5{ 5| @ 3 5{ . 5| @ 3 6{5 @ 3 {@3 3 . 5| @ 3 |@3 i{{ +{> |, @ 9{ . 5> i|| +{> |, @ 5> i{| +{> |, @ 5 G @ +9{ . 5,+5, +5,5 @ 45{ Dw +3> 3,> G @ 3> zklfk jlyhv qr lqirupdwlrq1 H{0 dplqh wkh judsk ri } @ {6 . +{ |,5 = lq wkh |}0 sodqh/ wkh wudfh lv } @ | 5 > zklfk kdv d plqlpxp dw +3> 3> 3,> lq wkh {}0sodqh/ wkh wudfh lv } @ {6 . {5 > zklfk kdv d plqlpxp dw +3> 3> 3,= Exw lq wkh sodqh | @ {> wkh wudfh lv } @ {6 > zklfk kdv qhlwkhu d pd{lpxp qru d plqlpxp dw +3> 3> 3,= Vr wkh ixqfwlrq kdv qr uhodwlyh h{wuhpd1 Qrwlfh wkh rulhqwdwlrq ri d{hv lq wkh iroorzlqj jxuh= Wklv lv d edfn ylhz1
PXOWLYDULDEOH FDOFXOXV
Li Ve +p> e, @ 3> wkhq [ 5 +p{ . e |, @ 3 [ [ [ |@3 +p{, . +e, [ [ {,p . qe @ + |= +5 , Htxdwlrqv +4, dqg +5, jlyh xv wkh iroorzlqj v|vwhp ri htxdwlrqv1 [ [ [ + {| {,e . + {5 ,p @ [ [ | {,p @ qe . + 651 S +{> |, @ 4333 . 57{ {5 . ;3| | 5 S{ +{> |, @ 57 5{ S| +{> |, @ ;3 5| 57 5{ @ 3 ;3 5| @ 3 45 @ { 73 @ | D fulwlfdo srlqw lv +45> 73,= S{{ +{> |, @ 5> S|| +{> |, @ 5> S{| +{> |, @ 3 Iru +45> 73,> G @ 5+5, 35 @ 7 A 3= Vlqfh S{{ +{> |, ? 3> S +45> 73, @ 4333 . 57+45, +45,5 . ;3+73, +73,5 @ 5:77= Wkh pd{lpxp sur w lv '5:7/7331 671 F+{> |, @ 5{5 . 6|5 5{| . 5{ 459| . 6;33
631
[ +p{ . e |,5 [ Vp +p> e, @ 5+p{ . e |,4 +{, [ @5 +p{5 . e{ {|, [ Ve +p> e, @ 5+p{ . e |,+4, [ @5 +p{ . e |, V+p> e, @
Li Vp +p> e, @ 3> wkhq [ 5 +p{5 . e{ {|, @ 3 [ [ [ +p{5 , . +e{, @ +{|, [ [ [ + +{,e @ +{|,= +4 , {5 ,p .
F{ @ 7{ 5| . 5 F| @ 9| 5{ 459 3 @ 7{ 5| . 5 3 @ 9| 5{ 459 3 @ 5{ | . 4 3 @ 5{ . 9| 459 3@ 8| 458 | @ 58 3 @ 5{ 58 . 4 Li | @ 58> { @ 45= +45> 58, lv d fulwlfdo srlqw1 F{{ +{> |, @ 7 F|| +{> |, @ 9 F{| +{> |, @ 5
Vhfwlrq <16
Pd{lpd dqg Plqlpd
5 Li | @ 6> { @ +6,5 @ 9= 6
Iru +45> 58,> G @ 7 9 7 @ 53 A 3= Vlqfh F{{ +{> |, A 3> 45 xqlwv ri hohfwulfdo wdsh dqg 58 xqlwv ri sdfnlqj wdsh vkrxog eh surgxfhg wr |lhog d plqlpxp frvw1 F+45> 58, @ 5+45,5 . 6+58,5 5+45 58, . 5+45, 459+58, . 6;33 @ 556: Wkh plqlpxp frvw lv '556:1 691 S +{> |, @ 69{| {6 ;|6 S{ +{> |, @ 69| 6{5 S| +{> |, @ 69{ 57|5 S{ +{> |, @ 3 69| 6{5 @ 3 69| @ 6{5 |@
4 5 { 45
S{ +{> |, @ 3 69{ 57|5 @ 3 69{ @ 57|5
4 5 { 45 5 { @ |5 = 6 |@
5 5 5 | 6 4 7 |@ |7 45 < 4 | @ |7 5: 4 7 | | @3 5: 4 6 | 4 | @3 5: 4 45
4 6 | 4@3 5: 4 6 | @4 5: | 6 @ 5: |@6
5 Li | @ 3> { @ +3,5 @ 3= 6 Wkh fulwlfdo srlqwv duh +9> 6, dqg +3> 3,= S{{ +{> |, @ 9{ S|| +{> |, @ 7;| S{| +{> |, @ 69 S{{ +9> 6, @ 69 S|| +9> 6, @ 477 S{| +9> 6, @ 69 G @ +69,+477, +69,5 @ 6;;; Khuh G A 3 dqg S{{ ? 3> vr wkhuh lv d uhodwlyh pd{lpxp dw +9> 6,= S{{ +3> 3, @ 3 S|| +3> 3, @ 3 S{| +3> 3, @ 69 G @ 3 3 695 @ 45<9
5 { @ |5 6 Xvh vxevwlwxwlrq wr vroyh wkh v|vwhp ri htxdwlrqv
|@
5<4
ru | @ 3 ru | @ 3 ru | @ 3 ru | @ 3
Vlqfh G ? 3> wkhuh lv d vdggoh srlqw dw +3> 3,= S +9> 6, @ 69+9,+6, +9,6 ;+6,6 @ 97; 549 549 @ 549 Vr 9 wrqv ri vwhho dqg 6 wrqv ri doxplqxp surgxfh d pd{lpxp sur w ri '549/3331 6;1 S +> u> v, @ +6u5 +4 u, . u6 , . +4 , +6v5 +4 v, . v6 , +d, S +=<> =8> =9, @ =<+6+=8,5 +4 =8, . +=8,6 , . +4 =<,+6+=9,5 +4 =9, . +=9,6 , @ =847; S +=4> =;> =7, @ =4+6+=;,5 +4 =;, . +=;,6 , . +4 =4,+6+=7,5 +4 =7, . +=7,6 , @ =7397 Wkh mxu| lv ohvv olnho| wr pdnh wkh fruuhfw ghflvlrq lq wkh vhfrqg vlwxdwlrq1 +e, Li u @ v @ 4 wkhq S +> 4> 4, @ 4/ vr wkh mxu| dozd|v pdnhv d fruuhfw ghflvlrq1 Wkhvh ydoxhv gr qrw ghshqg rq / exw lq d uhdo0olih vlwxdwlrq lv olnho| wr lq xhqfh u dqg v=
5<5
Fkdswhu <
+f, Zkhq S uhdfkhv d pd{lpxp/ S > Su > dqg Sv htxdo 31 S +> u> v, @ 6u5 +4 u, . u6 +6v5 +4 v, . v6 , 5 @ 6u 5u6 +6v5 5v6 , Su +> u> v, @ +9u+4 u, 6u5 . 6u5 , @ 9u+4 u, Sv +> u> v, @ +4 ,+9v+4 v, 6v5 . 6v5 , @ 9v+4 ,+4 v, S +> u> v, @ 3 zkhq u @ v= Vlqfh Su +> u> v, @ 9u+4 u,> dqg Sv +> u> v, @ 9+4 ,v+4 v,> wkhq S > Su > dqg Sv duh vlpxo0 wdqhrxvo| 3 dw wkh srlqwv +> 4> 4, dqg +> 3> 3,= Vr +> 4> 4, dqg +> 3> 3, duh fulwlfdo srlqwv1 S +> 3> 3, @ 3 zkloh S +> 4> 4, @ 4 Vlqfh S +> u> v, uhsuhvhqwv d suredelolw|/ 3 S +> u> v, 41 Wkxv/ S +> 4> 4, @ 4 lv d pd{lpxp ydoxh ri wkh ixqfwlrq1
<17
Odjudqjh Pxowlsolhuv
51 Pd{lpl}h i+{> |, @ 7{| . 5> vxemhfw wr { . | @ 57= 41 j+{> |, @ { . | 57
71
PXOWLYDULDEOH FDOFXOXV
7|5 6 @ 3 ;{| . 5 @ 3 6{ 5| 8 @ 3
7| 5 dqg @ 7{| 6 7|5 @ 7{| 6 4 7 3 @ | 5 . 7{| @ 7| |.{ 6 6
81 @
| @ 3 ru | @ 6{ Vxevwlwxwlqj lqwr 6{ 5| 8 @ 3> zh kdyh iru | @ 3> 6{ 8 @ 3> vr { @ 86 > dqg iru | @ 6{> 6{ 5+6{, 8 @ 3> vr { @ 8< dqg | @ 86 = 8 8 i +3,5 @ 3 >3 @ 7 6 6 5 8 8 833 8 8 i @ @7 > < 6 < 6 ;4 8 8 833 Wkh pd{lpxp lv i < > 6 @ ;4 9=5= 91 Plqlpl}h i+{> |, @ 6{5 . 7|5 {| 5> vxemhfw wr 5{ . | @ 54= 41 j+{> |, @ 5{ . | 54
51 I +{> |> , @ 7{| . 5 +{ . | 57,
51 I +{> |> , @ 6{5 . 7|5 {| 5 +5{ . | 54,
61 I{ +{> |> , @ 7| I| +{> |> , @ 7{ I +{> |> , @ +{ . | 57,
61 I{ +{> |> , @ 9{ | 5 I| +{> |> , @ ;| { I +{> |> , @ +5{ . | 54,
71
7| @ 3 7{ @ 3 { . | 57 @ 3
81 @ 7| dqg @ 7{ 7| @ 7{ |@{ Vlqfh {.|57 @ 3 dqg | @ {> zh kdyh 5{57 @ 3> vr { @ 45 dqg | @ 45= Wkh pd{lpxp lv i+45> 45, @ 7+45,+45, . 5 @ 8:;= 71 Pd{lpl}h i+{> |, @ 7{|5 > vxemhfw wr 6{ 5| @ 8= 41 j+{> |, @ 6{ 5| 8 51 I +{> |> , @ 7{|5 +6{ 5| 8, 61 I{ +{> |> , @ 7|5 6 I| +{> |> , @ ;{| . 5 I +{> |> , @ +6{ 5| 8,
71 9{ | 5 @ 3 ;| { @ 3 5{ . | 54 @ 3 9{ | dqg @ ;| { 5 9{ | @ ;| { 5 ; |@ { 4: Vxevwlwxwlqj lqwr 5{ . | 54 @ 3> zh kdyh 81 @
; { 54 @ 3> 4: vr { @ 4: 5 dqg | @ 7= 5 4: 4: 4: .7+7,5 +7,5 Plqlpxp lv i 5 > 7 @ 6 5 5 5{ .
@
<:< @ 577=:8= 7
Vhfwlrq <17
Odjudqjh Pxowlsolhuv
;1 Pd{lpl}h i+{> |, @ 45{| {5 6|5 > vxemhfw wr { . | @ 49=
5<6
dqg | . 5} 5{ . 5| @ |} {|
41 j+{> |, @ { . | 49
{| 5 . 5{|} @ 5{|} . 5|5 } 5| 5 } {| 5 @ 3 | 5 +5} {, @ 3
51 I +{> |> , @ 45{| {5 6| 5 +{ . | 49, 61 I{ +{> |> , @ 45| 5{ I| +{> |> , @ 45{ 9| I +{> |> , @ +{ . | 49, 71 45| 5{ @ 3 45{ 9| @ 3 { . | 49 @ 3 81 @ 45| 5{ dqg @ 45{ 9| 45| 5{ @ 45{ 9| : |@ { < Vxevwlwxwlqj lqwr { . | 49 @ 3> zh kdyh : { . { 49 @ 3> < vr { @ < dqg | @ := Pd{lpxp lv i+<> :, @ 45+<,+:, +<,5 6+:,5 @ 85;= 431 Pd{lpl}h i+{> |> }, @ {| . 5{} . 5|}> vxemhfw wr {|} @ 65= 41 j+{> |> }, @ {|} 65 51I +{> |> }> , @ {| . 5{} . 5|} +{|} 65, 61 I{ +{> |> }> , @ | . 5} |} I| +{> |> }> , @ { . 5} {} I} +{> |> }> , @ 5{ . 5| {| I +{> |> }> , @ +{|} 65, 71
| . 5} |} @ 3 { . 5} {} @ 3 5{ . 5| {| @ 3 {|} 65 @ 3
81 @
| . 5} { . 5} 5{ . 5| > @ > dqg @ |} {} {| | . 5} { . 5} @ |} {} {|} . 5{} 5 @ {|} . 5|} 5 5|} 5 5{} 5 @ 3 5} 5 +| {, @ 3 } @ 3 ru | @ {
{ 5 Vlqfh {|} @ 65> } @ 3 dqg | @ 3 duh lpsrvvleoh1 Vxevwlwxwlqj | @ { dqg } @ {5 lqwr {|} 65 @ 3> zh kdyh { @ 65> vr { @ 7> | @ 7> dqg } @ 5= {+{, 5 Pd{lpxp lv | @ 3 ru } @
i+7> 7> 5, @ 7+7, . 5+7,+5, . 5+7,+5, @ 7;= 451 Ohw { dqg | eh wkh wzr qxpehuv vxfk wkdw { . | @ 69= Pd{lpl}h i+{> |, @ {5 |> vxemhfw wr { . | @ 69= 41 j+{> |, @ { . | 69 51 I +{> |> , @ {5 | +{ . | 69, 61 I{ +{> |> , @ 5{| I| +{> |> , @ {5 I +{> |> , @ +{ . | 69, 71
5{| @ 3 {5 @ 3 { . | 69 @ 3
81 @ 5{| dqg @ {5 5{| @ {5 {+{ 5|, @ 3 Vlqfh { @ 3 jlyhv d vpdoohu ydoxh ri i+{> |, @ {5 | wkdq dq| srvlwlyh ydoxhv ri { dqg |> zh fdq dvvxph { 9@ 3> vr | @ {5 = Vxevwlwxwlqj lqwr { . | 69 @ 3> zh kdyh { 69 @ 3> 5 vr { @ 57 dqg | @ 45= {.
471 Ohw {> |> dqg } eh wkuhh srvlwlyh qxpehuv vxfk wkdw { . | . } @ 573= Pd{lpl}h i+{> |> }, @ {|}> vxemhfw wr { . | . } @ 573= 41 j+{> |, @ { . | . } 573 51 I +{> |> }> , @ {|} +{ . | . } 573,
5<7
Fkdswhu <
61 I{ +{> |> }> , @ |} I| +{> |> }> , @ {} I} +{> |> }> , @ {| I +{> |> }> , @ +{ . | . } 573, 71 |} @ 3 {} @ 3 {| @ 3 { . | . } 573 @ 3 81 @ |}> @ {}> @ {| |} @ {} } @ 3 +lpsrvvleoh, ru { @ | {} @ {| { @ 3 +lpsrvvleoh, ru } @ | Wkxv/ { @ | @ }=
531 F+{> |, @ 5{5 . 9|5 . 7{| . 43> vxemhfw wr { . | @ 43 41 j+{> |, @ { . | 43 51 I +{> |, @ 5{5 . 9|5 . 7{| . 43 +{ . | 43, 61 I{ +{> |> , @ 7{ . 7| I| +{> |> , @ 45| . 7{ I +{> |> , @ +{ . | 43, 71 7{ . 7| @ 3 45| . 7{ @ 3 { . | 43 @ 3 81 @ 7{ . 7| dqg @ 45| . 7{=
{ . { . { 573 @ 3 { @ ;3
7{ . 7| @ 45| . 7{ ;| @ 3 |@3
Wkxv/ { @ | @ } @ ;3= Wkh wkuhh qxpehuv duh ;3/ ;3/ dqg ;31 4;1 Ohw { eh wkh zlgwk dqg | eh wkh ohqjwk ri d hog vxfk wkdw wkh frvw lq grooduv wr hqforvh wkh hog lv 9{ . 9| . 7{ . 7| @ 4533 43{ . 43| @ 4533= Wkh duhd lv i+{> |, @ {|= 41 j+{> |, @ 43{ . 43| 4533 51 I +{> |, @ {| +43{ . 43| 4533, 61 I{ +{> |> , @ | 43 I| +{> |> , @ { 43 I +{> |> , @ +43{ . 43| 4533, 71
PXOWLYDULDEOH FDOFXOXV
| 43 @ 3 { 43 @ 3 43{ . 43| 4533 @ 3
81 43 @ | dqg 43 @ { |@{ Vxevwlwxwlqj lqwr wkh wklug htxdwlrq jlyhv 43{ . 43{ 4533 @ 3 53{ 4533 @ 3 { @ 93 | @ 93= Wkhvh glphqvlrqv/ 93 ihhw e| 93 ihhw/ zloo pd{l0 pl}h wkh duhd1
Vlqfh { . | @ 43> { @ 43= 43 odujh nlwv dqg qr vpdoo nlwv zloo pd{lpl}h wkh frvw1 551 i+{> |, @ 6{4@6 |5@6 > vxemhfw wr ;3{ . 483| @ 73/333 41 j+{> |, @ ;3{ . 483| 73/333 51 I +{> |, @ 6{4@6 | 5@6 +;3{ . 483| 73/333, 61 dqg 71 I{ +{> |> , @ {5@6 |5@6 ;3 @ 3 I| +{> |> , @ 5{4@6 |4@6 483 @ 3 I +{> |> , @ +;3{ . 483| 73/333, @ 3 {5@6 | 5@6 5{4@6 | 4@6 @ ;3 483 48| @{ 49 Vxevwlwxwh lqwr wkh wklug htxdwlrq1 48| . 483| 73/333 @ 3 ;3 49 | @ 4:; +urxqghg,
81
{@
48+4:;, 49
49: Xvh derxw 49: xqlwv ri oderu dqg 4:; xqlwv ri fds0 lwdo wr pd{lpl}h surgxfwlrq1
Vhfwlrq <17
Odjudqjh Pxowlsolhuv
571 Ohw { dqg | eh wkh glphqvlrqv ri wkh hog vxfk wkdw 5{ . 5| @ 533> dqg wkh duhd lv i+{> |, @ {|= 41 j+{> |, @ 5{ . 5| 533 51 I +{> |, @ {| +5{ . 5| 533, 61 I{ +{> |> , @ | 5 I| +{> |> , @ { 5 I +{> |> , @ +5{ . 5| 533, 71
| 5 @ 3 { 5 @ 3 5{ . 5| 533 @ 3
81 5 @ | dqg 5 @ {> vr { @ |= 5{ . 5{ 533 @ 3 7{ 533 @ 3 { @ 83 | @ 83= Wkxv/ Glphqvlrqv ri 83 p e| 83 p zloo pd{lpl}h wkh duhd1 591 Ohw { eh wkh udglxv u ri wkh flufxodu edvh dqg | wkh khljkw k ri wkh fdq/ vxfk wkdw wkh yroxph lv {5 | @ 583 =
5<8
81 Iurp wkh vhfrqg htxdwlrq/ {+5 {, @ 3 5 = { Li { @ 3> wkh yroxph zloo eh 3/ zklfk lv qrw srvvleoh1 { @ 3 ru @
Vxevwlwxwlqj { @ 5 lqwr wkh uvw htxdwlrq jlyhv 5 5 |.5 |@3 7 | . 5| @ 3 7 @| 7 @ = | Vlqfh @ {5 > | @ 5{= Vxevwlwxwlqj lqwr wklug htxdwlrq jlyhv {5 +5{, 583 @ 3 5{6 583 @ 3 {@8 | @ 43= Vlqfh j+4> 583, @ 3 dqg i+4> 583, @ 835 A i+8> 43, @ 483 > d fdq zlwk udglxv ri 8 lqfkhv dqg khljkw ri 43 lqfkhv zloo kdyh d plqlpxp vxuidfh duhd1 5;1 Ohw {> |> dqg } eh wkh glphqvlrqv ri wkh er{ vxfk wkdw wkh vxuidfh duhd lv
Wkh vxuidfh duhd lv i+{> |, @ 5 {| . 5 {5 =
{| . 5|} . 5{} @ 833 dqg wkh yroxph lv i+{> |> }, @ {|}=
41 j+{> |, @ {5 | 583 51 I +{> |, @ 5 {| . 5 {5 + {5 | 583 , 61 I{ +{> |> , @ 5 | . 7 { +5 {|, I| +{> |> , @ 5 { + {5 , I +{> |> , @ + {5 | 583 , 71 5 | . 7 { +5 {|, @ 3 5 { {5 @ 3 {5 | 583 @ 3 Vlpsoli|lqj wkhvh htxdwlrqv jlyhv | . 5{ 4{| @ 3 5{ 4{5 @ 3 {5 | 583 @ 3=
41 j+{> |> }, 833 @ 3 51I +{> |> }, @ {|} +{| . 5|} . 5{} 833, 61 dqg 71 I{ +{> |> }> , @ |} +| . 5}, @ 3 I| +{> |> }> , @ {} +{ . 5}, @ 3 I} +{> |> }> , @ {| +5| . 5{, @ 3 I +{> |> }> , @ +{| . 5{} . 5|} 833, @ 3
+4 , +5 , +6 , +7 ,
5<9
Fkdswhu <
Pxowlso|lqj htxdwlrq +4, e| {> htxdwlrq +5, e| |> dqg htxdwlrq +6, e| } jlyhv {|} {+| . 5}, @ 3 {|} |+{ . 5}, @ 3 {|} }+5| . 5}, @ 3= 81 Vxewudfwlqj wkh uvw htxdwlrq iurp wkh vhfrqg htxdwlrq jlyhv {+| . 5}, |+{ . 5}, @ 3 5{} 5|} @ 3 }+{ |, @ 3> { @ |= vr Vxewudfwlqj wkh wklug htxdwlrq iurp wkh vhfrqg htxdwlrq jlyhv }+5| . 5{, |+{ . 5}, @ 3 5{} {| @ 3 {+5} |, @ 3> | }@ = vr 5 Vxevwlwxwlqj lqwr wkh irxuwk htxdwlrq jlyhv | | . 5| |5 . 5| 833 @ 3 5 5 6| 5 @ 833 u 833 |@ 6 45=<3<< { 45=<3<< 45=<3<< } 5 9=797<= Wkh glphqvlrqv duh 451<4 p e| 451<4 p e| 9178 p1 631 Ohw {> |> dqg } eh wkh glphqvlrqv ri wkh er{1 Wkh vxuidfh duhd lv 5{| . 5{} . 5|}=
71 5| . 5} |} @ 3 5{ . 5} {} @ 3 5{ . 5| {| @ 3 {|} 5: @ 3
5| . 5} 5{ . 5} @ dqg @ = |} {} Wkxv/ 5| . 5} 5{ . 5} @ |} {} 5{|} . 5{} 5 @ 5{|} . 5|} 5 5{} 5 5|} 5 @ 3 5} 5 @ 3 ru { | @ 3 } @ 3 +lpsrvvleoh, ru { @ |= Htxdwlrqv +5, dqg +6, jlyh 5{ . 5| 5{ . 5} @ dqg @ = {} {| 5{ . 5} 5{ . 5| @ {} {| Wkxv/ 5{5 | . 5{|} @ 5{5 } . 5{|} 5{5 | 5{5 } @ 3 5{5 +| }, @ 3 5{5 @ 3 ru | } @ 3 { @ 3 +lpsrvvleoh, ru | @ }= Wkhuhiruh/ { @ | @ }= Vxevwlwxwlqj lqwr htxdwlrq +7, jlyhv {6 5: @ 3 {6 @ 5: { @ 6=
i+{> |> }, @ 5{| . 5{} . 5|} vxemhfw wr {|} @ 5:= 51 I +{> |> }, @ 5{| . 5{} . 5|} +{|} 5:, 61 I{ +{> |> }> , @ 5| . 5} |} I| +{> |> }> , @ 5{ . 5} {} I} +{> |> }> , @ 5{ . 5| {| I +{> |> }> , @ +{|} 5:,
+4 , +5 , +6 , +7 ,
81 Htxdwlrqv +4, dqg +5, jlyh
Zh pxvw plqlpl}h
41 j+{> |> }, @ {|} 5:
PXOWLYDULDEOH FDOFXOXV
Wkxv/ | @ 6 dqg } @ 6= Wkh glphqvlrqv wkdw zloo plqlpl}h wkh vxuidfh duhd duh 6 p e| 6 p e| 6 p1
Vhfwlrq <18
Wrwdo Gl huhqwldov dqg Dssur{lpdwlrqv
651 +d, Wkh vxuidfh duhd ri wkh er{ lv VD @ {| . 5{} . 5|}= Ohw wkh vlghv zlwk duhd {} eh wkrvh pdgh iurp wkh iuhh pdwhuldo +dorqj zlwk wkh erwwrp,1 Wklv jlyhv wkh frqvwudlqw 5{} . {| @ 71 Wkh yroxph ri wkh 733 er{ lv {|}/ wkxv lw zloo wdnh {|} wulsv wr wudqvsruw 73 733 +=43, @ wkh pdwhuldo dw d frvw ri {|} 1 Wkh {|} wrwdo frvw dovr lqfoxghv wkh frvw ri wkh pdwhuldo iru wkh hqgv ri wkh er{= +5|},+53, @ 73|}1 Wkxv/ wkh wrwdo frvw lv i+{> |> }, @
73 . 73|}= {|}
+e, Xvlqj d vsuhdg vkhhw/ { @ 5 |dugv> | @ 4 |dug> } @ 45 |dug1
<18
Wrwdo Gl huhqwldov dqg Dssur{lpdwlrqv
s 51 Ohw } @ i+{> |, @ {5 . | 5 = Wkhq g} @ i{ +{> |, g{ . i| +{> |, g|= 4 g} @ +{5 . |5 ,4@5 +5{, g{ 5 4 . +{5 . | 5 ,4@5 +5|, g{ 5 | { g{ . s g| @s 5 5 5 { .| { . |5 s Wr dssur{lpdwh 9=3:5 . :=<85 > zh ohw { @ 9> g{ @ =3:> | @ ;> dqg g| @ =38= 9 ; g} @ s +=3:, . s +=38, 95 . ;5 95 . ;5 9 ; @ +=3:, . +=38, 43 43 @ =335 i+9=3:> :=<8, @ i+9> ;, . } i+9=;, . g} s @ 95 . ;5 . =335 i+9=3:> :=<8, 43=335 s Xvlqj d fdofxodwru/ 9=3:5 . :=<85 43=3357= Wkh devroxwh ydoxh ri wkh gl huhqfh ri wkh wzr uhvxowv lv m43=335 43=3357m @ =3337=
5<:
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8 755 =
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6
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4;1 i+{, @ n{6 > ^5> 7` ] 5
7
676
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4
3
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4 =9654 h
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49 =4857 438
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4 @ h{ @4
3
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677
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Vhfwlrq 4414
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671 s+{> w, @ U O 3
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678
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3
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3
8
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p
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s 5. 5 5 >
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=66 @ +5{ {5 , =5<
@ 5+=66, +=66,5 5+=5<, . +=5<,5 =39
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SUREDELOLW\ DQG FDOFXOXV
531 i+{, @ 6{7 > ^4> 4, ] +d,
4
p
p 6{7 g{ @ {6 4
4 4 @4 6 @ p 5 p6 @ 5s p@ 65 4=59 +e, H+{, @ @ 4=8 +Iurp H{huflvh ;, S +4=59 ? { ? 4=8, ] 4=8 @ 6{7 g{ 4=59
4=8 @ {6
4=59
4 4 @ +4=59,6 +4=8,6 =537
Vhfwlrq 4415
H{shfwhg Ydoxh dqg Yduldqfh ri Frqwlqxrxv Udqgrp Yduldeohv
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4 3 4
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s Ydu+{, =77379
67<
4 571 i+{, @ 44
6 4. s > ^7> <` {
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4 ^< . 4; 9=74 9+9=74,4@5 ` 44
=7< 591 i+{, @
4 > ^4> 53` +oq 53,{ ]
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53
4
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53
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4
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53
4
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4 g{ 5 +oq 53,{
53
{ g{ 5 oq 53 4 53 {5 @ +9=67,5 5 oq 53 4 @
@
6<< +9=67,5 5 oq 53
59=73 s 59=73 8=47 vhf +f, S +9=67 8=47 ? { ? 9=67 . 8=47, @ S +4=5 ? { ? 44=7;, ] 44=7; 4 @ g{ +oq 53,{ 4=5 44=7; oq { @ oq 53 4=5 @
4 +oq 44=7; oq 4=5, oq 53
=:8
683
Fkdswhu 44
4 5;1 i+{, @ s > ^4> 7` 5 {
] +d, @
7
4
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651 i+{, @
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7
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4 { s g{ 5 { {
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5 7 7< {8@5 @ 8 4 < @
4
7< <
4 7< +65 4, 8 <
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631 i+{, @ 4=4;8 43< {7=8555 h=37<;79{ ]
4333
4
8
5 : 6
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H+{, @
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8=8 { 48
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8
3
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SUREDELOLW\ DQG FDOFXOXV
{ i+{,g{
Xvlqj wkh lqwhjudwlrq ixqfwlrq rq rxu fdofxodwru1 H+{, 443=;3 Wkh h{shfwhg vl}h lv derxw 4441
] +e, Ydu+{, @
8
3
5
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g{ 5
8=8 5 4 { {6 g{ 5 48 48 3 8 8=8 6 4 { {7 5 @ 78 93 3 ]
@
@
8
8=8 { 48
4 8=8 458 958 3 5 78 93
4=9343; s @ Ydu+{, 4=598
+f, S +{ , @ S +{ 4=;39 4=598, @ S +{ =874, ] =874 8=8 { @ g{ 48 3 =874 4 8=8 { {5 @ 48 63 3 8=8 4 @ +=874, +=874,5 3 48 63 =4;;9
Vhfwlrq 4416
671 i+{, @
Vshfldo Suredelolw| Ghqvlw| Ixqfwlrqv
=4<39 > ru ^49> 77` {=8345
684
4416
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Vshfldo Suredelolw| Ghqvlw| Ixqfwlrqv
51 i+{, @ 5 iru ^4158/ 41:8` 77 49 77 49
{
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@
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@
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4=8
]
77
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Ydu+{, @ @
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77 =4<39 5=7<;; 5 { @ 5;=;68; 5=7<;; 49 =4<39 @ +775=7<;; 495=7<;; , 5;=;68;5 5=7<;; 98=:8
71 i+w, @ =38h
iru ^3> 4,
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73 @ h=38w
s @ s Ydu+{, @ 98=:8 ;=4
53
4
5
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91 i+{, @ =4h=4{ iru ^3> 4, 53=:
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@
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53 @ h=4{ 43 @ h4 h5 =56
685
Fkdswhu 44
Lq H{huflvhv ;047/ xvh wkh wdeoh lq wkh Dsshqgl{ iru duhdv xqghu wkh qrupdo fxuyh1
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431 Duhd ehwzhhq } @ 5=46 dqg } @ =37 lv
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x@{ gx @ g{
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d{
{dh
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4
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gy @ dhd{ g{> y @ hd{ =
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g{ @ {h
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@ olp
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U
4 @ {hd{ hd{ . F d d{ . 4 .F @ dhd{
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SUREDELOLW\ DQG FDOFXOXV
e
3
{dhd{ g{
e d{ . 4 @ olp e$4 dhd{ 3 de . 4 4 @ olp . e$4 dhde d 4 d vlqfh hde jurzv pruh udslgo| wkdq de1 5 ] 4 4 5 d{ { dh g{ Ydu+{, @ d 3 U 5 d{ g{ xvlqj lqwhjudwlrq Zh uvw frqvlghu { dh e| sduwv1 @
Ohw
x@{
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gy @ {dhd{ g{
4 y @ {hd{ hd{ d iurp wkh suhylrxv lqwhjudwlrq e| sduwv1 gx @ g{
dqg
Vhfwlrq 4416
]
Vshfldo Suredelolw| Ghqvlw| Ixqfwlrqv
{5 dhd{ g{ 4 d{ d{ @ { {h h d ] 4 d{ d{ g{ {h h d 5 d{
@ { h 4 . d
]
4 {hd{ . d
]
{hd{ g{
hd{ g{
686
4 5 5 551 i+{, @ s h+{, @5 5 +{ , 4 5 5 i 3 +{, @ s h+{, @5 5 5 5 5 +{ , i 3 +{, @ s 6 5 h+{,5 @5 5 $ # 5 5 5 5 h+{, @5 +4,.+{,h+{, @5 +{, 4 5 33 i +{, @ s h+{,5 @ 5 6 5 5 5 5 h+{, @5 4 +{, 5 s i 33 +{, @ 6 5 h+{,5 @ 5 5
4 @ {5 hd{ {hd{ d 4 4 d{ 4 d{ . {h 5 hd{ h d d d
4 +{, 5 s i +{, @ 5 @5 5 6 +{, 5 h 33
Li i 33 +{, @ 3> wkhq 4
+{ ,5 @ 5 { @ { @ =
4 d{ 4 h 5 hd{ 5 d d
@
d5 {5 . 5d{ . 5 d5 hd{ ]
Ydu+{, @
4
3
Li { ? > i 33 +{, A 3= Li ? { ? . > i 33 +{, ? 3= Li { A . > i 33 +{, A 3=
{5 dhd{ g{ ]
e
4 d5
4 5 d 3 e d5 {5 . 5d{ . 5 4 @ olp 5 5 d{ e$4 d h d 3 5 5 d e . 5de . 5 4 5 @ olp . 5 5 e$4 d5 hde d d 5 4 @ 5 5> d d @ olp
e$4
{5 dhd{ g{
vlqfh hde jurzv pruh udslgo| wkdq d5 e5 . 5de . 5= Ydu+{, @
4 d5
Wkhuhiruh/ wkhuh duh srlqwv ri lq hfwlrq zkhq { @ dqg zkhq { @ . = 571 Iurp H{huflvh 56+e, dqg 56+f,/ ] 83 @ =8{h=8{ g{ 4=<<<<< dqg 3
] 3
83
@ =8{5 h=8{ g{ ;=33336
Iurp H{huflvh 56+d,/ zh kdyh wkh h{srqhqwldo glv0 wulexwlrq i+{, @ =8h=8{ iru { lq ^3> 4, zlwk d @ =8= Iru wklv glvwulexwlrq/ ] 4 ] 83 @ =8{h8{ g{ =8{h=8{ g{ 3
Wkxv/
u @
@3
+{ ,5 @4 5
4 4 @ {5 hd{ {hd{ {hd{ d d
+{,5 5
4 4 @ = d5 d
3
4=<<<<< 4=<<<<<
4 4 4 @ 5= vlqfh @ d d =8
687
Fkdswhu 44
631 Zh kdyh d qrupdo glvwulexwlrq zlwk @ 65=;> @ 4=4= Ohw { @ qxpehu ri rxqfhv ri mxlfh1
Dovr/ iru wklv glvwulexwlrq/ ] 4 =8{5 h=8{ g{ 5 Ydu+{, @ 3
]
83
3
=8{5 h=8{ g{ +4=<<<<<,5
;=33336 +4=<<<<<,5 7=3333:= s 7=3333: 5=33335
e
591 Xvh i+{, @ de{e4 hd{ zlwk d @ 7 dqg e @ 4=8= 4=84 7{4=8
i+{, @ +7,+4=8,{ h =8 7{4=8 @ 9{ h ] 4 4=8 {+9{=8 h7{ ,g{ +d, @ 3
] @
3
]
4
6
3
4=8 7{4=8
9{
h
g{
3
@ S +} A =4;, @ 4 S +} =4;, @ 4 =8:47 @ =75;9
S +d ? } ? d, @ 4=8
9{4=8 h7{ g{
6
4 5
S +} ? d, @ =58
=68;6= +e, 5
66, +e, S +{ A 66 65=; { 65=; ? @S 4=4 4=4
651 Zh kdyh d qrupdo glvwulexwlrq/ zlwk @ 87=73> @ 46=83=
Xvlqj wkh lqwhjudwlrq ihdwxuh rq d judsklqj fdofx0 odwru/ ]
65, +d, S +{ ? { 65=; 65 65=; @S ? 4=4 4=4 @ S +} ? =:6, @ =565:
4 4 4 @ 5= vlqfh @ d d =8
5=33335
SUREDELOLW\ DQG FDOFXOXV
Vlqfh wkh forvhvw ydoxh wr 158 lv 15847/ zh xvh } @ =9:= =9: ? } ? =9:
4=8
+{ =68;6,5 9{=8 h7{ g{
Xvlqj wkh lqwhjudwlrq ihdwxuh rq d judsklqj fdofx0 odwru/ 5 =38<49 =5765= 5;1 Zh kdyh dq h{srqhqwldo glvwulexwlrq zlwk phdq @ 8= 4 @ @8 d
=9: ?
{ 87=73 ? =9: 46=83
78=69 ? { ? 96=78 Wkhuhiruh/ S +78=69 ? { ? 96=78, @ 45 > dqg wkh plggoh 83( ri wkh fxvwrphuv vshqg ehwzhhq '78169 dqg '961781 671 Iru dq h{srqhqwldo glvwulexwlrq/ i+{, @ dhd{ iru ^3> 4,= Vlqfh d @ 5> i+{, @ 5h5{ iru ^3> 4`= +d, Wkh h{shfwhg sursruwlrq lv
d @ =5 =5{
+d, i+{, @ =5h
iru ^3> 4, ]
+e, S +5 ? { ? 9, @
5
9
=5h=5{ g{
9 =5{
@ h
=7
5 4=5
@h h =69<
@ +e, S 3 ? { ? 46 @
4 @ =8= 5 ]
4@6
3
5h5{ g{
4@6 @ h5{ 3 @
4 .4 h5@6
=7<
Vhfwlrq 4416
Vshfldo Suredelolw| Ghqvlw| Ixqfwlrqv
691 @ 6=5 iw/ @ =5 iw Li zh zlvk wr qg wkh plggoh 83(/ ru 183/ zh zdqw wr qg wkh ydoxh ri } fruuhvsrqglqj wr d sured0 elolw| ri 158 lq wkh vwdqgdug qrupdo glvwulexwlrq wdeoh1 Wklv lv derxw 19:81 Vlqfh wkh ghvluhg duhd lv v|pphwulf derxw wkh phdq/ zh xvh =9:8= { { 6=5 =9:8 @ =5 }@
6;1 Iru dq h{srqhqwldo glvwulexwlrq/ i+{, @ dhd{ iru ^3> 4,= 4 4 @ 57=;> d @ = d 57=; ] 4 4 {@57=; g{ h +d, S +{ 58, @ 57=; 58 ] e 4 {@57=; h g{ @ olp e$4 58 57=; e {@57=; @ olp h Vlqfh @
58
@ olp he@57=; . h58@57=; e$4
@ h58@57=; =69 ] 53 h{@57=; g{ +e, S +{ ? 53, @ 3
53 @ h{@57=; 3
@ h =88
.4
731 +d, Zh kdyh d qrupdo glvwulexwlrq zlwk @ 9=< dqg @ 7=9= 9 9=< S +{ A 9, @ S } 7=9 @ S +} =4<, =8; Qr/ wklv lv lqvx!flhqw hylghqfh1
9 =9 S +{ 9, @ S } =6 @ S +} 4;, 3
751 Xqlirup glvwulexwlrq rq ^65> 77`
Wkh odujhvw khljkw lv 61668 iw/ dqg wkh vpdoohvw khljkw lv 61398 iw1
53@57=;
+e, Zh kdyh d qrupdo glvwulexwlrq zlwk @ =9 dqg @ =6=
\hv/ e| wrgd|*v vwdqgdugv wkhuh lv vx!flhqw hyl0 ghqfh wr frqfoxgh wkdw Dqguhz Mdfnvrq vx huhg iurp phufxu| srlvrqlqj1
=468 @ { 6=5 { @ 6=5 =468 { @ 6=398/ 61668
e$4
688
i+{, @
4 4 @ iru ^65> 77` 77 65 45
4 +d, @ +65 . 77, @ 6; lqfkhv 5 +e, S +6; ? { ? 73, ] 73 4 @ g{ 6; 45 { 73 @ 45 6; 4 @ =4: 9 771 Iru dq h{srqhqwldo glvwulexwlrq/ i+{, @ dhd{ iru 4 4 ^3> 4,1 Vlqfh d @ 93<=8 > i+{, @ 93<=8 h{@93<=8 = +d, Wkh h{shfwhg qxpehu ri gd|v lv @
4 d
@ 93<=81 Wkh vwdqgdug ghyldwlrq lv
@
4 d
@ 93<=8= ]
4
4 {@93<=8 g{ h 93<=8 698 ] 698 4 {@93<=8 h @4 g{ 93<=8 3 698 {@93<=8 @4. h
+e, S +{ A 698, @
3
@ 4 . +h698@93<=8 4, @ h698@93<=8 =88
689
Fkdswhu 44
Fkdswhu 44 Uhylhz H{huflvhv 71 i+{, @ ]
7
4
4 +5{ . 7,> ^4> 7` 5:
7 4 4 5 +5{ . 7,g{ @ +{ . 7{, 5: 5: 4 @
4 +65 8, 5:
@4 Vlqfh 4 { 7> i+{, 3= Wkhuhiruh/ i+{, lv d suredelolw| ghqvlw| ixqfwlrq1 91 i+{, @ =4> ^3> 43` ]
43
43 =4 g{ @ =4{ 3
3
@ =4+43, 3 @ 4 i+{, 3 iru doo { lq ^3> 43`= Wkhuhiruh/ i+{, lv d suredelolw| ghqvlw| ixqfwlrq1 s ;1 i+{, @ n {> ^4/ 7` ] 4
7
s n { g{ @
] 4
7
5 n+; 4, 6 47 @ n 6 @
Vlqfh i+{, lv d suredelolw| ghqvlw| ixqfwlrq/ 47 n@4 6 6 n@ = 47 4 > ^5> 8` 431 i+{, @ 4 s {4 +d, S +6 { 8, ] 8 @ ^4 +{ 4,4@5 `g{ 6
+e, S +5 { 7, ] 7 @ ^4 +{ 4,4@5 `g{ 5
7 @ ^{ 5+{ 4,4@5 `5 s @75 65.5 =869 +f, S +6 { 7, ] 7 @ ^4 +{ 4,4@5 `g{ 6
7 @ ^{ 5+{ 4,4@5 ` 6 s s @75 66.5 5 =697 451 Li zh frqvlghu wkh suredelolwlhv dv zhljkwv/ wkh h{0 shfwhg ydoxh ru phdq ri d suredelolw| glvwulexwlrq uhsuhvhqwv wkh srlqw dw zklfk wkh glvwulexwlrq lv edodqfhg1 4 471 i+{, @ > ^7> <` 8 < ] < {5 4 g{ @ H+{, @ @ { 8 43 7 7
n{4@5 g{
7 5 6@5 @ n{ 6 4
8 @ ^{ 5+{ 4,4@5 ` 6 s @ 8 5+5, 6 . 5 5 =;5;
SUREDELOLW\ DQG FDOFXOXV
@ ]
;4 49 98 @ @ 9=8 43 43 43 <
4 +{ 9=8,5 g{ 7 8 < 4 6 @ +{ 9=8, 48 7
Ydu+{, @
@
4 +5=86 . 5=86 , 48
5=3;6 s @ Ydu+{, 4=776 4 5 4. s > ^4> 7` 491 i+{, @ : { ] 7 4 H+{, @ @ { +4 . 5{4@5 , g{ : 4 ] 7 4 @ +{ . 5{4@5 ,g{ : 4 7 4 {5 7 6@5 @ . { : 5 6 4 65 4 7 4 ;. @ : 6 5 6 5=738
Fkdswhu 44 Uhylhz H{huflvhv
]
68:
7
{5 +4 . 5{4@5 ,g{ +5=738,5 : 4 ] 7 5 5 6@5 { . { g{ +5=738,5 @ : : 4 7 6 7 8@5 { 5 . { @ +5=738, 54 68 4 4 7 97 45; . +5=738,5 54 68 54 68
Ydu+{, @
+g, S + { . , S +=66< { =;5:, ] =;5: @ +7{ 6{5 ,g{ =66<
=;5: @ +5{5 {6 , =66<
@ 5+=;5:,5 +=;5:,6 5+=66<,5 . +=66<,6 =944
=:8< s @ Ydu+{, =;:4 531 i+{, @
5
4;1 i+{, @ 7{ 6{ > ^3> 4` ] 4 {+7{ 6{5 ,g{ +d, @
]
]
4
3
@
+7{5 6{6 ,g{
4 7{6 6{7 6 7 3
7 6 6 7 : @ 45 @
46=9
=8;6 ]
4
5
: 45 5 ] 4 : @ +7{6 6{7 ,g{ 45 3 4 5 6{8 : @ {7 8 3 45 5 6 : @4 8 45
+e, Ydu+{, @
3
{5 +7{ 6{5 ,g{
=38<: s Ydu+{, =577 : +f, S 3 { ? 45 ] :@45 +7{ 6{5 ,g{ @ 3
:@45 @ +5{5 {6 , @5
: 45
=7;5
58
8{ +4 {6@5 ,g{ 445 4 ] 58 8 +{ {4@5 ,g{ @ 445 4 5 58 { 8 @ 5{4@5 445 5 4 958 4 8 @ 43 . 5 445 5 5
+d, @
3
@
8 +4 {6@5 ,> ^4> 58` 445
5
3
: 45
6
]
58
8{5 +4 {6@5 ,g{ +46=9,5 445 4 ] 58 8 @ +{5 {4@5 ,g{ +46=9,5 445 4 6 58 { 5 8 @ {6@5 +46=9,5 445 6 6 4 6 58 583 4 5 8 @ . +46=9,5 445 6 6 6 6 76=;8 s Ydu+{, 9=9
+e, Ydu+{, @
+f, S + { . , @ S +: { 53=5, ] 53=5 8 @ +4 {6@5 ,g{ 445 : 53=5 8 4@5 +{ . 5{ , @ 445 : 5 5 8 53=5 . s @ : s 445 53=5 : =8;
68;
Fkdswhu 44
Iru H{huflvhv 5505;/ xvh wkh wdeoh lq wkh Dsshqgl{ iru wkh duhdv xqghu wkh qrupdo fxuyh1
Wkxv/ @
551 Duhd wr uljkw ri } @ 4=86 lv 4 =<6:3 @ 3=96 ru 9=6(=
4 4 +63 . 43, @ +73, @ 53 5 5
dqg 635 5+43,+63, . 435 45 733 = @ 45 u 733 @ 8=:: 45
Ydu+{, @
571 Duhd wr ohiw ri } @ 4=36 lv =;7;8= Duhdv wr ohiw ri } @ 4=7: lv htxlydohqw wr duhd wr uljkw ri } @ 4=7: = 4 =<5<5 @ =3:3;= Duhd ehwzhhq lv =;7;8 =3:3; @ =:::: ru ::=::(=
SUREDELOLW\ DQG FDOFXOXV
+h, S + { . , @ S +53 8=:: { 53 . 8=::, @ S +47=56 { 58=::, ] 58=:: @ =38 g{ 47=56 58=::
@ =38{
591 @ 5=8> @ 3> } @ 3 . 5=8
47=56
Duhd wr ohiw ri } @ 5=8 lv
@ =38+58=:: 47=56, =8;
=<<6; ru <<=6;(=
5
5;1 Zh zdqw wr qg wkh }0vfruh iru 54( ri wkh duhd xqghu wkh qrupdo fxuyh wr wkh ohiw ri }1 Zh qrwh wkdw 54( ? 83(> vr } pxvw eh qhjdwlyh1 Wkh }0vfruh iru wkh ydoxh lq wkh wdeoh qhduhvw 154 lv } =;4=
h{ 651 i+{, @ s iru +4> 4, +d, Vlqfh wkh h{srqhqw ri h lq i+{, pd| eh zulwwhq {5 @
631 i+{, @ =38 iru ^43> 63` +d, Wklv lv d xqlirup glvwulexwlrq1 +e, Wkh grpdlq ri i lv ^43/ 63`1 Wkh udqjh ri i lv ~1381 +f,
dqg
+{ 3,5 5 > 5 s45
4 s @
s4 5
4 s > 5
i+{, lv d qrupdo glvwulexwlrq zlwk @ 3 dqg @ s4 = 5 +e, Wkh grpdlq ri i lv +4> 4,= Wkh udqjh ri i lv 3> s4 = +f,
+g, Iru d xqlirup glvwulexwlrq/ 4 +e . d, dqg 5 e5 5de . d5 = Ydu+{, @ 45 @
+g, Iru wklv qrupdo glvwulexwlrq/ @ 3 dqg @ s45 =
Fkdswhu 44 Uhylhz H{huflvhv
68<
6;1 i+{, @ =34h=34{ iru ^3> 4, lv dq h{srqhqwldo glvwulexwlrq1
+h, S + { . , @ 5S +3 { . , @ 5S 3 { s45
s4 > 5
Li { @
}@
s4 5
3
s4 5
S +3 { 433, ] 433 @ =34h=34{ g{
@ 4=33=
4 @4 h
S + { . , @ 5S +3 } 4=33, @ 5+=6746, =9;=
=96
S +Qr uhsdluv lq |hduv 406, @ S +Iluvw uhsdlu qhhghg lq |hduv 7058, ]
58
8 +4 w6@5 ,gw 445 7 58 8 @ +w . 5w4@5 , 445 7 5 8 58 . 7 4 @ 445 8 @
55=9; F
4 691 i+{, @ h{@9 iru ^3> 4, lv dq h{srqhqwldo 9 glvwulexwlrq1
+e, @
4 4 9
4 4 9
9 48/<58 +58,7 +58,6 +53,7 +53,6 . 7 6 7 6
@
84 =<44 89
+d, @
9 +{5 . {, iru ^53> 58` 48/<58 ] 58 9 +{5 . {,g{ +d, @ 53 48/<58 ] 58 9 +{6 . {5 ,g{ @ 48/<58 53 58 7 9 {6 { @ . 6 53 48/<58 7
731 i+{, @
8 +4 w6@5 ,> ^4> 58` 445
@
433 @ h=34{ 3
Wkxv/
671 i+w, @
3
@9
+e, S +{ ? , ] 55=9; @
9 +{5 . {,g{ 48/<58 53 6 55=9; { 9 {5 @ . 5 53 48/<58 6 9 48/<58 +55=9;,6 +55=9;,5 +53,6 +53,5 . 6 5 6 5
@ @9 ]
4
4 {@9 g{ h 9 9 ] e 4 {@9 h @ olp g{ e$4 9 9 e {@9 @ olp h
+f, S +{ A 9, @
e$4
9
4 @ olp h4 e@9 e$4 h 4 @ =6: h
=7; 751 Qrupdo glvwulexwlrq/ @ 5=7 j> @ =7 j> { @ whqvlrq S +{ ? 4=<, @ S
{ 5=7 4=< 5=7 ? =7 =7
@ S +} ? 4=58, @ =4389
693
Fkdswhu 44
SUREDELOLW\ DQG FDOFXOXV
771 +d,
sro|qrpldo ixqfwlrq +e, Q+{, @ =33447584{7 . =594:85{6 54=9<6;{5 . :5<=9<7{ 8599=66>
Wkh ixqfwlrq prghov wkh gdwd zhoo1 +f, Iurp sduw e/ Q+{, @ =33447584{7 . =594:85{6 54=9<6;{5 . :5<=9<7{ 8599=66 ]
<6=5 <=:
Vr/ ohw n @
4 = Wkhq 4::/847
Q+{,g{ 4::/847
V+{, @ nQ+{, @ 4=
+g, Xvlqj wkh lqwhjudwlrq ihdwxuh rq wkh fdofxodwru/ ] S +58 { 68, @ S +78 { 98, @ S +{ 88, @
68
58 ] 98
V+{,g{ =4;4 V+{,g{ =599
78 ] <6=5 88
V+{,g{ =66;
Iurp wkh wdeoh/ wkh dfwxdo suredelolwlhv duh 6343 =4:4 4:/88< 597: . 4;8< S +78 { 98, @ =58: 4:/88< 4;8< . 4<39 . 493; . 7<7 S +{ 88, @ 4:/88< S +58 { 68, @
=667 ] +h, H+{, @
<6=5 <=:
{V+{,g{ 79=7 |hduv
H{whqghg Dssolfdwlrq= H{srqhqwldo Zdlwlqj Wlphv
694
p =33447584 8 =594:85 7 54=9<6; 6 :5<=9<7 5 { . { { . { 8599=66{ 8 7 6 5 <=: <=: 4 =33447584 8 =594:85 7 54=9<6; 6 :5<=9<7 5 p . p p . p 8599=66p +=45;7, 8 7 6 5 4::/847 4 @ zkhq p 76=< |hduv/ 5 xvlqj wkh LQWHUVHFW ihdwxuh ri d judsklqj fdofxodwru1 ]
+i ,
p
V+{,g{ @
4 4::/847
791 Qrupdo glvwulexwlrq/ @ 73> @ 46> { @ wdnh S +{ A 83, @ S
{ 73 83 73 A 46 46
@ S +} A =::, @ 4 S +} =::, @ 4 =::<7 @ =5539
H{whqghg Dssolfdwlrq= H{srqhqwldo Zdlwlqj Wlphv 41 Li wkh ghqvlw| ixqfwlrq lv frqwlqxrxv/ wkh suredelolw| ri dq| hyhqw ri wkh irup +{ @ f, lv 3/ vr lw grhv qrw pdwwhu zkhwkhu zh xvh vwulfw ru lqfoxvlyh lqhtxdolwlhv1 51 Zh douhdg| nqrz wkdw wkh suredelolw| ri d jds ri 63 plqxwhv ru orqjhu lv derxw 1381 Ryhu 73 wulsv/ wkh h{shfwhg qxpehu ri zdlwv wkdw duh 63 plqxwhv ru orqjhu lv 73+=38, @ 5= ] 4 ] 53 53 4 w@43 4 w@43 gw @ 4 gw @ 4 hw@43 h h 61 S +w A 53, @ 43 43 3 53 3 @ 4 +h5 . 4, @ h5 =468 71 Iru wkh h{srqhqwldo vfkhgxoh/ ] 43 43 4 w@43 S +< ? w ? 43, @ h gw @ hw@43 @ h4 . h<@43 =36< 43 < < Iru wkh xqlirup vfkhgxoh/ 43 ] 43 4 4 < gw @ w @ 4 @ =4 S +< ? w ? 43, @ 43 43 43 < < 81 Iru wkh h{srqhqwldo prgho/ ] 45 45 4 w@43 S +; ? w ? 45, @ h gw @ hw@43 @ h45@43 . h;@43 =47; 43 ; ; Iru wkh xqlirup prgho/ doo lqwhuduulydo wlphv duh 43 plqxwhv/ vr wkh| doo phhw wkh 5 fulwhuld1 91 Wkh wulfn xvhg dw wkh KrwElwv vlwh lv wkh iroorzlqj= wkh| phdvxuh wlphv ehwzhhq vxffhvvlyh sdluv ri ghfd| hyhqwv> li wkh uvw zdlw lv vkruwhu wkdq wkh vhfrqg/ wkh jhqhudwru rplwv d 3/ dqg li wkh uvw zdlw lv orqjhu wkdq wkh vhfrqg/ lw rplwv d 41 Iru pruh ghwdlov/ vhh kwws=22zzz1irxuplode1fk2krwelwv2kz2kwpo1
Fkdswhu 45
VHTXHQFHV DQG VHULHV 4514
Jhrphwulf Vhtxhqfhv
51 d4 @ 7> u @ 5> q @ 8 Vlqfh q @ 8> zh pxvw qg d4 / d5 / d6 / d7 dqg d8 zlwk d @ d4 @ 7= d4 d5 d6 d7 d8
@7 @ 7 +5,54 @ 7 +5,64 74 @ 7 +5, 84 @ 7 +5,
@ 7 +5,4 @ 7 +5,5 6 @ 7 +5, 7 @ 7 +5,
@ 7 +5, @ ; @ 7 +7, @ 49 @ 7 +;, @ 65 @ 7 +49, @ 97
Wkh uvw yh whupv ri wklv jhrphwulf vhtxhqfh duh 7/ ;/ 49/ 65/ dqg 971
d4 @ 5:
54 4 4 4 4 @< d5 @ 5: @ 5: @ 5: 6 6 6 64 5 4 4 4 d6 @ 5: @6 @ 5: @ 5: 6 6 < 74 6 4 4 4 d7 @ 5: @ 5: @ 5: @4 6 6 5: Wkh uvw irxu whupv ri wklv jhrphwulf vhtxhqfh duh 5:> u @ 7 Vlqfh zh zdqw d8 > xvh q @ 8 lq wkh irupxod zlwk d @ d4 @ ; dqg u @ 7= d8 @ ; +7,84 @ ; +7,7 @ ; +589, @ 537; dq @ ; +7,q4
71 d4 @ 56 > u @ 9> q @ 6 Vlqfh q @ 6> zh pxvw qg d4 / d5 / dqg d6 zlwk d @ d4 @ 56 = 5 d4 @ 6 5 5 5 d5 @ +9,54 @ +9,4 @ +9, @ 7 6 6 6 5 5 5 d6 @ +9,64 @ +9,5 @ +69, @ 57 6 6 6 Wkh uvw wkuhh whupv ri wklv jhrphwulf vhtxhqfh duh 56 / 7/ dqg 571 91 d5 @ <> d6 @ 6> q @ 7 Vlqfh q @ 7> zh pxvw qg d4 / d5 / d6 / dqg d7 zlwk u @ dd65 @ 6< @ 46 = Wr qg d> xvh d5 @ <> u @ 46 > dqg q @ 5 lq wkh irupxod1 54 4 6 4 4 < @ d 6 4 < @ d 6 5: @ d < @ d
695
431 d4 @ 7> u @ 5 Vlqfh zh zdqw d8 > xvh q @ 8 lq wkh irupxod zlwk d @ d4 @ 7 dqg u @ 5= 84
7
@ 7 +5, d8 @ 7 +5, @ 7 +49, @ 97 q4 dq @ 7 +5,
451 d6 @ 9> u @ 6 Wr qg d> xvh d6 @ 9> u @ 6> dqg q @ 6 lq wkh irupxod1 9 @ d +6,64 9 @ d +6,5 9 @ d +<, 5 @d 6 Vlqfh zh zdqw d8 > xvh q @ 8 lq wkh irupxod zlwk d @ 56 dqg u @ 6= 5 5 5 +6,84 @ +6,7 @ +;4, @ 87 6 6 6 5 q4 dq @ +6, 6 d8 @
Vhfwlrq 4514
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Ydoxh dw 3
53 . {5 S +{, @ 83 . { S +4, +{, @ S
+5,
+{, @
S +3, @ =7
{5 . 433{ 53
S +4, +3, @ =33;
5
+83 . {,
5 +83 . {, +5{ . 433, {5 . 433{ 53 ^5 +83 . {,`
S5 +{, @ S +3, .
+83 . {,7
S +5, +3, @ =37365
S +5, +3, 5 =37365 5 S +4, +3, =33; {. { @ =7 . {. { @ =7 =33;{ . =35349{5 4$ 5$ 4 5
S5 +=6, @ =7 =33; +=6, . +=35349, +=6,5 @ =6<<7477 Wkh Wd|oru sro|qrpldo S5 +=6, dssur{lpdwhv S +=6, dv 16<<74771 Wkh dssur{lpdwh sur w zkhq 633 wrqv ri dssohv duh vrog lv 16<< wkrxvdqg grooduv/ ru '6<<1 Ilqglqj S +=6, e| gluhfw vxevwlwxwlrq jlyhv= S +=6, @
53 . +=6,5 =6<<7368:;8 83 . +=6,
Gluhfw vxevwlwxwlrq dovr hvwlpdwhv wkh sur w dv 16<< wkrxvdqg grooduv/ ru '6<<1 751
Ghulydwlyh
Ydoxh dw 3
F +{, @ h{@83 4 F +4, +{, @ h{@83 83 4 {@83 F +5, +{, @ h 5833
F +3, @ 4
4 83 4 F +5, +3, @ 5833 F +4, +3, @
4 F +5, +3, 5 F +4, +3, {. { @ 4 . 83 { . 4$ 5$ 4$ 4 4 5 S5 +8, @ 4 +8, . +8, @ =<38 83 8333
S5 +{, @ F +3, .
4 5833
5$
{5 @ 4
4 5 4 {. { 83 8333
Wkh Wd|oru sro|qrpldo S5 +8, dssur{lpdwhv F +8, dv 1<381 Wkh dssur{lpdwh frvw lv 1<38 groodu/ ru derxw '1<41
6:;
Fkdswhu 45
VHTXHQFHV DQG VHULHV
Ilqglqj F +8, e| gluhfw vxevwlwxwlrq jlyhv= F +8, @ h8@83 =<37;6:74; Gluhfw vxevwlwxwlrq hvwlpdwhv wkh sur w dv 1<38 groodu/ ru derxw '1<31 771
Ghulydwlyh D +{, @ D+4, +{, @
Ydoxh dw 3
9{ 4 . 43{ 9
D +3, @ 3 D+4, +3, @ 9
+4 . 43{,5 453 D+5, +{, @ +4 . 43{,6 S5 +{, @ D +3, .
D+5, +3, @ 453
D+5, +3, 5 D+4, +3, 9 453 5 {. { @3. {. { @ 9{ 93{5 4$ 5$ 4 5 5
S5 +=38, @ 9 +=38, 93 +=38, @ =48 Wkh Wd|oru sro|qrpldo S5 +=38, dssur{lpdwhv D +=38, dv 1481 Wkh dssur{lpdwh dprxqw ri wkh guxj lq wkh eorrgvwuhdp diwhu 138 plqxwhv lv 148 ploololwhuv1 Ilqglqj D +=38, e| gluhfw vxevwlwxwlrq jlyhv= D +=38, @
9 +=38, @ =5 4 . 43 +=38,
Gluhfw vxevwlwxwlrq lqglfdwhv wkdw 15 ploololwhu ri wkh guxj uhpdlq lq wkh eorrgvwhdp diwhu 138 plqxwhv1
4517
Lq qlwh Vhulhv
51 4 . =< . =;4 . =:5< . = = = lv d jhrphwulf vhulhv zlwk d @ d4 @ 4 dqg u @ =<= Vlqfh u lv lq +4> 4, > wkh vhulhv frqyhujhv dqg kdv vxp d 4 4 @ @ @ 43= 4u 4 =< =4 71 6 . < . 5: . ;4 . = = = lv d jhrphwulf vhulhv zlwk d @ d4 @ 6 dqg u @ 6= Vlqfh u A 4> wkh vhulhv glyhujhv1 91 ;4 . 5: . < . 6 . 4 . = = = lv d jhrphwulf vhulhv zlwk d @ d4 @ ;4 dqg u @ 46 = Vlqfh u lv lq +4> 4, > wkh vhulhv frqyhujhv dqg kdv vxp d ;4 ;4 576 @ = @ 5 @ 4 4u 5 4 6 6 ;1 45; . 97 . 65 . = = = lv d jhrphwulf vhulhv zlwk d @ d4 @ 45; dqg u @ 45 = Vlqfh u lv lq +4> 4, > wkh vhulhv frqyhujhv dqg kdv vxp d 45; 45; @ 4 @ 589= @ 4u 4 45 5 431
7 8
. 58 . 48 . = = = lv d jhrphwulf vhulhv zlwk d @ d4 @ 78 dqg u @ 45 = Vlqfh u lv lq +4> 4, > wkh vhulhv frqyhujhv dqg kdv vxp 7 7 d ; @ 8 4 @ 84 @ = 4u 8 4 5 5
Vhfwlrq 4517
Lq qlwh Vhulhv
6:<
4 4 . +4=34, 451 4 . 4=34 5 . = = = lv d jhrphwulf vhulhv zlwk d @ d4 @ 4 dqg u @ frqyhujhv dqg kdv vxp
d 4 @ 4 @ 4u 4 4=34
4
=34 4=34
4 4=34 =
Vlqfh u lv lq +4> 4, > wkh vhulhv
@ 434=
471 h . h5 . h6 . h7 . = = = lv d jhrphwulf vhulhv zlwk d @ d4 @ h dqg u @ h= Vlqfh u A 4> wkh vhulhv glyhujhv1
4 4 @ 4.4 5 4 4 4 8 4 . @ . @ V5 @ d4 . d5 @ 4.4 5.4 5 6 9 4 4 4 4 4 4 46 V6 @ d4 . d5 . d6 @ . . @ . . @ 4.4 5.4 6.4 5 6 7 45 4 4 4 4 4 4 4 V7 @ d4 . d5 . d6 . d7 @ . . . @ . . . 4.4 5.4 6.4 7.4 5 6 7 4 4 4 4 4 . . . . @ V8 @ d4 . d5 . d6 . d7 . d8 @ 4.4 5.4 6.4 7.4 8.4
491 V4 @ d4 @
4;1 V4 @ d4 @
4 :: @ 8 93 4 4 4 4 4 5< . . . . @ 5 6 7 8 9 53
4 4 @ 6 +4, 4 5
V5 @ d4 . d5 @
: 4 4 4 4 . @ . @ 6 +4, 4 6 +5, 4 5 8 43
V6 @ d4 . d5 . d6 @
4 66 4 4 4 4 4 . . @ . . @ 6 +4, 4 6 +5, 4 6 +6, 4 5 8 ; 73
V7 @ d4 . d5 . d6 . d7 @
4 4 4 4 4 4 4 736 4 . . . @ . . . @ 6 +4, 4 6 +5, 4 6 +6, 4 6 +7, 4 5 8 ; 44 773
4 4 4 4 4 . . . . 6 +4, 4 6 +5, 4 6 +6, 4 6 +7, 4 6 +8, 4 4 4 6374 4 4 4 . @ @ . . . 5 8 ; 44 47 63;3
V8 @ d4 . d5 . d6 . d7 . d8 @
6;3
Fkdswhu 45
531 V4 @ d4 @
VHTXHQFHV DQG VHULHV
4 4 @ +4 . 6, ^5 +4, . 4` 45
V5 @ d4 . d5 @
4 6: 4 4 4 . @ . @ +4 . 6, ^5 +4, . 4` +5 . 6, ^5 +5, . 4` 45 58 633
V6 @ d4 . d5 . d6 @
4 4 4 4 4 4 436 . . @ . . @ +4 . 6, ^5 +4, . 4` +5 . 6, ^5 +5, . 4` +6 . 6, ^5 +6, . 4` 45 58 75 :33
4 4 4 4 . . . +4 . 6, ^5 +4, . 4` +5 . 6, ^5 +5, . 4` +6 . 6, ^5 +6, . 4` +7 . 6, ^5 +7, . 4` 4 4 4 4 435: @ . . . @ 45 58 75 96 9633
V7 @ d4 . d5 . d6 . d7 @
V8 @ d4 . d5 . d6 . d7 . d8 4 4 4 4 4 . . . . +4 . 6, ^5 +4, . 4` +5 . 6, ^5 +5, . 4` +6 . 6, ^5 +6, . 4` +7 . 6, ^5 +7, . 4` +8 . 6, ^5 +8, . 4` 4 4 4 4 4 57/49< @ . . . . @ 45 58 75 96 ;; 46;/933 @
551 +d, Frqvlghu wkh qlwh vhtxhqfh 4
F +4 . u,
5
> F +4 . u,
6
> F +4 . 8,
q
> = = = > F +4 . u,
=
Wklv lv d qlwh jhrphwulf vhtxhqfh kdylqj uvw whup d @ d4 @ F +4 . u,4 dqg frpprq udwlr z @ +4 . u,4 = Wkxv/ d +zq 4, S @ Vq @ z4 k l 4 q +4 . u, 4 F +4 . u, @ 4 +4 . u, 4 l k +4 . u,4 +4 . u,q 4 @F +4 . u,4 ^4 +4 . u,` @F
+4 . u,q 4 u
@F
+4 . u,q 4 u
@F
+4 . u,q 4 = u +4 . u,q
q
. u, +4 +4 . u,q
+e, Wkh suhvhqw ydoxh ryhu dq lq qlwh dprxqw ri wlph lv jlyhq e| wkh lq qlwh jhrphwulf vhulhv S @ F +4 . u,4 . F +4 . u,5 . F +4 . u,6 . = = = zkhuh d @ d4 @ F +4 . u,4 dqg wkh frpprq udwlr z @ +4 . u,4 = Wklv vhulhv frqyhujhv wr 4
4
d F +4 . u, F +4 . u, F @ @ 4 @ 4 4z u 4 +4 . u, +4 . u, ^+4 . u, 4` Khuh/ S @
F u=
Vhfwlrq 4518
Wd|oru Vhulhv
6;4
5 571 Wkh khljkw wkdw wkh edoo uhwxuqv wr diwhu wkh uvw erxqfh lv jlyhq e| 43 67 / diwhu wkh vhfrqg erxqfh 43 67 / 6 diwhu wkh wklug erxqfh 43 67 / dqg vr rq1 Wkxv/ wkh glvwdqfh wkdw wkh edoo wudyhov ehiruh lw frphv wr uhvw lv jlyhq e| 5 6 6 6 6 43 . 5 +43, . 5 +43, . === = . 5 +43, 7 7 7 5 6 5 +43, 67 . 5 +43, 67 . 5 +43, 67 . = = = lv dq lq qlwh jhrphwulf vhulhv zlwk d4 @ d @ 5 +43, 67 @ 48 dqg u @ 67 = Wklv vhulhv frqyhujhv wr d 48 48 @ 4 @ 93= @ 4u 4 67 7 Vr/ wkh wrwdo glvwdqfh wudyhohg e| wkh edoo lv 43 . 93 @ :3 phwhuv1 591 Rq lwv vhfrqg vzlqj/ wkh shqgxoxp ere zloo vzlqj wkurxjk dq duf 73 +=;, fhqwlphwhuv orqj/ rq lwv wklug vzlqj 73 +=;,5 fhqwlphwhuv orqj/ rq lwv irxuwk vzlqj 73 +=;,6 fhqwlphwhuv orqj/ dqg vr rq1 Wkxv/ wkh glvwdqfh lw zloo vzlqj dowrjhwkhu ehiruh frplqj wr d frpsohwh vwrs lv jlyhq e| 73 . 73 +=;, . 73 +=;,5 . 73 +=;,6 . = = = = Wklv lv dq lq qlwh jhrphwulf vhulhv zlwk d4 @ d @ 73 dqg u @ =;= Wklv vhulhv frqyhujhv wr d 73 73 @ @ @ 533= 4u 4 =; =5 Vr/ wkh wrwdo glvwdqfh dowrjhwkhu ehiruh frplqj wr d frpsohwh vwrs lv 533 fhqwlphwhuv1 5;1 Wkh uvw wuldqjoh kdv vlghv 5 phwhuv lq ohqjwk/ wkh vhfrqg wuldqjoh kdv vlghv 4 phwhu lq ohqjwk/ wkh wklug wuldqjoh kdv vlghv 45 phwhu lq ohqjwk/ wkh irxuwk wuldqjoh kdv vlghv 47 phwhu lq ohqjwk/ dqg vr rq1 Wkxv/ wkh khljkw ri wkh s s s uvw wuldqjoh lv 6 phwhuv/ wkh khljkw ri wkh vhfrqg wuldqjoh lv 56 phwhu/ wkh khljkw ri wkh wklug wuldqjoh lv 76 s phwhu/ wkh khljkw ri wkh irxuwk wuldqjoh lv ;6 phwhu/ dqg vr rq1 Wkh wrwdo duhd ri wkh wuldqjohv/ glvuhjduglqj wkh ryhuodsv/ lv jlyhq e| #s $ #s $ #s $ s 4 4 6 6 6 4 4 4 4 +5, 6 . +4, . . . 5 5 5 5 5 7 5 7 ; s s s s 6 6 6 . . . 1 @ 6. 7 49 97 s Wklv lv dq lq qlwh jhrphwulf vhulhv zlwk d @ d4 @ 6 dqg u @ 47 = Wklv vhulhv frqyhujhv wr s s s d 6 6 7 6 @ 6 @ @ = 4u 6 4 47 7 Wkh wrwdo duhd ri wkh wuldqjohv/ glvuhjdugqj wkh ryhuodsv/ lv jlyhq e|
4518
s 7 6 6
vtxduh phwhuv1
Wd|oru Vhulhv
51 Wklv ixqfwlrq prvw qhduo| pdwfkhv ghqrplqdwru e| 71
4 4{ =
Wr jhw 4 lq wkh ghqrplqdwru/ lqvwhdg ri 7/ glylgh wkh qxphudwru dqg 67 6 @ 7{ 4 {7
6;5
Fkdswhu 45
Wkxv/ zh fdq qg wkh Wd|oru vhulhv iru e| 67 > dqg uhsodflqj hdfk { zlwk
{ 7=
67 e| vwduwlqj zlwk wkh Wd|oru vhulhv iru 4 {7
VHTXHQFHV DQG VHULHV
4 4{ >
pxowlso|lqj hdfk whup
67 6 @ 7{ 4 {7 6 6 { 6 { 6 6 { q 6 { 5 @ 4. . . . . . 7 7 7 7 7 7 7 7 7 6{q 6 6{ 6{5 6{6 @ q.4 7 49 97 589 7
Wkh Wd|oru vhulhv iru
4 4{
lv ydolg zkhq 4 ? { ? 4= Uhsodflqj { zlwk
4 ?
{ ? 4 ru 7 ? { ? 7= 7
Wkh lqwhuydo ri frqyhujhqfh ri wkh qhz vhulhv lv +7> 7, =
{ 7
jlyhv
Vhfwlrq 4518
71
Wd|oru Vhulhv
6;6
:{ : @ { 4 . 5{ 4 +5{, Xvh wkh Wd|oru vhulhv iru n @ 4=
4 4{ >
pxowlso| hdfk whup e| :/ dqg uhsodfh { zlwk 5{= Dovr/ xvh surshuw| +6, zlwk
:{ : @{ 4 . 5{ 4 +5{, @ { : 4 . { : +5{, . { : +5{, . { : +5{, . = = = . { : +5{, . = = = q @ :{ 47{5 . 5;{6 89{7 . = = = . +4, : 5q {q.4 . = = = 5
Wkh Wd|oru vhulhv iru
4 4{
6
q
lv ydolg zkhq 4 ? { ? 4= Uhsodflqj { zlwk 5{ jlyhv 4 ? 5{ ? 4 ru
4 4 A{A = 5 5
Wkh lqwhuydo ri frqyhujhqfh ri wkh qhz vhulhv lv 45 > 45 =
91
<{7 < @ {7 4{ 4{ Xvh wkh Wd|oru vhulhv iru
4 4{ >
dqg pxowlso| hdfk whup e| <= Dovr/ xvh surshuw| +6, zlwk n @ 7=
<{7 < @ {7 4{ 4{
@ {7 < 4 . {7 < +{, . {7 < +{, . {7 < +{, . = = = . {7 < +{, . = = = @ <{7 . <{8 . <{9 . <{: . = = = . <{q.7 . = = = 5
Wkh Wd|oru vhulhv iru
<{7 4{
6
q
kdv wkh vdph lqwhuydo ri frqyhujhqfh/ +4> 4, > dv wkh Wd|oru vhulhv iru
4 4{ =
;1 Zh qg wkh Wd|oru vhulhv ri oq 4 {5 e| vwduwlqj zlwk wkh Wd|oru vhulhv iru oq +4 . {, dqg uhsodflqj hdfk { zlwk {5 = { 5 { 6 { 7 q.4 q { +4, {5 5 5 5 { . . . . oq 4 5 @ 5 5 6 7 q.4 { {5 {6 {7 {q.4 @ 5 ; 57 97 +q . 4, 5q.4 Wkh Wd|oru vhulhv iru oq +4 . {, lv ydolg zkhq 4 ? { 4= Uhsodflqj { zlwk {5 jlyhv 4 ?
{ 4 ru 5 A { 5= 5
Wkh lqwhuydo ri frqyhujhqfh ri wkh qhz vhulhv lv ^5> 5, = 5
431 Zh qg wkh Wd|oru vhulhv iru h6{ e| vwduwlqj zlwk wkh Wd|oru vhulhv iru h{ dqg uhsodflqj hdfk { zlwk 6{5 = 4 5 6 q 4 4 5 h6{ @ 4 . 6{5 . 6{5 . 6{5 . . 6{5 . 5$ 6$ q$ q q 5q +4, 6 { < < . @ 4 6{5 . {7 {9 . . 5 5 q$ 5
Wkh Wd|oru vhulhv iru h6{ kdv wkh vdph lqwhuydo ri frqyhujhqfh/ +4> 4, > dv wkh Wd|oru vhulhv iru h{ =
6;7
Fkdswhu 45
VHTXHQFHV DQG VHULHV
451 Xvh wkh Wd|oru vhulhv iru h{ dqg surshuw| +6, zlwk n @ 8= 4 5 4 4 { . {8 {6 . . {8 {q . 5$ 6$ q$ 4 4 {q.8 . @ {8 . {9 . {: . {; . . 5 9 q$
{8 h{ @ {8 4 . {8 { . {8
Wkh Wd|oru vhulhv iru {8 h{ kdv wkh vdph lqwhuydo ri frqyhujhqfh/ +4> 4, > dv wkh Wd|oru vhulhv iru h{ =
471
9 @ 6 . {5
5 5 { 4 6 Xvh wkh Wd|oru vhulhv iru
4 4{ >
5
pxowlso| hdfk whup e| 5/ dqg uhsodfh { zlwk {6 =
9 @ 6 . {5
5 5 { 4 6 5 5 6 5 q 5 5 { { { { .5 . . 5 . .5 @54.5 6 6 6 6 +4,q 5 {5q 5 5 5 . @ 5 {5 . {7 {9 . . 6 < 5: 6q
Wkh Wd|oru vhulhv iru
4 4{
5
lv ydolg zkhq 4 ? { ? 4= Uhsodflqj { zlwk {6 jlyhv 4 ?
{5 ? 4 ru 6 A {5 A 6= 6
s s Wkh lqhtxdolw| lv vdwlv hg e| dq| { lq wkh lqwhuydo 6> 6 =
491
h{ h{ @ 5 Zh qg wkh
4 { 4 { h h 5 5 Wd|oru vhulhv iru 45 h{ e| vwduwlqj zlwk wkh Wd|oru vhulhv iru h{ dqg pxowlso|lqj hdfk whup e| 45 = 4 4 4 4 4 4 4 { 4 h @ 4 . { . {5 . {6 . . 5 5 5 5 5$ 5 6$ 5 4 6 4 q 4 5 4 4 { . @ . { . { . { . . 5 5 7 45 5q$
q$4 {q .
Zh qg wkh Wd|oru vhulhv iru 45 h{ e| vwduwlqj zlwk wkh Wd|oru vhulhv iru h{ > pxowlso|lqj hdfk whup e| 45 > dqg uhsodflqj { zlwk {= 4 4 4 4 4 4 4 4 4 5 6 q +{, . +{, . +{, . . +{, . h{ @ 4 . 5 5 5 5 5$ 5 6$ 5 q$ 4 +4,q.4 q 4 4 4 { . @ . { {5 . {6 . . 5 5 7 45 5q$
Vhfwlrq 4518
Wd|oru Vhulhv
6;8
Xvh surshuw| +4, zlwk i+{, @ 45 h{ dqg j+{, @ 45 h{ = h{ h{ 5 4 4 @ h{ h{ 5 5 # $ q.4 +4, 4 4 4 4 4 4 4 4 4 @ . . {. {5 . . {6 . . . {q . 5 5 5 5 7 7 45 45 5q$ 5q$ 4 @ { . {6 . 9 4 @ { . {6 . 9
4 8 4 : 4 . +4,q.4 q { . { . . { . 453 8373 5q$ 4 8 4 : 4 { . { . . {5q.4 . 453 8373 +5q . 4,$
Wkh qhz vhulhv kdv wkh vdph lqwhuydo ri frqyhujhqfh/ +4> 4, > dv wkh Wd|oru vhulhv iru h{ = 4;1 oq 4 8{5 @ oq 4 . 8{5 Xvh wkh Wd|oru vhulhv iru oq +4 . {, dqg uhsodfh { zlwk 8{5 = 5 6 7 q.4 +4,q 8{5 8{5 8{5 8{5 5 5 oq 4 8{ @ 8{ . . . . 5 6 7 q.4 q.4 5q.5 { 58 458 9 958 ; 8 @ 8{5 {7 { { 5 6 7 q.4 Wkh Wd|oru vhulhv iru oq +4 . {, lv ydolg zkhq 4 ? { 4= Uhsodflqj { zlwk 8{5 jlyhv 4 4 4 ? 8{5 4 ru A {5 = 8 8 Wkh lqhtxdolw| lv vdwlv hg e| dq| { lq wkh lqwhuydo s48 > s48 =
4.{ @ oq +4 . {, oq +4 {, 531 oq 4{ Wkh Wd|oru vhulhv iru oq +4 . {, lv jlyhq dv oq +4 . {, @ {
+4,q {q.4 {5 {6 {7 . . . . 5 6 7 q.4
Zh qg wkh Wd|oru vhulhv iru oq +4 {, e| vwduwlqj zlwk wkh Wd|oru vhulhv iru oq +4 . {, pxowlso|lqj hdfk whup e| 4> dqg uhsodflqj { zlwk {= 5
6
7
q
q.4
+{, +{, +{, +4, +{, . +4, . . +4, +4, 5 6 7 q.4 5 6 7 q.4 { { { { @{. . . . . . 5 6 7 q.4
oq +4 {, @ 4 +{, +4,
.
Xvh surshuw| +4, zlwk i+{, @ oq +4 . {, dqg j+{, @ oq +4 {, = 4.{ @ oq +4 . {, oq +4 {, oq 4{ 4 +4,q 4 4 4 4 4 4 {5 . . {6 . . {7 . . . {q.4 . @ +4 . 4, { . . 5 5 6 6 7 7 q.4 q.4 5 @ 5{ . {6 . 6 5 @ 5{ . {6 . 6
5 8 { . 8 5 8 { . 8
q
+4, . 4 q.4 5 : { . . { . : q.4 5 : 5{5q.4 { . . . : 5q . 4
6;9
Fkdswhu 45
VHTXHQFHV DQG VHULHV
551 Xvh wkh Wd|oru vhulhv iru h{ dqg uhsodfh { zlwk { wr qg wkh Wd|oru vhulhv iru h{ = h{ @ 4 . +{, . @4{.
{5 5
@4{.
{5 5
@4{.
{5 5
4 4 4 4 4 5 6 7 8 q +{, . +{, . +{, . +{, . . +{, . 5$ 6$ 7$ 8$ q$ {6 {7 {8 +4,q {q . . . . 9 57 453 q$ & % q { {5 {6 +4, {q5 . 4 . . . q$ 6 45 93 5 & % { {6 { {q5 { { 4 4 4 4 q$ 6 7 93 9 q.4 5
6 6 q5 { Frpsduh wkh vhulhv {6 4 {7 {93 4 {9 { q$ 4 q.4 zlwk wkh vhulhv {6 4 {7 {7; 4 {7 5 q5 6 { 7 q6 4 {7 = = = 1 Wkh vhfrqg vhulhv lv dq lq qlwh jhrphwulf vhulhv wkdw lv ohvv wkdq ru htxdo wr wkh 5 5 uvw vhulhv whup e| whup1 Wkh jhrphwulf vhulhv kdv d @ {6 4 {7 dqg u @ {49 > dqg vxpv wr 49{7{ 7;6{5 = Wkxv/ iru 3 ? { ? 4> % & {5 {5 { 49{ 7{5 { { {6 { {q5 4{. ?4{. 4 4 4 4 4 q$ 5 7; 6{5 5 6 7 93 9 q.4 5 {5 ?4{. > 5 ru 4{.
Iru ydoxhv ri { vx!flhqwo| forvh wr 3/
{5 {5 49{ 7{5 ? h{ ? 4 { . = 4 5 5 7; 6{ 5 49{7{5 7;6{5
4{.
dssurdfkhv 3/ dqg
{5 {5 49{ 7{5 4 4{. = 5 5 7; 6{ 5
5
Wkxv/ h{ 4 { . {5 = Iru { ? 3> d vlplodu dujxphqw fdq eh pdgh1
571 Wkh duhd lv jlyhq e| ]
4 5
5
h{ g{=
3 5
Ilqg wkh Wd|oru vhulhv iru h{ e| xvlqj wkh Wd|oru vhulhv iru h{ dqg uhsodflqj { zlwk {5 = 4 5 5 5 4 5 6 4 5 7 4 5 q { { { { . . . . . h{ @ 4 . {5 . 5$ 6$ 7$ q$
Vhfwlrq 4518
Wd|oru Vhulhv
6;:
Xvlqj wkh uvw yh whupv ri wklv vhulhv jlyhv ]
4 5
5
h{ g{
3
]
4 5
4 5 5 4 5 6 4 5 7 4 . {5 . g{ { { { . . 5$ 6$ 7$
4 5
4 7 4 9 4 ; 5 4 . { . { . { . { g{ 5 9 57
3
] @
3
4 4 6 4 8 4 : 4 < 5 @ {. { . { . { . { 6 43 75 549 3
4 4 4 4 4 @ . . . . 3 5 57 653 86:9 443/8<5 =8783
591 Wkh duhd lv jlyhq e| ] 4 7
Ilqg wkh Wd|oru vhulhv iru
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4 5
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e| xvlqj wkh Wd|oru vhulhv iru
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4 5
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7
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=8<3;
5;1 Wkh duhd lv jlyhq e| ] 3 5 @5
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4 4 5 s h{ @5 g{ @ s 5 5
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6;;
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3
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4 g{ s 5 @ @ @
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=4<48=
631 Wkh grxeolqj wlph q iru d txdqwlw| wkdw lqfuhdvhv dw dq dqqxdo udwh u lv jlyhq e| q@
oq 5 oq 5 @ 49=7:= oq+4 . u, oq =3758
Lw zloo wdnh derxw 4917: |hduv/ ru derxw 49 |hduv ; prqwkv1 Dffruglqj wr wkh Uxoh ri :3/ wkh grxeolqj wlph lv jlyhq e| Grxeolqj wlph
:3 :3 @ @ 49=7:= 433u 433+=3758,
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651 +d,
4 [
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5
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f5 @ f4
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f7 @ f6
Vxevhtxhqw dssur{lpdwlrqv zloo djuhh zlwk f7 dqg f8 wr wkh qhduhvw kxqguhgwk1 Wkxv/ { @ 6=9:=
@ 8=5694 Vxevhtxhqw dssur{lpdwlrqv zloo djuhh zlwk f6 dqg f7 wr wkh qhduhvw kxqguhgwk1 Wkxv/ { @ 8=57=
71
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f6 @ f5
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f7 @ f6
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f8 @ f7
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f6 @ f5
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f7 @ f6
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=4963; @ =8334 4<=8; =334<6 @ =8333 4<=587
Vxevhtxhqw dssur{lpdwlrqv zloo djuhh zlwk f8 dqg f9 wr wkh qhduhvw kxqguhgwk1 Wkxv/ { @ 6 lv d vroxwlrq lq ^6> 5` dqg { @ =83 lv d vroxwlrq lq ^3> 4` =
Vhfwlrq 4519
431
Qhzwrq*v Phwkrg
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6<4
471
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f5 @ f4
6 i +f4 , @ 5=:<<9 @4 3 i +f4 , 4=99:
f6 @ f5
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f7 @ f6
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f8 @ f7
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Ohw f4 @ 6=
Vxevhtxhqw dssur{lpdwlrqv zloo djuhh zlwk f7 dqg f8 wr wkh qhduhvw kxqguhgwk1 Wkxv/ { @ 5=3:=
451
i+{, @ h5{ . 6{ 7 i 3 +{, @ 5h5{ . 6
f5 @ f4
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i +f7 , <=4 437 @ =7:6; @ =7:6: i 3 +f7 , ;=48<
Vxevhtxhqw dssur{lpdwlrqv zloo djuhh zlwk f7 dqg f8 wr wkh qhduhvw kxqguhgwk1 Wkxv/ { @ =7:=
f5 @ f4
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f6 @ f5
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f7 @ f6
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f8 @ f7
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f9 @ f8
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f: @ f9
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f; @ f:
=34935 i +f: , @ =:<59 @ =:<34 3 i +f: , 9=7:8
Vxevhtxhqw dssur{lpdwlrqv zloo djuhh zlwk f: dqg f; wr wkh qhduhvw kxqguhgwk1 Wkxv/ { @ =:<=
i +3, @ 6 ? 3 dqg i +6, @ 73;=75;:< A 3 vr d vroxwlrq h{lvwv lq +3> 6, = Ohw f4 @ 3=
i+{, @ {5 h{ . {5 5 i 3 +{, @ {5 h{ . 5{h{ . 5{
i+{, @ 5 oq { . { 6 5 i 3 +{, @ . 4 { i +4, @ 5 ? 3 dqg i +7, @ 6=::58;;: A 3 vr d vroxwlrq h{lvwv lq +4> 7, = Ohw f4 @ 4=
f5 @ f4
i +f4 , 5 @4 @ 4=999: 3 i +f4 , 6
f6 @ f5
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f7 @ f6
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Vxevhtxhqw dssur{lpdwlrqv zloo djuhh zlwk f6 dqg f7 wr wkh qhduhvw kxqguhgwk1 Wkxv/ { @ 4=;4=
6<5
4;1
Fkdswhu 45
s 6 lv d vroxwlrq ri {5 6 @ 3=
571
i+{, @ {5 6 i 3 +{, @ 5{ s Vlqfh 4 ? 6 ? 5> ohw f4 @ 4= 5 @5 5 4 f6 @ 5 @ 4=:8 7 =3958 f7 @ 4=:8 @ 4=:65 6=8 5 437 @ 4=:65 f8 @ 4=:65 6=797 f7 @ f8 @ 4=:65> wr wkh qhduhvw wkrxvdqgwk/ Vlqfh s 6 @ 4=:65= s 48 lv d vroxwlrq ri {5 48 @ 3=
591
i+{, @ { 48 i 3 +{, @ 5{ s Vlqfh 6 ? 48 ? 7> ohw f4 @ 6= 9 @7 9 4 f6 @ 7 @ 6=;:8 ; =34896 f7 @ 6=;:8 @ 6=;:6 :=:8 4=6 437 @ 6=;:6 f8 @ 6=;:6 :=:79 f5 @ 6
551
s 633 lv d vroxwlrq ri {5 633 @ 3= i+{, @ {5 633 i 3 +{, @ 5{ s Vlqfh 4: ? 633 ? 4;> ohw f4 @ 4:= 44 @ 4:=657 67 =453<; f6 @ 4:=657 @ 4:=654 67=97; =34:37 @ 4:=654 f7 @ 4:=654 67=975 f5 @ 4:
f6 @ f7 @ 4:=654> wr wkh qhduhvw wkrxvdqgwk/ Vlqfh s 633 @ 4:=654=
s 6 454 lv d vroxwlrq ri {6 454 @ 3= i+{, @ {6 454 i 3 +{, @ 6{5 s Vlqfh 7 ? 6 454 ? 8> ohw f4 @ 7= 8: f5 @ 7 @ 8=4;; 7; 4;=96: @ 7=<8: f6 @ 8=4;; ;3=:79 =;3599 @ 7=<79 f7 @ 7=<8: :6=:49 =3397 f8 @ 7=<79 @ 7=<79 :6=6;< f7 @ f8 @ 7=<79> wr wkh qhduhvw wkrxvdqgwk/ Vlqfh s 6 454 @ 7=<79=
5
f7 @ f8 @ 6=;:6> wr wkh qhduhvw wkrxvdqgwk/ Vlqfh s 48 @ 6=;:6=
s 6 48 lv d vroxwlrq ri {6 48 @ 3= i+{, @ {6 48 i 3 +{, @ 6{5 s Vlqfh 5 ? 6 48 ? 6> ohw f4 @ 5= : @ 5=8;6 f5 @ 5 45 5=5668 f6 @ 5=8;6 @ 5=7:4 53=349 =3;:86 @ 5=799 f7 @ 5=7:4 4;=64; =336< @ 5=799 f8 @ 5=799 4;=576 f7 @ f8 @ 5=799> wr wkh qhduhvw wkrxvdqgwk/ Vlqfh s 6 48 @ 5=799=
f5 @ 4
531
VHTXHQFHV DQG VHULHV
5;1
i+{, @ {6 . <{5 9{ . 7 i 3 +{, @ 6{5 . 4;{ 9 Wr qg fulwlfdo srlqwv/ vroyh i 3 +{, @ 6{5 . 4;{ 9 @ 3= i 33 +{, @ 9{ . 4; i 3 +:, @ 48 A 3 dqg i 3 +9, @ 9 ? 3 vr d vroxwlrq h{lvwv lq +:> 9, = Ohw f4 @ := 48 @ 9=6; 57 4=5:65 f6 @ 9=6; @ 9=65 53=5; =39:5 f7 @ 9=65 @ 9=65 4<=<5 f5 @ :
Vhfwlrq 4519
Qhzwrq*v Phwkrg
6<6
Vxevhtxhqw dssur{lpdwlrqv zloo djuhh zlwk f6 dqg f7 wr wkh qhduhvw kxqguhgwk1 Wkxv/ { @ 9=65= Vlqfh i 33 +9=65, @ 4<=<5 ? 3> wkh judsk kdv d uhodwlyh pd{lpxp dw { @ 9=65= i 3 +3, @ 9 ? 3 dqg i 3 +4, @ 48 A 3 vr d vroxwlrq h{lvwv lq +3> 4, = Ohw f4 @ 3=
9 @ =66 4; =599: @ =65 f6 @ =66 4<=<; =39:5 f7 @ =65 @ =65 4<=<5 Vxevhtxhqw dssur{lpdwlrqv zloo djuhh zlwk f6 dqg f7 wr wkh qhduhvw kxqguhgwk1 Wkxv/ { @ =65= Vlqfh i 33 +=65, @ 4<=<5 A 3> wkh judsk kdv d uhodwlyh plqlpxp dw { @ =65= f5 @ 3
631
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8 3
Vlqfh f5 lv xqgh qhg/ ohw f4 @ =8=
6 @ =;6 < 4=7538 f6 @ =;6 @ =:8 4;=55: =3958 @ =:8 f7 @ =:8 48=:8 Vxevhtxhqw dssur{lpdwlrqv zloo djuhh zlwk f6 dqg f7 wr wkh qhduhvw kxqguhgwk1 Wkxv/ { @ =:8= Vlqfh i 33 +=:8, @ 48=:8 A 3> wkh judsk kdv d uhodwlyh plqlpxp dw { @ =:8= f5 @ =8
651 Wkh surfhvv vkrxog eh xvhg dw ohdvw xqwlo wkh vdylqjv surgxfhg lv htxdo wr wkh lqfuhdvhg frvwv lqfxuuhg/ ru zkhq V +{, @ F +{, = Wkhuhiruh/ vroyh V +{, F +{, = i +{, @ V +{, F +{, @ {6 . 8{5 . < {5 . 73{ . 53 @ {6 . 7{5 73{ 44 i 3 +{, @ 6{5 . ;{ 73 i 3 +{, @ 6{5 . ;{ 73 i +7, @ 76 ? 3 dqg i +8, @ 47 A 3 vr d vroxwlrq h{lvwv lq +7> 8, = Ohw f4 @ 7=
76 @ 8=3; 73 53=455 f6 @ 8=3; @ 7=;5 :;=38< 4=43<; @ 7=;3 f7 @ 7=;5 9;=58: =57; @ 7=;3 f8 @ 7=;3 9:=85 f5 @ 7
6<7
Fkdswhu 45
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8: +=35, 8: +=35, +4 . =35,45 933 +=35,5 k l 454 45 8: 4 . +45, +=35, +4 . =35, . +4 . =35,
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l6 @ =353:87;8 @ =353:87;8 @ =353:8:75
451:
45
5
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O*Krvslwdo*v Uxoh
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5
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{$3
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O*Krvslwdo*v Uxoh
6<8
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olp
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{
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olp
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4 5+<.{,4@5
4
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4 = 9
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