577 lines
20 KiB
Plaintext
577 lines
20 KiB
Plaintext
5XVVLDQ#3K\VLFV#-RXUQDO/#9RO1#77/#1R1#;/#5334
|
||
|
||
7+(#6$*1$␩(;#237,&$/#())(&7
|
||
11#91#.XSU\DHY#
|
||
|
||
8'&=#863145=86414;
|
||
|
||
,W# LV# VKRZQ# WKDW# WKH# 6DJQDF# SKHQRPHQRQ# REVHUYHG# H[SHULPHQWDOO\# LV# LQ# FRQIOLFW# ZLWK# VSHFLDO# +DQG# DOVR# JHQHUDO, UHODWLYLW\1#7KH#FDOFXODWLRQ#RI#WKH#PDJQLWXGH#RI#WKH#6DJQDF#HIIHFW#ZLWKLQ#WKH#IUDPHZRUN#RI#D#UHODWLYLVWLF#WKHRU\#UHVXOWV LQ# D# SDUDGR[1# 7KXV/# IRU# WKH# PRYLQJ# UHODWLYLVWLF# REVHUYHU/# WKH# VLPXOWDQHRXV# HYHQWV# RI# PHHWLQJ# RI# ±# UD\V# ZLWK# WKH UDGLDQW#DUH#QRQVLPXOWDQHRXV#IRU#WKH#HYHQWV#RI#PHHWLQJ#RI#WKH#UDGLDQW#ZLWK#±#UD\V1#+HUH/#WKH#LQGH[#%.%#GHQRWHV#D#UD\ SURSDJDWLQJ# LQ# WKH# GLUHFWLRQ# RI# PRWLRQ# RI# WKH# REVHUYHU/# ZKLOH# WKH# LQGH[# %±%# GHQRWHV# D# UD\# SURSDJDWLQJ# DJDLQVW# WKH PRWLRQ1#7KH#SDUDGR[#LV#UHVROYHG#ZLWKLQ#WKH#IUDPHZRUN#RI#WKH#WKHRU\#RI#D#OXPLQLIHURXV#HWKHU#DW#UHVW1
|
||
|
||
,1752'8&7,21
|
||
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
|
||
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
|
||
+RZHYHU/# WKLV# LV# SHUKDSV# WKH# UDUHVW# FDVH# LQ# VFLHQFH# ZKHUH# HUURQHRXV# SUHGLFWLRQV# RI# VSHFLDO# DQG# JHQHUDO# UHODWLYLW\ FRLQFLGH/# PHUHO\# E\# FKDQFH/# ZLWK# %FRUUHFW%# UHVXOWV# RI# WKH# WKHRU\# RI# DQ# HWKHU# DW# UHVW/# ZKLFK# LV# D# PLVXQGHUVWDQGLQJ1# $FWXDOO\/ WKH#6DJQDF#YRUWH[#RSWLFDO#HIIHFW#LV#QRW#LQ#WKH#FRPSHWHQFH#RI#HLWKHU#VSHFLDO#RU#JHQHUDO#WKHRU\#RI#UHODWLYLW\#±#WKH#WKHRU\#OHDGV#WR D# SDUDGR[/# DQG# WKH# SDUDGR[# LV# FRPSOHWHO\# UHVROYDEOH# ZLWKLQ# WKH# IUDPHZRUN# RI# WKH# WKHRU\# RI# HWKHU# DW# UHVW1# 7KHUHIRUH/# WKH DXWKRU#KDV#IRXQG#LW#SHUWLQHQW/#WR#DYRLG#PLVXQGHUVWDQGLQJ/#WR#VWUHVV#RQFH#DJDLQ#WKH#SXUHO\#HWKHUHDO#QDWXUH#RI#WKH#6DJQDF#HIIHFW WKDW#GHILHV#D#FRQVLVWHQW#H[SODQDWLRQ#ZLWKLQ#WKH#IUDPHZRUN#RI#ERWK#VSHFLDO#DQG#JHQHUDO#WKHRU\#RI#UHODWLYLW\1
|
||
|
||
41#1(*$7,21#2)#7+(#(;,67(1&(#2)#7+(#6$*1$&#())(&7#,1#63(&,$/#5(/$7,9,7<
|
||
7KXV/# IROORZLQJ# >5@/# ZH# VKDOO# FRQVLGHU# RQFH# DJDLQ# D# FLUFXODU# WUDMHFWRU\# RI# SURSDJDWLRQ# RI# UD\V# LQ# WKH# 6DJQDF H[SHULPHQW/# FRUUHVSRQGLQJ# WR# WKH# FDVH# RI# DQ# LQILQLWH# QXPEHU# RI# PLUURUV# LQ# D# ULQJ# LQWHUIHURPHWHU1# /HW# WKHUH# EH# DQ LQWHUIHURPHWHU#URWDWLQJ#LQ#D#ODERUDWRU\#IUDPH#RI#UHIHUHQFH#6#ZLWK#DQJXODU#YHORFLW\# +ω #FORFNZLVH#+VD\/#LQ#WKH#SODQH#RI#WKLV
|
||
|
||
6DPDUD# 'HSW1# RI# 31# 11# /HEHGHY# 3K\VLFDO# ,QVWLWXWH# RI# WKH# 5XVVLDQ# $FDGHP\# RI# 6FLHQFHV# +),$1,1# 7UDQVODWHG# IURP ,]YHVWL\D#9\VVKLNK#8FKHEQ\NK#=DYHGHQLL/#)L]LND/#1R1#;/#SS1#96±9:/#$XJXVW/#53341#2ULJLQDO#DUWLFOH#VXEPLWWHG#-XO\#37/#53331
|
||
|
||
;8;
|
||
|
||
43970;;;:2342773;03;8;'58133##©5334#3OHQXP#3XEOLVKLQJ#&RUSRUDWLRQ
|
||
|
||
SDJH,1# 7KH# UD\V# RI# OLJKW# HPLWWHG# E\# D# PRYLQJ# UDGLDQW# LQ# ERWK# GLUHFWLRQV# SURSDJDWH# LQ# YDFXXP# LQ# WKH# SODQH# RI# WKH LQWHUIHURPHWHU#SODWIRUP#+ZLWK#WKH#]0FRRUGLQDWH#RI#WKH#UD\#EHLQJ#FRQVWDQW=# ] = 3 ,#RYHU#D#FLUFOH#RI#UDGLXV#U#ZLWK#OLQHDU#YHORFLW\
|
||
F1#7KH#UD\V# +UHFDOO# WKDW# WKD\# DUH# HPLWWHG# DW# W3 = 3 ,/# LQ# YLHZ# RI# WKH# LQLWLDO# VWDWH# ϕ± (3) = 3 /# ZLOO# PHHW# DW# WLPH# W4 # DW# ZKLFK ϕ± (W4 ) = ±π 1# ++HUH/# ϕ(W ) # LV# WKH# FXUUHQW# FRRUGLQDWH# RI# WKH# UD\/# DQG# LW# LV# VXSSRVHG# WKDW# WKH# DQJXODU# FRRUGLQDWH# RI# WKH# UD\
|
||
YDULHV# FRQWLQXRXVO\# DQG# FDQ# WDNH# DQ\# ODUJH# SRVLWLYH# RU# QHJDWLYH# YDOXH,1# 1H[W# WLPH# WKH# UD\V# ZLOO# PHHW# DW# W5 # DW# ZKLFK
|
||
ϕ± (W5 ) = ±5π /#L1H1/#DW#D#PXOWLSOH#SRLQW#RI#WKH#LQLWLDO#VWDWH/#HWF1#+HUH/#WKH#LQGH[#%.%#GHQRWHV#D#UD\#SURSDJDWLQJ#LQ#WKH#GLUHFWLRQ
|
||
RI#URWDWLRQ#RI#WKH#SODWIRUP/#DQG#WKH#LQGH[#%±%#GHQRWHV#D##UD\#SURSDJDWLQJ#LQ#WKH#RSSRVLWH#GLUHFWLRQ1#2EYLRXVO\/#WKH#UHVXOW#GRHV QRW#GHSHQG#RQ#WKH#YHORFLW\#RI#WKH#UDGLDQW1#7KH#YHORFLW\#RI#OLJKW#LQ#WKH#IUDPH#RI#UHIHUHQFH#6#LV#HTXDO#WR#F#E\#GHILQLWLRQ1
|
||
+RZHYHU/#DV#WKH#DQJXODU#FRRUGLQDWH#RI#WKH#UDGLDQW# YDULHV# DV# +ωW /# WKH# PHHWLQJ# RI# WKH# ±# UD\V# ZLWK# WKH# UDGLDQW# +RU/ ZKDW#LV#VDPH#LQ#WKLV#FDVH/#WKH#PHHWLQJ#RI#WKH#UDGLDQW#ZLWK#WKH#±#UD\V,#ZLOO#KDSSHQ#DW#WKH#FRRUGLQDWH#+L1H1/#ODERUDWRU\,#WLPH#W± GHWHUPLQHG#E\#WKH#FRQGLWLRQ# ±FW± = ±5πU + ωW± U =
|
||
|
||
W±
|
||
|
||
=
|
||
|
||
5πU F B ωU
|
||
|
||
1#
|
||
|
||
#+4,
|
||
|
||
)URP# WKH# YLHZSRLQW# RI# WKH# LPPRELOH# REVHUYHU# 6/# WKH# HYHQWV# XQGHU# FRQVLGHUDWLRQ# DUH# QRQVLPXOWDQHRXV1# 7KH# UD\# IRU ZKLFK# WKH# GLUHFWLRQ# RI# E\SDVV# FRLQFLGHV# ZLWK# WKH# GLUHFWLRQ# RI# URWDWLRQ# FRPHV# WR# WKH# UDGLDQW# ODWHU# WKDQ# WKH# UD\# IRU# ZKLFK# WKH GLUHFWLRQ#RI#E\SDVV#LV#RSSRVLWH#+WKH#UDGLDQW#KDV#WLPH#WR#PRYH#RYHU#GLVWDQFH# +ωW± U ,1#7KHUHIRUH/#WKH#GLVWDQFHV#FRYHUHG#E\#WKH ±#UD\V#E\#WKH#LQVWDQW#WKH\#PHHW#ZLWK#WKH#UDGLDQW#ZLOO#DOVR#EH#GLIIHUHQW=
|
||
|
||
O±
|
||
|
||
=
|
||
|
||
±5πU
|
||
|
||
+
|
||
|
||
ωW± U
|
||
|
||
=
|
||
|
||
±5πU 4 ±
|
||
|
||
ωU F B ωU
|
||
|
||
|
||
|
||
1#
|
||
|
||
#+5,
|
||
|
||
,W# LV# HDV\# WR# EH# FRQYLQFHG# WKDW# KHUH# WKH# FRRUGLQDWH# YHORFLW\# RI# OLJKW# LQGHHG# GRHV# QRW# GHSHQG# RQ# WKH# YHORFLW\# RI# WKH UDGLDQW/#DQG/#WKXV/#WKH#UHODWLRQ# F± = O± 2 W± = ±F #LV#IXOILOOHG1# :H# VWUHVV# WKDW# WKLV# YHORFLW\# LV# D# SK\VLFDO# YHORFLW\/# WKRXJK/# IRU
|
||
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
|
||
2I# LQWHUHVW# WR# XV/# KRZHYHU/# DUH# WKH# LQVWDQWV# RI# PHHWLQJ# RI# WKH# ±# UD\V# ZLWK# WKH# UDGLDQW# +ZKLFK# DUH# QRW# LGHQWLFDO# E\ GHILQLWLRQ,# QRW# IURP# WKH# YLHZSRLQW# RI# WKH# ODERUDWRU\# REVHUYHU# 6/# EXW# IURP# WKH# YLHZSRLQW# RI# WKH# UHODWLYLVWLF# REVHUYHU# 6 ′ PRYLQJ#LQ#D#FLUFOH1#,W#DSSHDUV#WKDW#IURP#WKH#YLHZSRLQW#RI#WKH#REVHUYHU# 6 ′ /#WKH#±#UD\V#PHHW#ZLWK#WKH#UDGLDQW#VLPXOWDQHRXVO\#LQ FRPSOHWH# FRUUHVSRQGHQFH# ZLWK# WKH# (LQVWHLQ# LGHD# RI# WKH# UHODWLYLW\# RI# VLPXOWDQHLW\# RI# HYHQWV1# $FWXDOO\/# VXEVWLWXWLRQ# RI# WKH FRRUGLQDWHV#RI#HYHQWV#+4,#DQG#+5,#LQWR#WKH#/RUHQW]#WUDQVIRUP#+ZULWWHQ#LQ#F\OLQGULFDO#FRRUGLQDWHV,
|
||
|
||
O′ = γ (O − ωUW )/
|
||
|
||
U′ = U/
|
||
|
||
]′ = ]/
|
||
|
||
W′
|
||
|
||
=
|
||
|
||
γ
|
||
|
||
|
||
|
||
W
|
||
|
||
−
|
||
|
||
ωU F5
|
||
|
||
O
|
||
|
||
|
||
|
||
/#
|
||
|
||
#+6,
|
||
|
||
ZKHUH# γ = 42
|
||
|
||
4
|
||
|
||
−
|
||
|
||
ω5 U F5
|
||
|
||
5
|
||
|
||
DQG#O#LV#WKH#OHQJWK#RI#WKH#FLUFOH#DUF/#JLYHV#VLPXOWDQHRXV#HYHQWV=
|
||
|
||
O±′
|
||
|
||
= ±γ5πU/
|
||
|
||
W±′
|
||
|
||
=
|
||
|
||
γ 5πU 1# F
|
||
|
||
#+7,
|
||
|
||
7KXV/#WKH#PHHWLQJV#RI#WKH#±#UD\V#ZLWK#WKH#UDGLDQW#+QRW#WR#EH#FRQIXVHG#ZLWK#WKH#PHHWLQJV#RI#WKH#UDGLDQW# ZLWK#WKH#± UD\V># ODWHU# ZH# VKDOO# VHH# WKDW# WKH# ODWWHU# DUH# QRW# VLPXOWDQHRXV,/# IURP# WKH# YLHZSRLQW# RI# WKH# UHODWLYLVWLF# REVHUYHU# 6 ′ /# DUH
|
||
|
||
;8<
|
||
|
||
VLPXOWDQHRXV1#2QH#FDQ#FKHFN#WKDW#/RUHQW]#WUDQVIRUPV#GR#QRW#DOWHU#WKH#YHORFLW\#RI#OLJKW/#L1H1/# F±′ = O±′ 2 W±′ = ±F 1#7KXV/#ZH#VHH
|
||
|
||
WKDW#VSHFLDO#UHODWLYLW\#OHDGV#WR#QHJDWLRQ#RI#WKH#H[LVWHQFH#RI#WKH#6DJQDF#HIIHFW#LQ#WKH#IUDPHZRUN#RI#WKH#WKHRU\#LWVHOI1
|
||
|
||
&HUWDLQO\/# WKH# LQYHUVH# FRXUVH# RI# UHDVRQLQJ# JLYHV# WKH# VDPH# QHJDWLYH# UHVXOWV1# ,QGHHG/# IURP# WKH# YLHZSRLQW# RI# WKH
|
||
|
||
REVHUYHU# 6 ′ /#WKH#ODERUDWRU\#IUDPH#RI#UHIHUHQFH#6#URWDWHV#FRXQWHUFORFNZLVH#ZLWK#DQJXODU#YHORFLW\# −ω /#ZKLOH#WKH#SODWIRUP#RI
|
||
|
||
WKH# LQWHUIHURPHWHU# LV# DW# UHVW1# 7KH# UD\V# HPLWWHG# DW# WLPH# W′ = 3 # IURP# D# SRLQW# ϕ±′ (3) = 3 # ZLOO# PHHW# DW# WKH# WLPH# W4′ # DW# ZKLFK
|
||
|
||
ϕ±′ (W4′) = ±π 1#1H[W#WLPH#WKH#UD\V#ZLOO#PHHW#DW# W5′ #DW#ZKLFK# ϕ±′ (W5′ ) = ±5π /#L1H1/#DJDLQ#DW#D#SRLQW#PXOWLSOH#ZLWK#UHVSHFW#WR#WKH
|
||
|
||
LQLWLDO# RQH1# $V# WKH# UDGLDQW# LV# DW# UHVW# LQ# WKH# %IUDPH# RI# UHIHUHQFH# RI# WKH# SODWIRUP# %/# WKH# ±# UD\V# ZLOO# PHHW# ZLWK# WKH# UDGLDQW# +RU/
|
||
|
||
ZKDW# LV# WKH# VDPH/# WKH# UDGLDQW# ZLOO# PHHW# ZLWK# WKH# ±# UD\V,# VLPXOWDQHRXVO\/# L1H1/#
|
||
|
||
W±′
|
||
|
||
=
|
||
|
||
γ
|
||
|
||
5πU F
|
||
|
||
# +GHWHUPLQHG# IURP# WKH# HTXDWLRQ
|
||
|
||
±FW±′ = ±γ5πU − 3 ,1
|
||
|
||
7KLV# FRQVLGHUDWLRQ/# QDWXUDOO\/# DOVR# DJUHHV# ZLWK# WKH# (LQVWHLQ# SULQFLSOH# RI# WKH# UHODWLYLW\# RI# VLPXOWDQHLW\# RI# HYHQWV1
|
||
|
||
$FWXDOO\/#LI#ZH#FRQVLGHU/#IRU#H[DPSOH/#D#URG#PRYLQJ#LQ#D#IUDPH#RI#UHIHUHQFH#6#ZLWK#VRPH#YHORFLW\#9/#WKH#OLJKW#VLJQDO#KDYLQJ
|
||
|
||
HPLWWHG#IURP#WKH#PLGGOH#RI#WKH#URG#DQG#KDYLQJ#UHDFKHG#VLPXOWDQHRXVO\/#LQ#%WKH#IUDPH#RI#UHIHUHQFH#RI#WKH#URG/%#WKH#HTXLGLVWDQW
|
||
|
||
HQGV#RI#WKH#URG/#UHDFKHV/#LQ#6/#WKH#DSSURDFKLQJ#HQG#RI#WKH#URG#PRYLQJ#WRZDUGV#WKLV#VLJQDO#HDUOLHU#WKDQ#WKH#HQG#PRYLQJ#DZD\
|
||
|
||
+WKH# YHORFLW\# RI# WKH# VLJQDO# LV# FRQVLGHUHG# WKH# VDPH# LQ# ERWK# IUDPHV# RI# UHIHUHQFH,1# 7KDW# LV/# WKLV# FRQVLGHUDWLRQ# LV# FRPSOHWHO\
|
||
|
||
VLPLODU#WR#WKDW#ZLWK#D#URWDWLQJ#ULQJ#LQWHUIHURPHWHU1 +RZHYHU/#RQ#WKH#RWKHU#KDQG/#DFFRUGLQJ#WR#WKH#FORFN#RI#WKH#UDGLDQW#+EHLQJ#DW#UHVW#DW#WKH#RULJLQ#RI# 6 ′ ,/#WKH#HYHQWV#RI
|
||
|
||
PHHWLQJ#RI#WKH#UDGLDQW#ZLWK#WKH#±#UD\V#+QRW#WR#EH#FRQIXVHG#ZLWK#WKH#HYHQWV#RI#PHHWLQJ#RI#WKH#UD\V#ZLWK#WKH#UDGLDQW>#WKH#IRUPHU/
|
||
|
||
DV#ZH#ZHUH#FRQYLQFHG/#KDYH#WDNHQ#SODFH#VLPXOWDQHRXVO\,#DUH#QHYHUWKHOHVV#DVVRFLDWHG#ZLWK#GLIIHUHQW#WLPHV=
|
||
|
||
W±′
|
||
|
||
=
|
||
|
||
4 W±
|
||
|
||
F
|
||
|
||
∫
|
||
3
|
||
|
||
GV
|
||
|
||
=
|
||
|
||
W±
|
||
|
||
4
|
||
|
||
−
|
||
|
||
ω5 U 5 F5
|
||
|
||
=
|
||
|
||
5πU F B ωU
|
||
|
||
4−
|
||
|
||
ω5 U 5 F5
|
||
|
||
/#
|
||
|
||
#+8,
|
||
|
||
ZKHUH# GV = FGW
|
||
|
||
4
|
||
|
||
−
|
||
|
||
ω5 U 5 F5
|
||
|
||
# LV# WKH# YDOXH# RI# WKH# LQWHUYDO# DW# WKH# ORFDWLRQ# RI# WKH# UDGLDQW1# 7KH# UHVXOW# +8,# FDQ# EH# REWDLQHG
|
||
|
||
LPPHGLDWHO\# IURP#WKH# /RUHQW]#WUDQVIRUP#+6,# LI# ZH# VXEVWLWXWH# LQWR# WKH# HTXDWLRQ/# LQVWHDG# RI# WKH# FXUUHQW# SDUDPHWHU# O/# QRW# WKH
|
||
|
||
FXUUHQW#FRRUGLQDWHV#RI#WKH#UD\V/# ±5πU + ωW±U /#EXW#WKH#FXUUHQW#FRRUGLQDWHV#RI#WKH#PRYLQJ#UDGLDQW/# +ωUW± /#ZKLFK#GLIIHU#IURP
|
||
|
||
WKH#FRRUGLQDWHV#RI#WKH#UD\V#E\# ±5πU 1#6ROXWLRQ#+8,#LV#GLUHFWO\#RSSRVLWH#WR#VROXWLRQ#+7,1# ,W#LV#WKLV#VHFRQG#VROXWLRQ#+8,# WKDW#LV
|
||
|
||
FRQVLGHUHG/#IRU#H[DPSOH/#E\#WKH#DXWKRUV#RI#>5@#DV#D#VSHFLDO#UHODWLYLVWLF#HIIHFW#RI#WKH#WKHRU\1 ,W#VKRXOG/#KRZHYHU/#EH#QRWHG#WKDW#LQ#WKH#JHQHUDO#FDVH#WKH#LQYDULDQFH#RI#WKH#LQWHUYDO#+ V5 = V′5 ,#LV#QRW#WKH#QHFHVVDU\
|
||
|
||
DQG# VXIILFLHQW# FRQGLWLRQ# IRU# WKH# LQYDULDQFH# RI# WKH# YHORFLW\# RI# OLJKW# + F′ = F ,1# ,Q# DFWXDOLW\/# WKH# GHILQLWLRQ# RI# WKH# LQWHUYDO
|
||
|
||
LQYROYHV# WKH# SURGXFW# RI# DQ# XQNQRZQ# YHORFLW\# RI# OLJKW# E\# DQ# XQNQRZQ# WLPH/# DQG# WKLV# SURGXFW# FDQ# JHQHUDOO\# KDYH# DQ# LQILQLWH QXPEHU#RI#VROXWLRQV/#LQFOXGLQJ#WKH#VROXWLRQ# F′ ≠ F #+VHH#>8@,/#IRU#H[DPSOH=
|
||
|
||
[′ = [ −9W / 4 −9 5 2 F5
|
||
|
||
\′ = \/
|
||
|
||
]′ = ]/
|
||
|
||
W′ = 4−9 5 2 F5 /
|
||
|
||
F′
|
||
|
||
=
|
||
|
||
F − 9[ 2 FW 4−9 5 2 F5
|
||
|
||
1#
|
||
|
||
#+9,
|
||
|
||
7KLV#WUDQVIRUP#LV#SUHIHUDEOH#WR#WKH#/RUHQW]#WUDQVIRUP#DOVR#IRU#WKH#UHDVRQ#WKDW#WKH#/RUHQW]#WUDQVIRUP#GRHV#QRW#VDWLVI\ WKH# 6DJQDF# HIIHFW# QRU# WKH# SULQFLSOH# RI# FRUUHVSRQGHQFH1# $FWXDOO\/# IRU# 9 2 F << 4 /# # WKH# /RUHQW]# WUDQVIRUP# RI# WLPH/
|
||
( ) W′ = γ W − 9[ 2 F5 /# JRHV# RYHU# QRW# WR# WKH# FODVVLFDO# WUDQVIRUP# W′ ≈ W /# DV# LW# VKRXOG# EH/# EXW# WR# D# WUDQVIRUP# RI# WKH# IRUP
|
||
W′ ≈ W − 9[ 2 F5 #FRQWDLQLQJ#D#WHUP#RI#WKH#ILUVW#RUGHU#RI#VPDOOQHVV/# 9 2 F 1#)RU# [ 2 F >> 3 /#WKH#ODVW#WHUP#LQ#WKH#H[SUHVVLRQ#FDQQRW EH#QHJOHFWHG1#7KXV/#HYHQ#E\#WKLV#FULWHULRQ#RQO\/#WKH#/RUHQW]#WUDQVIRUP#RI#WLPH#FDQ#EH#FRQVLGHUHG#HUURQHRXV1
|
||
|
||
;93
|
||
|
||
51#1(*$7,21#2)#7+(#(;,67(1&(#2)#7+(#6$*1$&#())(&7#,1#*(1(5$/#5(/$7,9,7<
|
||
|
||
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
|
||
7KXV/#WKH#DXWKRUV#RI#>9@/#ZKHQ#FDOFXODWLQJ#WKH#FRPSRQHQWV#RI#WKH#PHWULF#WHQVRU#RI#D#URWDWLQJ#IUDPH#RI#UHIHUHQFH/#XVHG FODVVLFDO#UDWKHU#WKDQ#UHODWLYLVWLF#WUDQVIRUPV#RI#FRRUGLQDWHV#DQG#WLPH=
|
||
|
||
[ = [′FRV ωW′ − \′VLQ ωW′/ \ = [′VLQ ωW′ + \′FRV ωW′/ ] = ]′/ W = W′
|
||
|
||
#+:,
|
||
|
||
+WKRXJK/#VWULFWO\#VSHDNLQJ/#LW#LV#QHFHVVDU\#WR#FRQVLGHU#GLIIHUHQWLDO#WUDQVIRUPV#RI#FRRUGLQDWHV#DQG#WLPH,1#,W#LV#FOHDU#WKDW#LQ#WKLV FDVH#WKH#VTXDUHG#GLIIHUHQWLDO#RI#WKH#LQWHUYDO#LV#QRW#UHWDLQHG#+WKH#FRHIILFLHQW#RI# F5GW′5 #LV#QRW#HTXDO#WR#XQLW\,=
|
||
|
||
( ) GV5
|
||
|
||
=
|
||
|
||
4 −
|
||
|
||
ω5 F5
|
||
|
||
[′5 + \′5
|
||
|
||
|
||
|
||
F5
|
||
|
||
GW
|
||
|
||
′5
|
||
|
||
−
|
||
|
||
5ω ( [ ′G\ ′
|
||
|
||
−
|
||
|
||
\′G[′)GW′ −
|
||
|
||
G[ ′5
|
||
|
||
−
|
||
|
||
G\ ′5
|
||
|
||
−
|
||
|
||
G] ′5
|
||
|
||
|
||
|
||
( ) =
|
||
|
||
4
|
||
|
||
+
|
||
|
||
5
|
||
|
||
ω F5
|
||
|
||
|
||
|
||
\′
|
||
|
||
G[′ GW ′
|
||
|
||
−
|
||
|
||
[′
|
||
|
||
G\′ GW ′
|
||
|
||
|
||
|
||
−
|
||
|
||
ω5 F5
|
||
|
||
[′5 + \′5
|
||
|
||
|
||
|
||
F
|
||
|
||
5
|
||
|
||
GW
|
||
|
||
′5
|
||
|
||
−
|
||
|
||
G[′5
|
||
|
||
−
|
||
|
||
G\ ′5
|
||
|
||
−
|
||
|
||
G] ′5
|
||
|
||
|
||
|
||
#+;,
|
||
|
||
≠ GV′5 = F5GW′5 − G[′5 − G\′5 − G]′5 1
|
||
|
||
7KHUHIRUH/# WKLV# WUDQVIRUPDWLRQ# LV# D# PHUH# PLVXQGHUVWDQGLQJ1# :H/# IRU# H[DPSOH/# LQ# FDOFXODWLQJ# WKH# FRPSRQHQWV# RI# WKH PHWULF#WHQVRU#RI#D#PRYLQJ#IUDPH#RI#UHIHUHQFH/#XVH#QRW#WKH#FODVVLFDO#WUDQVIRUPV
|
||
|
||
G[ = G[′ − 9GW′/ G\ = G\′/ G] = G]′/ GW = GW′ /#
|
||
|
||
#+<,
|
||
|
||
EXW#WKH#UHODWLYLVWLF#RQHV
|
||
|
||
G[ = G[′ − 9GW′ / G\ = G\′/ G] = G]′/ GW = GW′ − 9G[′ 2 F5 /#
|
||
|
||
#+43,
|
||
|
||
4−9 5 2 F5
|
||
|
||
4−9 5 2 F5
|
||
|
||
DV# ZH# UHDOL]H# WKDW# LQ# WKH# ILUVW# FDVH# WKH# VTXDUHG# GLIIHUHQWLDO# RI# WKH# LQWHUYDO# ZRXOG# EH# H[SUHVVHG# LQ# WKH# VDPH# ZD\# DV/# IRU H[DPSOH/#LQ#>9@#RU#>:@/#WKDW#LV/#ZLWK#DQ#LPSURSHU#PHWULF/#DQG/#WKXV/#LWV#ZKROHQHVV#DQG#LQYDULDQFH#ZRXOG#EH#EURNHQ=
|
||
|
||
GV5
|
||
|
||
=
|
||
|
||
4
|
||
|
||
−
|
||
|
||
95 F5
|
||
|
||
|
||
|
||
F5
|
||
|
||
GW
|
||
|
||
′5
|
||
|
||
|
||
|
||
+
|
||
|
||
59G[′GW′ − G[′5
|
||
|
||
− G\′5
|
||
|
||
− G]′5
|
||
|
||
#+44,
|
||
|
||
=
|
||
|
||
4
|
||
|
||
+
|
||
|
||
5
|
||
|
||
9 F5
|
||
|
||
G[′ GW′
|
||
|
||
−
|
||
|
||
95 F5
|
||
|
||
|
||
|
||
F5
|
||
|
||
GW
|
||
|
||
′5
|
||
|
||
|
||
|
||
− G[′5
|
||
|
||
− G\′5
|
||
|
||
− G]′5
|
||
|
||
≠
|
||
|
||
GV ′5
|
||
|
||
=
|
||
|
||
F 5 GW ′5
|
||
|
||
− G[′5
|
||
|
||
− G\′5
|
||
|
||
− G]′5
|
||
|
||
1
|
||
|
||
2QO\#LQ#WKH#VHFRQG#FDVH#GLG#ZH#REWDLQ#D#SURSHU#UHODWLRQ#VDWLVI\LQJ#VLPXOWDQHRXVO\#WKH#SULQFLSOH#RI#LQYDULDQFH#RI#WKH YHORFLW\#RI#OLJKW/#WKH#SULQFLSOH#RI#UHODWLYLW\/#DQG#WKH#SULQFLSOH#RI#LQYDULDQFH#RI#WKH#GLIIHUHQWLDO#RI#WKH#LQWHUYDO1
|
||
|
||
;94
|
||
|
||
7KH# UHODWLYLVWLF# WUDQVIRUP# IRU# WKH# GLIIHUHQWLDO# RI# WLPH# LQ# JHQHUDO# UHODWLYLW\# FDQ# EH# IRXQG# LQ# D# YHU\# VLPSOH# ZD\1# ,W# LV QHFHVVDU\/#ILUVW/#WR#IRUP#WKH#VXP#+QHJDWLYH#LQ#RXU#FDVH,#RI#VTXDUHG#GLIIHUHQWLDOV#RI#WKH#VSDWLDO#FRRUGLQDWHV#RI#WKH#WUDQVIRUP
|
||
|
||
( ) −G[5
|
||
|
||
−
|
||
|
||
G\5
|
||
|
||
−
|
||
|
||
G] 5
|
||
|
||
=
|
||
|
||
− −5
|
||
|
||
|
||
ω F5
|
||
|
||
|
||
|
||
\
|
||
|
||
′
|
||
|
||
G[′ GW′
|
||
|
||
−
|
||
|
||
[′
|
||
|
||
G\ ′ GW′
|
||
|
||
|
||
|
||
+
|
||
|
||
ω5 F5
|
||
|
||
[′5 + \′5
|
||
|
||
|
||
|
||
F
|
||
|
||
5
|
||
|
||
GW
|
||
|
||
′5
|
||
|
||
−
|
||
|
||
G[ ′5
|
||
|
||
−
|
||
|
||
G\ ′5
|
||
|
||
−
|
||
|
||
G] ′5
|
||
|
||
/#
|
||
|
||
|
||
|
||
#+45,
|
||
|
||
DQG#WKHQ#VROYH#WKLV#HTXDWLRQ#LQ#FRPELQDWLRQ#ZLWK#WKH#HTXDWLRQ
|
||
|
||
F5GW5 − G[5 − G\5 − G]5 = F5GW′5 − G[′5 − G\′5 − G]′5 #
|
||
|
||
#+46,
|
||
|
||
WR#ILQG#WKH#VRXJKW0IRU#WUDQVIRUP#RI#WKH#GLIIHUHQWLDO#RI#WLPH1#7KXV/#ZH#KDYH
|
||
|
||
( ) GW = GW′
|
||
|
||
4
|
||
|
||
−
|
||
|
||
5
|
||
|
||
ω F5
|
||
|
||
|
||
|
||
\′
|
||
|
||
G[′ GW ′
|
||
|
||
−
|
||
|
||
[′
|
||
|
||
G\′ GW ′
|
||
|
||
|
||
|
||
+
|
||
|
||
ω5 F5
|
||
|
||
[′5 + \′5
|
||
|
||
#
|
||
|
||
#+47,
|
||
|
||
LQVWHDG#RI# GW = GW′ /#DV#LQ#>9@1#+5HFDOO#WKDW#LQ#>9@/#VLOHQWO\#DQG#ZLWKRXW#DQ\#UHDVRQLQJ/#WKH#WUDQVIRUP#RI#WKH#GLIIHUHQWLDO#RI#WLPH LV# WDNHQ# PHUHO\# DV# GW = GW′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′5 − G[′5 − G\′5 − G]′5 = GV′5 /# ZKLFK# VKRXOG# UHVXOW# LQ
|
||
QHJDWLRQ#RI#WKH#H[LVWHQFH#RI#WKH#6DJQDF#HIIHFW#LQ#JHQHUDO#UHODWLYLW\/#DV#ZDV#WR#EH#VKRZQ1 7KXV/#WKH#NH\#IHDWXUH#RI#WKH#6DJQDF#H[SHULPHQW#LV#WKDW#WKH#WLPH#LQ#D#URWDWLQJ#IUDPH#RI#UHIHUHQFH#FDQ#EH#PHDVXUHG#DW
|
||
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
|
||
|
||
5()(5(1&(6
|
||
# 41# 61#,1#9DYLORY/#&ROOHFWLRQ#RI#:RUNV/#9RO1#7#>LQ#5XVVLDQ@/#8665#$FDG1#6FL1#3XEOLVKHUV/#0RVFRZ#+4<89,1 # 51# $1#$1#/RJXQRY#DQG#<X1#91#&KXJUHHY/#8VS1#)L]1#1DXN/#849/#46:±476#+4<;;,1 # 61# 91#,1#9\VRWVN\/#91#,1#9RURQWVRY/#51#11#.X]PLQ/#HW#DO1/#,ELG1/#497/#63<±657#+4<<7,1 # 71# *1#%1#0DO\NLQ/#,ELG1/#49:/#66:±675#+4<<:,1 # 81# 11#91#.XSU\DHY/#,]Y1#9\VVK1#8FKHE1#=DYHG1/#)L]LND/#:/#;±47#+4<<<,1 # 91# 61#,1#%\FKNRY#DQG#$1#,1#%DNDO\DU/#7KH#/DVHU#*\URVFRSH#>LQ#5XVVLDQ@/#6RY1#5DGLR/#0RVFRZ#+4<:8,1 # :1# 91#$1#)RFN/#7KH#7KHRU\#RI#6SDFH/#7LPH/#DQG#*UDYLW\#>LQ#5XVVLDQ@/#,)0//#0RVFRZ#+4<94,1
|
||
|
||
;95
|
||
|