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h?2 +Q``2bTQM/BM; KmHiB/BK2MbBQM�H 6Qm`B2` b2`B2b Q7 i?2 T2@
`BQ/B+ 7mM+iBQM BM U8V Bb i?2 6Qm`B2` b2`B2b UJ�`Fb AA- kyyNc
S�TQmHBb- RNd3VN

.
*!" + = | det P|

,
!'
/ (P !+) exp

-
021!"% P !+

.
UeV

NhQ b22 i?2 T2`BQ/B+Biv- +QMbB/2` b?B7i Q7 UeV 7`QK (
*!! + iQ (

-
!! " Q!)

.
r?2`2 !) Bb �M �`#Bi`�`v p2+iQ` Q7 BMi2;2`bX A7 (

-
!! " Q!)

.
= (

*!! + 7Q` �HH bm+?
b?B7ibX (

*!! + Bb T2`BQ/B+ rBi? T2`BQ/B+Biv K�i`Bt QX 6`QK UeV-

(
-
!! " Q!)

.
= | det P |

,
!"
* (P !') exp

/
"2#

-
!! " Q!)

.!
P !'

0
.

r?2`2 P �M/ Q �`2 `2H�i2/ #v �M BMp2`b2 i`�MbTQb2

P = Q"% , UdV

h?2 2[mBp�H2M+2 Q7 U8V �M/ UeV bi2Kb 7`QK i?2 6Qm`B2` /m�H
Q7 i?2 SQBbbQM bmK 7Q`KmH� UJ�`Fb AA- kyyNc S�TQmHBb- RNd3c
S�TQmHBb �M/ SBHH�B- kyykVX

,
!+
$ ( !% " P!2) = | det Q|

,
!+
# (Q!2) &"2# !$! Q!+

9XR 1tT�M/BM; E2`M2Hb

q2 �`2 �#H2 iQ MQr /2b+`B#2 i?2 2tT�M/BM; F2`M2H T2`BQ/B+
7mM+iBQM 7Q` �`#Bi`�`v T2`BQ/B+Biv K�i`Bt Q �M/ F2`M2H - *!"+X
.2}MBiBQM 9XRX h?2 2tT�M/BM; F2`M2H T2`BQ/B+ 7mM+iBQM-
#,

*!" +- ;2M2`�i2/ #v � F2`M2H - *!" + Bb

#,
*!" + = ,

!'
-

/ !" " Q !+
3

0
U3V

�b 3 BM+`2�b2b- i?2 F2`M2H 2tT�M/X

AM #Qi? i?2 2tT�M/BM; +B`+mH�` TBHH#Qt �M/ 2tT�M/BM; +B`+mH�`
+QM2 2t�KTH2b- 3! Bb i?2 `�/Bmb Q7 i?2 +B`+H2X

6`QK UeV- i?2 +Q``2bTQM/BM; 6Qm`B2` b2`B2b Q7 i?2 2tT�M/BM;
F2`M2H Bb

#,
*!" + = | det P|32

,
!'
/ (3P !+) exp

-
021!"% P !+

.
. UNV

h?2 Qp2`H�TTBM; 2tT�M/BM; F2`M2Hb +�M #2 /2i`2M/2/ #v 2p�H@
m�iBM; i?2 K2�M p�Hm2 Q7 i?2 T2`BQ/B+ 7mM+iBQMX h?2 K2�M
p�Hm2 BM � 6Qm`B2` b2`B2b 2tT�MbBQM Bb i?2 x2`Qi? Q`/2` 6Qm`B2`
b2`B2b +Q2{+B2MiX h?Bb +�M #2 2p�Hm�iBM; #v BMi2;`�iBM; Qp2`
� bBM;H2 iBH2 7QHHQr2/ #v /BpBbBQM #v i?2 �`2� Q7 i?2 iBH2X

h?2 i?2Q`2K iQ 7QHHQr mb2b i?2 �`#Bi`�`BM2bb Q7 i?2 +?QB+2 Q7
iBH2b r?2M BMi2;`�iBM;X AM QM2 /BK2MbBQM- i?2 T2`BQ/ Q7 � T2@
`BQ/B+ 7mM+iBQM- b�v ( - Bb �`#Bi`�`vX q2 +�M +?QQb2 i?2 T2`BQ/
iQ #2 QM i?2 BMi2`p�H 0 & " < ( Q` "(/2 & " < (/2. h?2`2 Bb �
h?2 2tTQM2MiB�H i2`K ?2`2 #2+QK2b

exp
/
-2#

-
!! " Q!)

.!
P !'

0
= exp

-
"2#!!! P !'

.
exp

/
""2#

-
Q!)

.!
P !'

0
.

lbBM; UdV-

exp
-
""2#

-
!)! Q! P !'

..
= exp

-
""2#

-
!)! !'

..
= 1

bQ i?�i (
-
!! " Q!)

.
= (

*!! +X



k3 hBHBM; 1|Q`2b+2M+2 Q7 1tT�M/BM; E2`M2Hb BM � 6Bt2/ S2`BQ/B+ �``�v, :2M2`�HBxBM; i?2 6HQr2`@P7@GB72

bBKBH�` �`#Bi`�`BM2bb BM i?2 +?QB+2 Q7 � iBH2X hQ BHHmbi`�i2- +QM@
bB/2` � k. 7mM+iBQM rBi? ?2t�;QM�H `2THB+�iBQM �+`Qbb � TH�M2X
*?QQbBM; � iBH2 BM i?Bb 7mM+iBQM Bb HBF2 +?QQbBM; � +QQFB2 +mii2`X
�b BHHmbi`�i2/ BM 6B;m`2 R3- � ?2t�;QM�H +QQFB2 +mii2` +�M #2
mb2/X "mi � ?2t�;QM Bb MQi i?2 QMHv TQbbB#H2 iBH2X �b b?QrM BM
i?2 };m`2- � +QQFB2 +mii2` b?�T2/ HBF2 � T�`�HH2HQ;`�K +�M �HbQ
#2 mb2/ �b +�M �Mv +QQFB2 +mii2` i?�i b�iBb}2b i?2 T2`BQ/B+Biv
+QMbi`�BMibX

LQi�iBQM�HHv- BMi2;`�iBM; Qp2` �`2� C BM i?2 7QHHQrBM; i?2Q`2K
K2�Mb BMi2;`�iBQM Qp2` �Mv bBM;H2 iBH2X 6Q` �//BiBQM�H /2i�BHb-
b22 J�`Fb UJ�`Fb AA- kyyNVX

h?2Q`2K 9XkX h?2 K2�M p�Hm2 Q7 i?2 2tT�M/BM; F2`M2H 7mM+@
iBQM- #,

*!" +- Bb

'#,( := 1
| det Q|

!
!! )C

#,
*!" + '!"

= 32 | det P|
!
!!
-

*!" + '!" URyV

= 32 | det P|/ (!0) URRV

h?2 `2;BQM C Bb �Mv `2;BQM BM i?2 !" bT�+2 +Qp2`BM; � iBH2X

�b 3 BM+`2�b2b- i?2 /2i`2M/2/ bmK Q7 i?2 2tT�M/BM; F2`M2Hb BM
i?2 !" TH�M2 �TT`Q�+?2b � rBi?Qmi �Mv BMi2`2biBM; bi`m+im`2X

S`QQ7X h?2 2tT`2bbBQM BM URyV 7Q` i?2 K2�M Q7 #,
*!" + 7QHHQrb

7`QK i?2 !+ = !0 i2`K BM i?2 6Qm`B2` b2`B2b BM UNVX 1[m�iBQM
URRV 7QHHQrb 7`QK i?2 BMi2;`�H T`QT2`iv UkVX !

q2 +�M MQr /2}M2 i?2 /2i`2M/2/ T2`BQ/B+ 7mM+iBQMX

.2}MBiBQM 9XjX h?2 T2`BQ/B+ 2|Q`2b+2Mi 7mM+iBQM- 4,
*!" +- Bb

i?2 /2i`2M/2/ 2tT�M/BM; F2`M2H 7mM+iBQM- BX2X #,
*!" + KBMmb Bib

K2�MX
4,

*!" + = #, *!" + " '#,( .

h?2 +Q``2bTQM/BM; 6Qm`B2` b2`B2b Q7 4,
*!" + Bb bBKTHv i?2 6Qm`B2`

b2`B2b Q7 #,
*!" + BM UNV rBi? i?2 x2`Qi? Q`/2` 6Qm`B2` b2`B2b

+Q2{+B2Mi !+ = !0 i2`K `2KQp2/XRy

4,
*!" + = | det P|32

,
!'!!0

/ (3P !+) exp
-
021!"% P !+

.
. URkV

6`QK i?Bb 2tT`2bbBQM- r2 b22 i?2 6Qm`B2` b2`B2b +Q2{+B2Mib 7Q`
i?2 2|Q`2b+2Mi 7mM+iBQM �`2

5, [ !+] =
1

| det P|32/ (3P !+) ; !+ ! !0
0 ; !+ ! !0 URjV

Ry"v !0- r2 K2�M � p2+iQ` r?Qb2 QMHv 2H2K2Mib �`2 x2`QX

9Xk *QMp2`;2M+2

h?2 T2`BQ/B+ 2|Q`2b+2Mi 7mM+iBQM +�M KQbi BMi2`2biBM;Hv ;2M@
2`�i2 � M2p2` `2T2�iBM; T�ii2`M Q7 7�b+BM�iBM; b?�T2bX h?Bb
/Q2b MQi ?�TT2M r?2M- �b � 7mM+iBQM Q7 2tT�MbBQM- i?2 2|Q@
`2b+2Mi 7mM+iBQM `2�+?2b � HBKBi +v+H2 Q` }t2/ TQBMiX q2 MQr
2t�KBM2 r?2M i?Bb ?�TT2MbX

�bvKTiQiB+ *QMp2`;2M+2 Q7 i?2 1|Q`2b+2Mi 6mM+iBQM
iQ w2`Q

q2 }`bi 2bi�#HBb? r?2M i?2 2|Q`2b+2Mi 7mM+iBQM 2tT�M/BM;
F2`M2H �TT`Q�+?2b x2`QX

6B`bi /2}M2 i?2 E`QM2+F2` /2Hi� �b

6[ !+] :=
1

1 ; !+ = !0
0 ; !+ ! !0

h?2Q`2K 9X9X am{+B2Mi +QM/BiBQM 7Q` +QMp2`;BM; iQ i?2
K2�MX G2i * = 2XRR A7

3./ (3P !+) "*
,*%

3./ (!0)6[ !+] UR9V

i?2M #,
*!" + +QMp2`;2b iQ Bib K2�MX

#,
*!" + "*

,*%
'#,( .

�b � +QMb2[m2M+2
4,

*!" + "*
,*%

0.

AM Qi?2` rQ`/b- �b i?2 F2`M2Hb +QMiBMm2 iQ 2tT�M/- i?2 2|Q`2b@
+2Mi 7mM+iBQM �TT`Q�+?2b i?2 p2`v mMBMi2`2biBM; `2bmHi Q7 x2`Q
Qp2` i?2 2MiB`2 ("1, "2) TH�M2X

S`QQ7X �b 3 * %- �HH i2`Kb 3./ (3P !+) BM UNV i2M/ iQ x2`Q
r?2M UR9V Bb i`m2 2t+2Ti r?2M !+ = !0X h?2M UNV #2+QK2b

#,
*!" + "*

,*%
3. | det P|/ (!0) = '#,(

!

� bm{+B2Mi bKQQi?M2bb +`Bi2`BQM *QMp2`;2M+2 Q7 �M 2t@
T�M/BM; 2|Q`2b+2Mi 7mM+iBQM iQ x2`Q Bb �bbm`2/ r?2M i?2 F2`M2H
�/?2`2b iQ bKQQi?M2bb �M/ BMi2;`�#BHBiv T`QT2`iB2bX

RRh?2 i?2Q`2Kb ;Bp2M �`2 �TTHB+�#H2 BM �Mv /BK2MbBQM . X q2 ?�p2
+QM+2Mi`�i2/ QM k. bQ rBHH b2i . = 2 iQ �pQB/ +QM7mbBQMX h?Bb �HbQ
�TTHB2b iQ h?2Q`2K 9X8X



oQHmK2 j- Abbm2 R

9X �L�GuaAa kN

h?2Q`2K 9X8X *QMp2`;2M+2 iQ i?2 K2�M, h?2 7QHHQrBM;
i?2Q`2K �TTHB2b iQ * /BK2MbBQMbX 6Q` i?2 2t�KTH2b ?2`2BM-
* = 2X h?2 * /BK2MbBQM�H 7mM+iBQM- #,

*!" +- +QMp2`;2b iQ Bib
K2�M B7 Bib F2`M2H- - *!" +- Q#2vb i?2 7QHHQrBM; T`QT2`ivXRk

!
!!

22222
3
.4
)=1

7"#

7""#)

5
-

*!" +
22222 '!" = 8 < % UR8V

r?2`2 i?2 MQMM2;�iBp2 BMi2;2`b {0) |1 & 9 & *} Q#2v

.,
)=1

0) > * UReV

h?2 +?QB+2 Q7 !0 = [01 02 03 · · · 0. ]% Bb �`#Bi`�`v bQ HQM; �b
UR8V �M/ UReV �`2 b�iBb}2/X

h?2 T`QQ7 Bb ;Bp2M BM �TT2M/Bt eXRX

*Q`QHH�`v 9XeX 6Q` * = 1- h?2Q`2K 9X8 b�vb i?2 2|Q`2b+2Mi
7mM+iBQM rBHH +QMp2`;2 iQ x2`Q B7

!
!

2222 '
2

'"2
-(")

2222 '" = 8 < %

1t�KTH2 9XdX *QMbB/2` i?2 irQ /BK2MbBQM�H F2`M2H

-
*!" + = "("1)&"!

2
2 ,

r?2`2 i?2 `2+i�M;H2 7mM+iBQM Bb

"(") :=
1

1 ; |" | & 1
2

0 ; |" | > 1
2 .

URdV

h?2M UqQH7`�K�HT?�X+QK- kykyV
!
!!

22222
73

7"32
-

*!" +
22222 '!" =

! %

!1="%
"("1)'"1

! %

!2="%

22222
'3

'"32
&"!

2
2

22222 '"2
=

= 4
-
1 + 4&"3/2

.
= 8 < %

�M/ 01 + 02 = 0 + 3 > * = 2X h?2 +`Bi2`B� BM h?2Q`2K 9X8 �`2
K2i �M/ �bvKTiQiB+ +QMp2`;2M+2 Q7 i?2 2|Q`2b+2Mi 7mM+iBQM
iQ x2`Q Bb �bbm`2/X

�bvKTiQiB+ *QMp2`;2M+2 hQ � 6Bt2/ 6mM+iBQM

�b i?2 7QHHQrBM; 2t�KTH2 b?Qrb- i?2 2|Q`2b+2Mi 7mM+iBQM +�M
+QMp2`;2 iQ � }t2/ 7mM+iBQM Q7 !" �b b+�HBM; BM+`2�b2bX

1t�KTH2 9X3X *QMbB/2` i?2 QM2 /BK2MbBQM�H F2`M2H

-(") = &"!:(") UR3V

6B;m`2 RN, lbBM; i?2 2tTQM2MiB�H /2+�v F2`M2H BM UR3V
`2bmHib BM i?2 4, (")Ƕb b?QrMX �b 3 * %- i?2 7mM+iBQM
�TT`Q�+?2b i?2 b�riQQi? b?QrM #v i?2 i?B+F `2/ HBM2
;Bp2M #v URNVX U6Q` i?Bb THQi- ( = 275XV

r?2`2 :(") Bb i?2 >2�pBbB/2 bi2T 7mM+iBQMXRj h?2M 4,
*!" + +QM@

p2`;2b BM bi2�/v bi�i2 iQ i?2 bQHB/ `2/ b�riQQi? r�p27Q`K
b?QrM BM 6B;m`2 RNX

lim
,*%

4, (") =
1
2 " "

(
; 0 < " < ( URNV

� T`QQ7 Bb ;Bp2M BM �TT2M/Bt eXkX

�bvKTiQiB+ GBKBi *v+H2 S2`BQ/B+Biv BM 3

1|Q`2b+2Mi 7mM+iBQMb +�M #2 �bvKTiQiB+�HHv T2`BQ/B+ �b � 7mM+@
iBQM Q7 i?2 2tT�MbBQM b+�HBM; p�`B�#H2 3X

h?2Q`2K 9XNX lbBM; i?2 b+�H�` T2`BQ/B+Biv K�i`Bt Q = ( - i?2
QM2 /BK2MbBQM�H F2`M2H

-(") = " (")

`2bmHib BM � 2|Q`2b+2Mi 7mM+iBQM T2`BQ/B+ BM 3 rBi? � T2`BQ/ Q7
(, = 2( X

S`QQ7X h?2 QM2 /BK2MbBQM�H 6Qm`B2` b2`B2b 2tT`2bbBQM 7Q` i?2
Rk6Q` . = 5 �M/ !" = [4 0 1]! - 7Q` 2t�KTH2-3

$4
#=1

/%#

/!
%#
#

5
0

*!! + = /5

/!4
1 /!3

0
*!! + .

Rj1[m�H iQ QM2 7Q` TQbBiBp2 �`;mK2Mi �M/ x2`Q Qi?2`rBb2X



jy hBHBM; 1|Q`2b+2M+2 Q7 1tT�M/BM; E2`M2Hb BM � 6Bt2/ S2`BQ/B+ �``�v, :2M2`�HBxBM; i?2 6HQr2`@P7@GB72

2|Q`2b+2Mi 7mM+iBQM Bb

4, (") =
3

(

,
'!0

sinc
-3+
(

.
&"2#'!/% . UkyV

r?2`2
sinc(%) := sin(1%)

1%

Bb i?2 QM2 /BK2MbBQM�H 6Qm`B2` i`�Mb7Q`K Q7 "(")X 6Q` + ! 0-
i?2 +i? 6Qm`B2` b2`B2b +Q2{+B2Mi Bb

5, [+] = 3/
-3+
(

.
=

3

(
sinc

-3+
(

.
=

=
1
(

sin
* #',

%

+
1+

"2+�mb2 Q7 i?2 bBM i2`K- 5, [+] Bb T2`BQ/B+ rBi? `2bT2+i iQ 3
rBi? T2`BQ/ 2( X

5,+2% [+] = 5, [+]
aBM+2 �HH Q7 i?2 6Qm`B2` b2`B2b +Q2{+B2Mib BM UkyV �`2 T2`BQ/B+
rBi? T2`BQ/ 2( - r2 +QM+Hm/2 i?�i

4,+2% (") = 4, (").

�M/ i?2 2|Q`2b+2Mi 7mM+iBQM Qb+BHH�i2b �b � 7mM+iBQM Q7 3X !

h?2 T2`BQ/B+Biv Q7 4, (") Bb BHHmbi`�i2/ BM 6B;m`2 kyX h?2 7mM+@
iBQM Bb #QmM/2/ #v |4, (") | & 1 �M/ Bb THQii2/ Qp2` � bBM;H2
T2`BQ/X h?2 6B;m`2 #2;BMb rBi? �M �HH x2`Q 7mM+iBQM K�`F2/
rBi? i?2 MmK#2` 0X 3 BM+`2�b2b � #BiX 6Q` i?2 7mM+iBQM K�`F2/
1- i?2`2 Bb MQr � b?Q`i TQbBiBp2 TmHb2 �M/ i?2 `2K�BM/2` Q7 i?2
7mM+iBQM Bb x2`QX aBM+2 '4,( = 0- �HH Q7 i?2 7mM+iBQMb b?QrM
?�p2 x2`Q �`2�X h?2 TmHb2 �i i?2 Q`B;BM #2;BMb iQ bT`2�/ �b
3 BM+`2�b2b �b Bb b22M BM i?2 7mM+iBQMb K�`F2/ k i?`Qm;? dX
h?2M- �i 3- i?2 7mM+iBQM `2im`Mb iQ #2BM; B/2MiB+�HHv x2`QX h?2
b2+QM/ T?�b2 Bb b?QrM BM 6B;m`2 ky#X q2 #2;BM rBi? i?2 x2`Q
7mM+iBQM K�`F2/ 3 BM 6B;m`2 ky� r?B+? Bb �HbQ K�`F2/ 3 BM 6B;@
m`2 ky#X 3 BM+`2�b2b � #BiX h?2 7mM+iBQM K�`F2/ N Bb � b?Q`i
M2;�iBp2 TmHb2X �b 3 BM+`2�b2b- i?2 M2;�iBp2 TmHb2 rB/2Mb �b
Bb b22M BM i?2 7mM+iBQMb K�`F2/ N i?`Qm;? R8X 6mM+iBQM Re Bb
B/2MiB+�HHv x2`Q �M/ Bb i?2 b�K2 �b i?2 7mM+iBQM K�`F2/ y BM
6B;m`2 ky�X PM2 T2`BQ/ Bb +QKTH2i2 �M/- �b 3 BM+`2�b2b- i?2
M2ti B/2MiB+�H T2`BQ/ #2;BMbX

9Xj 1|Q`2b+2Mi 1t�KTH2b

h?2 KQbi BMi2`2biBM; Q7 2|Q`2b+2Mi 7mM+iBQMb �`2 i?Qb2 i?�i
?�p2 MQ }t2/ �bvKTiQiB+ +QMp2`;2M+2 T`QT2`iB2bX AM irQ /B@
K2MbBQMb- bm+? 2|Q`2b+2Mi 7mM+iBQMb +QMiBMm�HHv #HQQK BM �
MQM`2T2�iBM; K�MM2`X h?2 `2�/2` Bb 2M+Qm`�;2/ iQ pB2r i?2
QMHBM2 pB/2Qb ULv#�H- kyR9/c Lv#�H- kyR9+c Lv#�H- kyR9#c Lv@
#�H- kyR9�V U2bT2+B�HHv i?2 2tT�M/BM; TBHH#Qt +QM2 �M/ +B`+mH�`

U�V h?2 }`bi T�`i Q7 i?2 T2`BQ/B+Biv BM ,X

U#V h?2 b2+QM/ T�`iX

6B;m`2 ky, h?2 T2`BQ/B+Biv BM 3 Q7 4, (") 7Q` 2tT�M/BM;
`2+i�M;mH�` F2`M2Hb BM UkyVX h?2 T2`BQ/ BM 3 Bb 2( X h?2
7mM+iBQM 4, (") Bb �HbQ T2`BQ/B+ BM " rBi? T2`BQ/ ( X



oQHmK2 j- Abbm2 R

9X �L�GuaAa jR

+QM2 pB/2QbV iQ 7mHHv �TT`2+B�i2 i?Bb 2K2`;2M+2X a+`22M b?Qib
7Q` 2tT�M/BM; +B`+H2 �M/ +QM2 b?�T2/ F2`M2Hb �`2 b?QrM BM 6B;@
m`2b j- R9 �M/ R8X q2 MQr �M�Hvx2 T`QT2`iB2b Q7 i?2b2 irQ
F2`M2H ivT2bX

q2 }M/ mb27mH i?2 `�/B�H K2�bm`2b

; = +!"+ �M/ < = + !%+

r?2`2
+!"+ =

6
"21 + "22 . UkRV

q?2M i?2 p�`B�#H2 ; = ; ("1, "2) Bb mb2/ BM � irQ /BK2MbBQM�H
2tT`2bbBQM- r2 �bbmK2 Bi iQ #2 � irQ /BK2MbBQM�H 7mM+iBQM Q7
"1 �M/ "2 �b 2tT`2bb2/ ?2`2X .QBM; bQ �pQB/b r`BiBM; Qmi i?2
b[m�`2 `QQi 2tT`2bbBQM BM UkRV �i 2�+? mb�;2X h?2 p�`B�#H2
< = <(%1, %2) +�M bBKBH�`Hv #2 BMi2`T`2i2/ �b � 7mM+iBQM Q7 %1
�M/ %2X

1tT�M/BM; SBHH#Qt *B`+H2 1t�KTH2

h?2 2tT�M/BM; +B`+H2b 2t�KTH2 Bb � bT2+B�H +�b2 Q7 U3V 7Q` * =
2 /BK2MbBQMb r?2`2 i?2 F2`M2H -(!") Bb QM2 BMbB/2 � +B`+H2 Q7 mMBi
`�/Bmb �M/ Bb Qi?2`rBb2 x2`QXR9

-(!") = "
- ;
2
.
. UkkV

h?2M / (!0) Bb bBKTHv i?2 �`2� Q7 � mMBi `�/Bmb +B`+H2X 6`QK i?2
bB;M�H BMi2;`�H T`QT2`iv BM UkVX

/ (!0) =
!
!!
"

- ;
2
.
'!" = 1.

6Q` 2tT�M/BM; +B`+H2b- i?2 2tT�M/BM; F2`M2H 7mM+iBQM Bb

#,
*!" + = ,

!'
"

/ +!" " Q !++
23

0

bQ i?�i -
-
!!"Q !'
,

.
BM i?2 !+i? iBH2 Bb � +B`+H2 Q7 `�/Bmb 3 +2Mi2`2/

�i Q !+X

h?2 k. 6Qm`B2` i`�Mb7Q`K Q7 � mMBi `�/Bmb +B`+H2 BM UkkV Bb
UJ�`Fb AA- kyyNV

/ ( !%) = =1 (21<)
2<

r?2`2 =1 (·) Bb � }`bi Q`/2` "2bb2H 7mM+iBQM Q7 i?2 }`bi FBM/X
�bvKTiQiB+�HHv U�#`�KQrBix �M/ ai2;mM- RNdkV

=1 (21<)
2< "*

1*%
<"3/2

21 cos
/
21< " 3

41
0

UkjV

R9aT2+B}+�HHv- 7`QK i?2 /2}MBiBQM Q7 "( ·) BM URdV- 0 (!! ) = " (2/2) Bb QM2
7Q` &

2 < 1
2 X h?Bb Bb 2[mBp�H2Mi iQ 2 = +!! + =

6
!2
1 + !2

2 < 1 r?B+? /2}M2b �
+B`+H2 Q7 mMBi `�/BmbX

bQ i?�i- 7Q` !+ ! !0-

32/ (3P !+) = 32 =1 (213+P !++)
23+P !++

"*
,*%

1
21

7
3

+P !++3 cos
/
213+P !++ " 3

41
0

h?2 bm{+B2Mi +QM/BiBQM Q7 h?2Q`2K 9X9 7Q` +QMp2`;2M+2 Q7 i?2
2|Q`2b+2Mi 7mM+iBQM iQ x2`Q Bb i?2`27Q`2 MQi K2iX

SBHH#Qt 1tT�MbBQM QM � >2t�;QM�H :`B/ q2 ?�p2 v2i iQ
bT2+B7v � T2`BQ/B+Biv 7Q` i?2 2tT�M/BM; +B`+H2bX �bbmK2 i?2
+B`+H2 +2Mi2`b �`2 bT�+2/ ?2t�;QM�HHv BM �++Q`/�M+2 rBi? i?2
T2`BQ/B+Biv K�i`Bt BM UjVX 6`QK U9V r2 b22 i?�i

| det P| = 1
| det Q| =

2#
3!2

. Uk9V

h?2 2tT�M/BM; F2`M2H 7mM+iBQMǶb K2�M- 7`QK URyV- Bb i?2`27Q`2

'#,( = 32
/ 2#

3!2

0
1 =

2132
#

3!2

a+`22M b?Qib 7Q` i?2 2|Q`2b+2Mi 7mM+iBQM Bb b?QrM BM 6B;m`2 j
7`QK i?2 pB/2Q �p�BH�#H2 QMHBM2 ULv#�H- kyR9/VX

1tT�M/BM; *B`+mH�` *QM2b

a2i i?2 irQ /BK2MbBQM�H F2`M2H iQ � +B`+mH�` +QM2 Q7 mMBi ?2B;?iX

-
*!" + = (1 " ;)"

- ;
2
.

6Q` +B`+mH�`Hv bvKK2i`B+ 7mM+iBQMb- i?2 k. 6Qm`B2` i`�Mb7Q`K
#2+QK2b i?2 >�MF2H i`�Mb7Q`K UJ�`Fb AA- kyyNV bQ i?�i UqQH@
7`�K�HT?�X+QK- kykyV #2+QK2b

/ ( !%) = 21
! 1

0
; (1 " ;)=0 (21;<)'; Uk8V

=
>0 (21<)=1 (21<) " >1 (21<)=0 (21<)

4<2 .

r?2`2 >+ (·) �`2 ai`mp2 7mM+iBQMb Uq2Bbbi2BM- kykyV �M/ =+ (·)
�`2 "2bb2H 7mM+iBQMb Q7 i?2 }`bi FBM/X AM �TT2M/Bt eXj- r2
b?Qr i?�i

32/ (3P !+) "*
,*%

" 2
12

8
33

+P !++ cos
-
213+P !++ " 1

4
.
. UkeV

�b Bb i?2 +�b2 rBi? i?2 2tT�M/BM; +B`+H2b- i?2 +QM/BiBQM Q7
h?2Q`2K 9X9 7Q` +QMp2`;2M+2 iQ i?2 K2�M 7Q` i?2 2tT�M/BM;
+QM2b Bb i?2`27Q`2 MQi K2iX 6Q` �Mv }t2/ P �M/ !+ ! !0- i?2
HBKBi /Q2b MQi �TT`Q�+? x2`Q �b 3 BM+`2�b2b rBi?Qmi #QmM/X



jk hBHBM; 1|Q`2b+2M+2 Q7 1tT�M/BM; E2`M2Hb BM � 6Bt2/ S2`BQ/B+ �``�v, :2M2`�HBxBM; i?2 6HQr2`@P7@GB72

*B`+mH�` *QM2 1tT�MbBQM QM � >2t�;QM�H :`B/ �bbmK2
i?2 +QM2 +2Mi2`b �`2 bT�+2/ ?2t�;QM�HHv BM �++Q`/�M+2 rBi?
i?2 T2`BQ/B+Biv K�i`Bt BM UjVX h?mb r2 +�M mb2 Uk9VX aBM+2
=0 (0) = 1- i?2 pQHmK2 Q7 � +QM2 rBi? � mMBi +B`+H2 #�b2 �M/
mMBi ?2B;?i Bb- 7`QK Uk8V-

/ (!0) = 21
! 1

0
; (1 " ;)'; = 1

3 .

h?2 2tT�M/BM; F2`M2H 7mM+iBQMǶb K2�M- 7`QK URyV- Bb i?2M

'#,( = 32
/ 2#

3!2

0
1

3 =
2
#

3132

!2 .

�b r�b i?2 +�b2 7Q` i?2 +B`+mH�` TBHH#Qt- i?2 HBKBi /Q2b MQi
�TT`Q�+? x2`Q �b 3 BM+`2�b2b rBi?Qmi #QmM/X a+`22M b?Qib 7Q`
i?2 +QM2Ƕb 2|Q`2b+2Mi 7mM+iBQM �`2 b?QrM BM 6B;m`2b R9 �M/
R8 7`QK i?2 pB/2Q �p�BH�#H2 QMHBM2 ULv#�H- kyR9+VX

8 *QM+HmbBQMb

q2 ?�p2 BMi`Q/m+2/ i?2 B/2� Q7 T2`BQ/B+ 2tT�M/BM; F2`M2H �M/
2|Q`2b+2Mi 7mM+iBQMb �M/ ?�p2 b?QrM i?2v +�M /BbTH�v rB/2Hv
p�`B�Mi #2?�pBQ`b /2T2M/2Mi QM i?2 F2`M2H �M/ i?2 mM/2`Hv@
BM; T2`BQ/B+BivX 1t�KTH2b �`2 ;Bp2M Q7 2|Q`2b+2Mi 7mM+iBQMb
i?�i +QMp2`;2 iQ x2`Q- +QMp2`;2 iQ � MQM+QMbi�Mi }t2/ TQBMi
�M/ Qb+BHH�i2X q?2M i?2 2|Q`2b+2Mi 7mM+iBQMb ~m+im�i2 rBi?@
Qmi `2T2�iBM;- T�ii2`Mb `2KBMBb+2Mi Q7 +QMiBMm�H #HQQKBM; +�M
2K2`;2X aT2+B�H Q++m``2M+2b 7Q` � +B`+mH�` TBHH#Qt F27Q` Q7 i?2
i?`22 T2i�H ;2QK2i`v `2T`2b2MiBM; *?`BbiB�MBivǶb i`BMBiv �M/ i?2
~Qr2`@Q7@HB72X �HH 2K2`;2Mi T�ii2`Mb �`2 T2`BQ/B+ �M/ +�M #2
mb2/ 7Q` �`i7mH iBHBM;X

e �TT2M/B+2b

eXR S`QQ7 Q7 h?2Q`2K 9X8, *QMp2`;2M+2 iQ
i?2 K2�M

h?2 /2`Bp�iBp2 i?2Q`2K Q7 6Qm`B2` �M�HvbBb BM/B+�i2b
3
.4
)=1

7"#

7""#)

5
-

*!" +

?�b � 6Qm`B2` i`�Mb7Q`K Q7
3
.4
)=1

( ?21%) )"#
5
/ ( !%)

h?mb

/ ( !%) =

9
!!

(/:.
)=1

/%#

/!
%#
#

0
-

*!" +
)
&""2#!!

! !$'!"
:.

)=1 ( ?21%) )"#
.

�M/

|/ ( !%) | =

22229!!
(/:.

)=1
/%#

/!
%#
#

0
-

*!" +
)
&" -2#!!! !$'!"

2222
(21).̃ :.

)=1 |% |
"#
)

=

&

9
!!

2222
/:.

)=1
/%#

/!
%#
#

0
-

*!" +
2222 '!"

(21).̃ :.
)=1 |% |

"#
)

=

=
8

(21).̃ :.
)=1 |% |

"#
)

r?2`2

*̃ =
.,
)=1

0) . UkdV

*QMiBMmBM;

3. |/ (3 !%) | & 83.

(213).̃ :.
)=1 |% |

"#
)

�M/
3. |/ (3P !+) | & 83.".̃

(21).̃ :.
)=1 | (P)) !+ |"#

.

qBi? �HH Qi?2` T�`�K2i2`b }t2/- i?Bb 2tT`2bbBQM i2M/b iQ x2`Q
7Q` BM+`2�bBM; 3 r?2M

* " *̃ < 0

Q`- mbBM; UkdV-
.,
)=1

0) > *

eXk S`QQ7 Q7 h?2Q`2K 9X3, *QMp2`;2M+2 iQ �
}t2/ T2`BQ/B+ 7mM+iBQM

�TTHvBM; i?2 2tTQM2MiB�H F2`M2H BM UR3V iQ U3V 7Q` b+�H�` Q = (
;Bp2b

#, (") =
%,

'="%
exp

/
" " " +(

3

0
:(" " +()

= &"!/,
%,

'="%
&'% /,:(" " +()



oQHmK2 j- Abbm2 R

_161_1L*1a jj

Pp2` i?2 T2`BQ/ 0 & " < ( -

#, (") = &"!/,
0,

'="%
&'% /, .

lbBM; � ;2QK2i`B+ b2`B2b

#, (") =
&"!/,

1 " &"% /,

�M/
4, (") = #, (") "

3

(
.

1tT`2bb i?2 2tTQM2MiB�Hb �b � i`mM+�i2/ h�vHQ` b2`B2bX

4, (") =
1 " !

, + !2

2,2

%
, " % 2

2,2

" 3

(
.

�7i2` bQK2 K�MBTmH�iBQM

4, (") "*
,*%

3
*
" " %

2
+
+ !2

2
(

*
3 " %

2
+

"*
,*%

1
2 " "

(

r?B+? Bb i?2 /2bB`2/ `2bmHi BM URNVX

eXj S`QQ7 Q7 UkeV, *QM2 +QMp2`;2M+2

h?2 ai`mp2 7mM+iBQMb mb2/ BM UkeV +�M #2 /2}M2/ #v i?2B`
h�vHQ` b2`B2b Uq2Bbbi2BM- kykyV

>0 (.) =
2
1

%,
)=0

("1)) .2)+1

[(29 + 1)!!]2

�M/
>1 (.) =

2
1

%,
)=0

("1))+1.2)

(29 " 1)!!(29 + 1)!! .

h?2v ?�p2 i?2 7QHHQrBM; �bvKTiQiB+ #2?�pBQ` UqQH7`�K _2@
b2�`+?- kyky#V

>0 (.) "*
|( |*%

7
2
1.

sin
-
. " 1

4
. /

1 +@
/ 1
.2

00

�M/
>1 (.) "*

|( |*%

2
1

/
1 +@

/ 1
.2

00
.

GBF2rBb2- i?2 "2bb2H 7mM+iBQM ?�b i?2 �bvKTiQiB+ #2?�pBQ`
UqQH7`�K _2b2�`+?- kyky�V

=0 (.) "*
|( |*%

7
2
1.

cos
-
. " 1

4
. /

1 +@
/ 1
.2

00

�M/
=1 (.) "*

|( |*%

7
2
1.

cos
/
. " 31

4

0 /
1 +@

/ 1
.2

00

AM HB;?i Q7 i?2b2 #2?�pBQ`b- BMbT2+iBQM Q7 UkeV `2p2�Hb i?2
>1 (.)=0 (.) i2`K �bvKTiQiB+�HHv /QKBM�i2b i?2 >0 (.)=1 (.)
i2`K �M/

>1 (21<)=0 (21<) "*
1*%

2
12 <

" 1
2 cos

-
21< " 1

4
.
.

h?mb

32/ (3 !%) "*
,*%

" 2
12

8
33

<
cos

-
213< " 1

4
.

7`QK r?B+? UkeV 7QHHQrbX

_272`2M+2b

�#`�KQrBix- J �M/ A � ai2;mM- 2/bX URNdkVX >�M/#QQF Q7
J�i?2K�iB+�H 6mM+iBQMb, qBi? 6Q`KmH�b- :`�T?b- �M/
J�i?2K�iB+�H h�#H2bX oQHX 88X *Qm`B2` .Qp2` Sm#HB+�iBQMbX

�/�K�MiBQb UkyRjVX �p�BH�#H2 mM/2` i?2 *`2�iBp2 *QK@
KQMb �ii`B#miBQM@a?�`2 �HBF2 jXy lMTQ`i2/ HB+2Mb2X m`H,
?iiTb , f f +QKKQMb X rBFBK2/B� X Q`; f rBFB f 6BH2 ,
*?BM� @ #2BDBM; @ 7Q`#B//2M @ +Biv @ SRyyyR8d @
/2i�BHXDT;X

/� oBM+B- G2QM�`/Q UR9d3VX m`H, ?iiTb , f f +QKKQMb X
rBFBK2/B�XQ`;frBFBf6BH2,G2QM�`/Qn/�noBM+Bn@
n*Q/2tn�iH�MiB+mbn7QHBQnjydpXDT;X

.m/;2QM- . 1 �M/ _ J J2`b2`2�m URN39VX JmHiB/BK2MbBQM�H
.B;Bi�H aB;M�H S`Q+2bbBM;X oQHX RX S`2MiB+2@>�HH aB;M�H S`Q@
+2bbBM; a2`B2bX 1M;H2rQQ/ *HBzb, S`2MiB+2@>�HHX

62H/F�KT- G �- G * .�pBb- �M/ C q E`2bb URN39VX ǳS`�+iB+�H
+QM2@#2�K �H;Q`Bi?KǴX AM, CPa� e- TTX eRkĜeRNX

6BHBTQpB+- . a- C G oQH�FBb- �M/ G a �M/2`b2M URNNNVX ǳ1{+B2Mi
KQ/2HBM; �M/ �M�HvbBb Q7 BM}MBi2 T2`BQ/B+ �Mi2MM� �``�vb
#v i2i`�?2/`�H }MBi2 2H2K2MibǴX AM, �Mi2MM�b �M/ S`QT@
�;�iBQM aQ+B2iv AMi2`M�iBQM�H avKTQbBmKX oQHX 9X A111-
TTX k8y9Ĝk8ydX

6HQr2`b- _�v UkyyeVX �p�BH�#H2 mM/2` i?2 *`2�iBp2 *QK@
KQMb �ii`B#miBQM@a?�`2 �HBF2 jXy lMTQ`i2/ HB+2Mb2X m`H,
?iiTb , f f +QKKQMb X rBFBK2/B� X Q`; f rBFB f 6BH2 ,
h2KTH2@Q7@PbB`Bbn6HQr2`@Q7@GB72nykXDT;X

:Qmbb2iBb- :- � S 62`2bB/Bb- �M/ C * o�`/�tQ;HQm UkyyeVX
ǳh�BHQ`BM; i?2 �J* �M/ 1": +?�`�+i2`BbiB+b Q7 T2`BQ/B+
K2i�HHB+ �``�vb T`BMi2/ QM ;`QmM/2/ /B2H2+i`B+ bm#bi`�i2ǴX
AM, A111 h`�Mb�+iBQMb QM �Mi2MM�b �M/ S`QT�;�iBQM 89XR-
TTX 3kĜ3NX

>BHH- J P �M/ > : :�m+? URN3yVX ǳ.2i`2M/2/ +Q``2bTQM/2M+2
�M�HvbBb, �M BKT`Qp2/ Q`/BM�iBQM i2+?MB[m2ǴX AM, *H�bbB}@
+�iBQM �M/ P`/BM�iBQM- TTX 9dĜ83X

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	About This Journal
	The Purpose of the Journal
	Paper Submission Policies
	Other Journal Content

	From the Editors
	Sam S Rakover and Baruch CahlonSam S Rakover and Baruch CahlonWhen is Explanation Transitive? A Methodological Note
	Introduction
	Explanatory-transitivity
	Discussion
	Acknowledgments

	Robert J. Marks IIRobert J. Marks IITiling Efflorescence of Expanding Kernels in a Fixed Periodic Array: Generalizing the Flower-Of-Life
	Introduction
	Other Expanding Kernels 
	Properties of Expanding Kernels of Varying Periodicity
	 Analysis
	Conclusions
	Appendices

	Letters and Notes
	Eric HollowayDeciding a Bitstring of 1s is Non-Random is Impossible in General
	Jonathan BartlettProving the Derivative of sin(x) Using the Pythagorean Theorem and the Unit Circle
	J R MillerA Response to Clunn's Axioms of Morality
	Jorge FernandezIs Information Content a Single, Static Quantity?

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