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MQi�iBQM- � +QM+Bb2 `2@ pB2r Q7 KmHiB/BK2MbBQM�H 6Qm`B2` b2`B2b Bb �TT`QT`B�i2 U.m/@ ;2QM �M/ J2`b2`2�m- RN39c J�`Fb AA- RNNRc J�`Fb AA- kyyNVX h?2 6Qm`B2` b2`B2b ?�b i?2 7QHHQrBM; T`QT2`iB2b UJ�`Fb AA- kyyNV, Ç *QMp2`;2M+2 Bb BM i?2 K2�M B7 � T2`BQ/ Q7 i?2 T2`BQ/B+ 7mM+iBQM b�iBb}2b .B`B+?H2i +QM/BiBQMb +`Bi2`B�X Ç *QMp2`;2M+2 Bb mMB7Q`K B7 i?2 T2`BQ/B+�HHv `2THB+�i2/ F2`@ M2H Bb +QMiBMmQmbX Ç q?2M i?2`2 �`2 /Bb+QMiBMmBiB2b BM i?2 F2`M2H- i?2 6Qm`B2` b2`B2b +QMp2`;2b iQ i?2 �`Bi?K2iB+ KB/TQBMi Q7 i?2 /Bb+QM@ iBMmBivX h?2 2t�KTH2b BM i?Bb T�T2` �`2 7Q` QM2 �M/ irQ /BK2MbBQM�H T2`BQ/B+ 7mM+iBQMb �Hi?Qm;? i?2 i?2Q`v +�M #2 /2p2HQT2/ 7Q` �M �`#Bi`�`v /BK2MbBQMX hrQ /BK2MbBQM�H T2`BQ/B+Biv Bb /B+i�i2/ #v � 2$ 2 MQMbBM;mH�` T2`BQ/B+Biv K�i`Bt Q ;Bp2M �b Q = [ !)1 !)2] r?2`2 !)1 �M/ !)2 �`2 T2`BQ/B+Biv p2+iQ`bX AM QM2 /BK2MbBQM- i?2 T2`BQ/ Q7 � T2`BQ/B+ 7mM+iBQM Bb /2}M2/ #v � bBM;H2 b+�H�` r?B+? +�M #2 pB2r2/ �b � 1$1 K�i`BtX h?2 b+�H�` 2Mi`v BM i?2 K�i`Bt Bb i?2 QM2 /BK2MbBQM�H T2`BQ/- ( X AM irQ /BK2MbBQMb- � T�B` Q7 k. p2+iQ`b Bb `2[mB`2/ iQ /2}M2 T2`BQ/B+BivX AM * /BK2MbBQMb- * T2`BQ/B+Biv p2+iQ`b �`2 `2[mB`2/X 1�+? p2+iQ` Bb Q7 H2M;i? * U.m/;2QM �M/ J2`b2`2�m- RN39c J�`Fb AA- RNNRc J�`Fb AA- kyyNV- � k. 2t�KTH2 Bb b?QrM BM 6B;m`2 Rd r?2`2 K�tBK�HHv T�+F2/ +B`+H2b Q7 `�/Bmb ! ;2M2`�i2 T2`BQ/B+Biv p2+iQ`b !)1 = "####$ & 2 & 2 # 3 %&&&&' ; !)2 = "####$ "& 2 & 2 # 3 %&&&&' . r?2`2 ! Bb i?2 +B`+H2Ƕb `�/BmbX h?2 +Q``2bTQM/BM; T2`BQ/B+Biv K�i`Bt 7QHHQrb �b Q = ! 2 ( 1 "1# 3 # 3 ) . UjV 6B;m`2 Rd, >2t�;QM�H T2`BQ/B+Biv p2+iQ`b BHHmbi`�i2/ 7Q` K�tBK�HHv T�+F2/ +B`+H2bX � iBH2 BbQH�i2b � bBM;H2 T2`BQ/ Q7 i?2 T2`BQ/B+ 7mM+iBQM �M/- r?2M `2THB+�i2/ �++Q`/BM; iQ i?2 T2`BQ/B+Biv K�i`Bt- }HHb i?2 bT�+2 rBi?Qmi ;�TbX 6Q` ?2t�;QM�H T2`BQ/B+Biv- � +Q``2bTQM/@ BM; ?2t�;QM�H iBH2 Bb b?QrM BM 6B;m`2 8X 6Q` � ;Bp2M T2`BQ/B+Biv bi`m+im`2- M2Bi?2` Q Q` i?2 iBH2 b?�T2 Bb mMB[m2X h?Bb Bb BHHmb@ i`�i2/ 6B;m`2 R3 r?2`2 ?2t�;QM �M/ � T�`�HH2HQ;`�K iBH2b #Qi? ?�p2 i?2 b�K2 T2`BQ/B+Biv p2+iQ`bX � iBH2 +2Mi2`2/ �i i?2 Q`B;BM rBHH #2 `2THB+�i2/ QM i?2 ("1, "2) TH�M2 �i i?2 p2+iQ`b !)1 �M/ !)2X h?2 iBH2 rBHH �HbQ #2 `2THB+�i2/ �i �Mv BMi2;2` KmHiBTH2 Q7 i?2 T2`BQ/B+Biv p2+iQ`b- 7Q` 2t�KTH2 �i !)1 + !)2 �M/ 4!)1 " 3!)2X �Mv iBH2 `2THB+�iBQM QM i?2 ("1, "2) TH�M2 +�M #2 `2T`2b2Mi2/ #v i?2 +QK#BM�iBQM +1 !)1 + +2 !)2 r?2`2 +1 �M/ +2 �`2 BMi2;2`bX � KQ`2 +QM+Bb2 2tT`2bbBQM Bb +1 !)1 + +2 !)2 = Q !+ r?2`2 !+ Bb � irQ /BK2MbBQM�H p2+iQ` Q7 BMi2;2`bX !+ = ( +1 +2 ) . h?2 ?2t�;QM�HHv b?�T2/ iBH2 BM 6B;m`2 8 ?�b �M �`2� Q7 | det Q| = # 3 !2 2 . U9V oQHmK2 j- Abbm2 R 9X �L�GuaAa kd 6B;m`2 R3, 6Q` � ;Bp2M b2i Q7 T2`BQ/B+Biv p2+iQ`b- i?2 +?QB+2 Q7 iBH2b Bb MQi mMB[m2X h?2 T2`BQ/B+Biv p2+iQ`b BHHmbi`�i2/ �i i?2 #QiiQK Q7 6B;m`2 Rd +�M �HbQ /2b+`B#2 i?2 T�`�HH2HQ;`�K iBH2 b?QrM ?2`2X UPMHv irQ +QHmKMb Q7 i?2 T�`�HH2HQ;`�K iBH2 �`2 b?QrM ?2`2XV AM #Qi? +�b2b- i?2 �`2� Q7 i?2 iBH2- | det, |- Bb i?2 b�K2 UJ�`Fb AA- kyyNVX 6Q` � ;Bp2M T2`BQ/B+Biv bi`m+im`2 /2}M2/ #v i?2 T2`BQ/B+Biv K�@ i`Bt Q- � T2`BQ/B+ 7mM+iBQM rBi? � F2`M2H Q7 - *!" + +�M #2 r`Bii2M BM � bT�+2 rBi? +QQ`/BM�i2b !" �b UJ�`Fb AA- kyyNV . *!" + = , !' - *!" " Q !+ + U8V r?2`2 i?2 bmK Bb Qp2` i?2 b2i Q7 �HH BMi2;2` T�B`bX , !' = %, '1="% %, '2="% . LQi2 i?�i Ç i?2 F2`M2H Bb MQi +QMbi`�BM2/ iQ #2 x2`Q QmibB/2 Q7 � iBH2 �M/ +�M 2p2M 2ti2M/ Qp2` i?2 2MiB`2 !" TH�M2- �M/ Ç K�Mv F2`M2Hb +�M ;2M2`�i2 i?2 b�K2 T2`BQ/B+ 7mM+iBQM- . *!" +X 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+? 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Ukyky�VX dz"2bb2H 6mM+iBQMb Q7 i?2 6B`bi EBM/ǴX AM, qQH7`�K 6mM+iBQMb aBi2X m`H, ?iiT , f f 7mM+iBQMbXrQH7`�KX+QKf"2bb2H@ hvT26mM+iBQMbf "2bb2HCfyefykfyjfyRfX qQH7`�K _2b2�`+? Ukyky#VX dzai`mp2>ǴX AM, qQH7`�K 6mM+iBQMb aBi2X qQH7`�K�HT?�X+QK UkykyVX AMi2;`�iBQM bQHp2/ mbBM; qQH7`�K �HT?�X w?�Q- u- G 1 �iH�b- �M/ _ C J�`Fb AA URNNyVX dzh?2 mb2 Q7 +QM2@b?�T2/ F2`M2Hb 7Q` ;2M2`�HBx2/ iBK2@7`2[m2M+v `2T`2@ b2Mi�iBQMb Q7 MQMbi�iBQM�`v bB;M�HbǴX AM, A111 h`�Mb�+iBQMb QM �+QmbiB+b- aT22+? �M/ aB;M�H S`Q+2bbBM; j3- TTX Ry39Ĝ RyNRX m`H, ?iiT , f f `Q#2`iK�`Fb X Q`; f _1S_ALha f RNNy@ydnh?2lb2P7*QM2XT/7X https://commons.wikimedia.org/wiki/File:Floor_decoration_from_the_palace_of_King_Ashurbanipal.jpg https://commons.wikimedia.org/wiki/File:Floor_decoration_from_the_palace_of_King_Ashurbanipal.jpg https://commons.wikimedia.org/wiki/File:Floor_decoration_from_the_palace_of_King_Ashurbanipal.jpg https://commons.wikimedia.org/wiki/File:Flower_of_life_ephesos_square.jpg https://commons.wikimedia.org/wiki/File:Flower_of_life_ephesos_square.jpg https://commons.wikimedia.org/wiki/File:Flower_of_life_ephesos_square.jpg https://youtu.be/3pDmrWwPhpg https://commons.wikimedia.org/wiki/File:Cup_Idalion_Louvre_N3454.jpg https://commons.wikimedia.org/wiki/File:Cup_Idalion_Louvre_N3454.jpg https://commons.wikimedia.org/wiki/File:Cup_Idalion_Louvre_N3454.jpg http://youtu.be/VO-pWA1BSmg http://youtu.be/zojd8hC6d8E http://youtu.be/sT3ccZgLmnY http://youtu.be/sT3ccZgLmnY http://youtu.be/bXoP5F0s33c http://robertmarks.org/REPRINTS/1992-07_SomePropertiesOfTheGeneralized.pdf http://robertmarks.org/REPRINTS/1992-07_SomePropertiesOfTheGeneralized.pdf http://robertmarks.org/REPRINTS/1992-07_SomePropertiesOfTheGeneralized.pdf http://robertmarks.org/REPRINTS/1994-07_KernelSythesisForGeneralized.pdf http://robertmarks.org/REPRINTS/1994-07_KernelSythesisForGeneralized.pdf http://mathworld.wolfram.com/StruveFunction.html http://mathworld.wolfram.com/StruveFunction.html https://en.wikipedia.org/wiki/Heat_map https://en.wikipedia.org/wiki/Heat_map https://en.wikipedia.org/wiki/Penny_(United_States_coin) https://en.wikipedia.org/wiki/Penny_(United_States_coin) https://en.wikipedia.org/wiki/Penny_(United_States_coin) https://en.wikipedia.org/wiki/Triquetra https://en.wikipedia.org/wiki/Triquetra http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/06/02/03/01/ http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/06/02/03/01/ http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/06/02/03/01/ http://robertmarks.org/REPRINTS/1990-07_TheUseOfCone.pdf http://robertmarks.org/REPRINTS/1990-07_TheUseOfCone.pdf 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? News