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Page 1: C:/Documents and Settings/Zoltán/Asztal/blaweb.cs.elte.hu/blobs/diplomamunkak/mat/2008/nagy_zoltan... · 2008. 6. 16. · 8 ! 9#p+, 3qj ij 2 e(g) r q*(/,j y*~j x i + x j a',@ )-

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Page 2: C:/Documents and Settings/Zoltán/Asztal/blaweb.cs.elte.hu/blobs/diplomamunkak/mat/2008/nagy_zoltan... · 2008. 6. 16. · 8 ! 9#p+, 3qj ij 2 e(g) r q*(/,j y*~j x i + x j a',@ )-

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Page 3: C:/Documents and Settings/Zoltán/Asztal/blaweb.cs.elte.hu/blobs/diplomamunkak/mat/2008/nagy_zoltan... · 2008. 6. 16. · 8 ! 9#p+, 3qj ij 2 e(g) r q*(/,j y*~j x i + x j a',@ )-

: )� �çQZD)-���ÎZD� +����A'E��) ZD� )2/,B&J� Y*X��>A@&',J� Y*�C O�@&"I*(@&�! �"n+,�

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ex(n,R) n2 è£/��x"� �B9å��(��B9è>A/,"�=QêÅ[�äf�9>(@&�zB�>!�9ST@&#I>( � �',@�"ÌR� �"L W*!+0*!/,�Ñ8��(V�O��9"F 9"�"IåÌZD*~�9����C²*E3�J�@�B�åex(n,R) = o(n2)

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Page 4: C:/Documents and Settings/Zoltán/Asztal/blaweb.cs.elte.hu/blobs/diplomamunkak/mat/2008/nagy_zoltan... · 2008. 6. 16. · 8 ! 9#p+, 3qj ij 2 e(g) r q*(/,j y*~j x i + x j a',@ )-

JQ@&B�å2/,B�ånè»8,�AU�8,�AUÅ3� È*!VQ�N �æA=�@&"��(��B&@�#Ì#� ���>(/�"�=�/���#�/,'�1ÍB&>!�YST"� �#���/�B� �����R�R2.Y �B�å-�D/,B9è

ST/��÷æA/�R�RHJ���"Iåj+���/���/�J�/�*,[��@&��B�@&'E 9*(V�"�#IR� �"ï P#�%�.&/0*!#�/,'�1L.Y���D*(@&'E Y*X�9*<.�ZD',�AB���� æAV�#y jO�>A@&R���+�)-��"� �#4[

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EÅù§¦�ù ú±û�üý4þ�ÿ=�G�G∗ 0:."459X."45�Z2,���5464 # �à%����R%���9X<=�67�^���� # <�2�75Y��H�:%Â�T�676�67�)?�,2�7��/%e.

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ä-Ù¸ÆG = Cn

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�A/,B�CD*!�A+,B&+0.&/,�ð[ö:<'�*������DCÎ*TæAV�#43xJ�@&B9å *(/,� æA/,�(íQ��"�Z�ST@&Bxi < xi+1 1 ≤ i < n

è£>A/�[öóm'i = 1

è£>A/�*(/,�÷æA/��(í��Å)-ZÎ.&/,�(1 − x1) > xn(1 − x1) = d > (1 − x2)

9'1[Í+,�È 9'

n[Í/�B�å&/,"Q��/�*7)2ZN Y*(*E3�*!/�J��9*

(1 − x1) > (1 − x2) �)2ZD*ÈZ�B� �'�@&�D"�Z� 9#Y �>«*!V�"�#4[ µ "�=�V�#�8�Z"C�.9 ��M#Y �OÌæAV�# �'j�9��CÎ*X���«*)2ZD"�=�/,"

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(i − 1)[¨+��

i[m/,B�å&/�"���/0*!/0*�%&�A�(',/0.&/�*(.&/

xi+1

xi= (1−xi−1)

(1−xi)

[ ò�ómR�R�1&�Ñ#�%�.&/�*!#�/�',Z�#434J�@&B9å0 < x1 < ... < xn < 1

[�� 2.ÌZ�>«*!V��9��ZD�

0�(U��Îå�UyO�@&"�*!@&#� 9*<ZD��)2/�B&/,"�B�/,=�"�+�"�#43Å �#I#�@&>< -�D/�.&/�',/0*!+,�Aí�"�#ÌR�/,"6 �'

/�B�å&/�"���1&�A+,BLZD�±)2Z�"�=Q/,"6O�@&"I*(@&"ï)-/�B&/,"QB&/,=�/0*(*,3f 9)-ZDR�1&�|.ÌZD�(',@�"I*7�N�9*(�(',ZD#F �'2�9����C²*X�9�)-���A@Ì=�Z�# ST/,�D/�3ÖJ�Z��A',/,"îJ� ö�(UQ�Då&@&# #�%&'�%�*(*-.9 �"

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x1, ..., xi

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#Ií��D%&"IR�%&',10è»�(U��Îå�UHO�@&"�*!@�*z*X 9�N���D"��9"�#43� �#I#�@&>� �'±/,B�å&/�"���19*!��/�"��(+�B&����"�8±B� �>( �"�*X���D"��

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.Y �B�å(1−xn) = (1−xn−1) = 0

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/�����/�"I*()-@�"�=����«*#� �O�"��9"�#4[��/��D*(/,J�/0*�æAí�#43ÌJ�@�B�å

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/��(/0*!R�/,"xi = (1−xn+1−i)

�AU��Då&@&',���«*�.9���� ��('0*(.9 Q3� �'~/,>A/,=�/0*!ZÎ.&/,��Z�',@�)-@&>«S��AU��Då&@�',@�*A*B�>!�9ST@9*z#Y 9O�"���"�#43� 9J�@&�Ñ ÍST/��D*(+�*!/���í�"Q#�)��9>Ö*(/,�÷æA/��(í��ð[

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�ì���� ��(�A',V�"�#H/�B�å(ε1, ε2, ..., εn) ε

è»�(/�>(/,B�/�*E3� �)-/��Dåj*(/,� æA/,�(CÎ*!Z� -#�%�.&/0*!#�/�',1jST/��D*!+0*!/���/0è#9/�*?j

• εn

εn−1= xn

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ε3= x4

(1−x3)

• ε3

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• ε2

ε1= x2−ε2

(1−x1)

[óm�D/,B&/�"�=�1Ì/�"Ò#IZD8,�(ZD"�/,#F.9�9�N ��A'�*(.Y

εn

è *E3�/,�D+,>(JQ/�*!1Q3|J�@&B�å(xi − εi) > 0

*!/��÷æA/��(í�� æA%&")2ZD"�=�/,"

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��+��('�CD*!�Aí�#ë/,�±)2@&�«*L �'6UYæx′

i = xi − εi

�(U��Îå&@&'E�9�A*E[ã�ìZD',�AB���� æAV�#é)2/,B J�@�B�å� �".Y���D*(@&',"� �#P �'±/�B�å&/,�~+,�D�(ê�>Aê��(+�B&/,#4[

• X1

+,�X2

#�%&',%9*(*1−d = (1−x1)x2 = (1−x′

1)x′2

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• i+,�

i + 1#�%&'�%�*(*

1 < i < n/,�A/�*(+,"

1− d = (1− xi)xi+1 > (1− xi)xi+1 +

εixi+1− εi+1(1−xi)− εiεi+1 = (1−xi)xi+1− εiεi+1 = (1−x′i)x

′i+1

3� �', �' �'±+����Aê�>(ê��A+,B-"Q1�[

• Xn

+,�X1

#�%&'�%�*(*�#�/����|8,�! 9#yB� �>( �"I*!����"QZð3�J�@�B�åy"�/�) 8��(%&#I#�/�" �'W+����Aê�>(ê��A+,B�3+,�z/,���D/,"�*!)2@&"�=��9�A*Æ#Y 9O�V�"�#4[

:�'±#�/,�D�Å*(/,J��9*,3�J�@&B9åd = xn(1 − x1) ≤ x′

n(1 − x′1) = (xn − εn)(1 − x1 + ε1)

34 �', �'((1 − x1) + ε1)εn ≤ xnε1

[:��N 9#ÌCÎ*!�AV�#P�9*z/,'0*< Í#IZDST/¶æA/�',+��A*?j εn

ε1≤ (xn−εn)

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µ *A* R� ��D@&��=� ��Î*-O�@�"I*!@��! �"ëZD�()2/,>TæAíQ#cjï �'εè»@&#; �>(��"Iå��9>! ö.&@&"� Y*!#�@&'+C /�B�å&/,"Q��/�*(/0è

#9/�*x%��(�('�/,�A',@&>(@�'�.9 � �R� ���@&�D=� 9�Î*#Y �OÌæAV�#4[J�� �B�åÌZD�f ~#�%�.&/0*!#�/�',1�*m/���+�B<R�ZD',@�"IåÌCD*! �"ÌV�"Q#cj(x2 − ε2)x3...xn

(1 − x1)(1 − x2)...(1 − xn−1)≤

(xn − εn)

(1 − x1)

��/��D*(/�*(*(í�#43�JQ@&B�å x3...xn−1

(1−x2)...(1−xn−1)≤ 1

x2

3�/�',+,>«*�/,�D+,B�J� �R�/��N�9*TæAV�#43�J�@&B�å(x2−ε2)xn

x2≤ (xn − εn) ⇔ x2εn ≤ xnε2

[óm'FO�/,=�ZDBÒ�(',ZD"I*(+,"Ô ST/��D*!+0*!/���R�1&�~#�%�.&/�*!#9/,',ZD#�3�J�ZD�(',/�"c 9'

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%��(�('�/,�A',@&>(',���!��.9 9�εn

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=xnxn−1...x3

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1

x2

/�#�.ÌZD.9 ��D/,"Q�� �'�'E �� 9)-ZÎ*ÆZDB� �'�@&��"QZÅ �#� �>A*(V�"�#4[:é�(UQ�Då&@&',���x)>CÌ=Q@&�(CÎ*X���(��.9 9�I*!/,J��9*m@&�Îå� �"W#�@�"��A*(>(V�#�8�Z"C�*m#Y 9OQ*!V�"Q#�3I �JQ@&�Q �'Ö+����Aê�>(ê��A+0èB�/,#6"�/,) /,B�å&/�"���1Ì/�#43mJ�ZD�('�/,"

x1xn

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+,�D/,#6#IZD.&+0*!/,�D+�.&/��Ö 9'�+����Aê�>(ê��A+,BF"�%�.�/0è#9/,=�/0*(*E3m/-#�+�*W+,�D/," O�/,=�ZDBy"�/,) 8��(%&#I#�/�"I*E[ óm'�/�����/�"I*()-@&"Q= B [N�H#�%�.&/�*(#�/,',)2+�"Iå)2Z� 9*(*Æ �#IZDZ�"�=�VQ�N���AZ�ST/,�Î*!/�.&+��(í�"Q#Ì"Q/,#�34*(/,J��9*~ �'7���D��C²*X�9�ÖZ�B� 9'�[kÅù§¦�ù¨ú±û�üý4þ�ÿ

d(Pn+1) = d(Cn)

¤ .�7 # $�{�/%��52 ALð O�*!Z�)-���ÎZD�W#�@&"Q�A*!>AV�#�8,ZFC&RcC&�ÆZ�"Q=�V���VQ"�# )2Z�"�=�#9+�*�Z�>(��"IåÌR� �[ �� P ∗n+1/�B�å

nJ�@&�A�(',UFUQ*("� �# )2/�B�ST/,�D/,��1�@�OQ*!ZD)����ÎZD��3 B [ \ ��/,)2)- Í�(',/�>(ZD"I*!Z#�@&"��«*!>AV�#�8,ZFC�3� 9#�è

#9@&>Í �'�UQ*W#�+�*W.&+,B�O�@&"I*Tæ(�9*Í/�B�åÌ)��9�(�! 9�M%&�(�A',/�>! �B� 9�('�*A.9 Cn

#�%&>A>(/PZD�h#Y �O�VQ"�# /�B�å#9@&"��A*(>(V�#�8�Z"C�*,3x 9)-ZDR�/,"ö )-ZD"�ZD)-�9�DZÎ�±+����Aê�>(ê��A+,By+�O�O�/,"

d(Pn+1)3|+��7ST/,�('�CD*(/�*(*

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:�'�*�#Y �OQ*(V�#W*(/,J��9*,3ÌJQ@&B�å� ~ [ \ #�@�"��A*!>AV�#�8,ZFC&RcC&�4 ±��/�)-)- ì>! 9B� ��('0*X�9�(Z�*!>(íQ#Ì#,æA/� �=Yæ( Cn

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Cn

B&>(�9ST>! Q3� �'�'E ��Ñ ÍST/,�Î*!+0*!/,�D��/��ð3�J�@&B�åx1.9 �B9å

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ñjÙ¸Æ òvÈVÕ¦àQÕ+ò>ó â áfË,âQÑ�È�Ñ âÑáfÏ ÜfÏ7ßfÎIò�à Í ÜfÏ�ôÌαàÖÓGÈJÉÈßfÓ7âQÎò�õIÈJõIò=È|Ü áÆâ¸óPàÖÎIòvÈVÎ

:�'z/,�D1&',1�ST/£æA/,',/0*�*!+0*!/���/�Z�"Iå�@&)��9"2#IZ��A'E��)2@&�DJ� 9*TæAV�#h �#�%&>(%�#Ì>A/~+���VQ*! �#I>! 7 �#I>(ZD*(Z�#IV��+����Aê�>(ê��A+,B�/�*E[î: ~ [ ¶ *(+�*!/��~+�>A*(/,��)2+,R�/�";)-@��A*X 9"I*�C��Æ/,�D+,BïVQ*X 9#Ì#� ��ÆST@&B&�N 9�D#�@&',"IV�"�#43ST/��D*(/,J�/0*�æAí�#-*!/,J��9*~J�@&B9å

G = Pn+1

9'(n + 1)

è£8,�AU�8,�AUÅ3nè£+���ê UQ*,[ ñ n > 2

òGHZ�VQ*!��"6 ~ [ \ #�@&"��«*!>(VQ#�8,ZFC&>�C&�x ~ [ ~ *!+0*!/,�DR�/,"6R�/��N�9*A*!V�#43ÅJ�@&B9åy@&OQ*!ZD)��9�DZÎ�ì P)2/,B9èST/���/���1 �AU��Då&@&',���(�( �� 3�/�',+,>«*W �(U��Îå&@&# �(/�B&CÎ*!�(+�B&+�.&/��

d(Cn) = d(Pn+1)è»>(/�+,�D�(',��)-"�åÌZ

/�B�å&/�"���/0*!/�*hST/,��CD>A.9 H#IZDST/£æA/,'�',í�# y#I>(ZÎ*!ZD#ÌVQ�h+����(êQ>(ê��(+�B&/0*E[���/���%&� æAí�#6 �'�/,B�åÌ�A',/,>Aê��(+,B#9/,=Q.&+�+,>A*~/,'0*~ �'ì+����Aê�>(ê��A+,B&/0*

dn

è»"�/,�ð[���/,�DJ� ��A',"����Î.9 hJ�@&B9åx1 = (1− xn+1) = 1

3� #9%�.&/�*(#�/,',1-/,B�å&/�"���/0*!>(/�"�=��A',/,>(J�/,'ÆæAVQ*(V�"�#]j(1 − dn) Ú x2

(1 − dn) Ú (1 − xi)xi+1

J� i = 2, ..., (n − 1)

(1 − dn) Ú (1 − xn).

�m/,',/��(�(í�#ÔR�/ï �'Fi(d) = xi

STí�B�B�.&+,"�åI*E3� �)-/��Dåî d+����Aê�>(ê��A+,B STí�B�B�.&+,"�å&+,R�/�"

)2/�B� �=Yæ( h �'xi

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F1(d) = 1öF2(d) = (1 − d)öFi(d) = 1−d

1−Fi−1(d)

J� 2 < i ≤ n

[½~B�å� �"� �#I#�@&>� 9'±VQ*!@�����C�/�B�å&/�"���/0*< 9�N �OÌæ(��"

Fn(d) = d ñ ö ò03*(/,J��Y*dn

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sù�ø�ù�ú±û�üý4þ�ÿ � 7 Fn(d) = d���H0:. # c��9X."2¹%TY5�R%±)R\w�6��S�8:$��*:d59f4�� !$# %e%=�����P�:$9"�:%�9"��� ]�²��� # *6* !$# 7�./%'�/S����PY64Z��9"�,2�7a� dn

45�R./%'."45�P2�8:9N2:�H�R�P2?8�� A¤ .�7 # $�{�/%��52 A : R�ZD',@�"IåÌCD*!���A*Æ*!%&RQRL�D+,O�+,�AR�/,"L.&+�B&/,'�',í�#4[Æóm��1&�A',%&>zR�/��N�Y*�æAV�#43�J�@&B9å �D/,B�"� �B�å&@&RQR B�å&%�#ö

(0, 1)Z�"�*!/�>A.9 9���DV�)WR� j/,�AZ�#43x)- �æA=ö)-/�B&)WVQ*! 9*�æAV�#FJ�@&B9åï8,�( �#

/�B�å�@&�Då� 9"d)-/�B&@&�D=����M�D/,J�/0*E3� �)2/,�DåÌ>A/

xi ∈ (0, 1))2Z�"�=�/�"

iè»>(/9[��ì+�B&í��ÅR�/��N�9*TæAV�#43

JQ@&B�åP/,'7 d�D/,�A'7 ��D/,B&"� 9B�å&@&R�R )2/,B&@&�D=����Æ W.9 ��FC&�ÆB�å&%&#�%&#P#�%�',%�*A*E[

sùFEÅùm×xý]�G�M��ÿd ≥ 1 ⇒ Fn(d) 6= d A

¤ .�7 # $�{�/%��52 A �< d > 13Ì �#I#�@&>

F2(d) = x2 < 0 ±)��9�(@Ì=�ZD#h/�B�å&/,"��D/�*�)2Z� 9*(*,3&Z�"�"Q/,"

ZD"�=�VQ#�8,Z"C�.9 ��Fi(d) = 1−d

1−Fi−1(d)< 0

[�ã�/,J��Y*Fi(d) < 0 < 1 < d

3Æ/,R�R�/�"c 9'/��(/0*!R�/,"F 9'7���D��C²*X�9�ÖZ�B� 9'�[��

d = 13Å 9#I#�@&>

F2(d) = 03�Z�"�"Q/,"FZ�"�=QV�#�8,ZFC�.9 9�

Fn(d) = 03Š�)2Z�/,�D��/�"I*!)2@&"�=y �'

Fn(d) = d/,B�å&/�"���/0*!"�/�#4[

sùFkÅù�×xý]�G�M�MÿVl����P�:%'9"�: # 9X�w�� d ∈ (0, 1)<W��� # 9�gZ�52��GS5��b�67M�����P�:$9"�:%'1�,46�J��<W�:9X� ���

xi ∈ (0, 1) ∀i ∈ {1, 2, ..., n} A¤ .�7 # $�{�/%��52 A :<' /���1&'�1 ST/¶æA/�',/0*-#�@&)WRQZ�"� 9*(@&>(ZD#IV��WUQ*(@&" ZDB� �'�@&�÷æA Q3mJQ@&B�å #I>(ZD*(Z�#IV��+����Aê�>(ê��A+,B�>(/6 ST/��D*(+�*(/,�~ST/,"�"��9���ð[Õã|/�B�åÌí�#;ST/,�ð3zJ�@�B�åî���D��C²*X�9�(V�"Q#Ì#� ����('�/,)WR�/,"Ô#�+0*ZD�Då&/�"

d)2/,B&@���=��9�ÆZD�Æ��+0*!/,'�"�/�j

d(1) < d(2) [m:<'±/�B�å&/,"��D/�*(/,#IR�1&�� �=`CÌ=�C� �" /,#I#�@&>F2(d

(1)) = (1 − d(1)) > (1 − d(2)) = F2(d(2))

3�)- YæA=jZ�"�"�/�"HZ�"Q=�V�#�8,ZFC�.Y ��Fi(d

(1)) = (1−d(1))

1−Fi−1(d(1))>

(1−d(2))

1−Fi−1(d(2))= Fi(d

(2)) ñ i ≤ nò£è£>A/�[Ø��Z�J� ��A',"��9�D*(V�#

#9%&',R�/�"Å3±J�@�B�åîST/,�D*(+�*(/,��/�',+��(í�"�#ë�('�/,>(ZD"I*xi ∈ (0, 1)

3� �', �'ï)2Z�"�=Q/,"c"�/0.&/,'�1îO�@Yè'�ZD*(CD.�[

(i = n)è£>(/P ���#Y 9��)- 9'�.Y

Fn(d(1)) > Fn(d(2))è *Í#� �OÌæAV�#43� �)2Z�/,���D/,"�*!)2@&"�=

d(1) < d(2) è£"Q/,#�[ ã�/,J��Y*-8,�( �# dn

>(/,"Q=�/,�D#�/,'�Z�# ��/�)-)- �9�D*X 9�M)-/�B&#ICD.9�9"I*Í*!V��� �æ¶è=Q@&"��!�9B&B� ��ð[sùtsùm×xý]�G�M��ÿ

dn

�67Fn(d) = d

�����P�:$9"�R%�S5��9"_62L���PY64Z�:.J4ZY�7�\H9�>9"����c�6��� # *6* A

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¤ .�7 # $�{�/%��52 A :<'�* ST@&BEæAV�# ZDB� �'�@&��"�Z�3PJ�@&B�ådn < d < 1

�('�Z�"�*!+," *!/��÷æA/,�ACD*(Zj �'xi ∈ (0, 1)∀i ∈ {1, 2, ..., n}

ST/,�D*(+�*(/,��/�#�/�*,3�/,'�+,>«*Æ"�/�)?��/�J�/�*Ö 9'</���1&'�1h��/�)-)- ±)2ZN Y*(*dn

è»"�+,�Å"� �B�å&@�R�RHB�å&%&#4[ µ �()2+�*� 9'±/,B�å&/�"��D/�*!/�#�/�*�J� ��('�"���� æAV�#]j0 < F2(d) = (1 − d) < (1 − dn) = F2(dn) < 1

3�*!@�.9��RQR��2 �'i[¨/,B9å&/,"��D/�*(R�1&�

0 < Fi(d) = (1−d)1−Fi−1(d)

<(1−dn)

1−Fi−1(dn)= Fi(dn) < 1

3QST/,�DJ� ��('�"����Î.9 Fi−1(d) < Fi−1(dn)

è»/�*zZ�"Q=�V�#�8,ZFC�.Y �� [móm','�/,�� �'7�9���DCÎ*!���(VQ"�#Y 9*ÆZ�B� 9',@&�Î*!VQ#�[:ÔJ���>(@�) �D/,)2)� ì/,B9å�)-����V�*X��"�Z4 ���#� ���)- 9'E�9�!��RcC&�� ±*!+0*!/������D��C²*X�9�! �#�%�.&/�*(#�/,',ZD#�[

sùF�Åù�÷hþÌþ�øNüHùJæIÿd(n−1) < dn

A¤ .�7 # $�{�/%��52 A : B [N|-*(+�*(/,��R�1��|#�%�.&/�*(#�/,',ZD#F �'W���D��C²*X�9�,3�J�ZD�('�/,"

Pn

>(+,�A',B�>!�9S æ( P(n+1)

è"Q/,#4[ómB9å�U�*(*X ��

3.7*!+�*(/,�Ñ)2ZN 9*A*z Í#I>(ZD*(Z�#IV��Æ+,�D�(ê�>Aê��(+�B-#�%&>A%&#I>(/±Z��Æ"�1Q[

óm'�/,"P 7O�@&"�*!@&"2.ÌZD�(�(', �VQ*! ���VQ"�#� 7RQZ�',@�"IåÌCD*!���¨/�B�å�)-/�B&/,�D1&�D/,B&'�/�*A*��D+,O�+��(+�>(/�jdn > 1

2J� ñ n > 2ò03�J�Z��A',/,"H)-��>

n = 3è£>! WZ��

d3 =√

5−12

[��ZD�(',��)-@��D.Y -#IZ��

nè£/�#Ì>A/Í P#�@&"Q�A*!>AV�#�8,ZFCH�9�D*X 9��R�ZD',@&"�åÌCD*!@9*(*</,B9å&/,"��D/�*!/�#�/�*,3� �#�%�.&/0*Aè

#9/,',1-+����Aê�>(ê��A+,B&ZÑ�(',��)2@&#Y 9*z#Y �OÌæAV�#]jd3 =

√5−12

≈ 0.618 ú d2 + d − 1 = 0 û'ü�ýTûoþ�ÿ û ����� ÿ �d4 = 2

3≈ 0.667 ú (1 − d) = (2d − 1) → 3d − 2 = 0 ûoü�ýTûoþ�ÿ û ����� ÿ �

d5 ≈ 0.692 ú (1 − d)(2d − 1) = d(d2 + d − 1) → d3 + 3d2 − 4d + 1 = 0 ûoü�ýTûoþ�ÿ û ����� ÿ �d6 = 1√

2≈ 0.707 ú (1 − d)(d2 + d − 1) = d(3d2 − 2d) → 4d3 − 2d2 − 2d + 1 = 0 ûoü�ýTûoþ�ÿ û ����� ÿ �

d7 ≈ 0.717 ú (1 − d)(3d − 2) = (d3 + 3d2 − 4d + 1) → d3 + 6d2 − 9d + 3 = 0 ûoü�ýTûoþ�ÿ û ����� ÿ �d8 = 5+

√5

10≈ 0.724

[:

dn

�A@&>(@�'E 9*�#�@�>(���9*!@&��+,��)2@&"�@�*(@&"F"Q1�3Å/�',+�>A*±�D+�*!/�',ZD#HJ� 9*X�9>(+,>«*!+�#�/�3Ñ)2/,�Îå&/�*ì)2/,BZD�Ö*!V�=QV�"�#jJ� 9*X��>A@&',"QZ [

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sùF Åù¨ú±û�üý4þ�ÿdn < 3

4

8,2dn → 3

4A

¤ .�7 # $�{�/%��52 A :<'h���D��C²*X�9�z/����A12ST/,�D/7#�%�.&/0*!#�/�',Z�#H B [ ~ *!+�*(/,�DR�1�� 34 dn

�A',ZDB&@&>(U� �"L)2@9è"Q@�*!@�" "�%�.&/�#�/,=�+��!/ )WVQ*X 9*Tæ( Q3xJQ@&B�åÒ L*!+�*(/,�DR�/," "Q/,) �����DJ� 9*È/,B�å�/,"��D1&�(+�B6�(/�)L[6:#9@&"I.&/�>(B&/�"�8,Z���>�C��m�('�C��"CH���D��CÎ*!���«*±�9*AST@&B� ��D)- 9',J� 9*�æAV�#L #�%�.&/�*(#�/,',1&#�+,O�O�/,"�jÈ)2Z�"�=Q/,"d < 3

4

+,�D�(ê�>Aê��(+�B&ZÑ�('E�9)->( ��D+�*!/�',ZD#�@&�Îå� �"n3�)2/,�ÎåÌ>(/

d ≤ dn

[�ì���� ��(�A',V�"�#ÔV�B�å& �"�Z��H/,B�åÔ*(/�*(�(',1���/,B�/,�

d < 34

�(',��)-@9*E[ ��/,'�=YæAí�#ÔST/���+�O�CD*(/,"�Z± ~ [ \ #�@&"��A*(>(V�#�8�Z"C9*m �'�/�"�"�/,#È

d�A'E��)2"� �#±)2/,B9ST/,��/���1z�(U��Îå&@&'E�9�(�! 9� 39*(/,J��Y*

Fi(d) = xi�D/,B9å&/,"-=�/:��"�Z����D.Y 7 ±#�@&>(��R�R�Z�>(/,#IV�>A',ZFCÈ�('�/,>AZ�"�*E[x: ~ [N|±�D/,)2)��9R� �"ÍZ�B& �',@��D*(V�#43�J�@&B�å <>A/,#IV�>A',CD.È/,B9å&/,"��D/�*!/�#ÈR�Z�'0*!@&�ACD*�æ(�9#

Fi(d))-@�"�@�*!@�"h"�%�.&/�#�/,=�+��!+�*,3Ì �)2C�B

Fi(d) < 1[

�<Z�'��(B��9�÷æAV�#Í)-/�B�3��D/,J�/0*!�(+�B&/��Ö.&@&�D"� Yè£/93�JQ@&B�åFi(d) < 1 ∀i

è£>A/�*(/,�÷æA/��(í�� æA%&"Å[f:�)-@Yè"Q@�*!@�"L"�%�.&/,#�/�=�+,�(Z�*!VQ�N �æA=�@&"��(��B2)2Z� 9*(*Æ/,#I#�@&>~��+0*!/,'�Z�#P/�B�å

0 < h < 1J� Y*X��>(+�>A*(+,#�/

9'Fi(d)

�(@&>A@&'E Y*!"� �#43�)2/,�Îå�>A/ìST/,"�"������Ñ Í>(/�#IV�>('�CD.j/,B�å�/,"��D/�*!/�Z�"�#j 9�N �OÌæ(��"Å3�J�@&B�å(1 − d) = (1 − h)h

[}�) .&/,B9å�íQ#;+,�A',>A/�3zJ�@&B9å ST/,�Î*!+0*!/,�D/,'�+,�(íQ"�# �('�/,>(ZD"I*

(1 − d) > 14≥ (1 − h)h

�A'E�9)-CÎ*X �"�Z²è»)2+�>A*X 9"�ZÑ#�%&',/�O�/,#L#�%&'�*(Z|/,B�å&/�"���19*!��/�"��(+�BP)-Z� 9*(*�r4*(/,J��9*</,'W �'h/��(/�*ì"�/,)�9���DJ� 9*2ST/�"�"Å[ DÑ+0*!/,'�Z�# *(/,J��Y*j/,B9å

N�(',��)H3~)2/,�ÎåÌ>(/

i ≤ N/��(/�*(+,"

Fi(d) < 1*(/,� æA/,�Aí�� 3¨ �',@�"ÌR� 9"FN+1 = (1−d)

(1−FN )≥ 1

[öã�/,#IZ�"�*!�Aí�#Ò/,#I#�@&>2 3.1

#�@&"��A*(>(V�#�8�Z"C9*PN+1

è»>(/�3Ì �J�@&����/,B9å&/,"�/�#2 �'xi

�(U��Îå&@&#� ±ST/,"I*��(',��)2CD*!@9*(*Fi(d)

+,>«*!+�#Ì#�/���/,B�å&/�"���1Ì/�#J�

1 < i < (N + 1)3¨�D/,B�å&/�" *(@�.9�9R�R��

x1 = 1+,�

(1 − xN+1) = 1[öóm#I#�@&>�

�AU��Då�@&'E���h+�>A.&+�"Iå&/,�,3mJ�Z��A',/�" )2Z�"Q=�/,"ö�(U��Îå(0, 1]

è»R�/�/��(ZD#43x+,�xi

=�/+��"QC�8,ZFCYæ( H)-Z� 9*(*)2ZD"�=�/,"î+���/�"Ô �'F+����Aê�>(ê��A+,B O�@�"I*(@&�! �"

d3~#IZD.&+0.&/

N(N + 1)+���/�"Å3~ZÎ*(*P �'�@&"ÌR� �"

de = xN(1 − xN+1) = FN ≥ d �'

N=�/:��"�C�8�Z"CYæ( P)2ZN Y*(*E[7ã�/,J��9*

d ≤ dN

3 �)2ZD*ZDB� �'�@&�D"�ZÅ 9#Y �>A*(V�"�#4[

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:2.ST/£æA/,'�/�*!R�/�"ÔKM@&"�=Qå�3Ki�J�/�"Å3zãÆJQ@&)� 9�(�(+L+,��ãÆJ�@�)� 9�(�(/�"ë/�>(/�=�)2+,"Iå&+0*jZD=�+,'0.&/

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K3

B&>!�9ST@9*FST/,�('�CD*(1+��y"�/,) ST/,�('�CD*(1éJ���>(@�)-O� �>A*(C�8�ZFC&� B&>!�YST@&#I>�C&�ð[ �±+��(1&RQR

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K2 #,! %'./<=��9X."2�4 # �2:%e�R�Z4Z0:.�_T@+�9F�,^]�:%V�I4Z8:%�0,2:&]0,2:&=\����,2������)��045�:./%'."45�Z2�8:9N2:�H�R�P268��6�H�:9 A �Q�����Z�:1�,4��=0,2:&$0,2 # 2�7�%��w9°� # 4

X1, X2, ..., Xn

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• xxn ∈ E(K∗n)�∀x ∈ Xi \ {xi} ,∀i ∈ {1, 2, ..., (n − 1)}),

• xix ∈ E(K∗n)�∀i < n, x ∈ Xj, i 6= j < n

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• X1 \ {x1}�X2 \ {x2}

� A/A/A �Xn−1 \ {xn−1} # 2�7�%���9X� # 45��(<W���52�7 # �R�/%'S5�¹9"�����P�:�67b8:9N2:%'�R�P45%'&���¢�Kn−1

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~ [���R�>! �[Kn

#9@&"��A*(>(V�#�8�Z"CYæ(

X2 \ {x2}3Î[D[Î[D3

Xn−1 \ {xn−1}J� ���)- �'�@&#IR� ;/,�(ZD#43W��) é>(/�#ÌVQ>(',ZFCë)2ZN 9*A*L/,' �'

(n − 1)8��(U�8���"�/,)ÛST/��(',CÎ*!ZÑ

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d2(n)(1 − d(n−1)) + d(n) − 1 = 0

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9#I#�@&>Æ B [ õ ��/,)2)- 7�A',/,>AZ�"�*Æ)2Z�"Q=�/,"�+,�D�(ê�>Aê��(+�BÍ/,B�å�/,"��D1�3Q/�)-Z� 9*(*w(xi) = 1− d(n)

(i < n)/,�A/�*(R�/," +,�

w(xn) = 1[U��/���CD>A.9 7/,B9å

Xn

è»>(/<"Q/,)$ZD���Î/,�('�#�/,=�1W+���J�/�'<*! �>A*!@�'�C+����Aê�>(ê��A+,B�/�*E34#� �OÌæAV�#j ÍST/�"I*!ZÅ/,B�å&/�"���/0*!/0*E[

�ÅùFkÅù�� ý�����ý��J®�¬Iû]æÌÿm�G�b�=26�o@R%�8,2:\��4v.��Z�67:�m�5464 # �d(3) =

√5−12

≈ 0.61803 ��7�%m.��Z�67 # 9X%'�Z4=."2 Ad(4) ≈ 0.77222

d(5) ≈ 0.83948� # S5�Z*6*,�W�=4 # �2:%'�R�P4Z0:.�_v<�.'��%'%��=45�R./%'."45�P2�8:9N2:�H�R�P268��b�K)?�:$%e.��,2?�:%T�,4�*:�:¢1�:< 9"�,^$�:%45."2?��*6*:�U<�./$%��W���,45�H�7��/SW�����Z�:$9"�:%T�,4v<W���)?�:9"�:9"d�<W��� # 9�gZ�52�� A:$#�@&"��A*(>(V�#�8�Z"C�*W"�+�)-ZD��/�By���D*! �����"Q@&�! �RQRöST@&>A)��9R� �"öZ��ÈJ� ��A',"�@&�ACD*(J� 9*�æAVQ#�[F�< y/�B�åGB&>!�YS

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)2/�B&/,B�å&/�',ZD#43�*!/�J��9*xB9å&%&#�/ì �'

x2(1− d(G)) + x− 1 = 0/,B�å&/�"���/0*!"�/�#43� �#I#�@&>M/�'

ÍB�å&%&#P ����C�R�/,8��(�D+,�(íQ���A',@&�DB����Ñ d(G+)

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�><W���Z�Zg # %e%�8:9N2:�H�:�Z268��H�,464Z�:9�ëdijdik + djk ≤ 1

�m�5^ # 9 {i, j, k} = {1, 2, 3} A¤ .�7 # $�{�/%��52 A ��/,�D*(/,J�/0*!1�3¨J�@&B�å i = 1

3j = 2

3k = 3

[ GL/,B& �=�V�"Q#G∗ è£>( @&�Îå� �"#9@&"��A*(>(V�#�8�Z"C�*,3M)2/,�DåÌ>A/H �' +����(êQ>(ê��A+,B&/�# *!/��÷æA/��(CD*(Z�#

d12d13 + d23 ≤ 1/,B�å�/,"��D1�*Aè

�D/,"Q�(+,B�/�*E3�+��~"�/�) ST/��('�CD*!ZÑ G∗ B&>!�YS K3

è£ 9*E[DÑ/�B�å&/,"; J���>(@�) @&�A'�*X�9�Då

X1 = (a), X2 = (b1, b2), X3 = (c1, c2)[À������ ��(�A',V�#

)2/�B- ��(UQ�Då&@&#� 9*� �#�%�.&/0*!#�/�',1&#�+,O�O�/,"�jw(a) = 1

3w(b1) = d12

3w(b2) = (1 − d12)

3w(c1) = d13

3w(c2) = (1 − d13)

3�+��ì��/�B�å&/,"Q/,#y 9'W+���/�#ab1

3b1c2

3c2b2

3b2c1

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X2

+��X3

#�%&',%9*(*h �'�+,�D�(ê�>Aê��(+�B&/�*h/��(/0*!��/�B&/,�A/,"Ò8��(%&#I#�/�"I*(.&/�)-/�B&#Y 9O�J� 9*(V�"�#/�B�å

d12

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dijdik + djk ≤ 1�m�5^ # 9 i, j 8,2 k

�67{1, 2, 3, 4}

^���9X<=�67�45\H9"Y5c*:Y�75d��:9"�:<W�:."�(dijdik + djk − 1)(dijdil + djl − 1) ≤ (1− dkl)d

2ij

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G∗ B&>!�YS K4è£/�*,[

: *!+0*!/,�W/����A1ëST/��D*!+0*!/���>(/�"�=��A',/,>A+,"�/�# /���/,)2/�ZW �'�*FST/¶æA/,'�Z�#_#IZ 3hJ�@�B�åÕJ���>(@&) 8��(U�8���9�D*! ��ÆST/,�('�CD*(/�*(*P+��<+����Aê�>(ê��A+,B&/�Zì"Q/,) 8,Z�#I�DZ�#IV��-J���>A)� 9�A*P ���#�@9*!"� �#4[Õóm#I#�@&>j

5.6.*(+�*(/,�4�('�/,>(ZD"I*�#�@&"Q�A*!>AV����DJ� 9*�C7@��Då� �"2J��9>(@&)2O� �>«*!CD8,ZFC��mB&>!�YS¶3I)-/��DåÌ"�/�#� �'~+����Aê�>(ê��A+,B�/,Z/�B�å&/�"�#�+,"I*W/���+�>(Z�#F HJ���>A@&) #ICÎ.9��"I*7�('E�9)-@�*,3�+,�h HB&>(�9SM"�/,) ST/,�A',CD*ÈJ���>(@&)2�('�%&B&/0*E[ómR�R�1��ì öB&>!�9STRcC��~)-/�B�ST/,�D/,�D1ö#�@&"Q�A*!>AV�#�8,ZFC�*P"Iå&/,>AJ�/�*(í�"�#î �'F�9���DCÎ*X�9�(VQ"�#IJ�@&'�3ÆJ� Ò "Q/,B�å&/�=�Z�#Ò8��(U�8��(>! ïZ��D�D/,�A',#�/�=�1ï+,�D/,#�/�"é �' +����Aê�>(ê��A+,B&/0*2*!/0*!�('�1&�D/,B&/��("�/�# ���D��C²*!J� 9*�æAVQ#R�/934 �"�/�B�å&/,=QZ�#P8,�AU�8,�«*�C&�ÑSTí�B&B&/0*!�D/,"Ií��Å"�/,)ÛST/��(',CÎ*!J�/0*

K4

è£/0*E[:�)2/,"�"�å�ZDR�/,"WR���>A)-/��Då±J���>A@&) 8,�AU�8,�m���D*! ���ST/,�A',CÎ*!/0*(*x+,�D/,#±+,�D�(ê�>Aê��(+�B&/,ZQ8�Z�#I��ZD#IV��|J��9>«è)- ��«*~ ���#�@9*!"���"� �#43� �#I#�@&>Æ*!/�B�åÌí�#PST/�� 3QJ�@&B�å(d12d13 + d23 − 1)(d12d14 + d24 − 1) ≤ (1 − d34)d

212

[ ñ ö òDÑ/�B�å&/,"L/�#Ì#�@�>X1 = (a), X2 = (b1, b2), X3 = (c1, c2, c3), X4 = (g1, g2, g3)

[DÑ/�B�å&/,"Q/,#

G∗ #�@&)2O���/�)-/�"I*(/,>(+�R�/,"L �'±+���/�#j Í#�%�.&/0*!#�/,'�1&#cjab2, ac3, ag3, b1c2, b1g2, c1g1

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w(a) = 13

w(b1) = d123

w(b2) = (1 − d12)3

w(c1) = λd133

w(c2) = (1 − λ)d133w(c3) = (1 − d13)

3w(g1) = µd14

3w(g2) = (1 − µ)d14

3w(g3) = (1 − d14)

34 �J�@��λ+��

µ/,B�åÌíQ*A*!J� 9*�C&#P+,>«*!+�#�/,Zej

λ = d12d13+d23−1d12d13

3I+,�µ = d12d14+d24−1

d12d14

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d212

[��/��D*(+�*!/���íQ"�#�/�* ñ ö òW�9*(>(/�"�=�/,'0.&/y �'0*2#Y 9O�æAV�#43�J�@&B�åÒ/,'H �'j+,���Aê�>(êQ�(+,Bï��/�B� ��N�9R�R d34

3CDB�å;#�@&"Q�A*!>AV�#�8,ZFC&"�#IRcC&�ì+,�D/,#;/��(/�*(��/�B&/,�j/,�DJ� �B�å��9�!��.Y ���.9 ��FC&R� 9"ë�D+�*!>A/,J�@�'�*!VQ"�#é/�B�åK4

è£/�*�"�/�) ST/,�A',CÎ*!1öST/,�ÎSTUYæ«*-B&>!�YST@�*E3Æ 9J�@&�� �'L+,�D�(ê�>Aê��(+�B&/,#; �'y/���1&>A/L)2/,B� �=Q@�*(*! �#4[~ �

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:�' @&OQ*(Z�)- ��DZ²*X�9�(RcC&�± �=`CÌ=`Cé/,B9å&/,"��D/�*!/�#�/�*F*!/,J��9* ;�(',ZD"I*(/,#c�(',��)-��.9 9�È)2/,B&/�B�å&/¦è'�1Õ�(',��)WUÝ/�B�å&/,"Q��/�*A*!/���J�/��Då&/0*(*(/,�(CÎ*!J�/0*�æAí�#

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/�B�å&/�"�è�D/�*~�Y*!C�>(���!�E.9 ��ejñ ö ò 1

∆x(2) = 1 − d(T )

ñ öwö ò 1−x(k)

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J� 1 < k < t + 1

3ñ öwöwö ò x(t+1) = 1�m/,',/��(�(í�#FR�/Í õ [�ST/¶æA/�',/0*!J�/�'2J� ��(@&"Q�"C� �" 9'

Fi(d) = x(i) STí�B&B�.&+�"IåI*E3 9)-/��DåL d+����Aê�>(ê��A+,B2STíQB&B�.&+,"�å&+,R�/�"F)2/,B& �=Yæ( � 9'x(i) �AU��DåI*E[¨: ST/�"I*(ZÑ/,B�å&/�"���/0*!/�#ÌR�1���#� �OÌæAV�#43

JQ@&B�åF2(d) = ∆(1 − d)

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3�J� 1 < k < t + 1VQB�å� �"� 9#Ì#�@�> ñ öwö�ö ò��('�/,>(ZD"I* Ft+1(d) = 1

/�B�å&/�"���/0*H*!/��÷æA/��(í��± Tt

B�>!�9SÈ#I>(ZD*(Z�#IV��d = d(Tt)

+����(êQ>(ê��A+,B&+�>(/�3� �)2/��Dådè»>(/�/,B�åj>A/,#IV�>('�CD.jO�@&�DZ�"�@�)2/,B9å&/,"��D/�*(/�*z �=Å[

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)-��>ÖZ��A)-/�>A*,[: #�%�.&/�*!#9/,',1&#IR�/,"6Z�B� �'�@&�D"�ZÅST@&BEæAV�#43�JQ@&B�åL �'

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k = t/,�(/0*!+�" /�B�å&/,"��D1&�(+�B6�����ð[¢i�1�*E3x)2Z�"�*h�N�Y*!"�Z¨ST@�B9è

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)2Z�"Q=�/,"kè»>! Q[�óm)2ZN 9*A*E3

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#� �OÌæAV�#P)� �æA=H �'±�����DCÎ*!����ST/,���A1�#�@&>(���9*�æ(�9*,[ ÅùF ÅùG×xý]�G�M�MÿL�G�

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) ≤ d < (1 − 1∆2 )

%T�:9 @?�,2:\H9N���5464 # �W�67 Fk(d)2 # � # 76��%< # # % # ¢1Y5S�d6����<����>%��6�@+��.Uq 18:9¯45."26��*6*:�,4 A

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1−F2(d)>

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(1 − F2(d)) > 03~CDB�å

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=�/:��"�C�8�Z"CYæ(�9*fUYæA>! <R�/,CD>A.9 (∆−1) > (1−F2(d))∆ ⇔ ∆2(1−

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:�'±Z�"�=QV�#�8,ZFC&�z�D+,O�+��~O�/�=�Z�B2 �'±���D*! �����"�@��Ö>(/�#ÌVQ>(',ZFC&RcC&�Å#�%�.&/0*!#�/,'�Z�#43�JQZ��('�/,"HJ� 1 > Fk(d) > Fk−1(d)

3� �#I#�@&>(∆−1) (1−d)

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1−Fk−1(d)

39 �)2Z�R�1&�� 9'¨ZD"�=�V�#�8�Z"C&����D��C²*X�9�,3Fk+1(d) > Fk(d)#9%�.&/�*(#�/,',ZD#4[

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3VQB�å� �"� 9#Ì#�@�>

1 = Ft+1(d(T )) < Ft+1(d)3� �)-Z�/,�D��/�"I*()-@&"Q=����M.&@&�D"� Q[

GHZ�"�=�/�"I*z/,��1�#�+,�('�CD*(/�*A*!í�"�#j �'±�����DCÎ*!���MR�Z�'�@&"IåÌCÎ*X���(��J�@&'9[mã�/,B�åÌí�#PST/��ÑZ�"Q=�Z�>A/,#�*E3QJ�@&B�åd > 1− 1

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�� È*(/,J��Y*Æ GB&>!�YSÅS� Q3� �#I#�@&>

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*«è£�A',Z�"�*!/��<S� �#I>(ZD*(Z�#IV��<+,�D�(ê�>Aê��(+�B&/Í�D/,B� 9�N��R�Rd[ µ "�=�Z�>A/,#�*�R�ZD',@�"IåÌCD*(V�"�#43�*(/,B�åÌí�#HST/,�ð3JQ@&B�åP)-ZD"�=�/,"

∆è£>A/,B&VQ�N��>AZD�M*Aè£�A',ZD"I*(/,�ÆS� Í#I>(ZD*(Z�#IV��Æ+,�D�(ê�>Aê��(+�B&/7#IZ��A/,R�R

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ã�/�#ÌZD"I*(�(í�#�)2@&�«*Æ �'Fk(d)

�(@&>A@&'E Y*!@�*,3Q+��ÆJ� ��(@&"Q��CD*(�(V�#�%&�A�(',/± 9'ì/���/�)-/�ZD*Fk(d(Tt)))2/�B�ST/,�D/,��1W/���/,)2/,ZÎ.&/,�ð[

F2(d) = ∆(1 − d) < ∆(1 − d(Tt)) = F2(d(Tt))*!/��÷æA/,�Aí��MST/,�D*(/�.&+��(í�"�#Ò�('�/,>(ZD"I*

)2ZD"�=�/,"tè»>(/�[

Fk+1(d) = (∆ − 1) (1−d)1−Fk(d)

< (∆ − 1) (1−d(Tt))1−Fk(d(Tt))

= Fk+1(d(Tt)) �'LZD"�=�V�#�8�Z"C

�A',/�>(ZD"I*~Z�B& �'�3Q�('�Z�"�*!+,"H)2Z�"�=�/�"tè£>(/9[

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[fã�/,J��Y*dè£>(/Æ G|Q[;|ì��/,)Íè

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τè .9 ��ð[î:�'

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τ = (∆ − 1) (1−d)1−τ

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[Ý}Ö*!>(/�"�=�/�'�.&/d ≥ 1 − 1

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3Ö �)-ZÆ/����D/,"�*!)2@&"�= ST/��D*(/�.&+��(í�"�#I"�/,#4[Móm','�/,�� �'7���D��C²*X�9�A*ÆZDB� �'�@&�D*(V�#4[

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