æ#êhomepages.inf.ed.ac.uk/.../gielis/kragujevacfinal.pdf · |Âsy }m^u ðf¡ x }mn ludz é u#ê o...

9
x y R 2 r O x 2 + y 2 = r 2 a b 6= a ( x a ) 2 + ( y b ) 2 =1 x A p + y B p =1,

Upload: others

Post on 22-Jul-2020

1 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: æ#êhomepages.inf.ed.ac.uk/.../GIELIS/KRAGUJEVACFINAL.pdf · |ÂSY }m^U ðf¡ X }mn LUdZ é U#ê O Pfe _`ê* SY LUXP § 7Ud @ -O ]nSVKNP ëLê MfU3j}WÉO P-O m X }mn LUdZT SY LUXP

����������� ����������������� ��������

�������! #"%$'&#(%)*$+��$+,-,�&.�-/10324$+ 657$�298: ;�< #"=$'>+�-&#�?24>7$@,-,?�-/BAC$+�-��,-$@&EDF$��G�HI ;��$

JLKNM-O�PRQTSVUXWYSVZ;[-\@]^U`_aO�PcbBONUdZ^UXPcO�PfehgiUdKNjkKNWVe7lFUXm^Zn]nmoONUXWVUXP

�qpsr�tvu wyx3z%r�{^usp|}m^KN~�]nMfU���P+]nm^K�eq�f�X]nSVKNP�O�Pfe']nMfU���j}SYWVKN���fU.K�_�e����sm^�X�R��MfKN~.jfZ^KNPi��Z�����P�QTm^K;��]nM�O�Pfe�|fKNmn~c�C���#��[m^UdZnjkUd�X]nSY�LUXWY�L[���U' +�fKN]^Uc]nMfU7_IKNWYWVK;��SYP}�f¡'����MfURZ^U;O�m^�^M:_¢KNm£eqS¥¤kUXm^UXPf�dUdZCKNmE_9�}PfefO�~¦UXP@]oO�WT�dKNP@]nmoONZn]^Z§ UX]���UdUXP£]nMfU1j}MfUXPfKN~¦UXP-O3K�_iKNmn�+O�P}SV�sKNmFSYPfKNmn�+O�P}SV��[qK�_%O�P}SY~EO�]^UsKNm�SYP-O�P}SY~EO�]^Us]nM}SYP}�LZ#[LM-ONZ*K��d�X�}j}SVUde~EO�P+�c~¦UXPi��Zs~.SYPfe}Z;[-��M}SYWVU6¨9©}ª¬«dªn­N®n¯`©y°X±N®y¯^±N²�²�³@´fµ¢¨¢¶7±�°�·�®`µ4´v¯Xµ¸·v¹Yªº«�±N®�ªº«º«dª`´�¨¢µ¢­N¹%«ºµ9²�µ4¹»µ9¨¢³}¼+ªº« ©q­�«½ ª^ªX´y·�³@®º«º³fªn¼ ½ ¶F°;ª`¾À¿FÁYÁYÁÂ]nM}SYP}�LZ*O�P}SY~EO�]^U1O�Pfe�SYP-O�P}SY~EO�]^U�[f¾�ª�¼N¾�ª`¹9¹Vª`®o«1µ9´6¨9©}ª�¾*±N®`¹Y¼3­N´v¼¬¨9©+µ4«�¾*±N®º¹Y¼¾À©qª`®^ª`µ9´�¾�ª3¼�¾�ª`¹4¹%­�®nª ½ ±�³@´�¼C­N¹»µ4ÃNª ½ ¶ ·-©@¶Ä«`µ¢¯o­N¹=­N´�¼E²¦­�¨9©qª`²¦­�¨¢µI¯^­N¹%¹¥­N¾�ÅnÆ

ÇU¦O�SY~�]^KCZnMfK#��]nM-O�] MfKNPfUX���dKN~ § Z3O�Pfe�ZnMfUXWYWVZ;[��Xmn�qZn]oO�WVZÈO�Pfe��+O�WÉOÄÊqSVUdZ;[iËBÌ��sÍa~¦KNWVUd�X�}WVUdZ¬O�PfeÎ K;��UXm^Z;[�Zn]^UX~¦Z;[�]nSVZ^Z^�fUdZ*O�Pfe�jvKNWYWVUXPE��moO�SYPfZFK�_vj}WÉO�P@]^Z;[+UX]^��Á�O�Pfe.]nMfU�m^UXWÉO�]nSY��SVZn]nSV�1Z^j-ON�dU`Ía]nSY~¦U��}P}SY�LUXm^Z^USY]^Z^UXW¥_n[�SYPÏON�d�dKNm^efO�Pf�dU'��SY]nMÐZnSY~.SYWÉO�m6P-O�]n�}moO�WT�X�}mn��O�]n�}m^Uc�dKNPfeqSY]nSVKNPfZ;[�O�WYWse}K?ONZ^Z^�}~¦UcZnM-O�jkUdZ¦��SY]nMZnSY~.SYWÉO�m1�LUdKN~¦UX]nmnSV�;O�W%_¢KNmn~EO�W=e}UdZ^�XmnSYj}]nSVKNPfZ;Á

Ñ Ò pÓr�Ô3Õ×Ö1Õ�u�ØÙÕÂr-t�ÚÛu1Ü£Ý}ÞßÔ3à�ráw¬Þ:Õ�âoâ^ãá{^pÓr�ÔyÕ�ÞÐu�tvâowcägiUX]

xO�Pfe

ye}UXPfKN]^U.å�O�mn]^UdZnSÉO�P<�dK�Í�KNm^eqSYP-O�]^UdZBSYPROEj}WÉO�PfU

R2Ás��MfUXPR]nMfU3Ud @�-O�]nSVKNPRK�_F]nMfUy�XSYm^�XWVU�K�_

moONeqSY�fZr��M}SV�oM<SVZB�dUXP+]^UXm^Ude�O�]B]nMfU�KNmnSY��SYP

OSVZB��SY�LUXP § �

x2 + y2 = r2O�Pfe']nMfU.Ud @�-O�]nSVKNP<K�_�]nMfU

UXWYWYSYjfZ^UÈK�_FOÄÊ}UdZaO�Pfe

b 6= a��M}SV�^M'SVZ1�dUXP+]^UXm^UdeRO�]�]nMfUÈKNmnSY��SYPhSVZ���SY�LUXP § � (x

a

)2+

(yb

)2= 1

Á

|ÂSY���}m^U¦æ�¡��XSYm^�XWVUdZTO�PfehUXWYWYSYjfZ^UdZ

QBKNSYP}� § ON�oçC]^K¦g%O�~¦è�[}]nMfUÈ�XSYm^�XWVUdZTO�Pfe7]nMfU UXWYWYSYjfZ^UdZ;[�ONZ���UXWYWÀONZ�]nMfUÈZ^ @�-O�m^UdZsO�PfeCm^Ud�X]oO�P}��WVUdZ���M}SV�^MO�m^U3O�WVZoKEZnMfK#��P'SYPR|=SY���}m^U6æ�[vO�m^UyO�WYW%SYPf�XWY�fe}Ude'SYP']nMfUÈ_IO�~.SYWY�hK�_�]nMfU3Z^K�Í��;O�WYWVUde7��Z^�}jvUXm^UXWYWYSYjfZ^UdZÄ�N[-SaÁ�U�Á]nMfU j}WÉO�P-O�ms�X�}mn�LUdZs��SY�LUXP § �hå�O�mn]^UdZnSÉO�PRUd +�-O�]nSVKNPfZ1K�_À]nMfUB_IKNmn~

x

A

p

+

y

B

p

= 1, é æ#ê

æ

Page 2: æ#êhomepages.inf.ed.ac.uk/.../GIELIS/KRAGUJEVACFINAL.pdf · |ÂSY }m^U ðf¡ X }mn LUdZ é U#ê O Pfe _`ê* SY LUXP § 7Ud @ -O ]nSVKNP ëLê MfU3j}WÉO P-O m X }mn LUdZT SY LUXP

|ÂSY���}m^U¬ëq¡�Z^�}jvUXm^�XSYm^�XWVUdZTO�PfehZn�}jkUXm^UXWYWYSYjfZ^UdZ

é ��MfUXm^U § � p,A,B ∈ R+0 ì § KN]nM'�;ONZ^UdZ A = B = r

O�PfeA = a 6= b = B

]nM+�fZ § UXSYP}�£e}U;O�WY]���SY]nM § �]nMfUÈZoO�~¦U Ud @�-O�]nSVKNP�ê`Á�i�}mnP}SYP}�?]^K'·q±N¹Y­N®c¯^±Ní�±N®n¼Nµ4´v­�¨�ªº«

ρO�Pfe

θ[*Zn�f�oM�]nM-O�]

x = ρ · cos θO�Pfe

y = ρ · sin θ[���MfUXPîSYP

ONe}eqSY]nSVKNP:SYP@]nm^K@eq�f�XSYP}��O<�dK�U`ïE�XSVUXP+] m4é ��M}SV�^MÏO�WYWVK#�1ZE]^K�@SVUXWVe:~¦KNm^Uhj-O�mn]nSV�X�}WÉO�m6Z^�@~.~¦UX]nmnSVUdZ£K�_m^KN]oO�]nSVKNP7O�m^KN�}Pfe

O]nM-O�P6]nMfK�Z^Usm^UXWÉO�]^UdeE]^K¬]nMfU1ðy +�-ONeqmoO�P@]^Z�K�_�]nMfUBå�O�mn]^UdZnSÉO�P£�dK�Í�KNm^eqSYP-O�]^UBZn�qZn]^UX~6ê

O�Pfe���MfUXP�~¦KNm^UdK;�LUXmÈ��SY��SYP}�£��SYPfe}UXjkUXPfe}UXPf�dUL��]^K£]nMfU.U`Ê�jkKNPfUXP+]^Z K��d�X�}mnmnSYP}�hSYP�]nMfUy]nM}m^UdU�]^UXmn~¦ZÈK�_Ud @�-O�]nSVKNP é æ#ê`[}]nM}SVZ1Ud @�-O�]nSVKNP § Ud�dKN~¦UdZ

ρ =

{∣

cos mθ4

A

n2

+

sin mθ4

B

n3}−1/n1

, é ëLê

é ��MfUXm^U § � n1 ∈ R0, n2, n3 ∈ Rê`Á*\�KN~¦U U`Ê}O�~.j}WVUdZ#[q�}j£]^K�Z^�;O�WVU�[}K�_=j}WÉO�P-O�m1�X�}mn�LUdZ���SY�LUXP § �CO¬jkKNWÉO�m

Ud @�-O�]nSVKNP é ëLê`[i��MfUXm^U § �<SYP�U;ON�^Mñ�;ONZ^U�MfUXm^U A = BO�Pfe�_IKNmÈ��M}SV�oM_¢�}mn]nMfUXm;[im^UdZnjkUd�X]nSY�LUXWY�L[%SYP?�;ONZ^U

é OLꬡ m = 3, n1 = 4.5O�Pfe

n2 = n3 = 10[vSYPR�;ONZ^U é § êÈ¡ m = 4, n1 = 12

O�Pfen2 = n3 = 15

[vSYP�;ONZ^U é �#êÈ¡ m = 5

O�Pfen1 = n2 = n3 = 4

O�Pfe'SYPR�;ONZ^U é e-ꬡ m = 7, n1 = 10O�Pfe

n2 = n3 = 6[

O�m^UÈZnMfK#��PhSYP7]nMfUB_IKNWYWVK;��SYP}�Eòf���}m^U�Á

|ÂSY���}m^U¬óq¡��X�}mn�LUdZ é OLê`[ é § ê`[ é �#êsO�Pfe é e-ê���SY�LUXP § �CUd @�-O�]nSVKNP é ëLê

��MfUdZ^U��X�}mn�LUdZ7j}m^UX]n]��ßON�d�X�}moO�]^UXWY�ôe}UdZ^�XmnS § U<]nMfU<ZnM-O�jkUdZhK�_y]nMfU<Ì��}j}M-O�mhWY�}]^UX�}~õjkUX]nSVKNWVU é OLêO�Pfeñ]nMfU7Zn]^UX~¦Z�K�_ \��Xm^KNj}M@�}WÉO�mnSÉO�PfK@e}K�ZºO é § ê`[��F @�}SVZ^UX]n�}~ é �#êyO�PfeñöTONZnj § UXmnmn� é e-ê`[�m^UdZnjkUd�X]nSY�LUXWY�LÁ\@SY~.SYWÉO�mnWY�L[�_IKNm SYPfZn]oO�Pf�dU¦eqS¥¤�UXm^UXP+]¬ç@SYPfe}Z K�_�Zn]oO�m�ò-ZnM�dKNmnm^UdZnjkKNPfe�]^K7�X�}mn�LUdZÈ��SY�LUXP § �RUd @�-O�]nSVKNP é ëLê_IKNmA = B

O�Pfe£_¢�}mn]nMfUXm;[qm^UdZnjkUd�X]nSY�LUXWY�L[qSYP7�;ONZ^U é U#ê�¡ m = 5, n1 = 2O�Pfe

n2 = n3 = 7[fO�PfeCSYP7�;ONZ^U

é _`êT¡ m = 5, n1 = 2O�Pfe

n2 = n3 = 13Á

ë

Page 3: æ#êhomepages.inf.ed.ac.uk/.../GIELIS/KRAGUJEVACFINAL.pdf · |ÂSY }m^U ðf¡ X }mn LUdZ é U#ê O Pfe _`ê* SY LUXP § 7Ud @ -O ]nSVKNP ëLê MfU3j}WÉO P-O m X }mn LUdZT SY LUXP

|ÂSY���}m^U ðf¡��X�}mn�LUdZ é U#ê�O�Pfe é _`ê*��SY�LUXP § �7Ud @�-O�]nSVKNP é ëLê

��MfU3j}WÉO�P-O�m��X�}mn�LUdZT��SY�LUXP § �7jkKNWÉO�m�Ud @�-O�]nSVKNP é ëLê1�;O�PcSYP�Z^KN~¦U¬Z^UXPfZoU § U¬SYP+]^UXmnj}m^UX]^Ude<ONZ § UXSYP}�K § ]oO�SYPfUdeZn]oO�mn]nSYP}�h_9m^KN~Ó¨9©qª¦³�´fµ9¨T¯Xµ9®�¯X¹Vª£¯oª`´-¨�ªX®nªn¼<­L¨0 é ρ = 1

ê § �R]nMfU7¨¢®�­N´f«4°d±N®º²¦­�¨¢µI±N´?��SY�LUXP § �]nMfU3mnSY��M+]BM-O�Pfe'Z^SVe}U3K�_ é ëLê`[ é _IKNmBO�P@�c�^MfKNSV�dU�K�_Fj-O�moO�~¦UX]^UXm^Z A,B,m, n1, n2, n3

ê`ÁT�sPfe�[vZnSY~.SYWÉO�mnWY�L[SYPfZn]^U;ONe7K�_À]nM+�fZ�]nmoO�PfZn_¢KNmn~.SYP}�¦]nM}SVZ1�}P}SY]s�XSYm^�XWVU�[-O�WYW�j}WÉO�P-O�ms�X�}mn�LUdZ1e}UX]^UXmn~.SYPfUde § �£jvKNWÉO�m1Ud @�-O�]nSVKNPfZρ = f(θ)

[ é ��MfUXm^U § � f§ ONZnSV�;O�WYWY���;O�P § U�O�P@�'jkK�ZnSY]nSY�LU�m^U;O�WÀ_¢�}Pf�X]nSVKNP�ê`[%�;O�P § Uy]nM+�fZ ]nmoO�PfZn_¢KNmn~¦Ude

SYP+]^K¦]nMfU j}WÉO�P-O�ms�X�}mn�LUdZ1��SY]nMcjvKNWÉO�m1Ud @�-O�]nSVKNPfZ

ρ = f(θ) ·

{∣

cos mθ4

A

n2

+

sin mθ4

B

n3}−1/n1

. é óLê

��MfUB_IKNWYWVK;��SYP}�Eòf���}m^U�[f�}jh]^K.Z^�;O�WVU�[-ZnMfK;�1ZsSYP é § ê�O�Pfe é �#ê�Z^�f�^Mh]nmoO�PfZ�_IKNmn~EO�]nSVKNPfZ���SY]nMhj-O�moO�~¦UX]^UXm^Zm = 2.5, n1 = 1

1.3

O�Pfen2 = n3 = 2.7

O�Pfe�[�m^UdZnjvUd�X]nSY�LUXWY�L[m = 2.5

O�Pfen1 = n2 = n3 = 5

[qK�_=]nMfUm^K�Z^U`Í��X�}mn�LUÈK�_*QTmoO�PfeqS é OLê���SY]nMcUd +�-O�]nSVKNP ρ = f(θ) = cos(2.5θ)

SYP+]^Ky��Zn�}jkUXmnm^K�Z^UdZ#�NÁ

|ÂSY���}m^U¬÷q¡FO�m^K�Z^U`Í��X�}mn�LU é OLê1O�PfeC]���K6�dKNmnm^UdZnjkKNPfeqSYP}�6Zn�}jkUXmnm^K�Z^UdZ é § ê�O�Pfe é �#ê

��PfeZn]oO�mn]nSYP}�c_¢KNmÈSYPfZ^]oO�Pf�dU¦_¢m^KN~ø]nMfUh¹Y±^ùL­�®ºµ¢¨9©+²�µ¢¯E«�·�µ4®n­N¹ é OLê ρ = f(θ) = e0.2θ [ § �<]nmoO�PfZn_¢KNm�Í~EO�]nSVKNP é óLê ��SY]nMñj-O�moO�~¦UX]^UXm m = 4O�Pfe

n1 = n2 = n3 = 100SYP��;ONZ^U é § ê¬O�Pfe m = 10

O�Pfen1 = n2 = n3 = 5

SYPÏ�;ONZ^U é �#ê`[�m^UdZ^jvUd�X]nSY�LUXWY�L[��}jÐ]^K�Z^�;O�WVU�[�]nMfUh_¢KNWYWVK#��SYP}���N«º³#·}ª`®º«a·vµ4®n­N¹ú«L�RO�m^UK § ]oO�SYPfUde�Á

|ÂSY���}m^U¬ûq¡*OyWVKN�+O�mnSY]nM}~.SV�3Znj}SYmoO�W é OLê�O�PfeC]���KE�dKNmnm^UdZ^jvKNPfeqSYP}�£Zn�}jvUXm^Z^j}SYmoO�WVZ é § ê�O�Pfe é �#ê

ó

Page 4: æ#êhomepages.inf.ed.ac.uk/.../GIELIS/KRAGUJEVACFINAL.pdf · |ÂSY }m^U ðf¡ X }mn LUdZ é U#ê O Pfe _`ê* SY LUXP § 7Ud @ -O ]nSVKNP ëLê MfU3j}WÉO P-O m X }mn LUdZT SY LUXP

ÇÏM-O�]BSVZ § mnSVU Î �'Zn]oO�]^Ude<O § K;�LU.�dKNPf�dUXmnP}SYP}�?¯X³�®`ü�ªº«.µ4´�ý�þ íV·�¹Y­N´vªº«¦�;O�P�m^U;ONeqSYWY� § UyU`Ê�]^UXPfe}Ude�]^K¯X³�®ºüNªº«c­N´�¼'«º³�®9°X­L¯ºªº«£µ9´îÿÄþÈía«�·q­L¯oªº«X[ é O�Pfe?_IKNm3]nM-O�]�~EO�]n]^UXm.]^KRZn� § ~EO�P}S¥_IKNWVe}Z¦K�_TO�P+�ñeqSY~¦UXPfZnSVKNPfZO�Pfe��dK@eqSY~¦UXPfZ^SVKNPfZºê`Á<��PfZ^]^U;ONe?K�_�Z^�}jvUXm^�XSYm^�XWVUdZ.O�Pfe�Zn�}jkUXm^UXWYWYSYjfZ^UdZy]nMfUXP�K�_T�dKN�}m^Z^U.�N«º³#·}ª`®º«a·�©qª`®^ªº«+�O�Pfe��«`³;·}ªX®nª`¹9¹»µ¸·-«d±NµI¼�«+�C�dKN~¦UESYP@]^K'j}WÉO;�LÁ��sPfe�[FONZySYPñ]nMfU6�;ONZ^U£K�_1j}WÉO�P-O�m.�X�}mn�LUdZ;[FO�WVZ^Kc_¢KNm�Znj-ON�dU`Í�X�}mn�LUdZ.O�Pfe?_¢KNmyZn�}m�_ION�dUdZySYP�Znj-ON�dU�[À]nMfUCù+ªn±N²EªX¨¢®`µ¢¯o­N¹1¨¢®�­�´}«4°d±N®`².­L¨¢µ¢±N´f«E]nM+�fZ�K § ]oO�SYPfUde�SYP�O�P-O�WVKN�����SY]nM7_IKNmn~¬�}WÉONU é ëLê�O�Pfe é óLê�j}m^K;��SVe}U.³�´fµ ��¯^­�¨¢µI±N´}«¬±�° ¾�µI¼+ª¬®�­�´@ù+ªº«3±�° ´v­�¨¢³@®n­N¹=­�´�¼6­ ½ «`¨¢®n­L¯d¨�«o©q­º·}ªº«XÁ

|ÂSY���}m^U3��¡FZ^KN~¦UT_¢�}mn]nMfUXms�X�}mn�LUdZ1SYPcëNË���SY]nM'Ud +�-O�]nSVKNP é ëLê

|ÂSY���}m^U��q¡FZoKN~¦UÈO�P-O�WVKN�LKN�fZ�Zn�}m�_aON�dUdZ�SYPcóNË

|fKNmT~¦KNm^U�e}UX]oO�SYWVZ O�Pfeh_IKNmBO¦~¬�}WY]nSY]n�fe}U.K�_�U`ÊfO�~.j}WVUdZBK�_FP-O�]n�}moO�W�ZnM-O�jvUdZTK��d�X�}mnmnSYP}�£SYP<O�P}SY~EO�WVZO�Pfe.SYP.j}WÉO�P@]^Z;[+SYP.]nMfUXSYm��dUXWYWVZ�O�Pfe.]nSVZ^Z^�fUdZ;[�SYP¦ZnPfK#� Î O�çLUdZ�O�Pfe¦Z^j}SVe}UXm�Ía��U § Z;[+SYPE�Xmn��Z^]oO�WVZ*O�Pfe.SYP Î �}SVe�Í�X�}mnm^UXP+]^Z#[�UX]^��Á:UX]^��ÁY[�O�Pfe���M}SV�oMÏO�m^UhZoKON�d�X�}moO�]^UXWY��e}UdZo�XmnS § Ude § �ñZ^�f�^Mî]nmoO�PfZ�_IKNmn~EO�]nSVKNPfZ;[�UXSY]nMfUXmeqSYm^Ud�X]nWY� § �?KNPfUEK�_�]nMfUX~ SYPfeqSY��SVeq�-O�WYWY�KNm¬SYPfeqSYm^Ud�X]nWY� § �Z^KN~¦UEP-O�]n�}moO�W��dKN~ § SYP-O�]nSVKNPîK�_1O7_¢UX��K�_]nMfUX~h[fZ^UdUE�¸ëq[-ûq[-÷Ä��ÁF|fKNWYWVK#��SYP}���FSVUX]sb�UXSYPi[fSYPh]nMfUdZ^U j}� § WYSV�;O�]nSVKNPfZ�]nMfUBòfm^Zn]�P-O�~¦UdeRO��}]nMfKNm�O�Pfe7M}SVZ�dK;��KNmnçLUXm���UXmn]���UXSYmnSYPf�oç+ÊE�fZoUdeE]nMfUT]^UXmn~���«`³;·}ªX®4°d±N®`²y³�¹Y­��s_IKNm�]nMfUTUd +�-O�]nSVKNPfZ1O�Pfe6]nmoO�PfZ�_IKNmn~EO�]nSVKNPfZ

ð

Page 5: æ#êhomepages.inf.ed.ac.uk/.../GIELIS/KRAGUJEVACFINAL.pdf · |ÂSY }m^U ðf¡ X }mn LUdZ é U#ê O Pfe _`ê* SY LUXP § 7Ud @ -O ]nSVKNP ëLê MfU3j}WÉO P-O m X }mn LUdZT SY LUXP

WYSYçLU é ëLê6O�Pfe é óLê`Áá��Pî]nMfU'WÉONZn]6P-O�~¦UdeÏO��}]nMfKNm;��Z��BmoO�����UX��ON�`Ía]oO�WYç�����PßQBUdKN~¦UX]nmn�:O�PfeÏÌTO�]n�}moO�W��M}SYWVK�Z^KNj}M@�k�N[%K�_F��M}SV�^M<]nM}SVZ O�mn]nSV�XWVUySVZBOE��mnSY]n]^UXP<Ud�^MfK}[k]nMfUy]nmoO�PfZ�_IKNmn~EO�]nSVKNPfZÈO�PfeR]nMfU3�X�}mn�LUdZÈO�Pfe]nMfU�Zn�}m�_aON�dUdZÀ��SY�LUXP § � é ëLêÀO�Pfe é óLêÀO�Pfe�]nMfU�WYSYçLU�[@O�WY]^UXmnP-O�]nSY�LUXWY����UXm^U��;O�WYWVUde£QTSVUXWYSVZ�Ía]nmoO�PfZ�_IKNmn~EO�]nSVKNPfZ;[Í��X�}mn�LUdZ*O�Pfe�Í�Z^�}m�_ION�dUdZ�]^K�K é ]^K ��M}SV�^ME]nMfU�òfm^Zn]*O��}]nMfKNm*eqSVe�PfKN]Fj}m^KN]^UdZn];[L~EOd� § UsO�WVZoK eq�fU�]^KB]nMfU�_ION�X]]nM-O�]FMfU���ONZFPfKN]F]nMfUXm^U#ê`Á���P+����O;��SYP¦j-O�mn]nSV�X�}WÉO�m*SYPc�Yæ`��[+]nMfU1KNmnSY��SYPfZ*K�_k]nMfUsO § K;�LU1K § ZoUXmn�NO�]nSVKNPfZ*��UXm^U]nmoON�dUde § ON�^ç3]^K��FSVUX]ÀbTUXSYPi��Z%��Zn�}jkUXm^UX���LZ#� é ��Zn�}jvUXm^UXWYWYSYjfZoK�ae}Z#��K�_-m^UX�LKNWY�}]nSVKNP�êÀONZÀZ^KN�}m^�dU�K�_-SYPfZnj}SYmoO�]nSVKNPSYP7]nmn��SYP}�¦]^K.�LUdKN~¦UX]nmnSV�;O�WYWY�he}UdZ^�XmnS § UB]nMfUÈZnM-O�jkUdZ1K�_Àj}WÉO�P-O�msZ^Ud�X]nSVKNPfZ1K�_ § O�~ § K@KEZn]^UX~¦Z;Á

|ÂSY���}m^U� q¡���SVUX]�b�UXSYP

|ÂSY���}m^U¦æ��q¡s�XZo +�-O�m^UL� § O�~ § K�K

��WVZ^K}[vU�Á �fÁ�SYP����q[kûÄ�=_IKNmBZn�f�^Mc��Zn�}jkUXm�_¢KNmn~¦Z#�3K��d�X�}mnmnSYP}�£SYP<ëNË�Íaj}WÉO�PfUdZÈO�PfecSYP<óNË�Í�Znj-ON�dUdZ;[iO�mn���qÍ~¦UXP+]^ZÂ�dKNPf�dUXmnP}SYP}�B�NO�mnSVKN�fZ%j}mnSYPf�XSYj}WVUdZÀK�_qP-O�]n�}moO�WqKNj}]nSY~.SVZoO�]nSVKNPfZÀO�m^U*eqSVZ^�X�fZoZ^Ude�[#j}m^Ud�XSVZoUXWY� SYP¬m^UXWÉO�]nSVKNP]^K.]nMfUXSYm1�LUdKN~¦UX]nmnSV�;O�WÂZnM-O�jkUdZ;[}Z^KN~¦UX]nSY~¦UdZBO�]�WVUXP}��]nMRO�PfehZ^KN~¦UX]nSY~¦UdZs~¦KNm^U § mnSVU Î �LÁ*|ÂSYP-O�WYWY�L[-��U m^U`Í~EO�mnç]nM-O�]�]nMfU7ëNË�Í O�Pfe�óNË�ÍaM+��jvUXm § KNWYSV�C�LUXm^ZnSVKNPfZ�K�_s]nMfU7O § K;�LUCZn�}jkUXm�_¢KNmn~3�}WÉOq��Z.K�_�UXWYWYSYj}]nSV�£]���jvUSYPfe}UdUde7~¦UXmnSY]sZoUXmnSVKN�fZnWY�C]^K § UB]oO�çLUXPhSYP'�dKNPfZnSVe}UXmoO�]nSVKNPRONZ���UXWYWaÁ

÷

Page 6: æ#êhomepages.inf.ed.ac.uk/.../GIELIS/KRAGUJEVACFINAL.pdf · |ÂSY }m^U ðf¡ X }mn LUdZ é U#ê O Pfe _`ê* SY LUXP § 7Ud @ -O ]nSVKNP ëLê MfU3j}WÉO P-O m X }mn LUdZT SY LUXP

� Ò pÓr�Ô3Õ×Ö1Õ�u�ØÙÕÂr-t�ÚÛu1Ü£Ýqr�ÔyÕ�ÞÐu�tvâow {^p ÞßÔ3{ozvÔ Þ:Õ×w3Þ:Õ�â^â�ä

��MfU.j}�}mnjkK�Z^U�PfK;��SVZ ]^K7ZnMfK#��]nM-O�];[%KNP]^KNj?K�_�]nMfUh¯X³�®`ü�ªº«CO�Pfe]nMfU£«º³�®4°d­L¯oªº«6��SY�LUXP § �R_IKNmn~¬�}WÉONUWYSYçLUB]nMfUBKNPfUdZ § U`_¢KNm^U�[���M}SV�^Mh_9m^KN~ O3j}�}m^UXWY�6�LUdKN~¦UX]nmnSV�;O�W%jvKNSYP@]1K�_=��SVUX�ßZoO�]nSVZ�_¢�¦P-O�]n�}moO�Wi�dKNPfeqSY]nSVKNPfZKNP£]nMfUXSYm��X�}mn��O�]n�}m^UdZ�O�Pfe¦��M}SV�^MCP-O�]n�}moO�WYWY�Ee}KyK��d�X�}m�SYP6]nMfUs��KNmnWVe}Z�K�_ij}M@��Z^SV�dZ;[@�oMfUX~.SVZn]nmn�L[q�LUdKNWVKN���L[§ SVKNWVKN���L[YÁYÁYÁ*O�Pfe�[+_¢KNm*]nM-O�]�~EO�]n]^UXm;[qO�WVZ^KÈSYP6]nMfU1��KNmnWVe}Z�K�_%O�mn]�O�PfeE]^Ud�oM}PfKNWVKN��� é Z^UdUsU�Á �fÁ��¸óq[+ðN�9ê`[@K�_%O�WYWjkK�Z^ZnS § WVU1��]nM}SYP}�LZ#�TSYPfe}UdUde7O�WVZ^K�KN�}mT®^ª`¹Y­�¨¢µ9ü;µÉ«X¨¢µ¢¯È«a·}­L¯oª`í ¨¢µ4²EªÈ³@´fµ9ü�ª`®º«dªÈSY]^Z^UXW¥_%SVZ § ONZnSV�;O�WYWY�£e}UX]^UXmn~.SYPfUde§ �7ZnSY~.SYWÉO�m1�X�}mn��O�]n�}m^U¬�dKNPfeqSY]nSVKNPfZTO�Pfehe}UdZ^�XmnS § Ude § �CZnSY~.SYWÉO�m�_IKNmn~¬�}WÉONU�Á

��MfUh·-©+¶�«ºµI¯^­N¹�«a·}­L¯oª`í ¨¢µ4²Eªº«���M}SV�^MáO�m^U?µ¢¼@ªn­N¹1«`³ ½ ².­�´fµ °d±N¹Y¼�«'SYPÏ]nMfURZ^UXPfZ^UcK�_���O�P}�NÍa�LUXPáå�MfUXP���q[� q[�æ��Ä�F��UXm^U¦m^Ud�dUXP+]nWY��Zn]n�feqSVUde § �R]nMfU�WÉO�]n]^UXm�ÍaP-O�~¦Ude<]���KcO��}]nMfKNm^Z;[%SYP<j-O�mn]¬O�WVZ^KC]^KN�LUX]nMfUXmÈ��SY]nM|}moO�P}ç�S@ËTSYWYWVUXPyO�Pfe��cSYmoO��=UX]nm^K#�@SV�B�Yæ�æ�[@æ;ëÄ��Á%��M+�fZÂSYPÈj-O�mn]nSV�X�}WÉO�mÀZ^KN~¦U1«a·q­L¯ºª`í�¨¢µ9²Eª�²¦±;¼+ª`¹ú«�±n°��sªn±N®Iù+ªº«�%ªX².­��9¨¢®^ª é SYP�]nMfUyWYSY]^UXmoO�]n�}m^U¦O�WVZ^K6m^U`_¢UXmnm^UdeR]^K7ONZB|}mnSVUdeq~EO�P}PqÍ�giUX~EO��V]nm^U`Í�ösK § UXmn]^Z^KNPqÍaÇ�O�WYçLUXm.Znj-ON�dU`Í]nSY~¦UdZºê`[TZnMfK;�Ù�}jôWYSYçLUc��j-O�mhU`Ê}�dUXWYWVUXPf�dUL�<ONZ6SVe}U;O�W M@�@jkUXm^Zn�}m�_aON�dUdZCSYP��cSYP}çLK#�1Znç@SÈZ^j-ON�dUdZ;Á���MfUdZ^UZnj-ON�dU`Ía]nSY~¦UdZ.O�WVZ^KRO�m^U6O�]y]nMfUEKNmnSY��SYP�K�_1]nMfUEZ^K�Í��;O�WYWVUde� § SY� § O�P}�-�ºÍa]nMfUdKNmnSVUdZ#Ác��P�]nM}SVZ3m^UdZnjvUd�X];[�O�PfeZ^K6~¬�f�oM<~¦KNm^U¦O�j}j}m^KNj}mnSÉO�]^UXWY�L[igiUX~EO��V]nm^U�M}SY~¦Z^UXW¥_��fZ^UdeR]nMfU¬]^UXmn~¦Z���j}mnSY~.SY]nSY�LU¦O�]^KN~c�.O�Pfec���dK�Zn~.SV�UX���-�N[k]nMfUyWÉO�]n]^UXm KNPfUyUdZ^jvUd�XSÉO�WYWY�h��SY��SYP}�hO¦~¦K�Z^] § U;O��}]nS¥_¢�}W=SY~EO��LU.K�_���]nMfU § UX��SYP}P}SYP}�-�yK�_���]nMfUyWYS¥_¢UL�K�_=��KN�}ms�dK�Zn~¦K�Z#�NÁ

|ÂSY���}m^U¦æ�æ�¡ ��m^K�_IUdZ^Z^KNmsgiUX~EO��V]nm^U3K�_Â]nMfU"!�P}SY�LUXm^ZnSY]��CK�_�giUX�}�LUXP

|}m^KN~ ]nMfU£_¢KNmn~EO�WsjvKNSYP@]6K�_B�@SVUX�È[�O<ösK § UXmn]^Z^KNPqÍaÇ�O�WYçLUXm¦~¦UX]nmnSV�'KNPR

4SVZ.��SY�LUXP:SYP�WVK��;O�Ws�dK�Í

KNm^eqSYP-O�]^UdZ(x, y, z, t)

§ �

ds2 = −dt2 +

{

c(t) ·

[

1 +k

4

(

x2 + y2 + z2)

]}−2

·(

dx2 + dy2 + dz2)

, (k = +1, 0 or −1), é ð@ê��MfUXm^U § �£]nMfUBZnj-ON�dU`Í�ZnWYSV�dUdZ�K�_%]nMfU�_¢�}WYW�Znj-ON�dU`Ía]nSY~¦UÈO�]1O�P+�E��SY�LUXPC]nSY~¦U

tO�m^U$#sµIªX².­�´f´fµI­N´hÿÄþÈía«a·}­L¯oªº«

±�°¦¯^±N´f«`¨a­N´-¨T¯X³�®ºüÄ­L¨¢³@®^ªc(t)

Á%���'��Od�<K�_���SVZn�-O�WYSVZoO�]nSVKNP�SYP�ëNËsÍamoO�]nMfUXm3]nM-O�P<SYP<ðLË�Íam^UXj}m^UdZ^UXP+]oO�]nSVKNPi[Zn�f�oM<~¦UX]nmnSV�dZ¬O�m^U��;O�mnmnSVUde�_¢KNmBSYPfZn]oO�Pf�dU § �'Zn�}m�_aON�dUdZ K�_�m^UX�LKNWY�}]nSVKNP�SYP<óNË�Í�Znj-ON�dU ì é O�Pfe�]^KC�LK6_9m^KN~]nMfU��LUdKN~¦UX]nmn��K�_�Zn�f�oMZn�}m�_ION�dUdZB]^KC]nMfU.ösSVUX~EO�P}P}SÉO�P<�LUdKN~¦UX]nmn�RK�_��LUXPfUXmoO�W���O�mnjkUde<j}m^K�eq�f�X]^Z;[�Z^UdU�Yæ;ó}[qæXð��9ê`ÁÀ��MfU�]nmoO�PfZ�_IKNmn~EO�]nSVKNPfZ*K�_k]nMfU Î O�]F���f�XWYSVe}U;O�P£óNË�Í&����]nM-O��LKNm^U;O�PE~¦UX]nmnSV�

ds2 = dx2 +dy2 +dz2

SYP+]^KcóNË�Ía~¦UX]nmnSV�dZ¬K�_1O�P+���dKNPfZn]oO�P@]3�X�}mn��O�]n�}m^U § �R~¬�}WY]nSYj}WYSV�;O�]nSVKNPñ��SY]nM?O�j}j}m^KNj}mnSÉO�]^U._¢�}Pf�X]nSVKNPfZO�m^UcUdZoZ^UXP+]nSÉO�WYWY�:eq�fU']^K�ösSVUX~EO�P}P O�Pfe:�LKNPÏbTUXWY~.MfKNWY]('RSYP:]nMfUXSYmCZn]n�feqSVUdZ6KNPÐ]nMfUh_IKN�}PfefO�]nSVKNPfZCK�_�LUdKN~¦UX]nmn�C�@SÉOq[fONZ�]nMfUX�£m^UdZnjkUd�X]nSY�LUXWY�CO�j}j}m^KLON�oMfUdeh]nM}SVZ;[qM@�@jkKN]nMfUdZ^UdZsO�Pfe£_aON�X]^ZÈ�Yæ;÷q[�æ;ûÄ��Á ��U;O�mnSYP}�¦SYP

û

Page 7: æ#êhomepages.inf.ed.ac.uk/.../GIELIS/KRAGUJEVACFINAL.pdf · |ÂSY }m^U ðf¡ X }mn LUdZ é U#ê O Pfe _`ê* SY LUXP § 7Ud @ -O ]nSVKNP ëLê MfU3j}WÉO P-O m X }mn LUdZT SY LUXP

~.SYPfe�]nM-O�];[�SYP�]nMfU���KNm^e}Z¬K�_�å�MfUXmnPî�Yæ#�Ä��[���ÇÏM}SYWVU¦O�WY�LU § moO7O�PfeO�P-O�WY��ZnSVZBj}m^K#�@SVe}U�]nMfU3_IKN�}PfefO�]nSVKNPfZK�_�~EO�]nMfUX~EO�]nSV�dZ;[Â�LUdKN~¦UX]nmn�<SVZ3O�]È]nMfU6�dKNm^UL�N[%SY]yZ^UdUX~¦ZÈ��KNmn]nM+��M}SYWVU6_¢m^KN~ø]nSY~¦U¦]^K7]nSY~¦UE]^K7m^U Î Ud�X]Z^KN~¦UX��M-O�]CKNP § KN]nMÐ]nMfUdZoUh��m^U;O�]£]^U`Ê�]^Z;[�SYPÏj-O�mn]nSV�X�}WÉO�m6SYPÐ]nMfUcWYSY��M@]£K�_È]nMfU'KNmnSY��SYPfZCK�_B�LUdKN~¦UX]nmn�ONZ£ò-UXWVeôK�_3~EO�]nMfUX~EO�]nSV�dZ § ONZ^Ude KNPô]nMfU�j}M+�qZnSV�;O�W �LUdKN~¦UX]nmnSV�;O�WyU`Ê�jkUXmnSVUXPf�dUdZC]nM-O�]7M+�}~EO�PfZcM-Od�LU§ ��]nMfUXSYm;[��@SVZ^�-O�WsO�PfeîKN]nMfUXm;[�Z^UXPfZoO�]nSVKNPfZEO�Pfe�jkUXm^�dUXj}]nSVKNPfZ;Áî��PîZnj-ON�dU`Ía]nSY~¦Uc~¦UX]nmnSV�dZ.WYSYçLU é ð@ê�KNPfU~EOd�K § ZoUXmn�LU�[ÂKNP?]nMfUEKNPfUEM-O�Pfe�[=]nMfU6�dKNPq_¢KNmn~EO�W�e}U`_IKNmn~EO�]nSVKNPfZyK�_ Î O�]yZnj-ON�dU`Ía~¦UX]nmnSV�dZy]^Kh~¦UX]nmnSV�dZK�_��dKNPfZn]oO�P@].Znj-O�]nSÉO�W��X�}mn��O�]n�}m^U

(=, < or > 0)��M}SV�oM�~EOd���oM-O�P}�LUCSYP?]nMfU£�dKN�}m^Z^U£K�_s]nSY~¦U�[*O�Pfe�[

KNP�]nMfUhKN]nMfUXm.M-O�Pfe�[�ORM+��jkUXm § KNWYSV�C]���SVZn]¦SYP�]nMfUCWYSYPfU`Í�eqSYm^Ud�X]nSVKNPi[���M}SV�^Mî_9�}mn]nMfUXm¦~.SY��M@]¦SYPf�XWY�fe}UcOZnjkUdUde<K�_�WYSY��M@]�ÍaPfKNmn~EO�WYSVZoO�]nSVKNPi[À��M}SV�^M?O�WYW�SYP�O�WYW*O�~¦KN�}P@]¬]^K�°d±N®`².­�¹�¼@ª9°X±�®º²¦­�¨¢µI±N´}«¦±�°¦¨9©qªE¨9©qªn±N®^ª`²±�°�)�¶�¨9©�­;ù+±N®�­N«+*X³@µ¢¨�ª£­N¹»µÉÃNªE¨9©qª£­ ½ ±�ü�ª¦«º³#·}ª`®X¨¢®�­N´f«4°d±N®º²¦­�¨¢µI±N´}«.®^ª`¹Y­�¨�ª£¨a±h¨9©qª£ª(*X³}­�¨¢µI±N´î±�°�,�³}¯X¹»µ¢¼+ª^­N´¯Xµ9®�¯d¹Yªº«h±N®E«a·-©qªX®nªº«XÁ���Pñ]nM}SVZ3�dKNP@]^U`Ê�]���UCONe}eqSY]nSVKNP-O�WYWY�?m^UX~EO�mnçSYP��dKNP}PfUd�X]nSVKNP���SY]nM?]nMfU£Z^K�Í��;O�WYWVUdeZn�}jkUXm^�XSYm^�XWVUdZ�]nM-O�]�O�WYm^U;ONeq�ESYP6]nMfUT�LUXmn�.òfm^Zn]�]oO�WYç¦KNPCM}SVZ*�LUdKN~¦UX]nmn�L[}ösSVUX~EO�P}P7U`Ê�j}WYSV�XSY]nWY�¦~¦UXP@]nSVKNPfUde]nMfU��LUdKN~¦UX]nmnSV��]oO�P}�LUXP@]

2DÍaSYPfeqSV�;O�]nmnS¥Ê

x4+y4 = 1ONZ*O�m^UXWVUX��O�P+]*U`Êq]^UXPfZnSVKNP.K�_k]nMfU��F�f�XWYSVe}U;O�PE�XSYm^�XWVU

x2 + y2 = 1[}]nM@�fZs�dKNPf�dUXSY��SYP}�EK�_Â]nMfU¬ZnjvUd�XSÉO�WiösSVUX~EO�P}PqÍ�|ÂSYPfZnWVUXms~¦UX]nmnSV�dZ

ds =

{

n∑

i=1

(

dxi)p

}−1

p

;

é _IKNmsm^Ud�dUXP+]B��KNmnçcK�_�å�MfUXmnPRU�Á¸OqÁ�KNP'ö1SVUX~EO�P}PqÍ�|ÂSYPfZnWVUXmB�LUdKN~¦UX]nmn�'O�Pfe'K�_-�cSYm^KNP�U�Á¸OqÁ1KNP�g%O���moO�P}�LU�LUdKN~¦UX]nmn�L[�O�~¦KNP}�6KN]nMfUXm^Z1SYP7m^UXWÉO�]nSVKNPc]^K.b�SYW § UXmn];��Z��NO�mnSÉO�]nSVKNP-O�W=j}m^K § WVUX~ ë�óq[fZ^UdU U�Á �fÁs�Yæ����9ê`Á��] �dKN�}WVe § Uym^UX~.SYPfe}Ude�_IKNm SYPfZn]oO�Pf�dU�]nM-O�] ]nMfUC®n±�¨a­�¨¢µI±N´�­N¹À«`³@®9°d­L¯oªº«E±�°3þyª`¹Y­N³@´v­N¶�[�SaÁ�U�Áy]nMfK�Z^U.K�_

�dKNPfZn]oO�P@]s~¦U;O�P'�X�}mn��O�]n�}m^UH[ é j}WÉO�PfUdZBO�Pfeh�;O�]^UXPfK�ae}Z1_IKNm H = 0

[-Znj}MfUXm^UdZ�O�Pfe'�XSYm^�X�}WÉO�m��X��WYSYPfe}UXm^ZO�Pfe��}Pfeq�}WVK�ae}Z6O�Pfe�PfK�e}K�ae}Z�_IKNm

H 6= 0ê`[�e}K<U`¤�Ud�X]nSY�LUXWY��K@�d�X�}m.SYP���O�mnSVKN�fZ�ò-UXWVe}Z¦K�_T]nMfUCP-O�]n�qÍ

moO�WsZ^�XSVUXPf�dUdZ#Á:��MfU7�LUdKN~¦UX]nmnSV�;O�WYWY��P-O�]n�}moO�WT�X�}mn�NO�]n�}m^Uh�dKNPfeqSY]nSVKNPfZCO�WYWY�fe}Udeî]^K § U`_IKNm^U7�dKNPf�dUXmnPÐKNP]nMfU7KNPfU£M-O�Pfe�[FSYP�]nMfUXSYm¦ZnSY~.j}WVUdZ^]3_IKNmn~h[F]nMfU<¯^±�´}«`¨a­N´v¯X¶�±�°7¯X³@®`üÄ­�¨¢³�®^ªº«'WYSYçLU£]nMfU�²Eªn­N´ ¯X³@®`üÄ­�¨¢³�®^ªH é U`Ê�j}m^UdZ^Z^SYP}���}P}S¥_IKNmn~ Zn�}m�_aON�dU7]^UXPfZnSVKNP�êEKNmE]nMfU.�1­N³@«o«'¯X³�®ºü�­�¨¢³@®^ª

K é _IKNm6Z^�}m�_ION�dUdZ MSYPÐZoOd�

���f�XWYSVe}U;O�P�óNË�Í�Znj-ON�dU�[�KNm;[�ZnSY~.SYWÉO�mnWY�L[k]nMfU�Z^Ud�X]nSVKNP-O�WF�X�}mn��O�]n�}m^UK_IKNmT�LUXPfUXmoO�WFösSVUX~EO�P}P}SÉO�P<Znj-ON�dUdZ;[

��MfK�Z^Uy�dKNPfZn]oO�Pf�X�'ONZTZnMfK;��P § �hösSVUX~EO�P}PqÍa�LKNP<b�UXWY~.MfKNWY]('XÍ�g�SVU.O�~¦KN�}P@]^ZB]^KEZoO�]nSVZ�_¢��SYP}�.��]nMfU�OÄÊ�SVKN~K�_*_¢m^UdUy~¦K § SYWYSY]����;ê`[%O�Pfe�]nM+�fZÈO�m^U.U`ÊfO�~.j}WVUdZ K�_�m^U;O�WYS/'#O�]nSVKNPfZ¬K�_ § ONZnSV��Zn��~.~¦UX]nmnSVUdZ KNm Zn]nSYWYW�K�_��XWÉONZ�ÍZnSV�;O�W���O�mnSÉO�]nSVKNP-O�W1KNj}]nSY~EOqÁ���P�]nMfUCKN]nMfUXm.M-O�Pfe�[F]nMfU£�LUdKN~¦UX]nmnSV�;O�W1P-O�]n�}moO�W1�X�}mn��O�]n�}m^U7�dKNPfeqSY]nSVKNPfZ~¦UXP+]nSVKNPfUde § U`_IKNm^U��dKNPf�dUXmnP<]nMfU3m^U;O�WYS/'#O�]nSVKNPfZ¬K�_�Ud @�-O�WYSY]nSVUdZBSYPR]nMfU7±º·k¨¢µ4²¦­N¹Àµ9´vª(*X³}­N¹»µ9¨¢µaªº«.��M}SV�^M�O�m^Uç�PfK;��Pñ]^KcMfKNWVe § UX]���UdUXP���O�mnSVKN�fZ3Z^�;O�WÉO�m�Ía��O�WY�fUdeñ�X�}mn�NO�]n�}m^UdZ�K�_�O�WYW�Zn� § ~EO�P}S¥_¢KNWVe}Z#Ác��P�j-O�mn]nSV�X�}WÉO�m;[Zn�f�oM<SYPfUd +�-O�WYSY]nSVUdZ ��SY�LU�SYPfUX��SY]oO § WVU.m^UXWÉO�]nSVKNPfZ § UX]���UdUXP��X�}mn��O�]n�}m^UdZ ��M}SV�^M<_¢�}PfefO�~¦UXP+]oO�WYWY��m^UXWÉO�]^U�]^K]nMfU'µ4´-¨¢®`µ9´}«ºµI¯¦´v­�¨¢³@®^ª7K�_1]nMfUdZoUEZn� § ~EO�P}S¥_IKNWVe}Z é WYSYçLUE]nMfUXSYm�Z^Ud�X]nSVKNP-O�W��X�}mn�NO�]n�}m^UdZ#[À]nMfUXSYmyZ^�;O�WÉO�myö1SVU`Í~EO�P}P}SÉO�P��X�}mn��O�]n�}m^U

τ��M}SV�^M?O;�LUXmoO��LUdZÈ]nMfU�Z^Ud�X]nSVKNP-O�W*�X�}mn��O�]n�}m^UdZ;[%O�Pfe�[�~¦KNm^U¦�LUXPfUXmoO�WYWY�L[=O�WYW�]nMfUXSYm

m^Ud�dUXP+]nWY�7SYP@]nm^K@eq�f�dUde10=©qª`´-í ¯X³�®ºü�­�¨¢³@®^ªº«X[���M}SV�oMc]^KN�LUX]nMfUXm���SY�LU WYSYçLUÈ]nMfUÈËBÌT�1Í�Z^]nmn�f�X]n�}m^U¬K�_À]nMfUÈö1SVU`Í~EO�P}P}SÉO�P�~EO�P}S¥_¢KNWVe}Z�SYP+�LKNWY�LUde-êTO�PfeR�X�}mn�NO�]n�}m^UdZT��M}SV�oM'_¢�}PfefO�~¦UXP+]oO�WYWY�'m^UXWÉO�]^U¬]^KE]nMfU3Z^M-O�jvU¬��M}SV�^M]nMfUdZ^U'Zn� § ~EO�P}S¥_¢KNWVe}Z£ONZ^Zn�}~¦UcSYPî]nMfUXSYm£O�~ § SVUXP+]C��KNmnWVe�[�SaÁ�U�Á�ª(2�¨¢®`µ9´}«ºµI¯'¯d³@®`üÄ­�¨¢³�®nª`« é ONZ § UXSYP}�?U`Ê@Íj}m^UdZ^Z^Ude § ��]nMfU6Z^Ud�dKNPfe?_¢�}PfefO�~¦UXP+]oO�W�_IKNmn~

hO�Pfe?]nMfUEPfKNmn~EO�W��X�}mn��O�]n�}m^U£]^UXPfZ^KNm

R⊥[À]nMfU6~¦K�Zn]

§ ONZnSV�.K�_���M}SV�^MSVZB]nMfU7«3*X³}­N®^ªn¼'²¦ª^­N´�¯d³@®`üÄ­�¨¢³�®nªH2[iU`Êqj}m^UdZ^ZnSYP}�7]nMfU.Zn� § ~EO�P}S¥_IKNWVe���Z¬Z^�}m�_ION�dU�]^UXPqÍ

ZnSVKNP�ê`Á�|fKNm O�WYW=Z^�}m�_ION�dUdZMSYPR���f�XWYSVe}U;O�PóNË�Í�Znj-ON�dU�[vO�WYm^U;ONeq�h�F�}WVUXmTZnMfK#��UdeR]nM-O�]TO�WY��Od�qZ

K ≤ H2UX�LUXmn����MfUXm^U¦KNPM[%Ud +�-O�WYSY]��RMfKNWVeqSYP}�RO�]¬UX�LUXmn�'jkKNSYP+]¬S¥_�O�PfeKNP}WY�RS¥_n[%ON�d�dKNm^eqSYP}�c]^KhO£]nMfUdKNm^UX~ÛK�_

�RUX�fZnP}SVUXm;[M

SVZEUXSY]nMfUXmEj}WÉO�P-O�mCKNmEZnj}MfUXmnSV�;O�WaÁ ��MfUcZ^K�Í��;O�WYWVUde å�MfUXPqÍaSYPfUd +�-O�WYSY]nSVUdZ£�LUXPfUXmoO�WYSVZ^Uc]nM}SVZ���}WVUXm�ÍaSYPfUd @�-O�WYSY]����dKNPf�dUXmnP}SYP}�R]nMfU¦eqSY~¦UXPfZnSVKNPfZyO�Pfe�dK�eqSY~¦UXPfZnSVKNPfZ3K�_�]nMfU¦Zn� § ~EO�P}S¥_IKNWVe}ZÈSYP@�LKNWY�LUde�[�dKNPf�dUXmnP}SYP}�ñ]nMfUcç�SYPfeîK�_3O�~ § SVUXP@]7Znj-ON�dUdZ¦SYP@�LKNWY�LUde O�PfeÐ�dKNPf�dUXmnP}SYP}��]nMfUhç�SYPfe}ZEK�_ÈSYP+]nmnSYPfZ^SV�RO�PfeU`Êq]nmnSYPfZnSV�¬�X�}mn��O�]n�}m^UdZsSYP@�LKNWY�LUde�Ás��Pfe'O.Z^� § ~EO�P}S¥_¢KNWVe'SVZsZoO�SVec]^K § U6­N´�µI¼+ªn­N¹%«º³ ½ ²¦­N´fµ °X±�¹¥¼hSYP'Z^KN~¦UZnj-ON�dU���MfUXPCSY]�ON�X]n�-O�WYWY�6m^U;O�WYS/';UdZ�O�]�UX�LUXmn�LKNPfU K�_iSY]^Z�jvKNSYP@]^Z�]nMfU�Ud @�-O�WYSY]��6SYP£Z^�f�^M£�LUXPfUXmoO�WYWY�EMfKNWVeqSYP}�SYPfUd @�-O�WYSY]��LÁ3QTSY�LUXP�]nMfUXSYmBSYP+]nmnSYPfZ^SV�.Zn]nmn�f�X]n�}m^UdZ;[v��M}SV�^M<PfKNmn~EO�WYWY�RZn� § ~EO�P}S¥_¢KNWVe}ZBM-O;�LU�]nMfU¬]^UXPfe}UXPf�X�moO�]nMfUXm�PfKN]*]^Ky�^M-O�P}�LU�[�SVe}U;O�WvZ^� § ~EO�P}S¥_¢KNWVe}Z�]nM+�fZ�U`ÊqjvUXmnSVUXPf�dU�]nMfU1WVU;ONZn]�jkK�Z^ZnS § WVU1U`Êq]^UXmnP-O�Wv�X�}mn��O�]n�}m^U

Page 8: æ#êhomepages.inf.ed.ac.uk/.../GIELIS/KRAGUJEVACFINAL.pdf · |ÂSY }m^U ðf¡ X }mn LUdZ é U#ê O Pfe _`ê* SY LUXP § 7Ud @ -O ]nSVKNP ëLê MfU3j}WÉO P-O m X }mn LUdZT SY LUXP

��M}SV�^MÐ~.SY��M+] § UCSY~.jkK�Z^UdeîKNPî]nMfUX~ § �?]nMfU7_¢KNmn~¦Z.]nMfUX��ONZ^Zn�}~¦UhSYP���SY�LUXP:Zn�}mnm^KN�}PfeqSYP}����KNmnWVe}Z;[~¬�f�oM<WYSYçLUySYPRZ^KN~¦U�Z^UXPfZoU3U`ÊqjvUXmnSVUXPf�XSYP}�C]nMfUyWVU;ONZn]BjkK�Z^ZnS § WVUyZn]nm^UdZ^Z���M}SV�^M<]nMfUyWYSY�@SYP}�7�dKNPfeqSY]nSVKNPfZBSYP]nMfUdZ^UTO�~ § SVUXP+]1Z^j-ON�dUdZ*~EO;�.SY~.jvK�Z^U�KNP6]nMfU�Zn� § ~EO�P}S¥_IKNWVe}Z;[�]nMfUs�Xm^U;O�]n�}m^UdZ;[���M}SV�^MCM-O�j}jkUXPE]^K3WYSY�LU1SYP]nMfUdZ^U¬��KNmnWVe}Z;Á1��]sSVZBONZsZ^�f�^Mi[�P-O�~¦UXWY�'ONZ1SVe}U;O�WÀgiKNm^UXP+]('XÍaM@�@jkUXm^Zn�}m�_aON�dUdZBSYP4�'SYP}çLK;�1Znç�S¥Í�Znj-ON�dUdZ;[v]nM-O�]SYP�j-O�mn]nSV�X�}WÉO�m3O�WVZ^KC]nMfU�giUX~EO��V]nm^U`Ía�}P}SY�LUXm^Z^UdZ3P-O�]n�}moO�WYWY��UX~¦UXmn�LUde<SYP�]nMfU�m^Ud�dUXP+]¬Zn]n�feqSVUdZB~¦UXP@]nSVKNPfUde§ U`_¢KNm^U�Á���MfUBZn� § ~EO�P}S¥_IKNWVe���Z��X�}mn�NO�]n�}m^U �dKNPfeqSY]nSVKNPfZseqSVZ^�X�fZ^Z^UdeCO § K#�LU�[qWYSYçLUT]nMfU�¯o±N´}«`¨a­�´�¯X¶EKNm�SYP+��O�mnS¥ÍO�Pf�dU1K�_k�dUXmn]oO�SYPE�X�}mn�NO�]n�}m^UdZ�O�Pfe�]nMfU50=©qª`´-í�µ9´vª(*X³}­N¹»µ9¨¢µaªº«`[�ONZ���UXWYWvONZ�KN]nMfUXmFP-O�]n�}moO�W-KNPfUdZ�WYSYçLU1U�Á �fÁÂ]nMfU6Cµ9¹4¹»²¦±N®nªXí ¯^±N´v¼Nµ9¨¢µI±N´}«c�Yæ� Ä��[%UX]^��ÁY[iSYPZ^KN~¦U.Z^UXPfZoU¦O�WYWFU`Ê�j}m^UdZ^ZÈ�NO�mnSÉO�]nSVKNP-O�WFj}mnSYPf�XSYj}WVUdZ é Z^UdU.U�Á �fÁ¦O�WVZ^K�¸ë����9ê�O�WVKNP}�¦]nMfUÈWYSYPfUdZ1Zn]n�feqSVUde7SYPhj-O�mn]nSV�X�}WÉO�m § �7e����sm^�X�C��MfKN~.jfZ^KNPiÁ

��Õ�ÜdÕÀtvÕ�pyz=Õ�ã�Yæ`�3JfÁNQTSVUXWYSVZ;[76Cµ4«oÃ#³�´�¼NµYù+ª�«`³;·}ªX®ºüı�®º²Eª`´ ½ µ 8 ½ ­N² ½ ±#ª`«`ÁNÌ�UX�1Z^WVUX]n]^UXmÂK�_}]nMfU-��UXWY��SÉO�P9��O�~ § K@KB\�K@�XSVUX]��

æ;óq[iæ� � �ûq[�ë��ÄÍ�ë�ûqÁ�¸ëÄ�3JfÁ�QTSVUXWYSVZ;[�:×ù+ªX´vª`®`µ¢¯¦ù@ªn±N²EªX¨¢®`µ¢¯7¨¢®n­N´}«9°X±�®º²¦­�¨¢µI±N´�¨9©�­�¨T³�´fµ ��ªº«6­R¹¥­�®¢ù+ªC®�­N´�ù+ª7±�°¦´v­�¨¢³@®n­N¹�­N´�¼

­ ½ «`¨¢®n­L¯d¨�«º©�­º·}ª`«`[q�s~¦UXmnSV�;O�P7JLKN�}mnP-O�W�K�_;��KN]oO�P@�+ �� é óLê���P+��SY]^UdeC\@jkUd�XSÉO�W<�ÂO�jkUXm;[¸ë�����óq[-ó�ó�óÄÍ�ó�ó��qÁ�¸óÄ�3JfÁfQTSVUXWYSVZ#[=�BÁ>��UXSYmnSYPf�^ç@ÊCO�PfeC�1Á>��ONZn]nSÉONUXPfZ;[@?-³#·}ª`®(*X³}­L¼N®`µ¢¯X«È¾�µ9¨9©£®�­�¨¢µI±N´�­�¹k­N´v¼.µ9®º®n­�¨¢µI±N´�­N¹v«º¶N²�í

²EªX¨¢®`µIª`«`[7��m^K��dUdUdeqSYP}�LZ3K�_1]nMfU���]nM?�såA�!Zn��~.jvK�Z^SY�}~ KNP�\�KNWYSVeB�'K�e}UXWYSYP}��O�PfeñO�j}j}WYSV�;O�]nSVKNPfZ;[ë�����óq[-ë�û�ëÄÍ�ë�û�÷qÁ

� ðN�3JfÁ�QTSVUXWYSVZ;[9C}­N®`µ¢­L¨¢µ¢±N´v­N¹ ?-³#·}ª`®9°X±�®º²�³�¹¥­¯d³@®`ü�ªº«T°X±N®Eý�þø­N´v¼'ÿ�þ ùN®n­º·-©+µI¯h­N®X¨9«`[D��m^K��dUdUdeqSYP}�LZ.K�_]nMfU���]nMcÇKNmnWVeE�'�}WY]nS¥Í å�KNPq_IUXm^UXPf�dUyKNP<\@��Z^]^UX~.SV�dZ;[vå�� § UXmnPfUX]nSV�dZBO�Pfec��Pq_IKNmn~EO�]nSV�dZ;[flFKNWY�}~¦U3ly¡]nMfU3å�KN~.j}�}]^UXm \��XSVUXPf�dU¬O�Pfeh��P}��SYPfUdUXmnSYP}�f[vë�����ðf[iæ�æ� ÄÍoæ;ë�ðfÁ

�¸÷Ä�3JfÁvQTSVUXWYSVZ�O�Pfeh� ÁvQBUXmoO�]^Z;[F: ½ ±L¨a­N´fµI¯^­N¹}·fª`®o«�·}ªn¯d¨¢µ9ü�ªE±N´¦·v¹¥­N´�¨F«º©�­º·}ª¬²¦±;¼+ª`¹»µ9´@ù�[G�Fm^K@�dUdUdeqSYP}�LZTK�_]nMfU���P@]^UXmnP-O�]nSVKNP-O�Wiå�KNPq_IUXm^UXPf�dUTKNPCå�KN~.j}�}]nSYP}�f[få�KN~.~¬�}P}SV�;O�]nSVKNPfZ1O�PfeCå�KNP+]nm^KNW��%Ud�oM}PfKNWVKN��SVUdZ;[lFKNWaÁ}lB�`[fë�����ðf[-ë�û�÷ÄÍ�ëL�NëqÁ

�¸ûÄ�3JfÁvQTSVUXWYSVZ;[GHº´fü�ªX´-¨¢µ4´�ùE¨9©qª�0µ9®�¯d¹YªX[vQBUXP}SÉO�O�WI��m^UdZ^Z#[f�sP@]���UXmnji[�ë�����óqÁ���#�¬Ç�Áke����sm^�X�c��MfKN~.jfZoKNPi[KJ�´L��®�±�¾F¨9©�­N´�¼�Mi±N®`².[iå�O�~ § mnSVeq�LUN!�P}SY�LUXm^ZnSY]��O��m^UdZ^Z;[%å�O�~ § mnSVeq�LU�[

æ� qæ#��Á���Ä����ÍQP.Á�å�MfUXPi[R#sµIªX².­�´f´fµI­N´y«º³ ½ ²¦­N´fµ °X±N¹Y¼�«X[ÄSYPNSÈ­�´�¼ ½ ±;±Nà ±�°�þ µ T�ª`®^ª`´-¨¢µI­N¹U�sªn±N²EªX¨¢®`¶�[NlFKNWaÁ@æ�[ é Ude}Z;Á|�Á}ËTSYWYWVUXP'O�PfehgFÁflFUXm^Zn]nmoONUXWVUXP�ê`[-��WVZoUX�@SVUXm;[f��~¦Zn]^UXm^efO�~h[-ë������qÁ�� Ä����ÍQP.Á�å�MfUXPi[V6£©�­�¨y¯^­N´ ¾�ªR¼L±¾�µ¢¨9©XWÈ­�«º©7Y «'ª`² ½ ª^¼L¼Nµ9´@ù�¨9©qªn±N®^ª`²[Zo[�\�K�K��^MfK;� JfÁ���O�]nMiÁ�ó��q[

ë�����ðf[-ó���óÄÍ�ó�ó��qÁ�Yæ��Ä��\vÁFbBONUdZ^UXPi[��3Á�\\�U § UXçLK;��S^]'O�Pfe:gFÁ�lFUXm^Zn]nmoONUXWVUXPi[�#1ª`¹Y­�¨¢µI±N´}« ½ ªX¨¢¾�ªoª`´ôµ4´�¨¢®ºµ9´}«ºµI¯h­N´v¼ñª(2L¨¢®ºµ9´}«ºµI¯

¯X³�®`üÄ­�¨¢³�®nªº«X[F�BmoO���}�LUX��ON�ÈJfÁ_��O�]nMiÁfë�÷q[-ë�����óq[iæ;ó� ÄÍoæXð+÷qÁ�Yæ�æ`�3|�Á�Ë�SYWYWVUXPi[f\vÁ�bTONUdZoUXPi[=�ñÁ=�=UX]nm^K#�@S^]`Í��=KNmn�+O`^UX�hO�Pfe6gFÁ�lFUXm^Zn]nmoONUXWVUXPi[G:�´7µ9´vª(*d³q­N¹»µ¢¨¢¶ ½ ªX¨¢¾�ª^ª`´�µ9´fí

¨¢®`µ4´f«ºµI¯ ­N´�¼Eª(2L¨¢®ºµ9´}«ºµI¯B«d¯^­N¹Y­N® ¯X³@®`üÄ­�¨¢³�®^ªÈµ9´füÄ­�®ºµI­N´-¨9«%°d±N® ¯^±;¼Nµ9²Eª`´}«`µ¢±N´cý£ª`² ½ ªn¼L¼�µ4´@ùN«`[fJfÁ}QBUdKN~hÁ��M@��Z#Á-÷�ëq[-ë�����ðf[iæ��qæ`Íoæ�æ;ëqÁ

�Yæ;ëÄ��\vÁ�bBONUdZ^UXP:O�Pfe�gFÁÀlFUXm^Zn]nmoONUXWVUXPi[KHX¼+ªn­�¹4¹»¶ñª`² ½ ªn¼+¼+ªn¼?«a·}­L¯oª`í ¨¢µ4²Eªº«X[�J-Á �RO�]nMiÁ ��M@��Z#ÁÀð+÷q[�ë�����ðf[æXðR L�#Íoæ;÷qæ��qÁ

�Yæ;óÄ���BÁi�y��Ì�UXSYWYWa[;?kªX²yµ9ía#sµaª`²¦­N´f´fµI­N´cù+ª^±N²EªX¨¢®º¶cb1¾�µ¢¨9©R­º·L·v¹úµI¯^­�¨¢µI±N´}«�¨a±C®nªX¹¥­�¨¢µ9ü;µ¢¨¢¶�[����;ONe}UX~.SV�%��m^UdZ^Z;[ÌTUX�1P*KNmnçv[%æ� ���óqÁ

Page 9: æ#êhomepages.inf.ed.ac.uk/.../GIELIS/KRAGUJEVACFINAL.pdf · |ÂSY }m^U ðf¡ X }mn LUdZ é U#ê O Pfe _`ê* SY LUXP § 7Ud @ -O ]nSVKNP ëLê MfU3j}WÉO P-O m X }mn LUdZT SY LUXP

�YæXðN�¬Ç�Á>�ed}M}PfUXWa[�þ µ T�ª`®^ª`´-¨¢µI­N¹-ù+ªn±�²¦ªd¨¢®º¶Vf�¯d³@®`ü�ªº«gbF«º³�®4°d­L¯oª`«È­�´�¼¦²¦­N´-µ °d±N¹Y¼�«`[}\@]n�fe}UXP+]���O�]nMfUX~EO�]nS¥Í�;O�W%giS § moO�mn�L[-��~¦UXmnSV�;O�PE�RO�]nMfUX~EO�]nSV�;O�W�\�K@�XSVUX]��L[-ë������qÁ

�Yæ;÷Ä�3byÁ-�LKNP�bTUXWY~.MfKNWY]('L[ehÀª ½ ªX®.¼�µIªji�­�¨9«d­L¯`©qª`´�¾�ª`¹¥¯X©qªE¼@ª`®��sª^±N²EªX¨¢®ºµaª�kd³l��®`³�´�¼+ªy¹»µIªaù+ª`´mb�ª;Æúù+ÆFµ4´: ½ ©q­N´�¼q¹»³@´�ù+ª`´Ek3n�®�)�©+µ9¹Y±�«X±º·�©+µIªÈ³�´�¼O�sªn±N²EªX¨¢®`µIªX[-J��}P}��M-O�P}�LZ;[vå��qÊqM-O��fZ^UXPi[iæ� ��L��Á

�Yæ;ûÄ���BÁ�ö1SVUX~EO�P}Pi[7hÀª ½ ª`®s¼Nµaª-S ¶o·}±�¨9©qªº«;ª`´6¾�ªX¹¥¯`©}ªs¼+ªX®e�sª^±N²EªX¨¢®ºµaª-kd³V��®º³�´�¼+ª�¹úµaªIù+ª`´�[�ª;Æúù+Æ}µ9´E�1­N³@«o«`í¯`©}ª�M*¹/oL¯`©qªX´-¨9©qªn±N®`µaªgb #sµaª`²¦­N´f´}«d¯`©qªe#KoN³�²EªÈ³�´�¼Npcµ9´}Ã�±N¾À«ºÃ#µ4íc6�ªX¹»¨I[f�=UX� § PfUXm;[-giUXSYjm'dSY�f[=æ� ���ðfÁ

�Yæ#�#��\vÁ�\vÁFå�MfUXmnPi[qHº´-¨¢®n±;¼N³}¯d¨¢µI±N´Ð¨a±jS¬­N´�¼ ½ ±;±�ñ�°�þȵ T�ªX®nª`´�¨¢µ¢­N¹"�sªn±�²¦ªd¨¢®º¶�[�lFKNWaÁ�æ�[ é Ude}Z;ÁÂ|�ÁÂËTSYWYWVUXPO�Pfehg*Á}lFUXm^Zn]nmoONUXWVUXP�ê`[f��WVZoUX�@SVUXm;[f��~¦Zn]^UXm^efO�~h[vë������qÁ�Yæ��Ä�3|�Á�Ë�SYWYWVUXP7O�PfeCgFÁ@lFUXm^Zn]nmoONUXWVUXP é Ude}Z;Á�ê1¡�S¬­N´�¼ ½ ±#±�Ã.±n°�þȵ T�ªX®nª`´�¨¢µ¢­N¹ �sªn±�²¦ªd¨¢®º¶�[�lFKNWaÁ�ëq[q��WVZoUX�@SVUXm;[��~¦Zn]^UXm^efO�~ é ]^KEO�j}jkU;O�mºê`Á�Yæ� Ä�3� Á�JfÁvÇÏSYWYWY~¦KNm^U�[��sªn±N²EªX¨¢®`¶c­N´�¼Bi-±º·}±N¹¥±oùN¶c±�°�?-³ ½ ².­�´fµ °d±N¹Y¼�«"H>C%[ é Ude}Z#Á�|�ÁkËTSYWYWVUXP<O�Pfe�gFÁvlFUXm�ÍZn]nmoONUXWVUXP�ê`[v\@SYP}�+O�jvKNm^U�[=æ� � �ëq[iæ�æ`Íoæ;ûqÁ�¸ë��Ä��\vÁ�bsSYWVe}U § moO�Pfeq]<O�Pfeá�yÁ1�im^KN~ § Oq[9)1­N´�±º·v¨¢µ9²�³@²�b�pR­�¨9©qªX².­L¨¢µÉ«d¯`©qª.��®º³�´�¼N²�³@«`¨�ª`®ñ¼+ªº«rC}±N¹»í

¹úÃ�±N²�²Eª`´vª`´�[�\@jvUXç�]nmn�}~h[fbTUXSVe}UXW § UXmn�f[%æ� ���ûqÁ��\vÁ<��O�P+�h]nM-O�P}çqZ O�m^U3eq�fUy]^K���U;O+�ÂUdUX]^UXm^Z;[iJLUXP}P@�'ÌTK�Z^ZoON�^çRO�Pfe4�BmnSVZn]^K�_�giUXP�KN�R_IKNm�]nMfUyUdeqSY]nSYP}�K�_À]nM}SVZ�]^U`Ê�]�s

���}]nMfKNm;��Z�ONeqm^UdZ^ZoUdZ;¡

•JfÁ1QTSVUXWYSVZ#[�QBUXP}SV�;O�jáå�K}Á ÌyÁ l�ÁY[1g%O�P}�LURl*O�Pôö1�}�fZ § m^K�Ud�^çqZn]nmoO�O�]�æ�æ;ûq[���ë��qæ��q[s��P+]���UXmnjkUXPi[��UXWY��SY�}~hÁ

•\vÁBbBONUdZ^UXP éut ê'O�Pfe�gFÁTlFUXm^Zn]nmoONUXWVUXPi[9�.Áv!yÁBgiUX�}�LUXP�\�Ud�X]nSVKNP��s_CQBUdKN~¦UX]nmn�L[¦å�UXWVUdZn]nS �PfUXPqÍWÉO�O�Pßë�����B[K��ó����qæ�giUX�}�LUXP é b�UX�LUXmnWVUdU#ê`[���UXWY��SY�}~ ì � éut ê£�%UX~.jkKNmoO�mn�ÏONeqm^UdZoZ;¡X!�P}SY�LUXm^ZnSVefONee}UEQBmoO�P-ONefOq[%|-ON��ÁEå�SVUXPf�XSÉONZ;[=ËBUXj}];ÁEQBUdKN~¦UX]nmnSÉO7K�_��=KNjvKNWVKN��SÉOq[�æ����L��æ6QTmoO�P-ONefOq[À\@j-O�SYPi[=ONZjkK�Zn]�Í�e}K�� |ÂÇÐ�sÍal�WÉO�O�Pfe}UXm^UXPq��Á