33

math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

  • Upload
    ngonhan

  • View
    230

  • Download
    0

Embed Size (px)

Citation preview

Page 1: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

������������ �������������� �������������������������� ����!������"$#%�'&��

(*),+.-0/2143658709:/2;<3=-0)>@? ;<3BA0CED8-0/F;G-IHJAK1438L6MNH:OQP�R8S6T �QUWV'X P*Y X S6TZY[S6\^] UW_a`�_a` S V P8bNc _aU P*Y V S UW_a` S6T U Y �P*T U YedfPhgjilk0Y Um_ S VZnoVp_ S6qrPhg�sKP*b n Y[P8q U i8btS�uvc0i _4U P*Y V

w�xpyGz�{B|~}~}~}�|az�������� y (����%��� � y ( �

] Um_4`�X PoS�� X6U S6Y _4V�U Y � )l� Um_4` P8c _ bNP VZV P8g��8S6Y[S�T4i8b UW_�n ]�S X i8Y^P*��R U P*c V b n TaS�uoc U TaS_a` S X P�S:� X6U S�Y _4V�_ PJ�KS U Y�d�) > VpnoVZ_ S�q y ( ��UNV i8b V P X i8bNbWS6\�i Vpn�Vp_ S�q�P8g ? ;<3BA0CED8-0/2;G-IHHa�67ED8/2;<3=-E�:)��Jg _ S�Y�P*Y[Sl] U bWb��KS U Y _ S6TaS VZ_ S6\ U Y�i�gFi8q U b n P8g V c XB` s[TZP*�[bNS�q V T4i _a` S6T_a` i8Y�i VaU YE�*bNS�P*Y[Sh� U Y _4`EUNV�X i V S�P*Y[S�TaS6uvc U TaS V�_4` S X PoS�� X6U S6Y _4V P8g _4` S w�x�_ Peb U SU Y V P*q�S ��y2�~{:|~}~}~}�|B�.�4�:  i8Y[\¡P*Y[S�P*� _ i U Y V i8Y U Y[\ U R U \[c[i8b¢s[TaP*�EbNS6q£� n � U R U Y[� _4` S�p¤ Rhi8bNc[S V¥U Y � ) > �*i U Y^P*Y[S X i8Y^�*S _ T U \^P8g�\[S�Y[P*q U Y0i _ P*T V )�¦oP*q�S�P8g _4` S�q�P VZ_P*��R U P8c V uocES VZ_aU P*Y V§_ P�i Va¨ i8�KP*c _§V c XB` ilgFi8q U b n i8TaSh©ª �«�¬�V�_a` S6TZS�i8Y�i8bN�8P*T Um_4` q­] `EUNXB` ] U bWb®\[S _ S�Taq U Y[S   g¯P8T�S~i XB` i VaVaU �*Y[S�\ V S _ Phg

Rhi8bWc[S V Phg _a` S �Z¤�  ] ` S _4` S�T _a` S X P*TZTaS V sKP*Y[\ U Y[��° U P*s ` i8Y _4U YESls[TaP8�[bNS�q ` i VV P*bNc _4U P8Y V6  S UW_a` S6T U Yd±P*T U Y ��²

ª "�«³ P*T�R8ihbNc[S V P8g _a` S �p¤ g¯P*T�] `[UNXB`l_a` S Vpn�Vp_ S�q UNV�V P*bWc[�[bWS  8UWV�_a` S6TZS´i8Y�i8bW�*P*T Um_4` qg¯P8T´S:µ `[U � UW_4U Y[��i V P*bWc _4U P*Y ²

³ P*T U Y[\ U R U \[c0i8bKq�S�ql�KS6T V P8g V c XB` ilgFi8q U b n8 �Um_§UNV i8b V P�Y0i _ c[T4ihb _ P�i VZ¨ ©ª.¶ «¡· i8Y�]�S¥\[S VaX T U �KS _4` S V S _ Phgji8bWb V P*bNc _aU P*Y V6  P*T´S:R*S�Y UW_aV´VZ_ Tac X:_ c[TZS ²ª �«¬�VJ_4` S�s ` Tai V S�¸¹\[S�Y VaUm_�n P8g V P*bNc _4U P8Y V6º q�S~i8Y U Y[�8g¯c[b   i8Y[\ U g V P   ] ` i _'X i8Ye]�S

V i n i8�KP*c _§UW_B²»´` S�i _Z_ S�q�s _aV´_ P�i8Y V ]�S�T _a` S V S�uvc[S VZ_4U P*Y V§` i~R*S�bNS�\ _ P _4` S U Y _ TaPo\[c X:_4U P8Y�Phg�Y[S:]U \[S~i V i8Y[\ _a` S V S ` i~R*S��*S�Y[S6Tai _ S�\�Y[S�]¼uvc[S VZ_4U P*Y V )^�'Y V P*q�S�P8g _a` S6q ¬ S:µ�sKS X:_s[TZP*�*TZS VaV ] UW_4`EU Y _4` SeYES:µ _ \[S X i8\[S   i8Y[\ ¬$` i~R8SeTaS VZ_ T UNX:_ S6\�q n�V S6bmg _ P _a` S V S U Y_a` S _ S.µ _ �tS�bNP�]¥)¥½�TaP*�8TaS VZV¾U Yq�i _a` S6q�i _aUNX6V c V c0i8bNb n q�S6i8Y V s[TZP�R*S6Y�TZS V cEb _4V �¿�[c __a` S6TZSJi8TZS X i V S V ] ` S6TaSÀS�R8S6Y�i']�S6bNbhÁZc VZ_4U k0S6\ X P*Y�ÁZS X�_ c[TaS _4` TaP�] V YES�]�b U � `�_ P*Y _4` SVp_ Tac X�_ cETaSlP8g _a` S V c[�oÁZS X:_ ) yF³ P8T VaU q U bÂi8T´TZS~i V P*Y V�  ]�S6bNb�q�P _4U Rhi _ S6\ X P*q�s[c _ i _4U P*Y VX i8Y��KS ` S6bNs�g¯c[b<�h�[c _jX P*q�s[c _ i _4U P8Y V Y[P _ �0i V S�\lP*Y�iÀ\[S�S6slg¯S�S6b U Y[�´g¯P*T _4` S VZ_ Tac X:_ c[TZSP8g _4` S V c[�oÁZS X:_e` i~R8SÃ�*S�Y[S6Tai8bNb nf_ cETaY[S�\�P*c _�_ PÄ�KSÅi�]�i VZ_ SÃP8g _4U q�S8) � Æ c _�¬` i~R8S�YEP _´U Y X bNcE\[S6\ _4` P V S's[TZP*�[bWS6q V¥yFV c XB` i V�_4` S¥Ç U S�q�i8YEY�È n sKP _4` S VaUWV i8Y[\ _4` S

(

Page 2: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

Æ�U T XB`�� ¦o] U Y[YES6T _ P*Y���° n S6T X P8Y=ÁZS X:_ c[TZS � P*Y�] `[UNXB`�¬ \[PJYEP _ S.µ�stS X�_ g¯c[T _4` S�T�s[TZP*�*TZS VaV] Um_4`[U Y V P Va` P*T _ i _aU q�S VZX i8bWS8)»´` P*cE� `�_4` S Vp_ c[\ n P8g V P*bWc _4U P*Y V�U Y�d i8Y[\ U Y � q�i n bWP�P ¨ R8S6T n�VaU q U bNi8T y i8Y[\

U Y[\[S6S�\�]�S6TZS��KS6b U S:R*S6\�g¯P*T§i�bNP*YE� _aU q�S _ P��KS V P �: 0Um_ Y[P�] i8sEstS6i8T V´_4` i _§_4` S n i8TaSi X:_ c0ihbNb n R*S6T n \ U�� S6TaS�Y _ i8Y[\ _4` i _�_4` S _a` S6P*T n g¯P8T V P*bNc _aU P*Y V,U Y � ` i V q�c XB` q�P*TaSVp_ Tac X�_ cETaS _4` i8Y _4` i _ g¯P*T V P*bWc _4U P*Y V�U Y d�) »´` S�q�i U YÅTaS~i V P*Y g¯P*T _4`[UWV�V S6S6q Vl_ P�KS _4` i _¥U Y _a` S$T4i _4U P8Y0i8b X i V S _a` S VZnoVp_ S6q y ( � \ES�k0Y[S V i�Rhi8T U S _�nU Y _4` S V S6Y V S�Phgi8bW�*S6�ET4i UWX �*S6P*q�S _ T n8  ihY[\q�i8Y n P8g _a` S _ PoP*b V P8g _4` i _ \ UWVaX6U s[b U Y[S X ihY�KS�c V S6\¿� U YX P*Y _ T4i VZ_~ 0U Y _4` S U Y _ S6�*Tai8b X i V S¥]�S�\EP�YEP _§¨ Y[P�] ` P�] _ P�q�i ¨ S VaU �*Y U k X i8Y _ c V S�PhgV c XB`^_ P�P8b V )�È®P�]�S�R8S6T   g¯P*TJRhi8T U S _aU S V P8g�\[S6�*TZS6S��*TZS~i _ S6T _a` i8Y�� UW_'UWV P*Y[b neU Y¡bWP�]\ U q�S�Y VaU P*Y _4` i _ ]�S n S _J¨ Y[P�]�S6Y[P*cE� ` Phg _a` S¥�*S6P*q�S _ T n g¯P8T UW_�_ P��KS'c V S:g¯c[b<)

� Y U uvc[S6b n* j_a` S�ÈJihTa\ n � Um_a_ bWS�]�P�Po\Ãq�S _a` Po\ UNV c V S�g¯cEb��tP _4` g¯P*T U Y _ S��*T4ihb�i8YE\g¯P*TÀT4i _4U P8Y0i8b�s[TZP*�[bNS�q V � Um_ ]�i V \[S VZU �*Y[S�\g¯P*T U Y _ S��*T4ihb�s[TaP*�EbNS6q V �[c _¾X i8Y¡i8b V P��tSi8sEs[b U S6\ _ P�Tai _4U P*Y0i8b¢s[TZP*�[bNS�q VÀU Y�s[TaP�ÁZS X�_aU R8S V s0i X S   �tS X i8c V S _4` S6Y _a` S w�x�U Y y ( �i8TZS ` P8q�P*�8S6Y[S�P*c V i8Y[\ Um_ \[PoS V Y[P _ q�i _Z_ S�T§] ` S _4` S6T¾]�S _ TZS~i _®_4` S�Rhi8T U i8�[bNS VÀz��i VÀU Y _ S6�*Tai8b�P*T§T4i _aU P*Y[i8b<) »´` S6TZS UWV i��[T U S�g�\ UNVZX c VZVaU P8YP8g _a`[UNV q�S _a` P�\ U Y� ��   i8YE\i X P8q�s[TZS ` S�Y VaU R*S V c[TpR*S n�U Y��������<)°¾S6YEP _ S�� n�� _a` S�Rhi8T U S _�n \[S:k0Y[S6\�� n�_4` S�S6uvc0i _aU P*Y V$y ( � i8Y[\ebWS _���� �KS$i8Y n

Rhi8T U S _�n � U Tai _4U P*Y0i8bWb n S�uvc U Rhi8bWS6Y _�_ P � P�R*S6T � ) »´` S§s[TaP*�EbNS6q P8gKk0Y[\ U Y[� V P*bWc _4U P*Y VP8g y ( �¢U Y � UNV¢_4` S V i8q�S�i V�_4` i _ P8g0k0YE\ U Y[�®T4i _aU P*Y[i8bvstP U Y _4V P*Y ��  ] `[UNXB`$UWV i8bWq�P VZ__a` S V i8q�SJi V�_a` i _ P8g�k0Y[\ U Y[��Tai _4U P*Y0i8b sKP U Y _aV P*Y � � )%ÈÀS6Y X S y S.µ X S6s _ sKP VZVaU �Eb n g¯P*T� c[S VZ_4U P*Y y ° � ih�tP�R*S � P*Y[S$S.µEsKS X:_4V¥_4` S$sETaP*sKS6T _4U S V P8g _4` S$Tai _4U P*Y0i8b V P*bWc _4U P*Y V Phgy ( ��_ P �tSS VaV S6Y _4U i8bNb n \[S _ S6Taq U YES6\�� nf_a` Se� U T4i _aU P*Y0ihb´S6uvc U Rhi8bWS6Y X S X bNi VaV P8g � �i8YE\ _4` S�]�i n U YÅ] `[UWXB` i8bN�8S6�[Tai UNX �*S6P8q�S _ S6T VlX bNi VaVZU g n Rhi8T U S _aU S V�Va` P*c[bN\Ãs[TaP�R U \[Si _ bNS6i VZ_ i�k0T Vp_ TaP8c[� ` �*c U \[S _ P _a` S X bNi VaVZU k X i _4U P*YPhg,° U P*s ` i8Y _4U YES�sETaP*�[bWS6q V��_a` P*c[� `�_4` S n q�i U YEb n$Vp_ c[\ n � U Tai _4U P*Y0i8b0S�uvc U Rhi8bWS6Y X S¾P�R*S6T ¶ Tai _4` S6T _a` i8Y�P�R*S�T � )Æ c _�Um_ \[P�S V i _,_a` SÀq�P*q�S6Y _,V S�S6q _4` i _%_4` SÀ�*S6P8q�S _ T UWX§X bÂi VaVaU k X i _4U P8Y�YES6S6\ V�V P8q�Sq�P�\ U k X i _4U P*Y V i8YE\�TZS�k0Y[S�q�S�Y _4VÀU g UW_§UNV´_ P�k _§_a` S¥YocEql�tS�T � _4` S�P*TaS _4UNX TZS V cEb _4V i8YE\X P*Y=ÁZS X:_ c[TZS V )� UW_a` P*c _ bNP VZV P8gK�*S6YES6T4ihb UW_�n ]�S X i8Y�i VZV c[q�S _4` i _!��UNV i8� V P8bNc _ S6b n�U TZTaS�\[c X6U �[bNSh)

³ P*T U g � ` i V s[TZP*stS�T X P*q�sKP*Y[S�Y _4V \[S:k0Y[S�\�P�R8S6T � Um_�UNV S6Y[P8c[� `�_ P�i Va¨�_4` SJuocES V �_aU P*Y V i8�KP�R8Seg¯P*T�S6i XB` P8g _4` S¡s[TaP*sKS6T X P*q�stP8Y[S6Y _4V �¥i8Y[\ U g � UNV�_4` S¡c[Y U P*Y±PhgRhi8T U S _aU S V�X P*Y�ÁZc[��i _ S'P�R8S6T �¼_4` S6Y�i8Y n T4i _4U P8Y0i8bIsKP U Y _ P*Y � b U S V P*Y _a` S U Y _ S�T V S X �_aU P*Y Phg _4` S V S X P8Y=ÁZc[��i _ S V�  ] `EUNXB` UNV ies[TaP*sKS6T V c[��Rhi8T U S _�n P8g � )^¦ U Y X S�]�S X ihY\[S VaU Y[�*c[bNi8T U�" S � � n i�� U Tai _4U P*Y0i8b _ T4i8Y V g¯P*Taq�i _aU P*Y�\[S�k[Y[S6\¡P�R*S�T �� tUW_®UNV Y[i _ c[Tai8b_ P X P*Y X S6Y _ Tai _ S�P*Y _4` S X i V S¥] ` S6Y � UNV s[TZP=ÁZS X:_4U R8S�i8Y[\�Y[P*Y VZU Y[�*cEbÂi8T�)»´` S�\[S�k0Y UW_4U P*Y V i8Y[\ _4` S,uvc[S Vp_4U P8Y V i8�KP�R*S X i8Y��tS,�*S6YES6T4ihb U#" S6\ _ P¾ihYli8TZ� UW_ T4i8T n

i8bW�*S6�ET4i UWX Yvc[ql�KS6T´k0S�bN\�i8Y[\ _4` S¥T U Y[�$P8g U Y _ S6�*S�T V§U Y Um_ � _a` S�i8Y V ]�S6T V ihTaS'c V c0i8bNb n

Page 3: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

¨ Y[P�]§Y�P*T X P*Y�ÁZS X�_ c[TaS�\ _ P��KS¥S VZV S6Y _aU i8bWb n�_a` S V ihq�S�i V P�R*S�T � P*TJd  [_4` P8c[� `�_4` Ss[TZP�Phg V�X i8Y��tSÀR*S�T n q�c XB`�` i8TZ\[S6T�) y2Æ c _�_4` S6TaS¾i8TaS®S:µ X S�s _4U P*Y V ��g¯P*T�S.µ[ihq�s[bWS  v_4` Sq�P�\Ec[bÂi8T UW_�n P8g´S6bWb U s _4UWXlX c[TpR*S V P*Y[b n�` P*bN\ V P�R*S�T � ) ��»´` S�uvc[S VZ_4U P*Y V i8�tP�R8S X ihYi8b V Pe�KS�sKP V S�\ g¯P8TlP _4` S6Tlk0S�bN\ V Phg®Yvc[q��tS�T � _a` S6P8TaS _aUNX�U Y _ S6TZS VZ_ � U Y s0ihT _4UWX c[bNi8Tg¯P*T®k0Y UW_ S�k0S�bN\ V ihY[\eg¯P*T X P*q�sEbNS _aU P*Y V P8g�i8bW�*S6�[Tai UNX Yvc[ql�KS6T¾k0S�bN\ V�� i8Y[\e] ` S�YP*YES Vp_ c[\ U S V ° U P*s ` i8Y _4U Y[S´s[TaP8�[bNS�q V%UW_%UWV S VaV S6Y _4U i8b _ P X P*Y VZU \[S�T _a` S V SJP _a` S6T�k0S�bN\ Vi8b V P[) ¬ g � UNV \[S:k0Y[S�\�P�R*S�T�i'k0S6bW\ �  v_a` S V S _ P8gIstP U Y _4V P*Y � \[S�k0YES6\�P�R8S6T � ] U bWbi8bm]�i noV �KS�\[S�Y[P _ S�\�� n ��y���� ) ¬ g ��y�����UWV Y[P _ S6q�s _�n ]�S V i n�_4` i _ ��UNV ��3=M 7[L6MNH�;G-� ) ¬ Y _4` S V sKS X�U i8b X i V S®] ` S6TaS � �����= �_4` S X P*q�s[bWS _4U P*Y�P8g¢i8Y�i8bN�8S6�[Tai UNX Yvc[ql�KS6Tk0S�bN\ � i _'_a` S$s[bÂi X S�   ]�S�ihb V P V i ne_4` i _ �@UNV MN3�9aD8MGM����3=M 7EL6MNH�D8/��K) ³ TaP*q Y[P�]P*Y�]�S�\ES6Y[P _ S¥� ne�� i8Y n�X P8q�s[bWS _aU P*Y�Phg � � _a` c V¥�� q�S~i8Y V§� P*T V P*q�S �� )�¾Y[S�q�i�ÁZP*T�TaS6i V P*Ylg¯P8T X P*Y VaU \[S6T U Y[� V P*bWc[� U b UW_�n'U Y X P*q�sEbNS _ S�k0S�bN\ V i8Y[\ U Y�k0Y Um_ S

k0S�bN\ V¥UWV¾_4` i _ i�Y[S X S VaV ihT n^X P*YE\ UW_aU P*Yeg¯P*T y ( �®_ P��KS V P*bWc[�[bNS U Y ��  g¯P*T'S:µEi8q�s[bWS  UWV¾_4` i _¥Um_'UNV'V P*bNcE�[bNS U Y^S:R*S�T n ��� ) »´` S X P8Y[\ UW_aU P*Y�P8g V P*bNc[� U b Um_�nU Y¡S�R8S6T n�����UNVX P*q�sEc _ i _aU P*Y[i8bNb n \[S X�U \0i8�EbNS8� V S6S �o)���P*TaS�P�R*S6T   i _ bNS6i VZ_ g¯P8T�s[T U q�S V�� g¯P*T�] `[UWXB`_a` S VpnoVZ_ S�q y ( �j` i V �*PoP�\�TaS�\[c X:_4U P8Y�q�P�\ �� �_4` S§k0T VZ_�VZ_ S�s U Y�\[S X6U \ U Y[� V P*bWc[� U b UW_�nU Y �� �UNV'_ P�\[S X�U \[S�] ` S _a` S6T _4` S�TaS�\[c X S�\ VZnoVZ_ S6q UNV¥V P8bNc[�[bWS U Y _a` S�k0Y UW_ S�k[S6bN\� ³´y�� � P8g � S�bNS6q�S6Y _aV )

� S6P*q�S _ S6T V ihTaS¡c V S6\ _ P VZ_ cE\ noU Y[��Rhi8T U S _aU S V P�R8S6T�Y[P*Y � i8bW�*S6�[Tai UNX i8bNb n � X bWP V S�\k0S�bN\ V�� ��] ` i _ q�i ¨ S V ° U P*s ` i8Y _aU Y[S$s[TZP*�[bWS6q V'V sKS X6U i8b UNV¾_a` S�Yvc[q��tS�T � _a` S6P8TaS _aUNXY0i _ c[TZS®P8g _4` SÀk0S6bW\ V�� ) Æ c _�UW_,V S6S6q V�_a` i _ P*Y[b n i�g¯S:]�P8g _a` S®sETaP*sKS6T _4U S V sKS X c[b U i8T_ P V c XB` k0S�bN\ V i8TaS�c V S�g¯c[b U Y _4`EUNV�X P*Y _ S.µ _~ �V P _a` i _ i��*S�P*q�S _ S�T¥Y[S6S�\ Y[P _ bNS6i8TaYqlc XB` Yvc[q��tS�T � _a` S6P8T n'U Y�P8Ta\[S�T _ P´]�P8T ¨ P*Y�° U P*s ` i8Y _aU Y[S�s[TaP*�EbNS6q V )��¾Y _4` S�P _a` S6T` ihY[\   i$Yvc[ql�KS6T � _4` S�P*T UWVZ_ ]�P*c[bW\��KS'] UNV S _ P�bWS~i8TZY�uoc UW_ S¥i$bWP _ P8g��*S6P*q�S _ T n )° U P8s ` i8Y _4U Y[S,s[TaP*�EbNS6q V ]�S6TZS�k0T Vp_�U Y _ TaPo\[c X S�\$� n ° U P*s ` i8Y _ c V P8g > bNS:µEi8YE\[T U i  

_a` SJbNi VZ_ P8g _4` S¾�*TaS6i _ � TZS6S ¨ q�i _4` S�q�i _4UNX�U i8Y V6  ] ` P$b U R*S�\�i _�V P*q�S _aU q�S®�KS _ ]�S6S�Y� �8��Æ ) · )¢ihY[\�� �*� > )¹°l)W���[c _�` Sl]�i V�` i8Y[\ UNX ihs[stS�\�� n�` i�R U Y[��P*Y[b n P*Y[S�bWS _a_ S6Ti~Rhi U bNi8�[bWS _ P'TaS�s[TaS V S6Y _ Rhi8T U i8�[bWS V6  i8bNb _4` S�P _4` S6T V �KS U Y[�¾c V S6\ U Y X bÂi VZVaUWX i8b � TZS6S ¨�_ PTZS6s[TZS V S�Y _�V sKS X�U k X YocEql�tS�T V ) ¬ Y[\ U R U \[c0i8bv° U P*s ` ihY _aU Y[S%s[TZP*�[bWS6q V ]�S6TZS VZ_ c[\ U S6\l� nV c XB` �*TZS~i _ q�i _4` S6q�i _4UWX6U ihY V i V�³ S6TZq�i _6 �� c[bNS�T,i8Y[\ � i8c VaV ) Æ c _,UW_ ]�i V È U bW�tS�T _~º Vi8\E\[TaS VaV%_ P _4` S ¬ Y _ S�TaY0i _4U P8Y0i8b · P*Y[�8TaS VZV�U Y�(�� �*� ] `[UWXB`�Vp_ i8T _ S�\ _4` SÀ\ES�R*S�bNP*sEq�S�Y _P8g�i VZnoVZ_ S6q�i _4UWX'_4` S�P*T n )�È UNV�_ S6Y _4` s[TaP*�EbNS6q@i Va¨ S6\¿©

� U R8S6Y¥i§° U P8s ` i8Y _4U Y[S�S�uvc0i _4U P*Y'] UW_a` i8Y n Yvc[ql�KS6T�P8goc[Y ¨ Y[P�]§Y�uoc[i8Y��_aUW_aU S V i8YE\] UW_a` T4i _aU P*Y[i8b U Y _ S��*T4ihb�Yvc[q�S6T UNX i8b X PoS�� X6U S6Y _4V © _ P�\[S:R UWV SiÅs[TaP X S VaV i X�X P*TZ\ U YE� _ PÃ] `EUNXB`±Um_�X ihY±�tS\[S _ S6Taq U YES6\±� n iÅk0Y Um_ SYvc[q��tS�TÀPhg�P*sKS6Tai _4U P*Y V ] ` S _4` S�T _a` S¥S6uvc0i _aU P*Y UWV´V P*bWc[�[bNS U Y�T4i _aU P*Y0ihbU Y _ S6�8S6T V )

Page 4: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

��P VZ_ Phg _4` S,S~ihTab n ]�P*T ¨ P*Y�° U P*s ` i8Y _4U Y[S%S6uvc0i _aU P*Y V ]�i V�X P8Y X S�TaY[S�\l] UW_4` T4i _aU P*Y[i8bTai _4` S�T _4` i8Y U Y _ S6�*Tai8b V P*bNc _aU P*Y V ��s[TaS V c[q�i8�[b n È U bN�KS6T _ stP V S6\ _4`[UWV s[TaP8�[bNS�q U Y_ S6Taq V P8g U Y _ S6�8T4i8b V P*bWc _4U P*Y V �tS X i8c V S V c XB` i¾s[TZP X S VaV g¯P8T U Y _ S6�*Tai8b V P*bNc _aU P*Y V ]�P*c[bN\i8c _ P*q�i _4UWX i8bWb n sETaP�R U \[S _a` S X P*TZTaS V sKP*Y[\ U Y[��s[TaP X S VaV g¯P*T¾Tai _4U P*Y0i8b V P8bNc _aU P*Y V i8b V P  � n TaS VZ_ T UNX:_4U Y[� _ P _a` S V sKS X6U i8b X i V S�] ` S6Y _a` S�S�uvc0i _4U P*Y V i8TaS ` P*q�P*�*S�Y[S6P8c V ) ¬ Y_a` P V S X P*Y�k0\[S6Y _ \[i noV �KS�g¯P8TaS _a` S ³�U T Vp_ �ÅP*TZbN\��Ãi8T  8UW_ ]�i V i VZV c[q�S6\ _a` i _�V c XB` i8Ys[TZP X S VZV qlc Vp_ S.µ UWVZ_ ���[c _�U Y�(���� � �i _aU Áai V S�R U��X§VZ` P�]�S�\ _a` i _,_4`[UWV ]�i V�U q�stP VZVaU �[bNSh)¬ Y[\ES6S6\ ` S%S:µ `[U � UW_ S6\�i�sKP*b n Y[P*q U ihb w�y2� ��� {.|~}6}~}6| � �*�IV c XB`�_a` i _�_4` S6TaS X i8Y[Y[P _ S.µ UNVZ_i8YÃi8bN�8P*T Um_4` q ] `[UNXB` ] U bNb%\[S X6U \ES�g¯P*T�S�R8S6T n � U R*S6Y � ] ` S _4` S6T w � ��UWV�V P*bNcE�[bNSU Y U Y _ S��*S6T V )�È UWV s[TaPoP8g UNV s0i8T _ P8g _4` Sl�8TaS~i _ s[TaP8�*T4i8q P*Y�\[S X6U \[i8� U b UW_�n�U Y UW_aU i _ S6\� n ���P*\[S�b<���*PoP�\�i X6X P8c[Y _4V P8g UW_'X i8Y��tSlg¯P8c[Y[\ U Y � ( � �   s[s ������� ��� ��P*T � � �2) »´` SX P*TaTZS V sKP*Y[\ U Y[��uocES VZ_aU P*Y^g¯P*T¥T4i _aU P*Y[i8b V P*bWc _4U P*Y V'UWV�Vp_4U bWbjP*sKS6Y¿� ¬ i8q i8q�P*Y[� _4` SR8S6T n g¯S:]�] ` P��KS6b U S�R8S _4` i _ÀUm_ q�i n�` i�R8S�i$sKP VaUm_4U R*S¥i8Y V ]�S�T6)Æ c _ S�R8S6Y U g _a` Sei8Y V ]�S6T _ P _a` SeihY0i8bNP8�*c[S�P8g'È U bW�tS�T _~º V$_ S�Y _a` s[TZP*�[bWS6q g¯P*T

Tai _4U P*Y0i8b V P8bNc _aU P*Y V,UNV stP VaUW_aU R8S   P8Y[SJqlc VZ_ S.µEsKS X:_´_4` i _ i V S6s0i8Tai _ S¥ihbN�*P*T UW_a` q�] U bWb�KS�Y[S�S6\[S�\�g¯P*T�S6i XB` ¨oU Y[\ P8g´Rhi8T U S _�n ) »´` c V ]�S Va` i8bNb,Y[S6S�\ Y[P _ P8Y[S�i8bW�*P*T Um_4` q�[c _ i8Y U YEk0Y Um_�n Phg _a` S6q�) ¦�P�Yvc[ql�KS6T _4` S6P*T UNVZ_aV \[S6sKS6Y[\ÄP*Y _4` S�\ES�R*S�bNP*sEq�S�Y _� n �8S6P*q�S _ S6T V P8g�i8Y�i8\[S�uvc0i _ S X bÂi VZVaU k X i _aU P*Y�P8g�Rhi8T U S _4U S V ) > _�_4` S'q�P*q�S6Y _6 �V c XB`i X bÂi VaVaU k X i _4U P8Y UNV TaS6i V P8Y0i8�[b n�X P*q�s[bNS _ S¥g¯P*T X c[TZR8S V i8YE\ V c[TpgFi X S V6  �Ec _¾Um_JUWV®Vp_4U bWbg¯Tai8�*q�S6Y _ i8T n S�R8S6Y U YÅ\ U q�S6Y VZU P*Y �E� V PeYvc[ql�KS6T _a` S6P*T UNVp_4V�` i�R8S _ P X P*Y X S6Y _ T4i _ SP*Y X c[TpR*S V i8Y[\ V c[TZgFi X S V�  i8Y[\�P*Y X S6T _ i U Y�s0i8T _4UNX c[bÂihTab nlVaU q�s[bWS ¨vU Y[\ V Phg¿Rhi8T U S _�n�U Y`[U � ` S�T´\ U q�S6Y VZU P*YI)

��)�¿C0H�¥D=�B��H ��1.;G-I9:; AKMNH�D8-K5�/GC0H �§1ZD87 H:1����¡D8-0;G-Å38L:�:/F1:709:/F;<3=-0)�S _ � �tS�i�Rhi8T U S _�n \[S�k[Y[S6\^P�R*S6T � ) ¬ g �@UNV bWP X i8bNb n�V P*bWc[�[bWSli _ S:R*S6T n s[bNi X SlPhg��  ]�S V i n�_4` i _�UW_�V i _aUNV k0S V�_a` S�'D=�B��H�9B3=-K58;G/F;<3=-0) ¬ g ��y����%UNV Y[P _ S�q�s _�n�_a` S6Y �X S6T _ i U YEb n�V i _4UNV k0S V®_4` S�ÈJi VZV S X P*Y[\ Um_4U P8Y¿)´� ` i _ q�i ¨ S V®_a`[UNV TaS�q�i8T ¨ R8ihbNc0i8�EbNS UNV_a` i _´_4` S�È®i VaV S X P*Y[\ Um_4U P*Y UNV´X P*q�sEc _ i8�EbNS � _4` i _§UNV�  P*Y[S X ihY�\[S X6U \[S U Y�k0Y Um_ S6b nq�i8Y n Vp_ S6s V ] ` S _4` S�T�i¡� U R*S�Y �QV i _4UWV k0S V�_a` S�ÈJi VZV S X P*YE\ UW_aU P*Y¿) »´`[UNV g¯P*bWbNP�] Vg¯TZP*q _a` S¥Y[S:µ _À_ ]�P�bNS�q�q�i V )#������������ H:/ � L:HJD8-�D�L:��3=M 7o/ H:M��;G1.1BHa58709:;FL6MNH"!=D81.;<H:/��3$#®58;GO�H:-E�:;<3=-&% 5oH('%-IHa53)!8H:1�/GC0H*'%-0;G/ H�'�H:Mm5 ��� � ³�y(+*�-, �¿C0H:-/. y0+8�21 /GC0H�-07oO�L:H:1�3$#lA03=;G-0/G�¡3=-��5oH('%-IHa5�3)!8H:1 �31 �6D8/2;N�('�HB�

4 . y0+8�657+ � 498;: + �)<0{>=�?

@ C0H:14H®/¯C0H�9B3=-E�:/<D8-0/ : 5oH�A0H:-K5=�¥3=-0M��3=-�/¯C0H'5oH�A814HBH¥D8-K5�58;GO�H:-E�:;<3=-e3$#B� D8-K5$;N�9B3=OJAK7o/ D�L6MNH ,

Page 5: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

»´`[UWV g¯P8bNbNP�] V g¯TaP8q _4` S��ÃS U b X P*Y�ÁZS X�_ c[TaS V6  g¯P8TÀ] `[UWXB`V S6S ���t�[c _ ]�S~i ¨ S6T¾TZS V cEb _4V] `[UWXB` i8TaSei8\[S6uvc0i _ Seg¯P*T _a` Ses[TZS V S6Y _ i8s[s[b UNX i _4U P8Y ]�S6TaS ¨ Y[P�]§Y�qlc XB` S6i8Tab U S6T�)¦ U Y X S _4` S VaU Y[�*c[bNi8T´sKP U Y _4V P8g � b U S¥P*Yi�s[TZP*sKS6T V c[��Rhi8T U S _�n8 K_a` S6TZSli8TZS�i _ q�P VZ_: {$+ � <0{ P8g _4` S6q   ] ` S�TaS : {§UWV i8b V P X P*q�s[c _ i8�[bNSh) ¬ _ g¯P*bWbNP�] VÀ_a` i _JU g + S:µ X S6S6\ V iX P*q�sEc _ i8�EbNS��tP*cEY[\¡\[S6sKS6YE\ U Y[��P8Y[b n P*Y _a` Sl\[S��*TaS�S$i8Y[\¡\ U q�S6Y VZU P*Y�P8g � _4` S�Y� X P*Y _ i U Y V i$Y[P*Y VZU Y[�8c[bÂi8T�stP U Y _ \[S�k0YES6\P�R8S6T � )� P�] bNS _ � �KS^i8Y i8� V P*bNc _ S6b nfU TaTaS�\[c X�U �[bWSRhi8T U S _�n \ES�k0Y[S�\�P�R*S�T � ) ¬ g �

` i V �*PoPo\�TZS6\[c X:_4U P*Y�i _ ��  ] `[UNXB`�` i8s[sKS6Y V g¯P*T,i8bNb0�Ec _ i¥k0Y Um_ S X P*q�s[c _ i8�[bNS V S _ Phgs[T U q�S V ��  \ES6Y[P _ S _4` i _ TZS6\[c X�_4U P*Yl� n��� ) ¬ g �$UWV bNi8Ta�*S,S6YEP*c[� `¿ 8UW_ g¯P*bNbWP�] V g¯TZP*q _4` STZS6q�i8T ¨vV i8�tP�R8S _4` i _���� lX P*Y _ i U Y V i�Y[P*Y VZU Y[�8c[bÂi8T§sKP U Y _�� \[S�k0YES6\eP�R*S�T � ³´y��K� )»´` STaS V c[b _ ] `[UNXB` g¯P8bNbNP�] V�  ] `[UNXB`fUNV�¨ Y[P�]§Y i V È®S6Y V S6b º¹V ¢S�q�q�i _a` P*c[� `f_4` SU \[S~ilP8g _4` S�sETaPoP8g��*P�S V �0i XB¨�_ P�¦ U T ¬�V i*i X�� S:] _ P*Y   Y[P�] Va` P�] V§_4` i _ � X P*Y _ i U Y Vi$sKP U Y _� \[S:k0Y[S6\�P�R*S6T �� )#��������� � H:/ � L:H�D8-�D�L:��3=M 7o/ H:M��;G1.1BHa58709:;FL6MNH*!=D81.;<H:/��5oH('%-IHa5�3)!8H:1 � D8-K5�MNH:/��� L:H�;G/¯��14Ha58709:/2;<3=- q�P�\ � , + # �� 9B3=-0/<D8;G-E�DÃ-I3=-E�:;G-�A87oMmD81$A03=;G-0/ � 5oH('%-IHa53)!8H:1 � ³�y��K� /GC0H:- � 9B3=-0/ D8;G-E��D�-I3=-E�:;G-�A87�MmD81®A03=;G-0/ � 5oH('%-IHa5¡3)!8H:1 �� @ C03���H14Ha58709:/2;<3=-�q�Po\ � ;N� � �,¬ Y�R U S�]�P8g _4`[UWV6 �_ P�\[S X�U \[S$] ` S _4` S�T �rV i _4UNV k0S V'_a` S�È®i VaV S X P*Y[\ Um_4U P*Y¡P*Y[S�P*Y[b n` i V¥_ P XB` S XB¨�U Y[\ U R U \[c[i8bNb n�V P*bNcE� U b UW_�n�U Y � i8Y[\ U Y¡k0Y Um_ S6b n q�i8Y nÃ�� ) � i XB` Phg_a` S V S XB` S XB¨vV§X ihY��tS Va` P�]§Y _ Pl�KS¾i�k0Y UW_ SJs[TZP X S VZV6  c VZU Y[� U \[S6i V�VZU q U bÂihT _ P _a` P V SU Y _a` S¥s[TaPoP8g�P8g �S6q�q�i�(*)> gFi8q U b n � P8g'Rhi8T U S _4U S V�UNV�V i U \ _ P V i _4UWV g nÄ_4` S ¥D=�B��H ��1.;G-I9:; AKMNH U g¥S�R*S�T n

� X P*Y _ i U Y[S�\ U Y � ihY[\ \ES�k0Y[S�\±P�R*S�T � ] `[UNXB` V i _aUNV k[S V�_4` S�ÈJi VZV S X P*YE\ UW_aU P*Yi X:_ c0ihbNb n�X P8Y _ i U Y V i _ bNS6i VZ_ P*YESÀsKP U Y _ \ES�k0Y[S�\�P�R8S6T � ) > �*i U Y   i'gFihq U b n�� UNV�V i U \_ PÃi8\Eq Um_ @ HaD��ÅD:A�AK143��*;GO�D8/2;<3=- U g®S�R*S�T n � X P*Y _ i U Y[S�\ U Y � i8YE\ \[S:k0Y[S�\±P�R*S6T��  i8YE\ V c XB`_4` i _ ��y����´UNV Y[P _ S6q�s _�n8 K` i VÀ_a` S¥g¯P*bNbWP�] U Y[�$s[TZP*stS�T _�n ©�� U R*S�Yei8Y nk0Y UW_ S V S _ P8g�s[bÂi X S V ��i8Y[\ X P*TZTaS V stP8Y[\ U YE�Y[P*Y ��S6q�s _�n�V S _4V�� ��� ��y ���6� P8stS�YU Y _4` S� � i8\ UWX�_ P*sKP*bNP*� n* 0_4` S�TaS UWV i�stP U Y _�� U Y ��y���� ] `[UNXB` b U S V®U YS~i XB` P8g _4` S� � ) ¬ Y _4` S V sKS X6U i8b X i V Sl] ` S6Y � X P*Y VZUNVp_4V P8g,i VaU YE�*bNS�R8ihT U S _�n ��  i8YE\ ��y����®UNVY[P _ S�q�s _�n*  ]�S VZU q�s[b nV i n�_4` i _ � i8\[q UW_aV ]�S6i ¨ i8s[s[TZP�µ U q�i _aU P*YI) ¬ Y X P*Y _ Tai VZ__ P _a` S�ÈJi VZV S X P*Y[\ UW_4U P*Y   ] ` S _4` S6T � i8\Eq Um_4V ]�S~i ¨ i8sEs[TaP~µ U q�i _4U P*Y UNV§U Y��8S6Y[S�T4i8bY[P _§X P*q�s[c _ i8�[bNSh�Eg¯P*T§i8Y U q�sKP*T _ i8Y _ S:µ X S6s _aU P*Y   V S6S � ��� �<)»´` Seq�P Vp_�U q�sKP*T _ i8Y _ gFi8q U b U S V ] `[UWXB` i8TaS ¨ YEP�]§Y _ P ` i~R*SS UW_a` S6T�P8g _a` S V S

s[TZP*sKS6T _aU S V¡y ihY[\±] `[UNXB` i X�_ c[i8bNb n ` i�R8S¡�tP _4` � i8TaS _a` SgFi8q U b U S V Phg¥uoc[i8\[T UWX6V Phgi8Y n � U R8S6Y�\ U q�S�Y VaU P*Y¿� _4`EUNV ]�i V s[TZP�R*S6\�� n � U Y ¨ P�] Va¨vU g¯P*T,uoc[i8\[T UWX6V P�R8S6T � i8YE\� n ÈJi VZV S�g¯P8T�uvc0i8\ET UNX�V P�R8S6T$i8YÃi8Ta� UW_ Tai8T n ihbN�*S��[T4i UWX Yvc[ql�KS6T�k[S6bN\I) Æ c _ q�i8Y ngFi8q U b U S V6  S:R*S�Y�P8goR*S�T n¥VaU q�sEbNS�Rhi8T U S _4U S V6  \[P§Y[P _¢V i _4UWV g n S UW_a` S6T _a` S%ÈJi VZV S�½�T U Y X6U sEbNS

Page 6: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

P*T%]�S~i ¨ i8s[s[TZP�µ U q�i _aU P*YI) yF³ P8T�S.µ[ihq�s[bWS   Y[S Um_4` S�T�P8g _4` S�q ` P8bN\ V g¯P*T�YEP*Y VaU Y[�*c[bNi8TX c[� UNX'V c[TZgFi X S V ) ��¬ _ÀUNV�_4` S6TaS:g¯P*TaS¥Y0i _ c[Tai8b _ P�i VZ¨��� ��������� � � 3=1�D A8; !8H:- #�D8O$;GM� � 1 @ CED8/JD81BH�/GC0H�38L:�:/21.709:/2;<3=-E��/�3�/¯C0H 'D=�B��H��1.;G-I9:; AKMNH�D8-K5�/�3 @ HaD���D:A�AK143��*;GO�D8/2;<3=-��³ P*Tl]�S~i ¨ i8sEs[TaP~µ U q�i _4U P*Y _4` S6TaS UNV i¡Rhi8T U i8Y _ Phg _a`[UNV uvc[S Vp_4U P8YÄ] `[UWXB` q�i n �tS�KP _4` q�P8TaS U Y _ S�TaS Vp_4U Y[�li8YE\�S~i VaU S�T _ Pli8Y V ]�S6T6) ³ P*T�i8Y[P _a` S6T%]�i n P8g Vp_ i _aU Y[�¥]�S~i ¨i8sEs[TaP~µ U q�i _4U P*Y�P*Y � UNVÀ_ P V i n_4` i _¾U g ��y����®UNV Y[P _ S�q�s _�n�_a` S6Y UW_®UNV \[S6Y V S U Y_a` S�i8\[S6b UNX¾V s0i X S ��y2��%��� � ��y ���6� ) »´`[UNV´V c[�*�*S VZ_aVÀ_a` S'g¯P*bNbWP�] U Y[�[©��� ��������� �� 3=1¾D&A8; !8H:- � 1 3=16#�D8O$;GM� � 1 @ CED8/�9aD8-eL:HJ�6D8;F5�D�L:3=7�/%/GC0H�9:MN3��:7o14H3$# ��y���� ;G-e/¯C0H�D�5oH:M ;<9J��AEDo9BH ��y2�� �ÈÀP�]�S�R8S6T  ¾_4` S�TaS^i8TaS¡gFi8q U b U S V g¯P*T�] `EUNXB` � c[S Vp_4U P8Y (�\[PoS V Y[P _�V S6S�q _ P �KSÅiV S6Y VZU �[bWS�uvc[S VZ_4U P*Y _ Pei VZ¨ � _a` S V S�s[TZP*�0i8�[b n�U Y X bNc[\ES�g¯P*T�S:µEi8q�s[bWS�i8bWbjgFi8q U b U S V PhgRhi8T U S _aU S V P8g��*S6Y[S�T4i8b _�n sKS8)�¦�P�P*Y[S Va` P*c[bN\��0i XB¨ c[s � c[S VZ_aU P*Y^('] UW_4`��� ������������� 3=1 @ CED8/ �=;G-K5=�À3$# #�D8O$;GM ;<HB��;N�ÀH:;G/¯C0H:1¢AED81./�3$#���70HB�:/2;<3=-�(�D¾��H:-E�:;FL6MNH�670HB�:/2;<3=-/ 3�D=� ���»´` S�P*Y[b neVZnoVp_ S6q�i _aUNX P*� Vp_ Tac X�_4U P*Y _ P _4` S�È®i VaV S�½�T U Y X6U s[bNSl] `EUNXB`^UNV'¨ Y[P�]§Y

UWV�_4` S Æ T4ihc[S6T� �i8Y U YÃP8� VZ_ Tac X:_4U P8Y  j_4` P8c[� ` P*� VZ_ TZc X�_aU P*Y VlX i8YÃ�tS�g¯P8c[Y[\ U Y _4` Sb UW_ S�T4i _ c[TaS§] `[UWXB` i8TaS¾YEP _´Æ Tai8c[S�T � ��i8Y U YI) �S _�� �KS'i�9BH:-0/F1ZD8MK�:;GOJAKMNH'D8M AoH6L61ZD �_a` i _�UWV6  i VZU q�s[bNS�i8bN�*S��[T4iÀ] `[UNXB`$UWV k0Y Um_ S�\ U q�S�Y VaU P*Y0i8boP�R*S�T%iJk0S6bW\ � ] `[UWXB`$UNVjUm_4VX S6Y _ TZS8) � i XB`¡V c XB` i8bW�*S6�[Tai X P*Y VZUNVZ_aV6  g¯P*TÀk�µES�\��Qi8Y[\ %   P8g%i8bWb %�� %Åq�i _ T UNX S V] Um_4` S�bNS6q�S6Y _aV$U Y�i\ U R UNVZU P*Y�ihbN�*S��[T4i��­] UW_4`�X S6Y _ TZS � ) » ]�P X S6Y _ Tai8b VaU q�sEbNSi8bW�*S6�ET4i V P�R8S6T � i8TZS Ha�67o; !=D8MNH:-0/ U g _4` S n±` i~R8S _4` S V i8q�Sc[Y[\ES6Tab noU Y[� \ U R UNVZU P*Yi8bW�*S6�ET4iE) ³ P*Taq�i _aU P*Y�P8g _ S6Y V P*T�sETaPo\[c X�_aV P�R*S�T � � U R*S V�_4` S V S _ P8g´S�uoc U Rhi8bNS�Y X SX bÂi VZV S VÀ_4` S VZ_ Tac X:_ c[TZSlP8gji X P8q�qlc _ i _aU R*S¥�*TZP*c[s  KX ihbNbNS�\ _4` S �§1ZD87 H:1�A8143=7�A^P8g �i8YE\Å]§T Um_a_ S6Y Æ T y���� ) »´` S6TZS UWV i X i8Y[P8Y UNX i8b UNV P*q�P8Tas `[UWV q � © Æ T y�� =�"! �$# dg¯P*T¥S6i XB` � �%i8YE\ _4` S6TaS UNV i X i8Y[P*Y UNX ihb UWV P*q�P*TZs `[UNV q%�'& © Æ T yF���(!%)=�E| {?+*  ¢_4` SY[P8Y _ T U R U i8bI\ U R UNVZU P*Y�i8bW�*S6�ET4i$P�R*S6T � �KS U YE� _a` S X bNi VaVZUNX ihb¿uvc0i _ S�TaY U P*Y V )

�S _-, �tS�i8Y�S6bWS6q�S6Y _ P8g Æ T y ��� � _ S6Y V P*T U Y[� , ] UW_4` i8Y n��� � U R8S V T UWV S _ P�i8YS�bNS6q�S6Y _ Phg Æ T y����6�.  i8Y[\ _4`EUNV S�bNS6q�S6Y _¥UWVJ_ T U R U i8b¢g¯P*T'ihbNq�P VZ_ i8bNb �K) »´` S6TZS UNV i8YS.µ[i X�_ÀV S�uocES6Y X S

�/. Æ T y ����. 0 Æ T y ���6�1. �$# d . �E|\[cES _ P�ÈJi VZV S  0U Y�] `[UNXB`�_4` S _a`[U TZ\�q�i8s UWV�_4` S V c[q P8g _4` S(� � � UW_�_ S6bWb V c V ] ` S6YeiV S _ P8gjS�bNS6q�S6Y _aV6  P*Y[S U Y�S~i XB`Æ T y ���6� i8YE\ei8bWq�P Vp_ i8bNb _ T U R U i8b  [X i8Y��tS��*S�Y[S6Tai _ S�\g¯TZP*q V P*q�S¥S�bNS6q�S6Y _ P8g Æ T y���� )

2

Page 7: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

� P�] bWS _ � �KS�i X P8q�s[bWS _ S�Y[P8Y VaU Y[�*c[bNi8T¾Rhi8T U S _�n \[S:k0Y[S6\ÃP�R8S6T � i8Y[\ � i8Y��� 7�O�D�8D�D8M AoH6L61ZD�P*Y � � _a` i _ÀUNV�  i VZU q�s[bNS'ihbN�*S��[T4il] UW_a`�X S6Y _ TZS ��y �l� ] `[UWXB`` i V iÃ�*PoPo\ V sKS X�U i8b U#" i _4U P8Y i _ S�R8S6T n sKP U Y _ P8g � ) »´` Se�*TaP8c[s�P8g'S�uoc U Rhi8bNS�Y X SX bÂi VZV S V P8g > " c[q�i n i�i8bW�*S6�[Tai V P*Y � UNV \[S�Y[P _ S�\� n�Æ T y �l� ) ¬ g ��UNV ihY n sKP U Y _ Phg��  ] UW_a` k0S6bW\lP8gK\[S�k[Y UW_aU P*Y ��y ���:  ]�S´P*� _ i U Y�i VaU q�s[bWS�ihbN�*S��[T4i �ly ��� ] Um_4`lX S6Y _ TZS��y ��� � nlV stS X6U ihb U#"�U Y[�Ài _ � ) ³ P*T�ihbNbv�[c _ k[Y UW_ S6b n q�i8Y n ��  ]�S ` i~R*S � oy�$y � =�a�����g¯P*T¾i8bNb � � i8\ UNX sKP U Y _aV�� P*Y � ) »´` c V i�Y[S X S VZV i8T n�X P*YE\ UW_aU P*Yg¯P8T _4` S$S.µ UWVZ_ S6Y X SP8g�i$T4i _4U P8Y0i8bKstP U Y _ � P*Y � UNV�_4` i _ g¯P8T�S�R8S6T n � _4` S�TaS VZ` P*c[bW\��KS¥i�� � i8\ UWX sKP U Y _� � P*Y � V c XB`�_4` i _ � � ��y�$y � ���a�%� � g¯P*T§i8bWb ��} y � �¦ U q U bÂi8TZb n8  i�YES X S VaV i8T nX P*Y[\ UW_4U P*Y�g¯P8T � ] UW_4` ��y���� Y[P _ S6q�s _�n�_ P�i8\[q UW_ ]�S~i ¨i8sEs[TaP~µ U q�i _4U P*Y UWV�_4` i _�y � ��Va` P*c[bN\ ` P*bN\ g¯P*Tli8bNb > " c[q�i n i¡i8bN�*S��[T4i V � i8Y[\Äi8bWbi8\ES6b UWX sKP U Y _aV/� � � � ) ¬ Y�S~i XB`^X i V S _4`EUNV'UNV®_4` S �§1ZD870H:1�����D8-0;G- 9B3=-K58;G/2;<3=-0) ¬ _¥UNVX bNS~ihTab n c[Y0i � S X�_ S�\ U g%]�S�i8\[\ _ P � i X P*Y VZ_ i8Y _ i8bN�*S��[T4i � _4` i _¥UWV6  i8Y�S�bNS6q�S6Y _P8g Æ T y���� ),¦�P�] ` i _ ]�S�i8TZS�TZS~i8bWb n�U Y _ S�TaS VZ_ S�\ U Y UWV§Æ T y ��� #8Æ T y ��� )> bWb _4`[UWV®X i8Y�KS�s[c _®U Y _ P `[U � ` �ETaP�] bÂihY[�*c0i8�8S8) � R*S�Y] UW_a` P*c _ i8Y n�`�n stP _a` S �

V S V P8Y �� �_4` S�TaS UNV i8Y U Y=ÁZS X:_4U P*Y Phg Æ T y �l�lU Y _ P _a` S��S _ i8bWS X P ` P*q�P*bWP*� n �8TaP*c[sÈ ? y ��|������ �Ii8Y[\ U gjg¯P*T®S:µEi8q�s[bNS �¼UWV i X P*q�sEbNS _ S�YEP*Y VaU Y[�*c[bNi8T V c[TpgFi X S  t_4`EUNVJU Y��ÁZS X�_4U P*Y UNV i8Y UWV P*q�P*Tas `[UNV q�) ¬ g�]�S�]§T UW_ S

Æ T {�y �l��� ¨ S�T yFÆ T y ���1. Æ T y��l�Z��� ¨ S�T y È ? y ��|�������. È ? y���|������a�.|_a` S6TZS UWV ilk0b _ T4i _aU P*Y

Æ T y ��� � Æ T {�y ��� ��Æ T y �l�.}ÈÀS6TaS�P*Y[b n�_a` Sli8� VZ_ Tai X�_¾VZ_ TZc X�_ c[TaS$Phg Æ T y ��� #hÆ T {�y �l�ÀUWVJ¨ Y[P�]§Y¿�¢i8YE\ U Y�8S6Y[S�T4i8b_a` S6TZS UWV Y[P ¨ YEP�]§Y�]�i n P8g�k0Y[\ U Y[� > " c[q�i n iJi8bW�*S6�[Tai V ] `[UWXB` TZS6s[TZS V S�Y _ Y[P*Y _ T U R U i8bS�bNS6q�S6Y _aV Phg _4`EUNV uvc[P _aU S6Y _6 =_a` P*c[� `�U Y�i§s0i8T _4UNX c[bÂihT X i V S�È®i8T4ihT U � � � � ` i V S.µ `EU � Um_ S6\i Æ T4ihc[S6T� �i8Y U Y�P8� VZ_ Tac X:_4U P8Y X P8q U Y[�'g¯TaP*q V c XB` i8Y�i8bN�*S��[T4i�) ¬ Y X P*Y _ Tai VZ_6 �_4` S6TaSUWV i8Y UNV P8q�P*TZs `[UWV q

Æ T {�y �l� #hÆ T y ���1! È { y � i8b y �� #v���:| ½ UWX*y �� ����a�.|i8YE\ _a`[UNV�UWV$X P*q�s[c _ i8�[bNS U Y �tP _4` \ U TZS X:_4U P8Y V s[TaP�R U \ES6\ ½ UWX�y �� �����UNV$¨ Y[P�]§Y¿)yF³ P8T�\[S _ i U b V P8g _4`[UWV6 [V S6S � � �2) �»´` S�TaS UWV Y[P ¨ Y[P�]§Y Vpn�Vp_ S�q�i _aUNX ]�i n P8gE\[S _ S6Taq U Y U Y[�À½ UNX*y � ���� g¯P*T�i8TZ� UW_ T4i8T n

��  i8Y[\ _4` S�TaS UWV�Vp_ TZP*Y[�¾TZS~i V P*Y _ P V cEs[stP V S _a` i _�_4`[UWV�UNV TZS~i8bWb n i®Yvc[q��tS�T � _a` S6P8TaS _aUNXTai _4` S�T _4` i8Y�i'�*S6P8q�S _ T UWX s[TaP8�[bNS�q�) ¬ g � UNV \[S:k0Y[S�\�P�R*S�T �ry Tai _4` S�T _4` i8Y�P�R*S�T�i8Yi8TZ� UW_ T4i8T n i8bN�8S6�[Tai UNX Yvc[ql�KS6T�k[S6bN\ ��_a` S6TZS UNV i _ S�Y _ i _aU R*S�i8bN�*P8T UW_a` q   \[S6sKS6Y[\ U Y[�'P*Y

Page 8: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

_a` S Æ�U T XB`�� ¦o] U Y[YES6T _ P*Y���° n S6T X P*Y=ÁZS X�_ c[TaS   g¯P*T,\[S _ S6Taq U Y U Y[��i8Y�i8bW�*S6�[Tai UNX Yvc[ql�KS6Tk0S�bN\ � y \[S6sKS6YE\ U Y[�$P8Y ����V c XB`�_4` i _ ½ UNX*y � ����%� ½ UNX�y � ���:  i8Y[\ _4`EUNV q�i n�KS _a` S�T U � `�_ k[T VZ_JVZ_ S6s _ P�]�i8Ta\ V \[S _ S6TZq U Y U Y[��½ UNX�y � ���� �I�[c _ P*Y[S ` P*sKS V Y[P _~ �KS X i8c V S�S�R8S6Y±g¯P*T V PÃS6bNS�q�S�Y _ i8T n i^Rhi8T U S _�n i V i X c[� UWXV c[TZgFi X S�]�Sq�i n Y[S6S�\_ P ` i~R8S�� � © � � � ��( � � � ) ¬ _�V S�S6q V�_ P�q�S�b UN¨ S�b n�_4` i _ i��KS _Z_ S�T�i8sEs[TaP�i XB`Å_ P_a`[UNV uocES VZ_aU P*Y�] U bNb0�KS _4` TaP*c[� `�_4` S » i _ S X P8Y=ÁZS X:_ c[TZS V6  g¯P*T�] `[UWXB`�V S�S �E���[c _�_a`[UNVUWV iR*S�T n bNP*Y[��� _ S�Taq s[TZP V sKS X:_ )^È®P�]�S�R*S�T  �UW_lUWV c V c0i8bWb n stP VaVaU �[bNS _ P¡\[S _ S�Taq U Y[S½ UNX*y � ���� g¯P8T®ihY n s0i8T _aUNX c[bÂi8T � _a` i _ P*YES UWV´U Y _ S�TaS VZ_ S�\ U Y¿)��� ����������� +B�¾/¯C0H:14H�DAoH:-IH:1ZD8M¿D8M Ao3=1:;G/GCoO��ZH !8H:-^9B3=-��~HB9:/27o1ZD8M � #~3=1�5oH:/�H:1:O$;G-0;G-�A½ UNX*y � ���� #~3=1*!=D81:;<H:/F;<HB� � 5oH('%-IHa5�3)!8H:1�D8-^D8M AoH6L61ZD8;<9¥-07�O�L:H:1 '�H:Mm5 �

¢i8Y[� ` i V�X P*Y=ÁZS X�_ cETaS6\ _4` i _%U g ��UNV i¾Rhi8T U S _�n P8gt�*S6YES6T4ihb _�n stS§\[S:k0Y[S6\�P�R*S6T,i8Yi8bW�*S6�ET4i UWX YocEql�tS�T,k[S6bN\ � _a` S6Y _4` S�TaS UNV i¥k0Y UW_ S§c[Y U P*YfPhg¿s[TZP*stS�T V c[��Rhi8T U S _aU S VP8g �¼V c XB`e_4` i _ S:R*S6T n sKP U Y _ P8g ��y���� b U S VÀU Y�J) yF³ ihb _4U Y[� V�ºE_4` S�P*TaS�q   g¯P*T§] `[UWXB`V S6S �  ¢UNV'_4` S V stS X6U i8b X i V S�Phg _4`EUNV g¯P*T X cETZR*S V ) �f»´`[UNV Tai UNV S V i8Y[P _a` S6T¥uvc[S VZ_4U P*Y  VZU q U bÂihT _ P � c[S VZ_aU P*Y ���[c _ s[TaP8�0i8�[b n S~i VaU S�T6©��� ��������� � +B�¥/GC0H:14H�D8-�D8M Ao3=1.;G/¯CoO�#~3=1�5oH:/ H:1.O$;G-0;G-�A�½ UNX*y ��� @ C0H:14H � ;N��D�!=D �1.;<H:/��5oH('%-IHa5�3)!8H:1�D8-^D8M AoH6L61ZD8;<9¥-07oO�L:H:1 '�H:Mm5 �»´` S�TaSlihTaS¥R*S6T n g¯S:] gFi8q U b U S V g¯P*T´] `[UNXB`_a` S Æ T4i8cES6T � �i8Y U YP*� VZ_ TZc X�_aU P*Y X ihY

�KS�YEP*Y _ T U R U i8b*�Ec _ g¯P8T¢] `EUNXB`�Um_¢` i V �tS�S6Y VZ` P�]§Y _4` i _�UW_¢UNV¢_4` S�P*YEb n P*� VZ_ TZc X�_aU P*Y _ P_a` S¥ÈJi VZV S¥s[T U Y X�U s[bWS8) y ¦�S6S ` P�]�S�R*S�T � ( ���¿i8Y[\  0V c[�oÁZS X�_§_ P�¦ XB`[U Y " S�b º V´`�n sKP _a` S VZUNV6 � ������i8YE\ � ( � �<) ��¬ _®UNV �*S6Y[S�T4i8bWb n �KS6b U S:R*S6\ _4` i _À_a` S Æ T4i8c[S�T � �i8Y U Y�P*� VZ_ TZc X�_aU P*Y UNVU Y[\[S6S�\ _4` S�P*Y[b n P*� Vp_ TZc X�_aU P*Y _ P _4` S�È®i VaV S�s[T U Y X6U s[bNS�g¯P*T�T4i _aU P*Y0ihb V c[TZgFi X S V �_a` i _�UNV6 �V cETZgFi X S V � U T4i _aU P*Y[i8bNb n S�uvc U Rhi8bWS6Y _�_ P � ? P�R*S�T �� )��¾Y _4` S�P _4` S�T ` i8Y[\  ¦ ¨ P*TaP8�tP*�*i _ P�R y � ��� �   i8Y[\ V S6S�i8b V P � � � � ��` i V S:µ `[U � UW_ S�\Äi8Y P*� VZ_ Tac X:_4U P*Y _ P _4` SÈ®i VaV S�s[T U Y X�U s[bWS¾P*Y�i$� U S�bNb U s _aUNXÀV c[TpgFi X S¥] `[UNXB`�UNV \ES�k0Y Um_ S�b n YEP _®Æ T4i8cES6T � �i8Y U Y¿)��� ��������� � +B��/GC0H �§1ZD870H:1�����D8-0;G- 38L:�:/F1.7 9:/F;<3=- /GC0H 3=-0M��38L:�:/21.709:/2;<3=- /�3Ä/GC0H'D=�B��H�AK1.;G-I9:; AKMNH #~3=1�D8MGM¿7o-0;G1aD8/F;<3=-KD8M��Z3=1�D8MGM � D8-I3�� !=D81.;<H:/2;<HB� ��ÅS X i8Y�Phg X P*c[T V S�i Va¨ i VaU q U bÂi8T%uvc[S Vp_4U P*Y�g¯P*T�]�S~i ¨ i8s[s[TZP�µ U q�i _aU P*YI) »´` S¥q�i�ÁZP*T\ U � X c[b _�n�U Y�i8\E\[TaS VaVaU Y[� V c XB` uocES VZ_aU P*Y V g¯P*T�i�� U R8S6Y�gFi8q U b n � UNV%_4` i _ ]�S'\EPlY[P _¨ Y[P�] ` P�] _ P�\[S�\[c X S�i8Y nv_4`EU Y[��c V S�g¯cEb�g¯TaP*q _a` SlgFi X�_®_4` i _¾_4` S�TaS UWV Y[P Æ T4i8c[S�T ��ihY U Y�P*� Vp_ TZc X�_aU P*Y¿) »´` S®sETaPoP8g V P8g V c XB` TZS V c[b _aV i V i8TZS ¨ Y[P�]§Y�i8bNbE�[TaS6i ¨ c[s U Y _ P_ ]�P�s[i8T _aV ©ª � « > VZV c[q U Y[� _4` i _ ��U Y � V i _aUNV k0S Vj_a` S´ÈJi VaV S X P*Y[\ Um_4U P*Y   P8Y[S�k0Y[\ V i®Y[S X S VaV i8T n

ihY[\ V c�� X�U S�Y _jX P*Y[\ UW_4U P*Y�g¯P8T � _ P ` i~R*S�iJT4i _aU P*Y[i8bvstP U Y _~  P*T _ P'i8\[q UW_ ]�S~i ¨ihs[s[TaP~µ U q�i _aU P*Y¿)

Page 9: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

ª �'� « �'YES _4` S�Y VZ` P�] V,_a` i _j_a`[UNV YES X S VaV i8T n i8YE\ V cE� X6U S6Y _,X P*YE\ UW_aU P*Y UWV S6uvc U R8ihbNS6Y __ P _4` S Æ Tai8c[S6T� �ihY U Y X P8Y[\ UW_aU P*YI)

¬�¨ YEP�]±P8gIYEP�gFi8q U b U S V g¯P*Tj] `[UWXB`�Um_%` i V �KS6S6Y�stP VaVaU �[bNS _ P X i8TZT n P*c _�_a` S´k0T Vp_ s[i8T _P8g _a`[UNV sETaP*�*Tai8q�q�S��[c _ Y[P _�_4` S V S X P8Y[\¿) Æ c _�_a` S6TZS§i8TaS�gFihq U b U S V g¯P*T�] `[UWXB`�Um_j` i V�KS6S�Y�stP VZVaU �[bNS _ P�k0Y[\ i V cE� X6U S�Y _$X P*YE\ UW_aU P*Y�g¯P*T V P*bNcE� U b UW_�nÄy i8\[\ Um_4U P8Y0i8b _ P _4` SÈ®i VaV S X P*Y[\ Um_4U P*Y � ] `EUNXB` i8s[sKS~i8T V T4i _a` S6T�]�S~i ¨ �[c _ ] `[UWXB`�UNV \[S:k0Y UW_ S6b n�Vp_ TaP8Y[�*S6T_a` i8Y _a` S Æ T4i8c[S�T � �i8Y U Y X P8Y[\ UW_aU P*YI)§È®P�]�S:R*S�T  �V c XB` i X P8Y[\ UW_aU P*Y UNV c V c0i8bNb n Y[P _Y[S X S VaV ihT n i8Y[\ _4` S��*i8s VZ` P*c[bW\��tS�i _Z_ T U �[c _ S6\ _ P X bWc[q VaU YES VaV§U Y _a` S¥s[TaPoP8g )��� ������������� C0H:-�/¯C0H �§1ZD87 H:1����¡D8-0;G-�9B3=-K58;G/2;<3=-�;N�´/F1.; !�;FD8M 1 C03 @ 9aD8-�3=-IH�O�D���H7���H$3$#'/GCo;N��#�Do9:/ �

�E)��jH:/<D � #.7o-I9:/2;<3=-E��D8-K5 � �<��H:1.;<HB�:)�S _ � � � � �KS�i�Y[P*Y VZU Y[�*cEbÂi8TJi8Y[\^i8� V P*bNc _ S6b nU TaTaS�\[c X�U �[bWS�s[TZP=ÁZS X:_4U R8S$R8ihT U S _�nP8gt\ U q�S�Y VaU P*Y���\[S:k0Y[S6\�P�R8S6T _4` S´k0Y UW_ S´k0S�bN\ ��� � ³´y(+*�.  i8Y[\�\[S6Y[P _ SÀ� n��y0+8��_4` S³ TaP8�tS�Y U c V ihc _ P*q�P*TZs `[UNV q Phg � � U R8S6Y�� n

��y(+*� © y �� | � {.|~}6}~}6| � �*� �. y ��� | ��� { |~}6}~}6| � �� �:}³ P*T�i8Y n�������_4` S®kEµ�S6\�sKP U Y _aV P8g y���y(+*�Z� � ihTaSJsETaS X�UNV S6b n�_a` S¾sKP U Y _4V P8g � ] `[UWXB`i8TZS�\[S:k0Y[S�\�P�R8S6T � ³´y0+ � � � V cEs[stP V S _4` i _�_a` S6TZS�i8TaS . y(+ � � P8g _a` S6q�) > b _a` P*c[� `_a` S X P*Y _ S.µ _¾UNV�_ P _ ihbNb n \ U � S�TaS6Y _6  _a`[UNV´UWV i8bNq�P VZ_�_4` S¥g¯P*Taq�i8b UNV q P8g _4` S �S�g VZXB` S _ "³�U µ�S6\�½�P U Y _'_4` S�P*TaS�q  IVaU Y X Slg¯P*T¾�8S6P*q�S _ T UNX TaS~i V P*Y V S6i XB` P8g _4` S V S$kEµ�S6\^stP U Y _4V` i V qlc[b _aU s[b UNX�UW_�n�� (*) »´`EUNV i8Y0ihbNP*� n bNS�\¡�ÃS U b _ P X P*Y=ÁZS X�_ c[TaS _4` i _'_4` S�TaS Va` P*c[bN\�KS�i X P ` P*q�P*bNP8� n_a` S6P*T n i8s[s[b UNX i8�[bNS U Y _4`[UWV�X P*Y _ S.µ _ ) »´`[UWV ]�P*c[bW\ U q�sEb n_a` i __a` S6TZS¥]�S6TZS�k[Y UW_ S6b n q�ihY n�X P*q�s[bWS:µ�Yvc[q��tS�T V��¢x ¤®V c XB`�_4` i _

. y(+ � �,�?���x��

����¤��t{y 5 ( � x � �x ¤ g¯P*T§i8bWb � ���E| y � �

] ` S�TaS ,ÀxIUWV,_a` SJ\ U q�S�Y VaU P*Y�Phg _4` S®� _a`�X P ` P*q�P8bNP*� n �*TaP8c[s�P8g � ihY[\ _4` S ��x ¤ i8TaS_a` S XB` i8Tai X�_ S6T UNVZ_aUNX TaPoP _4V P8g _4` S�q�i8s U Y[\Ec X S�\¡� n!��y(+*� P*Y _4` Sl� _a`eX P ` P*q�P8bNP*� n )³ P*T�S~i XB` �j\[c[i8b Um_�n�U q�sEb U S V,_a` i _ ,Àx¿� ,�?�� <vx i8Y[\ _a` S � ?�� <vx#" ¤ i8TZS'ilstS�Taqlc _ i _aU P*YP8g _4` S + � #$�¢x ¤ ) ¬ g�]�S�\[S�k0YES _4` S�bNP X ihb " S _ i�� g¯cEY X�_aU P*Y&% y('.| � � � n S Um_4` S�T�P8g _4` SS�uoc U Rhi8bNS�Y _ TaS6bNi _4U P*Y V

bNP*�)% y*'Z�%�&��+�t{

. y(+ � �+' � #,� P*T ' % � y*'Z� # % y('Z���&��+�t{

. y0+ � �+' � |

Page 10: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

_a` S6Y y � ��UWV S6uvc U R8ihbNS6Y _§_ P

% y('Z��� �%{:y('.| � ������� � ?�� <0{6y('.| � �� y('.| � � � ?�y*'.| � ������� � ?��=y('.| � �

] ` S�TaS ��x y('.| � �J� � ¤ y ( 5 ��x ¤ 'Z� ) � i XB` �¢x y('.| � � qlc VZ_�` i~R8S X P�S:� X�U S�Y _aV¥U Y d  i8YE\ _a` SJihY0i8bNP8�*c[S´P8g _4` S®Ç U S6q�i8Y[Y `�n stP _a` S VZUNV�UNV�_4` i _ 4 �¢x ¤ 4 � + x =�? ) y2³ P*T%i'g¯c[bNbWS6Ti X�X P*cEY _ P8g��ÃS U b º¹V®X P*Y=ÁZS X�_ c[TaS V i8Y[\ _4` S U T®q�P _4U Rhi _aU P*Y  0V S6S _a` S�S.µ X S6bWbNS�Y _¾V c[TpR*S n� ��� �2) � > bNb _a`[UNV´` i V YEP�]��tS�S6Y�s[TaP�R8S6\   _4` S¥q�i U Y X P*Y _ T U �[c _ P*T´�KS U YE��°¾S6b U �*Y[Sh)� P�]�bNS _ � �KS�i�YEP*Y VaU Y[�*c[bNi8T´i8YE\�ih� V P*bWc _ S�b n�U TaTZS6\[c X6U �EbNS¾s[TZP=ÁZS X:_4U R*S¥R8ihT U S _�n

\[S:k0Y[S�\ P�R8S6T�i8Y�ihbN�*S��[T4i UWX Yvc[q��tS�T�k0S6bW\ � ) ¬ g � ` i V �*P�Po\±TZS6\[c X:_4U P*Y�i _ is[T U q�S���Phg � ]�S X i8Y�g¯P*TZq ����� �_4` SJTaS�\[c X:_4U P8Y�P8g � q�Po\��   ihY[\ ` S�Y X SJg¯P8Taq _4` S�¢x y('.| ����.� ) ³ P*T� U Y ¶   ]�S X ihY�YEP�] \[S:k0Y[S _4` SJ� _4` �*bWP*�0i8b � V S6T U S V�jxpy � |���� Phg �i V i�s[TaPo\[c X:_ P�R*S�T¾i8bWb¿s[bÂi X S V P8g �  0_a` S�gFi X:_ P*TÀi _ i�s[T U q�S���Phg��*PoP�\�TaS�\[c X�_aU P*Y�KS U Y[� y ��x y(+ <� | ����:�Z� <0{ ] ` S6TaS +���� P*Taq�� =�� � ) »´` SJTac[bWS V g¯P8T,g¯P*TZq U Y[� _4` S®gFi X�_ P*T Vi _�_a` S§s[T U q�S V P8gI�[i8\�TZS6\[c X�_4U P*Y�i8Y[\�i _%_4` S U YEk0Y Um_ S´s[bNi X S V,X i8Y��tS´g¯P*c[YE\ U Y � �E( �2)»´` S V S � V S6T U S V P8g X P*c[T V S§\[S6sKS6Y[\�P*Y � i V ]�S�bNb0i V P*Y � ) ¬ Y�s0i8T _4UWX c[bNi8T  � y � |����UWV ÁZc VZ_§_4` S " S _ i�� g¯c[Y X�_4U P*Y�P8g _4` S�i8bW�*S6�[Tai UNX YocEql�tS�T´k0S6bW\ � )» P X i8bWbIi�g¯c[Y X�_aU P*Y w�y � � i y �*bNP*�[i8b � " S _ i��<g¯c[Y X:_4U P8Y�P*T � V S�T U S V�X ihTaT U S V ] UW_4`�UW_

X S6T _ i U Y U q�s[b UNX i _4U P*Y V ©� w�y � � qlc VZ_ �tS _a` S s[TaPo\[c X:_ Phg�i ° U T UNXB` bWS _eV S6T U S V i8Y[\ sKP VaVZU �[b n V P8q�S� i8q�q�i � g¯c[Y X:_4U P*Y V6  i8Y[\ _4` S ` i8bmg ��s[bNi8Y[SlP8g�i8� V P*bNc _ S X P*Y�R*S�Ta�*S�Y X S�g¯P*T _4` S° U T UWXB` bNS _�V S�T U S V qlc Vp_®` i�R8S _a` S'g¯P*Taq���� ��� À] Um_4` � � U Yd´)

� »´` S´° U T UNXB` bWS _�V S�T U S V qlc Vp_ �KS�S:µ�s[TZS VZVaU �EbNS´i V i8Y � c[bWS6Tjs[TZP�\Ec X�_ � �� oy�� <� �] ` S6TaS _a` S � ihTaS'T4i _4U P8Y0i8btg¯cEY X�_aU P*Y V )

� w�y � � q�c VZ_�` i~R8SÅihY ihY0i8b nv_aUNXeX P*Y _4U Yvc0i _4U P8Y _ P _4` S^S�Y _4U TZS�� ��s[bNi8Y[S¡i V iq�S6TZP*q�P*Tas `EUNX g¯cEY X�_aU P*Y   g¯P8T§i8bNbtP8g�] ` P V S�stP8bNS V � UNV´U Yd�)

� »´` S6TaSJqlc VZ_ �tSJi�g¯c[Y X:_4U P8Y0i8bKS6uvc0i _aU P*Y�TZS6bNi _4U Y[� w�y � � i8Y[\ w�y � � 5 ( 5 � � )

� »´` S " S6TZP�S V P8g w�y � �®U Y _4` S X T UW_aUNX i8b VZ_ T U s � 5 ( 8 ��� 8 � ¥qlc Vp_ b U S�P*Y��� ��� 5 {? )

¬ Y�P*c[T X i V S   _4` S¥k0T Vp_®_ ]�P U q�s[b UNX i _4U P*Y V ihTaS _ T U R U i8b2�0i8Y[\�g¯P*T _ c[Y0i _ S6b n P*Y[S UNV Y[P _S.µEsKS X:_ S�\ _ P$s[TaP�R*S _a` S'bÂi Vp_�_4` TaS6S   �[c _ P8Y[b n�_ P Vp_ i _ S _4` S�q i V´X P*Y=ÁZS X:_ c[TZS V ) »´` SbNi VZ_ P*Y[S UWV®_4` S$Ç U S6q�i8Y[Y¡È n stP _a` S VZUNV�  ] `[UNXB` ihs[stS6i8T V®_ P��KSlP*c _ P8g%TaS6i XB` S�R*S�YU Y _a` S VaU q�s[bWS VZ_¥X i V S   ] `[UWXB`ÃUWV¥_a` S X bÂi VaVaUWX i8b%Ç U S�q�ihY[Y " S _ i�� g¯cEY X�_aU P*Y¿��i8Y[\ _4` S_a`[U TZ\�ihY[\�g¯P*c[T _a`�` i�R8S V P$gFi8T´P*Y[b n �KS6S6Y�s[TaP�R8S6\ U Y�i$g¯S:]�gFi~R*P8c[T4i8�EbNS X i V S V )

( �

Page 11: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

��� ������������� D8-�3=-IH�H �*/�H:-K5^/GC0H�M ;N�:/�3$# � #~3=1 @ Co;<9BC D8-KD8M�=/2;<9�9B3=-0/2;G-07ED8/F;<3=-D8-K5�/¯C0H #.7o-I9:/F;<3=-KD8M�Ha�67[D8/F;<3=-Å9aD8-eL:H�AK143)!8Ha5 � �¯+:/j��HBH:O��¾M ; ��H:M��/GCED8/�D8- ¾AK1B3�3$#�3$#D8-KD8M�=/2;<9�9B3=-0/2;G-07ED8/2;<3=- @ ;GMGM�9aD81:1 �D�AK143�3$#�3$#¥/¯C0H�#.7o-I9:/F;<3=-KD8M�Ha�67[D8/F;<3=- @ ;G/GC�;G/ , �¬ _À` i V �KS6S6Y V i U \�i8�KP*c _´_4` S " S _ i�� g¯c[Y X�_4U P*Y V P8g�i8bN�8S6�[Tai UNX Yvc[q��tS�T§k[S6bN\ V§_a` i _

¸ _4` S " S _ i�� g¯c[Y X�_4U P*Y ¨ Y[P�] V S:R*S�T nv_4`[U Y[��i8�tP8c _´_4` S'Yvc[q��tS�T�k0S6bW\¿�E]�S�ÁZc VZ_À` i~R*S _ Ps[TZS�Rhi U b�P*Y Um_À_ P _ S6bWb�c V6º ) ¬ g _4`[UWV§UNVÀV P   ]�S ` i~R*S�Y[P _Àn S _ c[Y[bWP XB¨ S6\ _a` S _ TaS6i V cETaS �` P8c V Sh) > s[i8T _ g¯TaP8q _4` S X bNi VaVZUNX i8b*g¯P*Taq�c[bÂi�] `[UWXB` TZS6bÂi _ S V����Ä_ P� � yF��� i8bNb _a` i _�` i VV PlgFi8T��KS6S6Y�sETaP�R*S�\i8TZS X S�T _ i U Y�TZS V cEb _4V P8g Æ P*TaS�b � 2 �I] `EUNXB` TaS�bÂi _ S _4` S'�tS ` i~R U P*c[TP8g� � y � � Y[S~ihT � � ( 5 � g¯P*T U Y _ S��*S6T V � � ( _ P _a` S�� ���8TaP*c[s V P8g� ��®) ¬ ]�P*c[bW\l�tSTZS6bNc X�_ ihY _�_ P X bÂi U q _4` i _�_a` S � V S6T U S V P8g i®Rhi8T U S _�n ��X P*Y _ i U Y V i8bNb _4` S U YEg¯P8Taq�i _4U P*Y] `[UWXB` P*YES�]�P*c[bW\b UW¨ S _ P ` i~R*S�i8�KP*c _§_4` S�Yvc[ql�KS6T � _4` S�P*TaS _4UWX i8b�s[TZP*sKS6T _aU S V P8g � ��[c _ P*Y[S�q U � `�_¾` P8stS _4` i _ ] ` S6YÅi�q noVZ_ S6T U P*c V Yvc[ql�KS6T _ c[TaY V c[s U Y _a` S VZ_ c[\ nP8g�° U P8s ` i8Y _4U Y[S¾s[TZP*�[bWS6q V P*Y �� [V P8q�S� � V S6T U S V´X P*Y _ i U Y V§U YEg¯P*TZq�i _4U P8Y�i8�KP*c _ÀUW_ )¦�c[sEstP V S�g¯P*T X P8YvR8S6Y U S6Y X S _4` i _ � UNV \[S�k0YES6\¡P�R*S�T ��  i8Y[\�bNS _®UW_aV \ U q�S6Y VZU P*Y

�KS � ) � R8S6Y�g¯P8T�R8ihT U S _4U S V ] UW_a` ,�{%� � ]�S¥\[P�Y[P _ S:µ�sKS X:_ i�s[TaPo\[c X:_ b UW¨ S� . y�� � # � � P*T �

. y��K����� � ���t{ 5 (

��5 (�� y � �

_ P��tSeYES X S VaV i8T U b n ih� V P*bWc _ S�b nÄX P8YvR8S6TZ�*S6Y _ ) Æ c _�U Y V P*q�S X P*Y _ S:µ _aV�_a` S6TZS UNV iTZS V sKS X:_ i8�EbNS^S:µ�s[TaS VaVZU P*Y ] `[UNXB` UNV g¯P*TZq�i8bWb n S6uvc U R8ihbNS6Y _�_ P P*YESÅP8g _4` S V S   ] UW_4`i8sEs[TaP*sET U i _ S�q�Po\ U k X i _aU P*Y V P8g _a` S�gFi X:_ P*T V i _�_4` S��0i8\�s[T U q�S V ) »´` S U \[S~i _a` i _V c XB` i8Y S:µ�s[TZS VZVaU P8Y VZ` P*c[bW\ ` i~R*S�Yvc[ql�KS6T� _a` S6P*TZS _aUNX�VZU �*Y U k X i8Y X S��*P�S V �0i XB¨Å_ P¦ U S6�*S�b y g¯P*T¾�*S�Y[S6Tai�Phg�uvc0ih\[T4i _aUNX g¯P8Taq VB� i8YE\ÅÈ®i8Ta\ n i8Y[\� UW_Z_ bWS�]�P�Po\ y g¯P*TJ] ` i __a` S n�X i8bWbNS�\ _4` S��:;G-�A87�MmD81À��H:1:;<HB� � ) �ÀVaU Y[� _4` S U \[S6i V ih�tP�R*S   ]�S'i8TaSJbNS6\ _ PlTZS6s[bNi X S_a` S Vp_ c[\ n P8g _a` S¾s[TZP�\[c X�_4V'y � � � n i Vp_ c[\ n P8g _a` SJ�KS ` i~R U P8c[T�P8g ?�� <0{6y � |���� i8YE\ ?�� <�?=y � |���� Y[S~ihT � � �0) Æ�n \Ec0i8b Um_�n* 8_a`[UNV�UNV�_a` S V i8q�S�i V�VZ_ c[\ noU YE� �{6y � |���� Y[S~i8T� � (�i8Y[\ ?�y � |��l� Y[S~i8T � � ��) »´` S U Y�g¯P*Taq�i _4U P*Y�\[S6T U R*S�\ U Y _4`[UWV ]�i n ihs[stS6i8T V_ PÀTZS6bNi _ S _ P _4` S%½ UNX i8Ta\¥�*TZP*c[s¥P8g ��  \[S:k0Y[S6\�i V¿_4` S,�8TaP*c[s¥P8go\ U R UNV P8T V \[S:k0Y[S6\�P�R*S6T� q�P�\Ec[bNP�b U YES~i8T�S6uvc U Rhi8bWS6Y X S8) Æ�n�X P*Y VaU \ES6T U Y[� VaU qlc[b _ i8Y[S�P*c V b n �tP _4` � i8Y[\ Um_4V½ UNX i8Ta\�Rhi8T U S _�n¡y¯_a` S�i8�tS�b U i8Y�Rhi8T U S _�n ] `EUNXB` s0i8T4ihq�S _ T UWV S V \ U R UNV P*T V ihbN�*S��[T4i UWX i8bWb nS�uoc U Rhi8bNS�Y _�_ P " S�TaP¡q�P�\[cEbNP�b U YES~i8T¥S6uvc U Rhi8bNS�Y X S �:  P*Y[S X P*Y X bNcE\[S V�_a` i _ �{6y � |����qlc VZ_ �KS�i VaV P X6U i _ S�\¡] UW_4`�_4` S�½ UWX i8TZ\^Rhi8T U S _�n i8Y[\ ?=y � |���� ] UW_4`¡_a` S��*TaP*cEs^Phg\ U R UNV P8T V q�Po\[c[bWP¾i8bW�*S6�[Tai UNX S�uoc U Rhi8bNS�Y X S � _a` i _jUWV6  ] Um_4`�_4` S � �S6TZP*Y��p¦oS�R*S�T U �8TaP*c[sP8g � ) »´` S V S®TaS6q�i8T ¨oV q�P _4U R8i _ S _4` SÀ]�S~i ¨ g¯P*TZq V Phg _4` S Æ�U T XB`�� ¦v] U Y[Y[S�T _ P*Y�� ° n S6TX P*Y=ÁZS X:_ c[TZS y g¯P*T¿] `[UWXB`�V S�S � � i8Y[\ _4` S X i V S�� � (,P8g _a` S » i _ S X P*Y�ÁZS X�_ c[TaS��tS�bNP�]¥)³ P*T _4` S VZ_ TaP*Y[�®g¯P*Taq V�y ] `[UNXB` � U R*S�S:µ�s[TZS VaVZU P*Y V g¯P*T _a` S�bWS~i8\ U Y[� X PoS�� X6U S6Y _4V Phg _4` STZS6bNS:R*S�Y _ ¢i8cETaS6Y _'V S6T U S V S:µ�s0i8Y VZU P*Y VB�®` S�c[T UWVZ_aUNX i8TZ�*c[q�S6Y _4V i8TaS�bNS VZV¾X P8YvR U Y X6U Y[�[�

(*(

Page 12: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

�[c _ P*YES X i8Y�g¯P*TZqlc[bNi _ S X P8Y=ÁZS X:_ c[TZS V g¯P8T _a` S V S X PoS�� X�U S6Y _aV � n i Va¨vU YE��] ` i _ P _a` S6Tq noVp_ S6T U P*c V Yvc[ql�KS6T Vj_ c[TaY$cEs U Y _a` S V i8q�S X P*Y _ S.µ _ ihY[\ Va` P8c[bN\ _a` S6TZS�g¯P*TZS§i8s[sKS~i8TU Y _a` S'g¯P*Taq�c[bÂi8S®g¯P*T _4` S¥bWS~i8\ U Y[� X PoS�� X�U S6Y _aV )»´` Sl]�S~i ¨ g¯P*TZq P8g _a` S » i _ S X P*Y=ÁZS X�_ c[TaS$i VaV S�T _aV¾_4` i _J_4` SlP8Ta\[S�TJP8g _4` SlstP8bNS

P8g ? �¾y � |��l� i _ � � � � ( UNV S6uvc0i8b _ P _a` S�T4ihY ¨ P8g _4` S��*TaP*cEsÄPhg X bÂi VZV S V Phg� � X�noX bWS V P*Y � \[S:k0Y[S6\P�R8S6T �   q�Po\[c[bWP�ihbN�*S��[T4i UWX S6uvc U Rhi8bNS�Y X Sh� Um_ÀUNV i�Y[i _ c[Tai8b�*S�Y[S6Tai8b U�" i _aU P*YÄP8g _a` S X i V Se� � (g¯P8T�] `[UWXB`±_4` S ` S6c[T UNVZ_aUNX�V�` i~R*S�ÁZc VZ_ �KS6S�YVZ` P�]§Y¿) ³ P*T¢i´q�P*TaSj\ES _ i U bNS�\�i X6X P*c[Y _ P8gE�KP _a` Phg _a` S V S  8U Y X bWc[\ U Y[� _4` S X P*Y=ÁZS X�_ cET4i8bg¯P*TZqlc[bNi8SJg¯P*T _a` S¥bNS~ih\ U Y[� X P�S:� X6U S�Y _4V6  V S6S ���E( �¢P*T�� � 2 �2)��� ����������� � CED8/,;G- #~3=1.O�D8/2;<3=-¡D�L:3=7�/ � ;N�$9B3=-0/<D8;G-IHa5�;G-e;G/G� � �<��H:1:;<HB� �»´` S�TaS UWV$U Y _4` S�b Um_ S�T4i _ c[TaS�i��tS6i8c _aU g¯c[b�S�\ U k X S�P8g X P*Y=ÁZS X:_ c[TZS   b U � `�_ b nÅV c[s��

sKP*T _ S6\^� n S�R U \ES6Y X S   ih�tP*c _'_4` S��KS ` i~R U P*cET�Phg _4` S jx�y � |��l� i _�U Y _ S6�*Tai8bjsKP U Y _aV )»´` S§s[T U Y X�U s0ihb0i8T XB`EUW_ S X�_aV Phg _4`EUNV S6\ U k X S®ihTaS Æ S U b U Y V P*Y  �Æ bNP XB` i8Y[\ �¥i _ P[) Æ S U b U Y��V P*Y º V'X P*Y=ÁZS X�_ c[TaS V TZS6bÂi _ S _ P _4` S$P*TZ\[S6T¥ihY[\�bNS~ih\ U Y[� X PoS�� X6U S6Y _4V P8g _4` S �ihc[TaS�Y _V S6T U S V S.µEs[i8Y VaU P*Y V P8g _4` S �x y � |���� i8�tP8c _§U Y _ S��*S6T�R8ihbNc[S V P8g ��� U Y _4` S6q _a` S'bNS6i8\��U Y[� X P�S:� X6U S�Y _4V ihTaS _ TaS6i _ S�\�i V S6bWS6q�S6Y _4V P8g ¶�� #v� � ) y2³ P*T,i¥g¯c[bNb i X�X P*cEY _�V S�S � � � �P*T � � ���2) ��Æ bWP XB` i8Y[\ �¥i _ P y � � ��i8Y[\ � ��� �¾` i~R*S Vp_ TaS�Y[� _a` S6Y[S�\ _4` S V S X P*Y=ÁZS X�_ c[TaS V� n_ TaS6i _4U Y[� _4` S$bNS~ih\ U Y[� X PoS�� X6U S6Y _4V i V S6bNS�q�S�Y _4V P8g ¶�� ) Æ c _'¬ \[P�Y[P _ �tS�b U S:R*S_a` i _ i8Y nv_4`[U Y[��b UN¨ S _4` S¥g¯cEbNb Vp_ P*T n�` i V´n S _ �KS6S�Y�TZS�R8S~i8bWS6\¿)

�[) � 7o1�!8HB�:)»´` S�q�P VZ_�U q�sKP*T _ i8Y _$U Y�Rhi8T U i8Y _ P8g'i X cETZR*S UNV�Um_4V �*S6Yvc V ) ¬ Y _4` S�bÂi8YE�*c0i8�*S�Phgi8bW�*S6�ET4i UWX �8S6P*q�S _ T n P�R*S�T ¶  �X cETZR*S V P8g´�*S�Yvc V$� ihTaS X i8bNbWS6\Ä1ZD8/2;<3=-KD8M  �X c[TpR*S V Phg�*S�Yvc V (�i8TZS X ihbNbNS�\ÄH:MGM ; AK/F;<9�i8Y[\ X c[TZR8S V P8g��*S�Yvc V �*TaS6i _ S�T _a` i8Y�($i8TZS�3$# AoH:-IH:1aD8M/�.A0H:) Æ c _ Y[P _ S _a` i _ g¯P*T%i¥Yvc[ql�KS6T _4` S6P*T UNVZ_ ihY�S�bNb U s _4UWX�X c[TZR8S UWV i X c[TZR8S®P8gt�*S�Yoc V(�] Um_4` i\ UNVZ_aU Y[�8c UNVZ` S6\ sKP U Y _ � $P*Y UW_~  �tP _a` �tS U Y[�\[S�k0YES6\ P�R*S6T _4` S��*TZP*c[Y[\k0S�bN\ � ) »´` S�S � S X:_ P8g _4`[UWVlUNV�_a` i _$_a` S�stP U Y _4V P8Y i8Y S6bNb U s _aUNX�X cETZR*S�g¯P*TZq i8Yi8�KS6b U i8Y��*TZP*c[s�] UW_a`�� li V�UW_aV�U \[S6Y _aUW_�n S6bWS6q�S6Y _~ ¢_4` S V c[q P8g ��{ i8Y[\ � ? �tS U Y[�_a` S�P _4` S�T " S6TaP�P8g _a` S�g¯c[Y X�_aU P*Y y \ES�k0Y[S�\ c[s _ P�qlc[b _aU s[b UNX i _4U P*Y� n i X P*Y Vp_ i8Y _.�] Um_4` stP8bNS V i _�%{ i8Y[\ � ? i8Y[\�i " S6TZP�i _� ~)> X i8YEP*Y UNX i8b¿\ U R UNV P8T´P*Yi X c[TZR8S��ÅPhgj�*S�Yoc VJ�$` i V \[S��*TaS�S 5 ��� ` S�Y X S�� n�_4` S

Ç U S�q�i8YEY�� ǧP XB`�_a` S6P*TZS6q�� UWV � U T4i _4U P8Y0i8bNb n S6uvc U Rhi8bWS6Y _ P�R8S6T _4` S��*TaP8c[Y[\�k0S6bW\ _ Pi X P*Y UNX ) »´` S�ÈJi VZV S�sET U Y X6U sEbNS ` P*bW\ V g¯P*T X P*Y UNX6V�  i8Y[\ _4` S�TaS�g¯P8TaS¥g¯P*TÀi8bNb X c[TZR8S V Phg�*S�Yvc Vl� � _4`[UWV � U R*S V i X P*q�s[bNS _ S�i8Y V ]�S�T _ P � cES VZ_aU P*Y y > � i _�_4` S��KS6� U Y[Y U Y[��Phg_a` S V S¾Y[P _ S V ) Æ c _´UW_ \[PoS V Y[P _ � U R*S'i8Y�i8Y V ]�S6T _ P � c[S Vp_4U P*Y y2Æ�� )0�JR8S6T ��  i�R*S�T nVZU q�s[bNS'ihY V ]�S6T _ P � c[S Vp_4U P*Y y2Æ´��UNV i V g¯P8bNbNP�] V ©

( �

Page 13: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

�� � ��� � � ��� H:/�� | � {B| � ? L:H�-I3=- � H:143�H:MNH:O�H:-0/¯��3$#Jd , + #¥/¯C0H$Ha�67ED8/2;<3=-� z ? � � { z ?{ � � ?4z ?? � �

;N�¡��3=M 7EL6MNH�;G-�d 1 /¯C0H:- ;G/$CED=�ÅDÄ��3=M 7o/2;<3=-�#~3=1 @ Co;<9BC HaDo9BC�� xÂz ?x ;N�ÃD�L:��3=M 7�/�H:M�L:3=7o-K5oHa5�L 4 � �� { � ? 4 ,¦ U S6�*S�b�� � ��� ` i V � U R8S6Y ihY i8Y V ]�S�T _ P � c[S VZ_4U P*Y yFÆ´� P�R8S6T�i8Ta� UW_ Tai8T n i8bN�*S��[T4i UNXYvc[ql�KS6T´k0S�bN\ V�  i8Y[\�ÇÀi8� ` i~Rhi8Y�� � ��� ` i V �*S�Y[S6Tai8b U�" S�\¦ U S6�8S6b º V ]�P*T ¨�_ P�uvc0ih\[T4i _aUNXg¯P*TZq V´U Y�q�P8TaS¾Rhi8T U i8�[bWS V )»´` S ¨ Y[P�]§bNS6\E�*SÅPhglP*Y[S^T4i _4U P8Y0i8bJsKP U Y _ P*Y ��S�Y0i8�[bWS V c V�_ P _ T4ihY V g¯P*TZq �

� U T4i _aU P*Y[i8bNb nJU Y _ PJi®b U YES8� V P _a` S6TZS UNV iJs0ihT4i8q�S _ T UNX%V P*bNc _aU P*Y¥] `[UWXB` � U R8S V S:µ�s[b UNX6Um_ b ni8bWb _4` S®sKP U Y _aV P8Y �\[S:k0Y[S�\�P�R*S6T _4` S®�*TZP*c[Y[\�k0S6bW\¿) »´`[UWV i8Y V ]�S6T V � c[S Vp_4U P*Y y2·´� )¬ g � UWV i X c[TpR*S�P8g��*S�Y[S6Tai8b _�n sKS�\ES�k0Y[S�\eP�R*S�T¾i8Yi8bN�*S��[T4i UNX Yvc[ql�KS6TÀk0S6bW\ �  

��P8Ta\[S�bNb X P*Y=ÁZS X�_ cETaS6\^i8YE\ ³ ihb _4U Y[� V s[TaP�R8S6\ _4` i _ � y����®UWV k0Y Um_ Sh��i8Y[\^i�Yvc[ql�KS6TP8gjP _4` S�TJs[TZP�Phg VJ` i~R*Sli8s[sKS~i8TZS6\ VZU Y X S _4` S�Y¿) Æ c _¾Um_ \EP�S V Y[P _JV S6S�q _a` i _ i8Y n Phg_a` S6q S6Y0i8�EbNS'P*Y[S _ P X P*q�s[c _ S � y����. E_4` P8c[� `�V P*q�S¾P8g _a` S6q X P*q�S _ i8Y _ i8b U�"6U YE�*b nX bNP V S8) ³ P*T§i V c[TpR*S n Phg V S�R*S�T4i8b V c XB` s[TZP�P8g V� [V S�S � ( ���2)��� ��������� �� +B��/¯C0H:14H�D8-ÃD8M Ao3=1.;G/GCoO #~3=1�9B3=OJAK7�/F;G-�A � y���� @ C0H:- � ;N�$De9:7o1�!8H3$#"AoH:-IH:1ZD8M¿/�BA0H$5oH('%-IHa5�3)!8H:1�D8-�D8M AoH6L61ZD8;<9¥-07�O�L:H:1 '�H:Mm5 � �»´` S VZ_ c[\ n P8g�T4i _4U P8Y0i8b�sKP U Y _aV P*Y�S6bNb U s _aUNX�X c[TpR*S V�UNV YEP�] i�q�i�ÁZP*T U Y[\[c VZ_ T n* 

i8bWq�P VZ_ S6Y _4U TaS6b n�V S6s0ihT4i _ S¾g¯TaP*q _4` S VZ_ cE\ n P8g�P _a` S6T�° U P*s ` i8Y _4U YES®s[TZP*�[bWS6q V ) ¬ g �UWV ihY�S�bNb U s _4UWX�X c[TZR8SJ\[S:k0Y[S6\�P�R8S6T�i8Y�i8bN�*S��[T4i UNX Yvc[ql�KS6T%k0S6bW\ �  �_a` S®�8TaP*c[s � y����UWV¾X i8bWbNS�\ _4` S �^3=1Z5oH:MGM � � H:;GM A8143=7�AK) ��P*TZ\[S6bWb�s[TZP�R8S6\ _4` i _ � y����®UWV k0Y Um_ S6b n �8S6Y��S�T4i _ S6\¿�E�ÅS U b º V�X P8Y _ T U �[c _4U P*Y�]�i V´_ P$S.µ _ S6Y[\ _4`EUNV TZS V cEb _´_ P�ihbNb > �KS6b U i8Y�Rhi8T U S _aU S V )»´` ihY ¨vV%_ P��i " cET yFV S�S � � � � ��_a` S _4` S�P*T n P8g _4` S _ P*T VZU P*Yls[i8T _ P8g _a` S���P*Ta\ES6bNb � �ÅS U b�*TZP*c[s UNV Y[P�] TZS~i V P*Y0i8�[b n�X P*q�s[bWS _ Sh�0�[c _ g¯P*T _4` S¥Y[P8Y�� _ P*T VZU P*Y�s0i8T _ ihbNb _a` i _ ]�i V¨ Y[P�]§Y��KS�g¯P8TaS¥(�� 2 �'UWV%_a` i _ � y����jX i8Y��KS§S6ql�KS6\[\ES6\ U Y _ P�i X S6T _ i U Y��8TaP*c[s�] `[UWXB`UWV k[Y UW_ S6b n �*S6Y[S�T4i _ S6\i8Y[\ X P*q�s[c _ i8�[bNSh) »´` S�s[TZP X S VaV®U Y�R*P8bWR*S�\   ] `[UNXB`�UNV§¨ Y[P�]§Yi VÀ_4` S�q�S _4` Po\Phg U YEk0Y UW_ S¥\[S VaX S6Y _~  �*PoS V �0i XB¨�_ P ³ S�Taq�i _ � g¯P*TÀc V S U Y� 2 ¬´Va` ihbNbU bNbNc VZ_ Tai _ S UW_ �KS6bWP�] U Y�i's0i8T _4UNX c[bÂihTab n�VaU q�sEbNS X i V Sh) Æ�n q�S~i8Y V P8g _4`EUNV s[TZP X S VaV P*Y[SX i8Y�i8bW]�i n�V´X P*q�s[c _ S¥ihY�c[s[sKS6T´�KP*c[Y[\�g¯P*T _a` S'T4ihY ¨ P8g _a` S ��P*TZ\[S6bWb � �ÅS U bt�8TaP*c[sP8g�i8Y n s0i8T _aUNX c[bÂi8T �   i8Y[\ _a` S�cEs[stS�T��KP*c[YE\ _4` c V P*� _ i U Y[S�\ X i8YÅg¯TZS6uvc[S6Y _ b n �tSVZ` P�]§Y _ P¥�KS§S6uvc0i8b _ P _a` SÀi X�_ c0i8b0Tai8Y ¨ � n S.µ `EU � Um_4U YE�'S6Y[P*cE� ` S�bNS�q�S�Y _aV P8g � y���� )¬ _ ]�i V i8b V P X P*Y�ÁZS X�_ c[TaS�\ _a` i _$_a` S�\ U�� S6TZS6Y X S��tS _ ]�S�S6Y _a` S�c[s[sKS6T$�KP*c[Y[\ _4` c VX P*q�sEc _ S�\Ãi8YE\ _4` S�i X�_ c[i8b�T4i8Y ¨ ]�i V i8bm]�i noV ihYÅS:R*S6Y U Y _ S��*S6T   �[c _ ihs0i8T _ g¯TZP*q_a`[UNV�_4` S�i X�_ c[i8b�Tai8Y ¨ ]�i V q noVZ_ S6T U P*c V ) »´`[UNV Y[P _ ] ` P*bNb n�V i _aUNV gFi X�_ P8T n^VZ_ i _ S�Phg

(��

Page 14: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

i � i U T V%` i V �tS�S6Y�Tai8\ UWX i8bWb nlXB` i8Y[�*S�\�� n�_a` S Æ�U T XB`�� ¦o] U Y[Y[S6T _ P*Y � ° n S�T X P*Y=ÁZS X�_ cETaS  _a` S']�S6i ¨ g¯P*Taq P8g�] `[UNXB`�UNV \[S VaX T U �KS6\i _´_4` S�S�Y[\�P8g _a`[UNV´V S X:_4U P8Y¿)¬ Y[P�] _ c[TaY _ P _4` S VaUW_ c0i _aU P*Y U Yl] `[UWXB` � UWV i X cETZR*S§P8g �8S6Yvc V (´\[S:k0Y[S6\�P�R*S�T �

�[c _ YEP _ Y[S X S VZV i8T U b nlX P*Y _ i U Y U Y[�¥i'stP U Y _ \[S�k0YES6\�P�R8S6T � ) �S _�� �tS _a` S��*i X P8� U i8YP8g �   \[S:k0Y[S�\�i V i X c[TpR*SJ] ` P V S®stP U Y _4V i8TaS U Y�P*Y[S���P*Y[S X P8TaTaS V sKP*Y[\[S�Y X S®] UW_4`�_4` S\ U R UNV P8T V P8g�\[S��*TaS�S � P8Y � q�Po\[c[bWP�b U Y[S6i8T§S6uvc U R8ihbNS6Y X S8) »´` S�Y �ÃUWV ihb V P�i X c[TZR8SP8g%�8S6Yvc V ($\[S�k[Y[S6\�P�R*S�T �   i8Y[\ �%y����®X P8Y _ i U Y V¾_4` SlsKP U Y _ ] `[UNXB`¡X P*TaTZS V sKP*Y[\ V_ P _a` S _ T U R U i8bt\ U R UWV P*T�)%¦�P �ÅUWV i8Y�S6bWb U s _4UNX®X c[TpR*S U Y�P*c[T V S6Y V S8)· P*Y�R*S�T V S6b n8 �U g[]�S�kEµ�i8Y�S6bNb U s _aUNX�X c[TZR8S � \[S:k0Y[S�\$P�R8S6T � ]�S X i8Y X P*Y VaU \[S6T _4` S

S�uoc U Rhi8bNS�Y X S X bÂi VZV S V�y g¯P*T¾� U T4i _aU P*Y[i8b¢S6uvc U Rhi8bWS6Y X S$P�R*S6T ��� P8g X cETZR*S V � P8g,�*S�Yoc V(�\ES�k0Y[S�\ P�R8S6T � ] `[UWXB`�` i~R*S � i V �*i X P*� U ihY¿) ³ P*T�Yvc[ql�KS6T _a` S6P*T n* j_a` S�P*Y[b nP*YES V P8g U Y _ S�TaS VZ_ i8TaS _a` P V S¥] `EUNXB`�X P*Y _ i U Y�sKP U Y _4V \ES�k0Y[S�\�P�R8S6TÀS~i XB`�X P*q�s[bWS _4U P*Y� � ) »´` S V S�g¯P*Taq i X P*q�qlc _ i _4U R*S _ P*T VaU P8Y �*TaP8c[s  jX i8bNbWS6\ _a` S �0D8/ H ���[CED #�D814H !�;<9BCA8143=7�A i8Y[\�c V c0i8bWb n \ES6Y[P _ S6\^� n�� � _a` S U \[S�Y _4UW_�n S6bNS�q�S�Y _ P8g _4`[UWV �*TaP*cEs UWV¾_4` SX bÂi VZV ] `[UNXB`�X P*Y _ i U Y V�� UW_aV S�bWg   i8Y[\ Um_�X P8Y VaUWVZ_4V Phg _a` P V S � ] `[UWXB`�` i~R*S � i V�*i X P*� U i8Y�i8Y[\�] `[UNXB`�X P*Y _ i U Y�i�sKP U Y _ \[S:k0Y[S�\eP�R*S6T � ) yF»´` S VaU q�s[bWS Vp_ S:µEi8q�s[bNSP8g�i$Y[P*Y _ T U R U ihb¿S6bWS6q�S6Y _ P8g�i » i _ S �p¦ ` ihgFi8TZS�R UNXB` �*TaP8c[s UWV�_4` S X c[TZR8S

� z � � z{ � � z? � � ] UW_4` �*i X P*� U i8Y � � � { � 2 � � ? ���E} �»´` c V g¯P8T X c[TZR8S V P8g§�*S�Yvc V ( _a` S » i _ S��p¦ ` ihgFi8TaS:R UNXB` �*TZP*c[s UNV � n \[S:k0Y UW_aU P*Y _4` SP*� VZ_ TZc X�_aU P*Y _ P _a` S�ÈJi VZV S�s[T U Y X6U s[bNSh)¦�c[sEstP V S U Y�s0i8T _aUNX c[bÂi8T _4` i _À_a` S¥S6bWb U s _4UNXÀX cETZR*S �^UNV \[S:k0Y[S�\P�R*S�T � i8Y[\ ` i V

_a` S'g¯P*Taq�? ��y¯z 5 �~{Z�6yGz 5 � ?.��y¯z 5 �

] ` S�TaS _4` S¥\ UWVZ_aU Y[�*c UNVa` S6\�stP U Y _´UNV�_ i ¨ S�Y _ P��KS _4` S¥sKP U Y _ i _§U YEk0Y Um_�n ) »´` S _4` TaS6SsKP U Y _aV¾yF�.x<|B��� P8Y �e` i~R8S¾P*Ta\ES6T �o� _a` S n i8TaS X i8bWbNS�\ _4` S � �.58; !�;N�:;<3=-$A03=;G-0/G�:) » P$i8Y nTai _4U P*Y0i8btstP U Y _�y � | �E� P*Y � _a` S6TZS�S.µ UWVZ_ � {:| � ?�| �

i8YE\ �v{.| � ?~| �

V c XB`�_a` i _

� x y � 5 �Bx¯����� ?x g¯P*T � � ( | � | �E�

` S�TaS _a` S'� x i8TZS¾TaS6i8bNb n S6bNS�q�S�Y _4V P8g � � #v� � ? �[c _§Um_�UWV�X P8YvR8S6Y U S6Y _§_ P _ TaS6i _�_4` S6qi V�V uvc0ihTaS �<g¯TaS�S U Y _ S6�*S�T V )��ÅS�q�c VZ_�` i~R*S�� { � ? �

� � ? g¯P*T V P*q�S U Y _ S6�*S�T��  

i8YE\�� �����o{�� ? �) · P8YvR8S6T V S6b n�_4` S¥S�uoc[i _4U P*Y V

�?x � � x yGz 5 �.xG�fy � � ( | � | � � i8Y[\ ��� � � { � ? �

g¯P*T¾i8Y n � | � x ] Um_4` � { � ? �� � ? \[S:k0Y[S�i X cETZR*S�� � � y � {:| � ?�| �

� i8YE\Åi

g¯P*cET � _ P ��P8Y[SJq�i8s�� . � ) ¬ g�� y�����UWV Y[P _ S�q�s _�n* [Um_4V�U q�i8�8SJc[Y[\ES6T _4`[UWV q�i8s UNV i

( �

Page 15: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

X P V S _ P8g � �%y ���´U Y �%y ���:  i8Y[\�]�S�P*� _ i U Y�i8bWb V c XB`X P V S _4V®U Y _4`[UWV ]�i n ) »´` c V ]�SX P*c[bN\�k0YE\ ��y�����U g¢]�S X P*c[bW\�\ES X6U \[S'] `[UNXB` �Ãi8TaS V P*bNcE�[bNS U Y � )> g _ S6T�i XB` i8Y[�*S�P8g,R8ihT U ih�[bNS V ]�S X i8YÅi VaV c[q�S _a` i _¥_4` S �Bx ihTaS U Y�d´)�°¾S�k0Y[S

_a` S Ao3�365®AK1.;GO�HB�®g¯P*T � i V,_4` P V S§] `[UNXB` \EP�Y[P _ \ U R U \[S y2�~{ 5��?.��y2��? 5��6yF�5��~{a� �

_a` S6Y Um_�UNV Y[P _´` i8Ta\ _ P Va` P�] _4` i _ � UWV bNP X ihbNb nlV P*bWc[�[bNSJi _ i8bWb �*P�Po\�s[T U q�S V�U g¢i8YE\P*YEb n�U gKi8bWb _4` S´� x i8TaS´c[Y UW_4V i _ i8bNbo�*PoP�\$sET U q�S V )�¦�P _a` S6TZSÀi8TaS�P8Y[b n k0Y UW_ S6b n q�i8Y n�^] ` P V S V P*bWc[� U b UW_�n�U Y �@UNV i _ ihbNb ` ihTa\ _ P�\[S X�U \[Sh)»´` S X c[TpR*S V �fP*� _ i U YES6\ U Y _a`[UNV ]�i n i8TZS X i8bNbWS6\ � �69B3)!8H:1.;G-�A=��P8g ��  i8Y[\ _4` S

s[TZP X S VZV P8g§P*� _ i U Y U Y[� _4` S�q UWV�X i8bNbWS6\ i � �:5oHB��9BH:-0/F) »´` S n g¯P*Taq!i�*TZP*c[s cEY[\[S6TqlcEb _4U s[b UWX i _aU P*Y¥P8g _4` S X P*TZTaS V stP8Y[\ U YE� _ T U s[bNS V§y � {:| � ?6| �

� ) »´` S´k0Y UW_ S V c[�E�*TaP*cEs

X P*Y VaUWVZ_aU Y[�$P8g _4` P V S�� � X P�R*S�T U Y[� V ] `[UWXB` i8TZS'S�R8S6T n ] ` S�TaS�bNP X i8bWb n�V P8bNc[�[bWS UNV´X i8bNbWS6\_a` S � � �¿H:M O�H:1�A8143=7�AK) ¬ _�UNV S6i VaU b n¡X P*q�sEc _ i8�EbNS8��i8Y[\ VaU Y X S _4` S6TaS UNV i X i8YEP*Y UNX i8bS�ql�tS�\[\ U Y[��P8g ��y���� # � �%y ���ÀU Y _ P _a` S � � ¦�S6bWq�S�TÀ�8TaP*c[s  t_4`EUNV s[TZP�R U \[S V i8Yec[s[sKS6T�KP*c[Y[\�g¯P*T _4` S�T4i8Y ¨ Phg �%y ��� ) »´` S�\[S VZX S6Y _ sETaP X S VZV¾X ihY�KS X P8Y _aU Yvc[S�\  t_a` P*c[� `] Um_4`�V P*q�S�] ` i _ �*TaS6i _ S�TÀ\ U � X c[b _�n �[g¯P*T � ��\ES VaX S6Y _4V®V S�S � (*( �2)��¾Y[S X i8Y�i8b V P X i8TZT nP*c _ � ��\[S VaX S6Y _4V g¯P*T _4` S¥q�P*TaS'�8S6Y[S�T4i8b�S�bNb U s _4UWX®X c[TZR8S

�? ��z � � z ? � � z �� �

�[c _'U Y¡P*TZ\[S6T _ P�\[P _4`EUNV P*YES$TaS6uvc U TaS V'U YEg¯P*TZq�i _4U P8Yi8�KP*c _J_4` S V s[b Um_a_4U Y[��k[S6bN\¡Phg_a` S¥T U � `v_§` i8Y[\ VaU \[S8)»´` S�]�S6i ¨ g¯P8TaqrP8g _a` S Æ�U T XB`�� ¦o] U Y[Y[S6T _ P*Y � ° n S�T X P*Y=ÁZS X:_ c[TZS Vp_ i _ S V®_a` i _À_4` S

Tai8Y ¨ P8g _4` S ��P*Ta\ES6bNb � �ÅS U b��*TZP*c[sÃPhg´i8YÅS6bNb U s _aUNXlX c[TZR8S � UNV S6uvc0i8b _ P _4` S�P*TZ\[S6TP8g _a` S " S6TZP�P8g �{6y � | �¢� i _ � � (*� _a` S X P*Y�ÁZS X�_ c[TaS�i8b V P$� U R*S V ihY�S.µ�s[b UWX6UW_ g¯P*Taq�c[bÂig¯P*T _4` S�bWS~i8\ U Y[� X PoS�� X6U S6Y _ P8g _4` S�stP�]�S6T V S6T U S V S:µ�s0i8Y VaU P8Y¡i _À_4` i _ sKP U Y _ ) � P _ S_a` i _%_4`[UWV stP U Y _%UNV%U Y _4` S X T UW_aUNX i8b VZ_ T U s  vV P _4` i _%_a` S X P*Y=ÁZS X�_ cETaSÀs[TaS�� V cEs[stP V S V,_4` Si8Y[i8b nv_4UWX�X P*Y _4U Yoc[i _4U P*YeP8g �{6y � | ��� ) > _ sETaS V S6Y _¥_4` S�TaS$ihTaS _ ]�P�]�S�bNb ��c[Y[\ES6T Vp_ PoP�\X i V S VJU Y�] `[UNXB` i8Y0i8b no_aUNX'X P*Y _4U Yvc0i _4U P8Y UNV§¨ Y[P�]§Y¿©´] ` S�Y � � �� tV P _4` i _ �ÅX ihY�KS�s0i8Tai8q�S _ T UNV S6\l� n q�S~ihY V P8gKq�P�\[cEbÂi8T¢g¯c[Y X:_4U P8Y V6  ihY[\l] ` S6Y � ih\[q Um_4V�X P*q�s[bNS.µqlcEb _4U s[b UWX i _aU P*YI) ¬ Y X P*Y V S6uvc[S�Y X S  *_4` S V S _ ]�P X i V S V i8TZS,b UN¨ S6b nÀ_ P§�KS%S~i VZU S6T _a` i8Y _4` S�*S�Y[S6Tai8b X i V S8�o�[c _ S:R*S6Y ` S6TZS ¬ \[P¥Y[P _ S:µ�sKS X:_ qlc XB` g¯c[T _4` S�T%s[TaP*�8TaS VZV�U Y _4` S§Y[S.µ _\[S X i8\ES8) ¬ Y�S~i XB` Phg _4` S V S _ ]�P X i V S V6 8U g[P8Y[S�i VaV cEq�S V�_4` S Æ�U T XB`�� ¦v] U Y[Y[S�T _ P*Y�� ° n S6TX P*Y=ÁZS X:_ c[TZS�P*Y[S X i8Y \ES6T U R*S�ihY�i8bN�8P*T Um_4` q g¯P8T�k0YE\ U Y[� _4` S ��P*Ta\[S�bNb � �ÅS U b��8TaP*c[si8YE\ _4` S®P*TZ\[S6T%P8g _4` S » i _ S � ¦ ` ihgFihTaS�R UWXB` �*TZP*c[s¿��i8Y[\ U Y _4` S§k0T VZ_ Phg _4` S _ ]�P X i V S V_a`[UNV i8bW�*P*T Um_4` q ` i V �KS6S6Y U q�s[bNS�q�S�Y _ S6\�� n � S6�KS6b<) y ¦oS6S�� ( 2 �2) � � UW_a` P*c _ c VZU Y[�_a` S Æ�U T XB`�� ¦o] U Y[Y[S�T _ P8Y�� ° n S6T X P8Y=ÁZS X:_ c[TZS   È®S�S6�*Y[S�T§bNP*Y[�li8�*P$s[TZP�\[c X S6\�il]�i n Phg�*S�Y[S6Tai _4U Y[��i�sKP U Y _ P8Y � ] ` S6YES�R*S�T �Q� � i8Y[\ �ÅUNV q�P�\[cEbÂi8T��0i8Y[\ � TZP VaV i8YE\

� i8� U S6T y � ( � ��i8Y[\�� (�� � ��` i~R*S Va` P�]§Y _a` i _§_4`EUNV sKP U Y _§` i V§U YEk0Y Um_ S'P*TZ\[S6T§s[TZS X6UWV S�b n

( �

Page 16: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

] ` S�Y � y ( | ������ � ) Æ c U bW\ U YE��P*Y _4` S U T¾]�P*T ¨K  �'P*b n Rhi8� U Y y¯V S6S � ( ��� �J` i V'VZ` P�]§Y_a` S'g¯P*bNbWP�] U Y[�[) �� � ��� � � � 7�A�A03���HÅ/¯CED8/�/GC0H �HBH�A8-IH:1�A03=;G-0/�CED=� ;G- '%-0;G/�H 3=1Z5oH:1��^/GC0H:-�/GC0HA8143=7�A �%y���� CED=�¥1ZD8- ��(�D8-K5 �!y �¢� ;N��'%-0;G/ H ,�'P*b n Rhi8� U Y � � � � ` i V i8b V P P*� _ i U YES6\ V cE� X�U S6Y _�X P*YE\ UW_aU P*Y V g¯P8T��KP _a` ��y���� i8YE\� y�����_ P��KS¥k[Y UW_ S8) »´` S'g¯P*bNbWP�] U Y[�lTZS V c[b _§UWV \Ec[S _ P � S ¨ P�Rhi8TJihY[\�½%bNi _ S�T6) �� � ��� � � � + #�/¯C0H�3=1Z5oH:1e3$# Ày � | �¢� ;N�¡365�5Ã/¯C0H:-�H:;G/GC0H:1 �%y ��� ;N��;G- '%-0;G/�H¡3=1� y���� ) � * ;N��;G- '%-0;G/ H #~3=1lH !8H:1 &Ao3�365�3=1Z58;G-KD81 � ,¬ g ��X i8Y��KSÀs[i8T4i8q�S _ T U�" S6\$� n q�P�\Ec[bÂi8T�g¯c[Y X:_4U P8Y V g¯P*T V P8q�SÀi8T UW_4` q�S _4UNX�V c[�E�*TaP*cEsP8g¿¦ ?=yF����_4` S�Y�i8Y[i8b nv_4UWX´X P*Y _aU Yvc0i _aU P*Y�i8Y[\ _4` S§g¯cEY X�_aU P*Y0ihb[S6uvc0i _aU P*Y�g¯P*bWbNP�]¥�*�[c __a` S6TZS UWV Y[P _ S�R8S6Y�i�s[bÂihc VaU �[bNS X P*Y=ÁZS X:_ c[TZS U \[S6Y _aU g noU Y[� _a` S � ] `[UWXB` ` i�R8S _a`[UNVs[TZP*sKS6T _�n8  i8Y[\ _4` S�TaS UNV YEP ¨ Y[P�]§YeihY0i8bNP8�*c[S¾P8g�È®S�S6�*YES6T º VÀX P*Y Vp_ TZc X�_aU P*Y¿)¬ Y _4` S X P8q�s[bWS:µ�qlc[b _aU s[b UNX i _4U P*Y X i V S   ] ` i _§UNV§¨ Y[P�]§Y UNV i V g¯P*bWbNP�] V )

�� � ��� � � � � H:/ � L:H�D8-Å;GO�D A8;G-KD81 ��67ED�581ZD8/2;<9 '�H:Mm5�D8-K5 � D8- H:MGM ; AK/F;<9�9:7o1�!8H5oH('%-IHa5�D8-K5�D�58O$;G/2/F;G-�A�9B3=OJAKMNH ��O$7oM /2; AKM ;<9aD8/F;<3=-Å3)!8H:1 �/, + # ®y ( | ���������1 /GC0H:-ª � «��%y���� ;N��'%-0;G/�H��ª �'� « #~3=1�H !8H:1 JAK1:;GO�H ��� ��/¯C0H � �NAED81./�3$# � y���� CED=�'/GC0H�3=1Z5oH:1,AK14Ha58;<9:/�Ha5�L $/GC0H

�´;G1B9BC�� � @ ;G-0-IH:1./�3=-3� ? �H:1�9B3=-��~HB9:/F7�14H ,ÈÀS6TaS yFUÂ�jUNV \[c[S _ P · P*i _ S V i8YE\�� U bWS V�  ihY[\ y¯UNUG��_ Plǧc[� U Y¿) ³ P*T,i8Y�i X6X P*c[Y _ P8g _4` Ss[TZP�Phg V6 ¿V S6S � � � �<) �¥i _ "l` i V �*S6YES6T4ihb U#" S6\ yFUÂ� i8YE\^s[i8T _ P8g y¯UNUG��_ P��KS ` i~R U P8c[T'P�R*S6Ti8Y�i8�KS6b U i8Y�S.µ _ S6Y VaU P8Y�P8g ��  �[c _ ] Um_4`�_4` S V i8q�S � i V �KS�g¯P8TaS8)¬ Y��*S�Y[S6Tai8b�]�S�\[PeYEP _l¨ Y[P�] ` P�] _ P X P8q�s[c _ S � ) ¬ _�UNV�X P*Y=ÁZS X�_ c[TaS�\ _a` i _

Um_'UNV i8bm]�i noV k0Y Um_ Sh��i8YE\ U Y[\[S�S6\ _4`[UWV i VaV S6T _4U P8Y X i8Y��tS$TZS6��i8TZ\[S6\Åi V s0ihT _ P8g _4` SÆ�U T XB`�� ¦o] U Y[YES6T _ P*Y���° n S6T X P*Y=ÁZS X:_ c[TZS   g¯P*T _4` S§g¯P8Taqlc[bNiJg¯P*T _4` SÀbWS~i8\ U Y[� X P�S:� X�U S�Y _P8g _a` S�sKP�]�S6T V S6T U S V g¯P*T �{6y � | ��� i _ � � ( X P*Y _ i U Y V�_4` S�P*Ta\ES6TlP8g � y���� i V igFi X:_ P*T�) ¬ g U YE\[S6S�\ _4`[UWV P*TZ\[S6T UNV k[Y UW_ S  �_a` S6Y Um_ qlc Vp_ �KS�i V uoc[i8TaSh��g¯P*T · i VaV S6b V` i V sETaP�R*S�\ _4` S�S.µ UNVZ_ S6Y X S�Phg´i�Y[P*Y VZU Y[�8c[bÂi8T Va¨ S:] � Vpn q�q�S _ T UNX � U b U Y[S6i8T®g¯P8TaqQP*Y� ] Um_4` Rhi8bWc[S VÀU Y �$# d´) »´` c V k0Y Um_ S�Y[S VZVJU q�s[b U S V´_a` i _®U g � X P*Y _ i U Y V i _ q�P VZ_��5 (�S�bNS�q�S�Y _aV P8gjP*TZ\[S6TÀS:µEi X:_ b n� g¯P*T V P*q�S¥s[T U q�S ��_a` S6Y UW_ i X:_ c0i8bWb n�X P*Y _ i U Y VY[P V c XB` S6bNS�q�S�Y _4V � ` S6Y X S�i8YÅS6bWS6q�S6Y _ ] `EUNXB`ÃUWV¥¨oU bNbWS6\^� n �ÃUNV'_ T U R U ihb   i8Y[\ _4` SX c[TZR8S V P8gK�*S6Yvc V ( U Y _4` i _ S6uvc U Rhi8bWS6Y X S X bNi VaVjX P*Y _ i U Y$stP U Y _4V \ES�k0Y[S�\�P�R8S6T � ) ³ P*Tc V S¥bÂi _ S6T   ]�S VZ_ i _ S _4` S X i V S ��� ��i V i�bNS6q�q�iE)

( 2

Page 17: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

#���������� � 7�A�A03���He/¯CED8/ � y���� ;N�&'%-0;G/ HÅD8-K5Ä/¯C0HÅ�6703=/F;<H:-0/�3$#e/¯C0H �����¿H:M O�H:1A8143=7�AÄ3$# � L �;G/G����3=M 7EL6MNH�H:MNH:O�H:-0/¯�'CED=�$3=1Z5oH:1�D8/�O�3��:/ � ��/¯C0H:-¡/GCED8/§�6703=/2;<H:-0/�;N�Do9:/F7[D8MGM��/F1:; !�;FD8M ,»´`[UWV TaS V c[b _ ] U bNbts[bÂi n i X TZc X6U i8b¿TZP*bNS U Y� 2�ihY[\ ��)��) � H:-IH:1ZD8M ;G/F;<HB��D�L:3=7o/%�:7o1 #�Do9BHB�:)�JR*S�T ¶ ig¯c[bWb X bÂi VZVaU k X i _aU P*Y Phg V c[TZgFi X S V�X ihYÄ�KS�g¯P*c[YE\ U Y � ( �<) > k0T Vp_�X P�ihT V SX bÂi VZVaU k X i _aU P*Y UWV � U R8S6YÄ� nÃ_4` S��$365�D8;G1ZDÃ58;GO�H:-E�:;<3=-��   ] `[UWXB` g¯P*T V c[TZgFi X S V�X ihY_ i ¨ S _a` S�Rhi8bWc[S V 5���|4�E| (P8T ��) � ` i _ ihb V P V S�S6q V�_ P �KS VZU �*Y U k X i8Y _ g¯P*T _4` SYvc[ql�KS6T _4` S�P*T nÃy i8Y[\ X c _4V i X TZP VaVÀ_4`[UWVJX bNi VaVZU k X i _4U P*Y ��UNV ] ` S _4` S�T _4` S V c[TpgFi X S UNVH:MGM ; AK/F;<9 � _4` i _®UNV�  ] ` S _a` S6T¾P�R8S6T ¶ _a` S6TZS UWV i�q�i8s � . : g¯P*T V P*q�S X c[TZR8S :] ` P V S��*S6Y[S�T4i8b�k0�[TaS UNV i X c[TZR8S�P8g��*S�Yoc V (*) »´` S X i V S�] ` S�Y _4` S�q�i8s � . :UWV \[S�k[Y[S6\^P�R*S6T _a` S$�*TaP8c[Y[\¡k0S6bW\ � ihY[\ : ` i V �*S�Yvc V���UWV \ UWVaX c VaV S�\ U Y 2E� U Y_a`[UNVlX i V S _a` S�° U P8s ` i8Y _4U Y[S�sETaP*�[bWS6q V g¯P*T � i8TaS�P*YEb n P8g U Y _ S�TaS VZ_ ] ` S�Y : y����UWV Y[P*Y " S6TaP  �U Y¡] `[UWXB`ÃX i V S : X i8Y^�KS U \[S6Y _4U k0S6\�] UW_a`�� { )�� ` S6Y : ` i V �*S�Yoc V�*TZS~i _ S6T _a` i8Y (  ®_a` S�q�ihs � . : UNV S VaV S6Y _4U i8bNb n c[Y U uvc[S�i8Y[\ UW_ ihY[\ : i8TaS_a` S6TZS�g¯P*TZS��KP _4` \ES�k0Y[S�\�P�R*S6T � ) Æ�nó i8b _aU Y[� V6º¿_4` S6P*TZS6q   : y�����UWV�_a` S6Y k[Y UW_ S8�_a` c V S~i XB` sKP U Y _ P8g ��y���� b U S V P8YlP*Y[S�PhgIiÀk0Y UW_ S V S _ P8g0k0�ETaS V�  i8Y[\ Um_jUNV S6Y[P8c[� `l_ PVp_ c[\ n�_4` S V Sh) ¬ Y X P*Y _ T4i Vp_~  ]�S ¨ Y[P�] Y[P _4`EU Y[��i8�tP8c _�_a` S X i V SJ] ` S6Y : UNV S6bNb U s _aUNX )»´` S V c[TZgFi X S V ] Um_4` � � 5�� i8TZS�s[TZS X6UWV S�b n^_4` S¡1.7oMNHa5e�:7o1 #�Do9BHB� � _a` i _$UWV6 

_a` P V S¥] `[UWXB` i8TZS�� U T4i _4U P8Y0i8bNb n S6uvc U R8ihbNS6Y _ P�R*S�T ¶ _ P � { � : g¯P*T V P*q�S X c[TpR*S : )> q�P*YE� _a` S V S   � n gFihT _a` SÀq�P Vp_%U Y _ S�TaS VZ_4U Y[��i8TaS _4` S¥1ZD8/F;<3=-KD8M0�:7o1 #�Do9BHB�   ] `[UNXB` i8TaS� U T4i _aU P*Y[i8bNb n S6uvc U Rhi8bWS6Y _´_ P � ? P�R8S6T ¶ )¦�c[TpgFi X S V ] Um_4` � � � gFi8bWb U Y _ P$g¯P*c[T�gFi8q U b U S V ©

� > �tS�b U i8Y V c[TZgFi X S V ) »´` S V S'i8TaS _4` SJi8Y0i8bWP*�*c[S V�U Y _ ]�P$\ U q�S�Y VaU P*Y V P8g�S�bNb U s _4UWXX c[TZR8S V�  ihY[\ _4` S�TaS UNV YEP�TaS~i V P*Y _ P�\[P8c[� _¾_a` i _'_4` S U TJYocEql�tS�T � _4` S�P*TaS _4UNX i8bsETaP*sKS6T _4U S V´VaU q�s[b n �8S6Y[S�T4i8b U#" S _a` P V S�Phg�S�bNb U s _4UWXJX c[TZR8S V )

� ��� V c[TpgFi X S V6 �U Y X bNc[\ U Y[� U Y^s0i8T _4UWX c[bNi8T �'c[q�q�S6T V c[TZgFi X S V )�¦�P*q�S��[c _ Y[P _ihbNb ��� V cETZgFi X S V i8TaS¥S6bWb U s _4UWX )

� � Y[T U uvc[S V�V cETZgFi X S V6  ] ` P V S�Yvc[ql�KS6T _4` S�P*T n^` i V �KS6S�YÃR*S�T n b Um_a_ bWS VZ_ c[\ U S6\¿)� Y[T U uvc[S VÀV c[TZgFi X S V i8TZS�YES X S VaV i8T U b n S�bNb U s _4UWX )

� � U S6bWb U s _4UWX®V c[TZgFi X S V )¦�c[TpgFi X S V ] Um_4` � � (�i8TZS'Y[S X S VZV i8T U b n S6bNb U s _aUNX )

( �

Page 18: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

¦�c[TpgFi X S V ] UW_a` � � ��i8TaS X i8bWbNS�\��:7o1(#�Do9BHB��3$# AoH:-IH:1aD8MK/�.A0H � ] `[UWXB`�U Y�q�i _a` �S�q�i _aUNX�V�UNV �8S6Y[S�T4i8bWb n i�\[S6TZP*��i _ P*T n s ` Tai V Sh) > �KP*c _�_a` S6q _4` S�TaS UNV�X c[TZTaS6Y _ b nY[P _4`[U Y[� _ P V i n �KS n P*Y[\ ¢i8YE� º¹V§X P*Y=ÁZS X:_ c[TZS Vp_ i _ S6\ U Y� ��)¬ Y _a` S®YES:µ _�_ ]�P V S X�_aU P*Y V,¬�Va` i8bNb0P*c _ b U Y[S´] ` i _�X i8Y�i _ s[TaS V S6Y _ �KS V i U \�i8�KP*c _

Tai _4U P*Y0i8b V cETZgFi X S V i8Y[\ � � V c[TpgFi X S V TaS V stS X�_aU R*S�b n � _a` S V S�i8s[sKS~i8T _ Pe�KS _a` S _ ]�Pq�P VZ_¾U Y _ S�TaS Vp_4U Y[��gFi8q U b U S V P8g V c[TpgFi X S V g¯P8T _4` S$Yvc[ql�KS6T� _a` S6P*T UNVp_ ) ¬ Y^�KP _4`¡X i V S Vq�i8Y n Phg _4` S¾q�P Vp_ TaS X S6Y _ TZS V cEb _4V \[S6sKS6Y[\�P*Y�P8Y[S¾P*T��KP _a` P8g _ ]�P$q�i�ÁZP*T X P*Y�ÁZS X �_ c[TaS V )��¾Y[S¾P8g _a` S V S y g¯P*T _4` S¾TZS~i V P*Y�� U R*S6Y�YES~i8T _a` S'S6YE\�P8g � ��UWV�_a` SJk0Y Um_ S�Y[S VZVP8g _4` S » i _ S � ¦ ` ihgFi8TZS�R UNXB` �8TaP*c[sI� _4` S�P _a` S6T UNV ¦ XB`[U Y " S6b º¹V È n stP _4` S VaUNV�  ] `[UWXB` ]�SY[P�] \[S VaX T U �KS8) ¬ _ � U R8S V i X P*Y=ÁZS X:_ c[Tai8b�i8Y V ]�S6T _ P _4` S�g¯P*bNbWP�] U Y[�$uvc[S Vp_4U P*Y¿©�� U R*S�Yk0Y UW_ S�b n q�i8Y n sKP*b n YEP*q U i8b V�w�{�y¯z^�.|~}~}~}�|Bw��[y¯z���U YÄd � z ��] Um_4` sKP VaUm_4U R*S�bWS~i8\ U Y[�X P�S:� X�U S�Y _aV6 �UWV�_4` S6TaS�i8Y$i8TZ� UW_ T4i8T U b n bÂi8TZ�*S U Y _ S6�8S6T ��i _ ] `[UNXB`l_a` S n i8bWb _ i ¨ S�sET U q�SRhi8bWc[S V4²�»´` S6TZS�ihTaS _ ]�P�P*��R U P*c V P8� VZ_ Tac X:_4U P8Y V�_ P _4`[UWV ©

� �¾Y[S¥P*T´q�P*TZS'P8g _a` S w�x�y¯z^� q�i n�V s[b UW_´U Yed � z �<)

� »´` S6TaS�q�i n �KS�i�s[T U q�S �ÃV c XB`Ã_4` i _ g¯P8T�ihY n Rhi8bWc[S�P8g ��q�Po\ � i _ bNS6i VZ_P8Y[S¥P8g _a` S w�x�y � ��UNV \ U R UNVaU �[bNS®� n�� )

· bWS~i8TZb nÀ_4` S V S X P*YE\�P*� Vp_ TZc X�_aU P*Y X i8Y¥P*YEb n¾` i8s[sKS6Y¥g¯P*T � ��� \[S�� yFw�xG� )�¦ XB`[U Y " S6b º¹VÈ n sKP _a` S VZUNV�UWV�_4` i _�_a` S V S¥i8TaS _a` S¾P*YEb n P*� Vp_ Tac X�_4U P*Y V © U Y�P _4` S�T�]�P8Ta\ V� [U g¢Y[S UW_a` S6TP8g _a` S6q ` i8sEstS�Y V�_4` S6Yl]�S X i8Y XB` P�P V S´i8Y�i8TZ� UW_ T4i8T U b n bÂihTa�*S�� V P _4` i _ S:R*S6T n$w¢x y � �UWV i�sET U q�S8) »´` S6TZS�ihTaSlRhi8T U P*c V q�P*TaS X P*q�s[b UWX i _ S6\eRhi8T U i8Y _4V Phg _a`[UNV¾`�n sKP _4` S VaUWVy¯U Y X bNc[\ U Y[��P*Y[S V�U Y^P _a` S6T�i8bN�8S6�[Tai UNX Yvc[ql�KS6T¥k0S�bN\ V4�:  �Ec _�_a` S n i8bWbjg¯P*bNbWP�] gFi U Tab nS6i VaU b n g¯TZP*q _a` S `�n sKP _a` S VZUNVÀU Y UW_aV P*T U � U Y[i8btg¯P*TZq�)� P�P*Y[S U Y `[UWV T U � `�_ q U YE\�]�P*c[bW\¡i _a_ S6q�s _J_ P�s[TZP�R8S�¦ XB`[U Y " S�b º V È n sKP _4` S VaUWV �

U Y[\[S6S�\�P*YES U Y VZ_ ihY X S UNV�_4` S�Y[P _ P*T U P*c V b n'U Y _ T4i X:_ i8�[bWS _ ] U Yls[T U q�S V s[TZP*�[bNS�q   ] `[UWXB`UWV¾_4` S V sKS X�U i8b X i V S�] ` S�Y _a` S w�x i8TZS _4` S _ ]�PsKP*b n Y[P*q U ihb V§z � (�i8YE\ z 5 (*)Æ c _ s[TaP8�0i8� U b UNVp_4UWX i8TZ�*c[q�S6Y _4V'V c[�*�*S VZ_'_a` i _'_4` S `vn sKP _4` S VaUWV'UNV¾U YegFi X�_'_ Tac[Sh) > __a` S�R8S6T n bNS6i VZ_�UW_ ]�P*c[bW\ �tS�stS�TZR8S6T V S _ PbNPoP ¨ g¯P*T X P*cEY _ S6T��S:µEi8q�s[bWS V�_ PTZS V cEb _4V] `[UWXB`�` i�R8S��KS6S�Y�sETaP�R*S�\ V cE�oÁZS X:_®_ P�¦ XB`EU Y " S6b º V È n sKP _4` S VaUWV )

2E)���D8/F;<3=-KD8Mt�:7o1(#�Do9BHB�:)� ³ TaP*q _a` S�Yvc[ql�KS6T� _4` S6P*TZS _4UWX sKP U Y _ Phg´R U S�]  �_4` S6TaS�i8TaS _ ]�P ¨vU Y[\ V P8g§T4i _aU P*Y[i8bV c[TZgFi X S8©

� ½�S6Y X�U b V P8g X P*Y UNX�V6  � U R*S�Y�� n i8Y�S�uoc[i _4U P*Y�P8g _a` S'g¯P*Taq

� y��¢| � � z ? � � {6y��¢| � � z ?{ � � ?=y��¢| � � z ?? � � y � �

( �

Page 19: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

] ` S6TaS _a` S � x y �¢| � � ihTaS ` P*q�P*�*S�Y[S6P8c V stP8b n Y[P*q U i8b V P8g _a` S V ihq�S�\[S��*TaS�S8)½�S6Y X�U b V P8g X P*Y UNX�V�X i8Y��tS X bNi VaVZU k0S�\ U Y�q�P*TZS®\[S _ i U bIi X6X P8Ta\ U Y[� _ P _4` SJYocEq ��KS6T´Phg��0ih\�k0�ETaS V )

� °¾S�b�½�S " " P V c[TZgFi X S V Phg�\[S6�8TaS6S �   ] ` S�TaS � 8 � 8 �E)J�JR*S�T ¶  tV c XB` i V c[T �gFi X S UWV P*� _ i U Y[S6\�� n �[bNP�] U Y[��c[s y � 5 � � sKP U Y _4V P8g � ? U Y��*S6Y[S�T4i8bKsKP VaUm_4U P8Y¿)¬ _¾UNV¾¨ Y[P�]§Y _4` i _ °¾S6bj½�S " " P V c[TpgFi X S V P8g,\[S6�*TZS6S�� � � V i _4UWV g n_a` S$È®i VaV SsET U Y X6U sEbNS§i8Y[\�]�S6i ¨ i8s[s[TZP~µ U q�i _4U P8Y¿� U Y[\ES6S6\ _4` P V S®Phg�\[S6�8TaS6S ��YES X S VaV i8T U b nX P*Y _ i U Y T4i _4U P8Y0i8b%stP U Y _4V )¡°¾S6b�½�S " " P V c[TZgFi X S V P8g§\ES6�*TZS6S ��P*T�( ` i~R*S�Y[PihS VZ_a` S _aUNX q�S�T UW_aV ihY[\ ` i~R*S�i _a_ T4i X:_ S6\Ãb UW_Z_ bWS�i _a_ S�Y _4U P8Y¿� UW_�V S�S6q V�V S�Y VaU �[bNS_ P U �*Y[P8TaS _4` S6qrc[Y _4U b _4` S�s[TZP*�[bWS6q V´X P*q U Y[�lg¯TZP*q _a` P V S¥P8g�\[S6�8TaS6S V ��i8YE\� ` i~R*S��tS�S6Y V P*bmR*S6\I) »´` S�°¾S�b%½�S " " P V c[TpgFi X S V P8g´\ES6�*TZS6S ��i8TZS _a` S�Y[P*Y��VZU Y[�8c[bÂi8T X c[� UWX¥V c[TpgFi X S V6  ] `[UNXB`�` i~R*S�i8Y�S6Y[P*TZq�P8c V �[c _ bÂi8TZ�*S6b n�U TaTZS6bNS:R*S�Y _b UW_ S6T4i _ c[TZS   i8Y[\ _a` P V S�P8g0\[S��*TaS�S �'i8TZS _a` S�Y[P8Y VaU Y[�*c[bNi8T U Y _ S�T V S X�_aU P*Y V P8g _ ]�Puvc0ih\[T UWX6V,U Y � � ) ³ P8T `EUNVZ_ P*T UWX i8b0TZS~i V P*Y V�  i _Z_ S�Y _aU P*Y ` i V �KS6S6Y X P*Y X S6Y _ Tai _ S�\P8Y _4` S$°¾S�bj½�S " " P V c[TZgFi X S V Phg�\ES6�*TZS6S �E�¿�Ec _'_4` S�s[TaP*�EbNS6q V sETaS V S6Y _ S�\Ã� n_a` P V S�Phg�\[S��*TaS�S ��i8TZS'Y[S X S VZV i8T U b n�VZU q�s[bNS�T6)

�ÅS X P*Y VaU \[S6T�k0T VZ_ stS�Y X6U b V P8g X P*Y UNX6V�  ihY[\�i VaV cEq�S _4` i _¾y � �jUNV \[S:k0Y[S�\�P�R*S6T �� _a` S�i8TZ�*c[q�S6Y _ g¯P*T$ihY i8Ta� Um_ Tai8T n i8bW�*S6�[Tai UNX Yvc[ql�KS6Tlk0S�bN\�Y[P _ �tS U Y[�S VaV S6Y _4U i8bNb n\ U � S�TaS�Y _ )j�ÃS X i8Y�TaS6uvc U TaS _4` S X P�S:� X6U S�Y _4V P8g _4` S � x�y �¢| � ��_ Pl�KS U Y�d´),¦ U Y X S _4` SÈ®i VaV SJs[T U Y X�U s[bWS ` P*bN\ V g¯P8T X P*Y UNX�V6 vUW_,UWV S�Y[P*c[� `�_ P XB` PoP V S ����� | � � �, U Y V c XB`i�]�i n�_4` i _�y � �,UNV bNP X i8bWb n$V P*bNc[�EbNS¾i _ � | � i8YE\�i8bNb _4` S'Po\[\�sET U q�S V ] `[UNXB` \ U R U \[Si8Y n P8g _4` S � x�y � | �, � ) > V,UW_�VZ_ ihY[\ V6 o_4`[UWV i8s[sKS~i8T V�_ P U Y�R*P8bWR*S®i8TZ�*c U YE� U Y�i X6U T X bWS8�_a` S']�i n�_ P�q�i ¨ S _a` S�i8Ta�*cEq�S�Y _ TZS V stS X�_ ih�[bNS UNV i V g¯P8bNbNP�] V )> VZV c[q�S _a` i _¾y � ��UNV S:R*S6T n ] ` S6TZS�bWP X i8bNb n�V P*bNcE�[bNSh) Æ�n i8� V P*Ta� U Y[� V c Um_ i8�[bWS´gFi X �

_ P*T V,U Y _ P _4` S z�x   ]�S X i8Y�S6Y V c[TaS _4` i _%_a` S � x y��¢| � � ihTaS V uvc0i8TZS � g¯TZS6SJi8Y[\ X P*s[T U q�Sh)» P^i XB`[U S:R*S _4`EUNV6  ]�S ` i~R*S _ P�\[TZP*s _4` S X P*Y[\ Um_4U P8Y _4` i _$_4` S � xpy���| � � i8TaS�ihbNb�Phg_a` S V i8q�S�\[S6�8TaS6Sh�'�[c _�UW_�UWV�Vp_4U bNb _ TZc[S _4` i _�_a` S U T�\[S��*TaS�S V i8TZSei8bWb§S�R*S�Y�P*T�i8bWbPo\[\¿)´°¾S�Y[P _ S�� n�� _a` S V S _ P8gj�[i8\s[bNi X S V6  ] `[UNXB`�_ c[TZY V P8c _§_ P X P*Y VZUNVZ_ P8g � | �� _a` S$s[T U q�S V ] `[UWXB` \ U R U \[S _a` S$\ UWVaX T U q U Y0ihY _ P8g � y �¢| � � � {:y��¢| � � � ?�y �¢| � � i8Y[\ _4` Ss[T U q�S V ] `[UNXB` \[P¥Y[P _ S:µ X S�S6\ _4` S´\[S��*TaS�S´P8g _4` i _ s[TaPo\[c X:_ ) �S _ ��KS _4` S V s0i X S´Phgi8bWb�s0i U T V P8g X P8s[T U q�S U Y _ S��*S6T V � | �,   ] UW_a`_4` S _ P*sKP*bWP*� n�U Y[\[c X S6\e� n�_4` S�s[bÂi X S VP8g � ��i8YE\ÅbWS _ ��tS _a` S V c[� V S _ P8g� X P8Y VaUWVZ_4U Y[��P8g _4` S�sKP U Y _aV i _ ] `[UNXB` y � �JUNVbWP X i8bNb n$V P*bWc[�[bWS'i _ S:R*S6T n s[bNi X S U Y � ) Æ�n�`�n sKP _a` S VZUNV6  UWV Y[P _ S6q�s _�n �0i8Y[\ UW_�UNVP*sKS6Y U Y�J) » P�P8� _ i U Y V P*bWc[� U b UW_�n�U Y ��  ]�S ` i~R*S _ P XB` PoP V S � | �, U Y V P _a` i _y � ��UWV bNP X ihbNb n¾V P*bWc[�[bNS�i _ S~i XB` �*PoP�\�s[T U q�S � j] `[UWXB` \ U R U \[S V P*Y[S�P8g _a` S � x y�� | �, � �g¯P*T V P*bNc[� U b Um_�n i _À_4` S�P _a` S6TÀ�*PoP�\�s[T U q�S VÀUNV´_ T U R U i8b   i8Y[\�]�S ` i~R*SlihbNTaS6i8\ n�_ i ¨ S�Y

(��

Page 20: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

X i8TaS�P8g _4` S��0i8\�s[bNi X S V ) ¢S _$�hy �¢| � � �tS _4` S U TaTZS6\[c X�U �[bWS�gFi X�_ P8T�P8g _a` i _ P*Y[S�Phg_a` S � x y �¢| � � g¯P*T�] `[UNXB`�� 4 �8y�� | �, �:  i8Y[\ _ P¥kEµ U \ES~i V i VZV c[q�S _a` i _��hy �¢| � � \ U R U \[S V� ?�y���| � � � ` S6TaS �8y��¢| � ��UWV cEY U uvc[SÀ�tS X i8c V S � �\[P�S V YEP _ \ U R U \[S _4` SJ\ UWVaX T U q U Y0ihY _ Phg� �� { � ? ) »´` S X P*Y[\ UW_4U P*Y�P8g�bNP X i8b V P*bWc[� U b UW_�n i _ � UNV

y � y�� | �, � � {6y � | �$ �.|B�hy � | �, �a� ���� � ( y 2 �] ` S�TaS _4` S�P*c _ S�T®�ET4i XB¨ S _'UWV§_a` SlÈ U bW�tS�T _ÀVZn ql�KP*b<)´¦�P�i�Y[S X S VaV i8T n�X P8Y[\ UW_aU P*Y�g¯P*T_a` S V P*bWc[� U b UW_�n P8g y � ��UWV�_4` i _ ihbNb _a` S X P*YE\ UW_aU P*Y V b UN¨ S� y � y�� | �$ � � {6y � | �, �.|B�8y�� | �, �a� ¾� � ( y � �

` P8bN\ VaU qlc[b _ i8Y[S�P*c V b n g¯P8T V P*q�S y � | �, ��U Y   ] ` S�TaS _4` S¥s[TZP�\[c X�_ÀUNV´_ i ¨ S6Y�P�R*S6Ti8bWb � ] `[UNXB` \ U R U \[S �hy � | �, � )� ` i _lUNV c[YES:µ�stS X�_ S6\ UNV�_a` i _l_a`[UNV�_ c[TZY V P*c _�_ Pe�KS�c V S�g¯c[b   �tS X i8c V S�P8g _4` S

g¯P*bWbNP�] U YE��bNS�q�q�iE) »´` S¥s[TaPoP8g�P8g _4` S'bWS6q�q�i UNV�Vp_ T4i U � `�_ g¯P8TZ]�i8Ta\  EVaU Y X S _4` S¥g¯cEY X �_aU P*Y����tS ` i~R*S V b UW¨ S�i�uvc0i8\[Tai _4UWX TZS VZU \[c[S VZn ql�KP*b,i8Y[\ X i8YÃ�KS�S:Rhi8bNc0i _ S6\Å� n i� c X b U \ES~i8Y�i8bW�*P*T Um_4` q s[TZP X S VZV R8S6T n b UN¨ S _4` i _ ] `EUNXB`�UNV c V S�\�g¯P8T V c XB`�VZn ql�KP*b V )#�������� � � H:/ w�y��¢| � �:|���y �¢| � � L:H�C03=O�3�AoH:-IHB3=7��§A03=M�h-I3=O$;FD8M ��;G-¡d � �¢| ��� 1 @ ;G/GC\[S�� y2w�� H !8H:- , � H:/ � L:H�D '%-0;G/�H$��H:/¥3$#®AKMmDo9BHB��3$# � @ Co;<9BC 9B3=-0/<D8;G-E� � | � D8-K5D8MGM¢/GC0HÀAK1.;GO�HB� @ Co;<9BC58; !�;F5oH�/GC0H$58;N��9:1.;GO$;G-KD8-0/´3$# w�� , � 3=1�D8- �9B3BAK1.;GO�H � | �,;G-�d 1 @ 1:;G/�H

� y � � w´|�� � � | �, �%� � y2w�y � | �, �.|���y�� | �, �a� y � �@ C0H:14H�/GC0H3=7o/�H:1L61ZDo9 ��H:/�3=- /¯C0H�1.; A=Co/¥;N��/GC0H J;GMmL:H:1./'� =O�L:3=M´D8-K5�/¯C0H�AK1B3658709:/;N��/ D���H:-�3)!8H:1�D8MGM AK1.;GO�HB� � -I3=/�;G- � �:709BCe/GCED8/ � 4 ��y�� | �$ � , �¿C0H:-�� y�� | �, � ;N�9B3=-0/F;G-07 3=7��¥;G-¡/GC0H�/ 3BA03=MN3�A��3=- 1 D8-K5�9B3=OJAK7�/<D�L6MNH ,¬ Y _a`[UNV TaS V c[b _ ]�S _ i ¨ S w � � �� {:|��Q� � �%]�S�Y[P _ S�\ i8�KP�R8S _4` i _ \[S�� y � � {p��UNVY[S X S VaV ihT U b n S�R*S�Y¿) ¬ _ g¯P*bNbWP�] V'_a` i _ i�Y[S X S VaV i8T nÃX P*YE\ UW_aU P*Y�g¯P*T _4` S V P*bNcE� U b UW_�n Phgy � �ÀUWVJ_4` i _J_4` S6TaS UWV i�sKP U Y _$y � | �, �®U Y V c XB`¡_4` i _ � y � | �, �§� � (lg¯P*TJi8bNb��] `[UWXB`X i8Y�KS��8S6Y[S�T4i _ S6\�g¯TZP*q y � �§U Y _4`[UWV ]�i n ) »´`[UWV®X P*Y[\ Um_4U P*Y UNV§X P*q�s[c _ i8�[bWS  i8YE\ Um_%UNV c[Y V c[TasET UNVZU Y[� y¯_a` P*c[� ` Y[P _ P8�vR U P*c VB��_a` i _,UW_�_ c[TaY V P*c _,_ P��KS§S6uvc U R8ihbNS6Y __ P _a` S Æ T4ihc[S6T� �i8Y U Y X P*YE\ UW_aU P*Y�g¯P*T y � � )¬ g�P*YESJi VZV c[q�S V ¦ XB`[U Y " S�b º V È n sKP _a` S VZUNV6 o_4`[UWV%X P*Y[\ UW_4U P*Y UNV i8b V P V cE� X6U S6Y _ ) ³ P*T

V c[s[sKP V S _a` i _ � | �, ` i~R*S¾�KS6S�Y V P XB` P V S�Y _4` i _,_4` S6TaS UNV P*Y[b n P8Y[SJ�*PoPo\�s[T U q�S � ] `[UWXB` \ U R U \ES V´�hy � | �, � � _4` S�Y _a` S's[TZP�\[c X�_§U Y y � � TZS6\[c X S V´_ P _a` S'bNS:g _´` i8Y[\ VZU \[SP8g y 2 �.  i8YE\ V P y 2 �%` P*bN\ V g¯P*T _4`[UWV s[T U q�S8) � P�] XB` P�P V S'i8Y�P*stS�Y V S _ � � V c XB`_a` i _ly � �´` P*bW\ VÀ_a` TaP8c[� ` P*c _ � g¯P*TÀS~i XB`¡�hy �¢| � � �K� n i V b U � `v_ b n q�Po\ U k0S�\�R8S6T VaU P8Y

� �

Page 21: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

P8g�¦ XB`[U Y " S6b º¹V È n sKP _a` S VZUNV ]�S X i8Y XB` PoP V S y � | �, �®U Y � V P _a` i _ S:R*S�T n��hy � | �$ �UWV§_a` S�s[TaPo\[c X:_ P8g�P*Y[S��*PoP�\�s[T U q�S¥ihY[\sKP VaVZU �[b n�V P8q�S'gFi X:_ P*T V®U Y � ) > V®� Tac[Y V_a` TaP*cE� ` i8bWb U TZTaS6\Ec X6U �[bNS§gFi X�_ P*T V P8g � { � ? �

  � �TacEY V�_a` TaP*cE� ` i8bWb _4` P V SJsET U q�S V g¯P*T

] `[UWXB` ]�S ` i�R8S _ P§R8S6T U g n�y 2 � ) »´` c V´y � ��UWV S�R*S�T n ] ` S�TaS�bWP X i8bNb nJV P*bNc[�EbNSjg¯P*T _4` S�s0i U T� | �,   ihY[\ _4` S6TaS:g¯P*TaS��*bNP8�0i8bNb n¡V P8bNc[�[bWS8)�� UW_a` q U Y[P*T¥q�P�\ U k X i _4U P*Y V6 �_a` S V ihq�Si8TZ�*c[q�S6Y _lVZ` P�] V$_4` i _�yFV c[�oÁZS X:_�_ P�¦ XB`EU Y " S6b º V È n sKP _a` S VZUNVB��_a` S Æ Tai8c[S�T � ��i8Y U YP*� VZ_ TZc X�_aU P*Y UWV i8b V P _a` S¥P*Y[b n P*� Vp_ TZc X�_aU P*Y _ P�]�S~i ¨ ihs[s[TaP~µ U q�i _aU P*Y¿)¬ g _4` S6TaS UNV YEP Æ T4ihc[S6T� �i8Y U Y�P*� VZ_ TZc X�_aU P*Y  o_a`[UNV�X P*Y Vp_ TZc X�_aU P*Y�k0YE\ V�U Y�k0Y UW_ S6b n

q�i8Y nÄX P*Y UWX6V�U Y _a` SesKS6Y X�U b§] `[UNXB`�X P*Y _ i U Y±T4i _aU P*Y[i8b§stP U Y _4V ) Æ c _~ ´V P*q�S�] ` i _c[YES:µ�stS X�_ S6\[b n*  S�R*S�Y U g,]�S ¨ Y[P�] V P*q�S X P8Y UNX�V P8g _a` S�stS�Y X6U b�] `[UWXB` i8TZS V P*bWc[�[bWS  ] Um_4` P*c _ ¦ XB`EU Y " S6b º V È n stP _a` S VZUNV ]�S�\[P�Y[P _�¨ YEP�] ` P�] _ Pe�*S6YES6T4i _ S�q�P*TZS V c XB`X P*Y UNX�V )��� ��������� � � � ; !8H:-�DÀA0H:-I9:;GMt3$#¾9B3=-0;<9B�®D8-K5 '%-0;G/�H:M��O�D8- $9B3=-0;<9B�§;G-�/GC0H,A0H:-I9:;GMHaDo9BC�3$# @ Co;<9BCe9B3=-0/<D8;G-E�'1ZD8/2;<3=-KD8MoA03=;G-0/¯� 1 9aD8- @ H AoH:-IH:1ZD8/ H #.7�1./GC0H:1�9B3=-0;<9B�l3$#'/GC0HA0H:-I9:;GM @ Co;<9BC9B3=-0/ D8;G-e1ZD8/F;<3=-KD8M�A03=;G-0/G� @ ;G/¯C03=7o/�7��:;G-�A �¿9BCo;G- � H:M�� � BA03=/¯C0HB�:;N� �¬ g�]�S ¨ YEP�] S�R*S�YP*YES�Tai _4U P*Y0i8btstP U Y _ P*Yi�°¾S�b¢½�S " " P V c[TZgFi X S � Phg�\[S��*TaS�S �

P*T��   ]�S X i8YP*� _ i U Yi8Y U YEk[Y UW_�n Phg X c[TpR*S V P8gj�8S6Yvc V¾� S~i XB` P8g�] `[UWXB` b U S V P*Y �� _a` P*c[� `�_4` S n ] U bWb �tS VZU Y[�8c[bÂi8T�i8Y[\�g¯P8T�\[S��*TaS�S�� UW_ ] U bNb c V c0i8bWb n Y[P _ �KS _ Tac[S _a` i _S6i XB` sKP U Y _ P8g ��y���� b U S V P8Y�i _ bWS~i Vp_ P8Y[S X c[TZR8SJP8g _4` SÀgFi8q U b n ) Æ c _ ] UW_a` P*c _,V c XB`ilsKP U Y _6 �_4` S'�tS VZ_ ]�S X i8Y�\[P UNV�_ P$k0Y[\�P*Y � i�gFi8q U b n Phg X c[TZR8S V P8g��*S6Yvc V (*) > _k0T VZ_�VaU � `v_6 EUm_ ]�P*c[bW\ V S�S6q _a` i _�U Y _a` S V S X6U T X cEq Vp_ i8Y X S V YEP _4`[U Y[��TZS V S6ql�[b U Y[� _4` Si8TZ�*c[q�S6Y _ i8�KP�R*S X i8Y��KS®ihs[s[b U S6\¿�*g¯P8T�ihY�S VaV S6Y _4U i8b X P*q�stP8Y[S6Y _ P8g _4` i _ ihTa�*c[q�S6Y _]�i V´_4` i _§X P*Y UWX6V´V i _4UWV g n�_a` S¥ÈJi VaV S¥s[T U Y X�U s[bWS   i8Y[\ _4`[UWV´UNV YEP _´_ TZc[S'g¯P*T X cETZR*S V Phg�*S�Yvc V (8)%È®P�]�S:R*S�T��S6q�q�i��$s[TZP�R U \[S V c V ] UW_a` i$]�i n TaP*cEY[\ _a`[UNV P8� VZ_ i X bWS8)»´` S¥i8TZ�*c[q�S6Y _4V´U Y�R*P*bmR*S6\�i8TZS'i8s[sEb UNX i8�[bWS _ P V P*q�S V c[TpgFi X S V � ] `[UNXB` i8TZS'Y[P _

Y[S X S VaV ihT U b n T4i _aU P*Y0ihb   �[c _ ] `[UNXB` ihTaS�S�bNb U s _aUNX ] Um_4` i§k0�[T4i _4U P8Y � . � { ) · P*Y VaU \[S6Ti�sKS6Y X6U btP8g X cETZR*S V ����P8g¢�*S�Yvc V (   S6i XB` P8g�] `[UWXB`�UNV i � � X P�R8S6T U Y[�lPhg UW_4V �*i X P8� U i8Y� ��) ¬ g�]�S X ihY XB` PoP V S�� U Y V c XB` i�]�i n¡_4` i _ ��� UWV S�R*S�T n ] ` S�TaS�bWP X i8bNb n�V P*bNcE�[bNSi8YE\Åi _ bNS6i VZ_'` i8bWg _4` S$S6bNS�q�S�Y _4V P8g _4` S�� �p¦�S�bNq�S6TJ�8TaP*c[s�P8g � ��i8TZS V P*bWc[�[bNS y i8Y[\U g,]�S�i VaV c[q�S _a` S$k0Y Um_ S6YES VaV P8g _4` S�TZS6bNS:R*S�Y _�» i _ S��p¦ ` ihgFi8TaS:R UNXB` �*TaP*cEs �: �_4` S6Y UW_] U bNb�g¯P*bWbNP�] g¯TaP*q ¢S�q�q�i � _a` i _ ��� UNV�V P8bNc[�[bWS8) ³ P*T _4`[UWV q�i XB`EU Y[S�T n _ P ` i~R8Si8Y n�XB` ihY X S�P8g�]�P*T ¨vU Y[�   ]�S�qlc Vp_ �KS�ih�[bNS _ P U q�s[bWS6q�S6Y _¾_4` S � � \[S VZX S�Y _ P*Y � �U Y�i�q�i8Y[YES6T�] `[UWXB`�UWV c[Y U g¯P8Taq U Y��¢) »´`EUNV q�P*TaS'P*T´bWS VZV TaS6uvc U TaS V � � _ P ` i~R8S Um_4V� � \ U R UWVaU P8Y�stP U Y _4V \[S:k0Y[S�\P�R*S�T ��y � � ihY[\ _a` S6TZS�g¯P*TZS _ P ` i~R*S _4` S'g¯P*TZq

�? � y¯z 5 �~{�y � �a��y¯z 5���?=y � �a�6yGz 5 �

y � �a� y � �

��(

Page 22: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

] ` S�TaS _a` S �Bx y � � i8TZS U Y ��y � � ���[c _ i8Y�i8\E\ UW_aU P*Y0ihb _ T UWXB¨   � U R*S�Y U Y � ���   S6Y0i8�EbNS V,_4` Sq�S _a` P�\ _ P��KS¥c V S�\S:R*S6Y U g � � ` i V ÁZc VZ_ P*YES � ��\ U R UWVaU P*Y�sKP U Y _§U Y ��y � � )»´` S´\[S _ i U b V Phg _4`[UWV q�S _4` Po\�i8TZS�c[Y0i _Z_ Tai X�_aU R*S   �[c _�_a` S VZ_ T4i _ S6� n�UNV i V g¯P*bNbWP�] V )

yF³ P8T´i�g¯c[bNbIi X6X P*c[Y _~ 0V S6S � (�� �2) � � UW_a` P*c _ bNP VZV P8g��*S6Y[S�T4i8b UW_�n ]�S X i8Y�i VaV c[q�S _a` i __a` S �Bx y � � ihTaS U Y d � � �<) ³ P*T�i8Y n s0ihT _4UWX c[bNi8T U Y _ S��*T4i8b§Rhi8bNcES � �P8g �  ´_a` S�0ih\s[bNi X S V g¯P*T _a` S$S6uvc0i _aU P*Y y � � i8TZS _4` S$�0ih\^sEbÂi X S V g¯P*T _a` S VZnoVp_ S6q _ P*�*S _4` S�T¾] UW_4`_a` S�s[T U q�S V ] `[UNXB` \ U R U \ES�P8Y[S�P8g _4` S �.x�y � � 5��Z¤=y �� � ) ¬ _ ]�i V S:µ�s[bNi U YES6\ U Y �` P�] _ P U q�s[bWS6q�S6Y _J_a` S�� ��\ES VaX S6Y _ g¯P*T y � � )��ÅS Va` ihbNbjS:R*S6Y _ c0i8bNb n c V S�¦ XB`[U Y " S6b º¹VÈ n sKP _a` S VZUNV¥_ P XB` PoP V S �� V P _4` i _�_a` S�R8ihbNc[S$P8g�S~i XB`ÅU TZTaS6\Ec X6U �[bNSlgFi X:_ P*T'Phg´i8Y n�Bx y � � 5��Z¤8y � � i _ � � � UNV�_a` S§s[TaPo\[c X:_ Phg V P*q�S´�[i8\�s[T U q�S V g¯P*T _4` S Vpn�Vp_ S�q ] UW_4`P*YES��*PoP�\ sET U q�S8)e�ÅS X ihbNb _4` S�bNi _a_ S6Tli��¿9BCo;G- � H:M¢AK1.;GO�H8� _a` P*c[� `�Um_lUWV ie�*PoP�\s[T U q�S�g¯P*T _a` S VZnoVp_ S6q  �Um_$UNV i��[i8\ s[T U q�S�g¯P8T _4` S X c[TZR8S�P8� _ i U Y[S6\�� n ]§T UW_aU Y[�� � � U Y y � � ) »´` S�S � S X�_ P8g�TaS Vp_ T UNX�_aU Y[� � U Y _a`[UNV ]�i n�UWV´_4` i _ ]�S ¨ Y[P�] _a` P V S� � X P�R*S�T U Y[� V P8g y � � g¯P*T � � ��§] `[UWXB` i8TZS'bNP X i8bWb n�V P*bWc[�[bWS¥i _ i8bNbI�*P�Po\�s[T U q�S V g¯P*T_a` S X c[TZR8S8� _4` S n g¯P8Taq i�k[Y UW_ S��*TZP*c[sÅP8g � � X P�R*S�T U YE� V � �� P8g � ��] `EUNXB` \[PoS V Y[P _\[S�stS�Y[\P8Y _a` S XB` P UWX S¥P8g ��Às[TaP�R U \ES6\ UW_´V i _4UWV k0S VÀ_a` S X P*Y[\ UW_4U P*Y�i8�KP�R*S8)»´`[UWV �8TaP*c[s X S6T _ i U YEb neX P*Y _ i U Y V¾_4` S�P8T U � U Y0i8b ����i8Y[\ _4` S � � X P�R8S6T U Y[� V ] `[UWXB`

X P*TaTZS V sKP*Y[\ _ P _4` S � � \ U R UWVaU P8Y�stP U Y _4V ) »´` S�Y[S.µ _®VZ_ S6s   ] `[UWXB`�U YvR8P*bWR8S V i V P*s `[UWV �_aUNX i _ S6\ ihY0i8b noVZUNV P8g _4` S � ��\[S VaX S�Y _ sETaP X S VZV i8Y[\Äi8b V P�TaS6uvc U TaS V c V$_ P U Y _ TZP�\Ec X Si8\E\ UW_aU P*Y0ihbI]�S�bNb XB` P V S6Y¡�0ih\s[T U q�S V g¯P*T _4` S VZnoVZ_ S6q  IUWV§_ P�k0YE\ebNP X i8b X P*Y[\ Um_4U P*Y VP*Y�� Ji _´_4` S¥�0i8\�s[T U q�S V P8g _4` S Vpn�Vp_ S�q£] `EUNXB` S6Y V c[TaS _4` i _ g¯P*T � � �

� _a` S¾P*YEb n S6bNS�q�S�Y _4V P8g _a`[UNV �*TaP*cEs�] `[UNXB` i8TZS¾bNP X ihbNb nlV P*bWc[�[bNS¾i _ i8bWb _a` S¾�0ih\sEbÂi X S V P8g _a` S VZnoVZ_ S6q!b U S U Y _4` S V c[�E�*TaP*cEsÃ�*S�Y[S6Tai _ S�\Ã� n�_4` S�P*T U � U Y[i8b ���ihY[\ _4` S � � X P�R*S�T U YE� V �*S6Y[S�T4i _ S6\�� n�_4` S � � \ U R UNVZU P*Y�stP U Y _4V � ihY[\

� _a` S¥P*T U � U Y[i8b ��� UWV bNP X i8bWb n�V P8bNc[�[bWS¥i _ i8bWb _4` S¥�0ih\�sEbÂi X S V P8g _4` S VZnoVp_ S6q�)»´`[UWVjUNV Y[P _ ihbW]�i noV sKP VZVaU �EbNS8� U g UW_�UNV�U q�stP VZVaU �[bNS  =_a` i _ s[TZP�R U \[S V ihYlP*� Vp_ Tac X�_4U P*Y _ P_a`[UNV q�S _4` P�\�P8g%i _Z_ i XB¨ P8Y _a` S�s[TZP*�[bWS6q _a` P*c[� ` Y[P _ Y[S X S VZV i8T U b n�_ P _a` S V P*bWc[� U b �Um_�n P8g _a` S VZnoVZ_ S6q�) ¬ Y�*S�Y[S6Tai8b _4`EUNV P*� Vp_ Tac X�_4U P*Y UWV Y[P _ qlc XB`�VZ_ TZP*Y[�*S�T _a` i8Y _4` SÆ Tai8c[S�T � ��i8Y U YÃP*� VZ_ Tac X:_4U P*Y   i8Y[\ U Y V P8q�S X i V S Vl_4` S n i8TZS�s[TZP�Rhi8�Eb n�_4` S V i8q�S8�V P _4`[UWV�UWV Y[P _�_ PoP V S6T U P*c V i�[bNS�q UWVa` P*Y _4` S�q�S _4` Po\¿) ¬ g´]�S�i XB`[U S:R*S _a` S _ ]�Ps[TZP*sKS6T _aU S V i8�KP�R*S _4` S6Y V P8bNc[� U b Um_�n i _�_a` S¥¦ XB`EU Y " S6b¿s[T U q�S V,_ c[TZY V P*c _�_ P$�KS¾i8c _ P �q�i _4UWX ) yF»´`EUNV�UNV i8Y�S.µEi8q�sEbNS,P8g[] ` i _�V S�S6q V�_ P®�KS�i§T4i _a` S6T��*S6YES6T4ihbos ` S6Y[P8q�S�Y[P*Y  _a` i _§_4` S¥Yvc[ql�KS6T§P8g�TaS�bÂi _ S6\ _4`[U Y[� V ] `[UWXB` �*P$]§TaP8Y[��q�c VZ_ �tS¥S:R*S6YI) � � Um_4` P*c[TS�Y[bÂi8TZ�*S6\ V S _ P8g��0i8\ s[bNi X S V g¯P8T _a` S VpnoVZ_ S�q   ]�S�YEP�] XB` P�P V S � _ P V i _4UNV g n¡_4` SbWP X i8b X P*YE\ UW_aU P*Y V i8Y[\ _4` S�¦ XB`[U Y " S�b X P8Y[\ UW_aU P*Y U Y _4` S�s[TaS:R U P*c V s0i8Tai8�*T4ihs ` ) Æ�n�S6q�q�i �  �_4` S X cETZR*S y � ��UWV�V P*bWc[�[bWS�g¯P*T _4`[UWV Rhi8bNc[S§P8g ��) Æ c _%U Y X P*Y _ Tai VZ_,_ P'] ` i _

� �

Page 23: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

` ihs[stS�Y V g¯P*T¥stS�Y X6U b V P8g X P*Y UWX6V� �_a`[UNV¥¨vU Y[\ÅP8g�i8Ta�*cEq�S�Y _ i8sEstS6i8T V¥_ P�s[TaP�R U \ES�Y[PU YEg¯P*Taq�i _aU P*Y�i8�tP8c _ ]�S6i ¨ i8s[sETaP~µ U q�i _4U P*Y¿)��� ��������� � �� D8-�3=-IH§O�3658; # ¥/GC0H§O�H:/¯C0365�D�L:3)!8H���3¥/GCED8/�;G/ @ 3=1 ��� @ ;G/GC03=7�/�D8- D=�B�:7oOJAK/2;<3=-¡D�L:3=7�/,/¯C0H ���p58; !�;N�:;<3=-�A03=;G-0/G�$3$# � � �

� Y�g¯P*T _ c[Y0i _ S6b n^Um_lUNV Y[P _l¨ YEP�]§Y y i8Y[\�s[TZP*�0i8�Eb n^UNV Y[P _ S:R*S�Y _ Tac[S ��_4` i _ i8Yi8TZ� UW_ T4i8T n °¾S6bt½�S " " P V c[TpgFi X S¾Phg�\[S6�8TaS6S � X P8Y _ i U Y V i�sKS6Y X6U bKP8g X c[TZR8S V P8g¿�*S6Yvc V (P8g _4`[UWV s0i8T _aUNX c[bÂi8T _�n sKS � ihY[\ _a` S VZUW_ c0i _4U P*Ylg¯P*T,°¾S6b ½�S " " P V c[TZgFi X S V P8g¿\[S��*TaS�S �UWV qlc XB` ]�P8T V S  ¿U Y _4` i _J_4` S�Y0i _ c[T4ihb X c[TZR8S V¾_ P X P*Y VaU \ES6T¾ihTaS���� X P�R8S6T U Y[� V T4i _a` S6T_a` i8Y � � X P�R*S6T U Y[� V )È®P�]�S�R8S6T   g¯P*T�°¾S6b[½�S " " P V c[TpgFi X S V P8gK\[S6�*TZS6S � V P*q�S�s[TaP*�8TaS VZV�` i V �KS6S�Y�q�i8\ES8)

¦EihbN�KS6Ta�8S6T�i8YE\�¦ ¨ P*TZP*�KP*��i _ P�R � � � � ` i~R8S VZ` P�]§Y _a` i _j_a` S Æ Tai8c[S6T� �ihY U Y$P*� VZ_ TZc X �_aU P*Y UWV®_a` S$P*Y[b n P8� VZ_ Tac X:_4U P8Y _ P�]�S~i ¨ i8s[s[TZP�µ U q�i _aU P*YI) > V g¯P*T _4` S�È®i VaV S$s[T U Y��X�U s[bWS  j_4` S�sETaS V S6Y _�VaUm_ c0i _aU P*Y UWV i V g¯P*bWbNP�] V ) > °¾S6b�½�S " " P V c[TpgFi X S�P8gÀ\[S��*TaS�S �\[S:k0Y[S�\P�R*S�T _a` S�i8bN�8S6�[Tai UNX Yvc[q��tS�T§k[S6bN\ � ` i V´_4` S'g¯P*TZq

� © w%{6yGz |~}~}~}�|az��%� w ?=yGz |~}~}~}�|az

��%� �E|

] ` S�TaS _4` S X P�S:� X6U S�Y _4V P8g w�{ ihY[\ w ? ihTaS U Y � ) Æ�n i¡b U Y[S6i8T _ Tai8Y V g¯P*Taq�i _4U P*Y\[S:k0Y[S�\eP�R*S6T®i8Y�S:µ _ S�Y VaU P*Y ��{ P8g�P�\[\�\[S��*TaS�S�P�R*S�T �   ]�S X ihY V S�s0i8Tai _ S�P � P*Y[SP8g _a` S�Rhi8T U i8�[bWS V �*i8Y[\$P�R8S6T ��{ ]�S X i8Ylk0Y[\�iJsKS6Y X6U b�P8g X cETZR*S V ���ÀP8g �*S�Yvc V (´P*Y �g¯P*T¢] `[UWXB` S~i XB` � � ` i V P*Y[S � ��\ U R UWVaU P*Y¥stP U Y _ \[S:k0Y[S�\$P�R8S6T ��{6y � � ) �®VZU Y[� _4` S _ T UWXB¨\[S VaX T U �KS6\ U Y�� ���   i8Y[\ V c[�oÁZS X�_�_ P$i8Y�P8� VZ_ Tac X:_4U P8Y _�n s UNX i8bKg¯P8T _4` SJq�S _4` P�\  oUW_�X ihYY[P�]��tS Va` P�]§Y _4` i _ �@X P*Y _ i U Y V i�stP U Y _ \[S�k[Y[S6\ÅP�R8S6T ��{ ) > Vp_ Tai U � `�_ g¯P*TZ]�i8Ta\�*S�P*q�S _ T UNX i8TZ�*c[q�S6Y _~  ] `[UNXB` \[P�S V Y[P _ TZS6b n P8Y � �KS U Y[�ei8YÃi8bN�8S6�[Tai UNX Yvc[ql�KS6Tk0S�bN\   YEP�] VZ` P�] V�_a` i _ � X P*Y _ i U Y V iÃsKP U Y _ \ES�k0Y[S�\�P�R8S6T � ) � YEg¯P*T _ c[Y0i _ S6b n_a` S�P�R8S6T4ihbNb,i8Ta�8c[q�S�Y _4V i8TaS V P X P*q�s[b UWX i _ S6\ y i8YE\ V Pec[Y[Y0i _ c[Tai8b �J_4` i _�UW_�UNV Y[P _X bNS~ihT�] ` S _a` S6T _4` S´P*� VZ_ Tac X:_4U P*Y _ P _4` S´q�S _a` Po\ UWVjVZ_aU bWb VaU q�sEb n�_4` S Æ Tai8c[S�T � ��i8Y U YP*� VZ_ TZc X�_aU P*Y _ P _a` S V P*bWc[� U b UW_�n P8g � P�R8S6T � �0�[c _ S�R8S6Y U g UW_§UNV´Vp_ TZP*Y[�*S�T  0UW_§UWV Y[P _qlc XB`�Vp_ TaP8Y[�*S6T�)» P�c V S�i8Y[\ _a` S6Y X P8bNbÂihs V S$i�k[S6bN\�S.µ _ S6Y VZU P*Y U Y _4`[UWV ]�i n¡UWV i�\[S�R UWX Sl] `[UWXB`

s[TZP*�0i8�Eb n�` i V i�Yvc[ql�KS6TJP8g,c V S V ) ³ P8T V c XB` i X P*bNbNi8s V S VZ_ S6s _ P��KS�g¯S~i VaU �EbNS  K_4` S\[S��*TaS�SlP8g _a` S�k[S6bN\S:µ _ S�Y VaU P*YeY[S�S6\ V¾_ P��KS�s[T U q�S _ P _4` S�\[S��*TaS�SlP8g _a` S�Rhi8T U S _�n �i8YE\ _a`[UNV bWS~i8\ V P8Y[S _ P�s ` Tai V S _4` S V i8q�S'sETaP*sKS6T _�n�V P*q�S�] ` i _ \ U�� S6TZS6Y _ b n )��� ��������� ��� � H:/ � L:H�D !=D81.;<H:/��5oH('%-IHa5�3)!8H:1¥D '�H:Mm5 � 1 -I3=/�-IHB9BHB�B�6D81.;GM��3$#'D-07oO�L:H:1���/¯C0HB3=14H:/F;<9 �=;G-K5 , � 3=1 @ CED8/ #�D8O$;GM ;<HB��3$# � ;N�¾;G/,/F1.7 H�/GCED8/%; # � 9B3=-0/ D8;G-E�¥D� �a9 �9:MNHl3$#�5oH�A814HBH¥(�5oH('%-IHa5�3)!8H:1 � /¯C0H:-e;G/´9B3=-0/<D8;G-E��D¥A03=;G-0/�5oH('%-IHa5�3)!8H:1 � �

���

Page 24: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

> VJVZ_ i _ S�\¡i8�KP�R*S  t_4`[UWV®UWVÀ_ Tac[S�g¯P*T¾°¾S�b�½�S " " P V cETZgFi X S V P8g�\ES6�*TZS6S �E) ³ P*TÀstS�Y X6U b VP8g X P8Y UNX�V'UW_¥UNV'U Y¡�*S6YES6T4ihbjgFi8b V S   S�R8S6Y^g¯P*T¥i8bN�8S6�[Tai UNX Yvc[q��tS�T¾k0S6bW\ V � ) ³ P8T¥°¾S6b½�S " " P V c[TpgFi X S V P8g§\[S��*TaS�S � _4` S�uvc[S Vp_4U P8Y UNV P8stS�Y¿© ¬ S:µ�stS X�_�Um_�_ Pe�tS _ TZc[S�g¯P*Ti8bW�*S6�ET4i UWX Yvc[ql�KS6T´k0S6bW\ V � �Ec _ gFi8b V S¾g¯P*T´�*S6YES6T4ihb¿k0S6bW\ V )> Rhi8T U i8Y _ P8g _4` S'q�S _a` Po\�ih�tP�R*S X i8Y��tS¥i8sEs[b U S6\ _ P$\ U i8�*P8Y0i8b X c[� UWX'V c[TpgFi X S V

� ©� z � � { z{ � � ?4z? � � z | y ( ���V c[�oÁZS X:_¾_ P�P*Y[S�R8S6T neX P8c[Y _ S6T U Y _ c UW_4U R*S X P*Y[\ UW_4U P*Y   ] `[UWXB`¡UNV§_4` i _ �  t_a` S�k0S�bN\ePhg\[S:k0Y Um_4U P8Y�P8g ��  qlc VZ_ Y[P _�X P*Y _ i U Y _4` S's[T U q Um_4U R8S X c[�KS�TZP�P _4V P8g�c[Y UW_�n )j�ÅT Um_ S �U Y _a` S'g¯P*Taq

� z � � { z { � � � | � ?4z ? � � z � � � y (*( �] ` S�TaS � UWV i _ P*c[T�\ UWV sKP V i8b2)��ÅS�YEP�] ` i�R8S _ ]�P¾stS�Y X6U b V P8g X c[TZR8S V P8g[�*S6Yvc V (   S~i XB`P8g ] `[UWXB`�UWV i�� 5 ��� X P�R8S6T U Y[�¥P8g UW_aV �*i X P*� U i8Y¿�vihY[\$]�S ` i~R*S _ P�i8sEs[b n�_4` S´q�S _a` P�\VZU qlc[b _ i8YES6P*c V b nl_ P��KP _a`�X cETZR*S V ) »´`[UNV,U Y _ TaPo\[c X S V�X P*Y VaU \[S6Tai8�[bWS®ih\[\ UW_aU P*Y[i8b X P*q �s[b UNX i _4U P*Y V6  g¯P*T$] `[UWXB`±V S�S � � � �2�Ài8Y[\ _4` S�P*� VZ_ Tac X:_4U P*Y _ P _a` Sq�S _a` Po\  �_a` P*c[� `]�S~i ¨   UNV§X S6T _ i U YEb n�Vp_ T UNX�_ b n�VZ_ TZP*Y[�*S�T _4` i8Y _4` S Æ Tai8c[S�T � ��i8Y U Y�P*� VZ_ TZc X�_aU P*YI) »´` STZS~i V P*Y�g¯P*T _a` S X P8Y[\ UW_aU P*Y�P*Y � UNV�_a` i _ P _a` S6Tp] UNV S _4` S X c[TZR8S V�y (8( � ]�P*c[bW\�i8\[q UW_X P*q�sEbNS:µ�qlc[b _4U s[b UWX i _aU P*Y   i8Y[\ _4` S'bÂi _Z_ S6T�i X�_aV P*Y _4` S � 5 � �p¦�S�bNq�S6T��8TaP*c[s V � _4` c V_a` S'P*Ta\ES6T�P8g _a` S'bÂi _Z_ S�T�]�P*c[bN\�i8bm]�i noV �KS'i8Y�P�\E\�stP�]�S6T´P8g �   ] ` S6TaS6i V§UW_�` i V�_ P�KSJTZS6\[c X S6\ _ P���g¯P*T _4` S¾q�S _4` Po\ _ P�]�P*T ¨ ) y ÈÀS6TZSJP*Y[SÀgFi X�_ P*T ��i8T UWV S V �KS X ihc V S¾Phg_a` S���� \ U R UNVZU P*Y�stP U Y _4V P8Y _4` S �*i X P*� U i8Y¡\[S�k[Y[S6\�P�R*S�T ��y � 5 � � ) � �¾Y _a` SlP _a` S6T` ihY[\  8U Y _a`[UNV i8Ta�8c[q�S�Y _ ]�S�P*YEb n Y[S�S6\$i8sEs[b n ¦ XB`EU Y " S6b º V È n stP _a` S VZUNV�_ P _a` S VZU Y[�8bNSsKP*b n Y[P8q U i8b z  �V P _4` i _¥Um_'X ihYÅ�KS�TaS6sEbÂi X S6\^� n ° U T UWXB` bNS _~º VJ_4` S6P*TZS6qQP*Y�s[T U q�S VU Y�ihT UW_a` q�S _4UWX s[TZP*�*TZS VaVZU P*YI)> bWb _4`[UWV TZS6bNi _ S V,_ P � c[S Vp_4U P*Y y > � ) ³ P*T � c[S Vp_4U P8Y y2Æ���_a` S®P8Y[b n�¨ Y[P�]§Y�TZS V cEb _4V

i8TZS�g¯P*T�uvc0i8\[T UNX�V6  g¯P*Tj] `[UWXB`�V S6S _4` S§TZS6q�i8T ¨vV ihg _ S�T »´` S6P8TaS6q@(*) ¬ _%V S�S6q V TZS~i V P*Y��i8�EbNS _ P�i Va¨ ] ` S _4` S�T _4` S�TaS UNV i8Y^i8Y[i8bNP*�8P*c V TZS V cEb _ g¯P*T'P _a` S6T ¨vU Y[\ V P8g�T4i _aU P*Y[i8bV c[TZgFi X S V � _a`[UNV�UNV i8Y[P _4` S�TÀsETaP*�[bWS6q¼g¯P*T�] `[UWXB`�_a` S'k0T Vp_§VZ_ S�s VZ` P*c[bW\�sETaP*�0ih�[b n �tS_ Plc V SJYocEq�S�T UNX i8b V S~ihT XB`�_ Pl�8S6Y[S�T4i _ S¾ils[bNi8c VaU �[bNS X P*Y=ÁZS X:_ c[TZS8) ³ P*T _a`[UNV s[c[TZstP V S  P*YES´Y[S6S�\ V,_ P¥S:µEi8q U Y[S´i Vpn�Vp_ S�q ] UW_4` Y[P _�_ PoP'q�i8Y n s[i8T4i8q�S _ S6T V �oi8Y[\ _4`EUNV bNS6i8\ V_ P _a` S'g¯P*bNbWP�] U Y[�luvc[S VZ_aU P*Y¿©��� ��������� � � � 3=1�/GC0H��:7o1 #�Do9BH � A8; !8H:-ÄL y ( ��� @ ;G/GCÅ/¯C0H � x ;G- d 1 ;N��/GC0H:14H�DA03=M�=-I3=O$;FD8M � ;G- /¯C0H 4 � x 4 �:7 9BC /¯CED8/¥; # � ;N����3=M 7EL6MNH�;G- d@/¯C0H:-f;G/'CED=���:7 9BCÄD��3=M 7o/2;<3=- #~3=1 @ Co;<9BCHaDo9BC��:7�O$O�D8-K5�;N��D�L:��3=M 7�/�H:M��L:3=7o-K5oHa5�L � �»´` S U \[S~ihb�i8Y V ]�S6T _ P � c[S Vp_4U P8Y y<·´� ]�P*c[bW\e�KS _ P�s[TaP�R U \[Sli�� U T4i _aU P*Y0ihb�q�i8s

� . � ? \[S�k[Y[S6\^P�R*S6T � )�ÈÀP�]�S:R*S6T  ¢UW_¾X i8Y��tS Va` P�]§Y _4` i _'V c XB` i�q�i8seS:µ UWVZ_4V� �

Page 25: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

g¯P*T$YEP*Y VaU Y[�*c[bNi8T X c[� UNX�V c[TZgFi X S V �QU g®i8Y[\ÄP*Y[b n U g ��y�����UWV Y[P _ S6q�s _�n i8YE\ �X P*Y _ i U Y V il\ U R UWV P*T�\[S�k[Y[S6\�P�R*S6T � ] `[UWXB`�X P*Y VaUNVp_4V P8g �   �lP*T�2 VZ¨ S�]�b U Y[S V ) y2³ P*T°¾S�bt½�S " " P V c[TZgFi X S V P8g�\[S��*TaS�S �  �_4` S V S X P*YE\ X P*YE\ UW_aU P*Y�qlc Vp_ �KS®TZS6s[bNi X S�\�� n�_4` SVp_ i _ S6q�S6Y _§_4` i _ � X P8Y _ i U Y V i�\ U R UWV P*T´\ES�k0Y[S�\P�R*S�T � ] `[UWXB`X P8Y VaUWVZ_4V P8g�P*Y[S¥P*Tq�P*TaS VZ¨ S:] b U Y[S V ) �$³ P*T ·�`��i _ S6bWS _§V cETZgFi X S V6  ] `[UWXB`�` i�R8S _a` S'g¯P*Taq

z�?=y¯z ? 5 �Bz ?{ ��� � y¯z�?~|az �] ` S�TaS �¾UNV i�Y[P8Y�� V uoc[i8TaS�i8Y[\ � UWV®` P8q�P*�8S6Y[S�P*c V§X c[� UNX8 KUW_§UWV§¨ YEP�]§Y _4` i _À_4` S6TaSUWV iek0Y UW_ S V S _ PhgJs0ihT4i8q�S _ T UNX�V P*bWc _4U P*Y V�y S~i XB`ÄU Y � U Y ` P*q�P*�*S�Y[S6P*c V R8ihT U ih�[bNS VB�V c XB`Ã_4` i _ S~i XB` stP U Y _ Phg ��y ���¾UNV TaS�s[TaS V S6Y _ S6\�� n P*Y[S�P8g _a` S6q  �_4` P*cE� `ÃU YÃi8YU YEk0Y Um_�n P8g�\ U�� S6TZS6Y _ ]�i noV ) Æ c _®U Y��*S6Y[S�T4i8b¢q�P*TaS _a` i8Y�P*Y[S¥s0i8Tai8q�S _ T UNX'V P8bNc _aU P*YUWV Y[S�S6\[S�\   i8YE\s0i8Tai8q�S _ T UNX¥V P8bNc _aU P*Y VÀU Y�P*Y[b n ��Rhi8T U i8�[bWS V§X i8YEY[P _ sEbÂi n i�c V S�g¯c[bs0ihT _§U Y _a` S¥s[TaP X S VaV )��� ��������� � � +B��/GC0H:1BH�DeMmD81>AoH:1�9:MmD=�B��3$#�9:7[L6;<9$�:7o1 #�Do9BHB� � ;F5oHaD8MGM� 1 /GC0H�9:MmD=�B��3$#D8MGM¿-I3=-E�:;G-�A87oMmD81�9:7EL6;<9'�:7�1 #�Do9BHB� �B#~3=1 @ Co;<9BC�D8-KD8MN3�Ao3=7��¥14HB�:7oM /¯�'C03=Mm5 �»´` S,uvc[S Vp_4U P8Y�P8gEs0i8T4ihq�S _ T UWX�V P*bWc _4U P*Y V�UWV b U Y ¨ S6\ _ P _4` S U \[S~i§P8g � �aHa�67o; !=D8MNH:-I9BH:)

�S _ � �KS�i R8ihT U S _�n \[S:k0Y[S�\ P�R*S�T � � _a` S6Y Ç ��S6uvc U Rhi8bNS�Y X S UWV \[S�k[Y[S6\�i V�_4` Sk0YES VZ_ S�uvc U Rhi8bWS6Y X S´TaS6bNi _4U P*Y V c XB`$_4` i _�_ ]�P¥stP U Y _4V � U R*S�Y$� nl_4` S V i8q�S�s0i8Tai8q�S _ T UNXV P*bNc _aU P*Y¥ihTaS%S6uvc U R8ihbNS6Y _ ) > b _ S�TaY0i _4U R8S6b n* �UW_¢UWV¿_4` S�k0Y[S VZ_ S6uvc U R8ihbNS6Y X S%TaS�bÂi _aU P*Y V c XB`_a` i _ g¯P8Tji8Y n q�i8s � © � { . � ihY[\lsKP U Y _4V �%{.| � ?�  ihbNbo\[S�k0YES6\�P�R*S6T �� 8_4` S�stP U Y _4V� y �%{Z� i8Y[\ � y � ?.� ihTaS�S6uvc U Rhi8bWS6Y _ ) > �*PoP�\�\[S~ihb UWV®¨ Y[P�]§Y¡i8�KP*c _ Ç ��S6uvc U Rhi8bNS�Y X SP*Y X cE� UNX�V cETZgFi X S V � U Y s0ihT _4UWX c[bNi8T  %Um_�UNV$Va` P�]§Y U Y � ��� � _4` i _�_a` S X bWP V c[TZS�P8g'i8YÇ ��S�uoc U Rhi8bNS�Y X S X bÂi VaV�U Y ��y2���UWV�X P8q�s[c _ i8�[bWS   i8Y[\ _a` i _�_4` S X bNP V c[TZS V P8g _ ]�PÇ ��S�uoc U Rhi8bNS�Y X S X bÂi VZV S V ihTaS¥S Um_4` S�T _4` S V ihq�S¥P*T´\ UWV ÁZP U Y _ )��� ��������� � � +B�À/GC0HÀ��H:/�3$# � �aHa�67�; !=D8MNH:-I9BH¾9:MmD=�B��HB�'3=-�Dl-I3=-E�:;G-�A87oMmD81J9:7[L6;<9§�:7o1��#�Do9BHB'%-0;G/�H � � D8- /GC0H:1BH�L:H�/ @ 3e58;N�:/2;G-I9:/ � �aHa�67�; !=D8MNH:-I9BH�9:MmD=�B��HB� @ Co;<9BCeCED !8H�/GC0H�6D8O�Hl9:MN3��:7o14H¥;G- ��yF�� ���) �����:7o1(#�Do9BHB��D8-K5 ��7�O$O�H:1J�:7o1 #�Do9BHB�:)��� V c[TpgFi X S V i8TZS _4` S VaU q�sEbNS Vp_¥¨vU Y[\^P8g,R8ihT U S _�n i8�KP*c _ ] ` P V S�Yvc[q��tS�T � _a` S6P8TaS _aUNXs[TZP*sKS6T _aU S V R*S6T n b Um_a_ bWS UWV�¨ YEP�]§YI� U YE\[S6S�\ _a` S n�Vp_4U bNbEsETaS V S6Y _ q�i8Y n cEY V P*bmR*S�\�s[TZP*���bWS6q V S�R*S�Y _ P _a` S§�*S6P*q�S _ S6T�) »´` S6TZSJihTaS U Y�k0Y UW_ S6b n q�i8Y n gFi8q U b U S V P8g ��� V cETZgFi X S V �_a` S VZU q�s[bNS VZ_ P8g _a` S6q   i8YE\ _4` S�P*Y[b n P*Y[S�] `EUNXB` ] U bNb��KS X P8Y VaU \[S6TZS6\ U Y _4` S�s[TZS V S�Y _i8T _4UWX bNS  �X P*Y VZUNVp_4V P8g0i8bWb*Y[P*Y VaU YE�*c[bÂihT�uvc0ihT _4UWX%V c[TpgFi X S V ) > Y U q�sKP*T _ i8Y _�V sKS X�U i8b _�n stSP8g � � V c[TpgFi X S V$X P*Y VZUNVp_4V Phg _4` S ��7oO$O�H:1$�:7�1 #�Do9BHB�   i�s ` T4i V S�] `[UWXB`�X i8Y X i8TZT nS UW_4` S6T´P8g _ ]�P�TaS�bÂi _ S6\�q�S6i8Y U Y[� V ©

� �

Page 26: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

� »´` S�uvc[P _aU S�Y _ P8g´ihY > �KS6b U i8Y V c[TZgFi X S � � n^_a` S�i8c _ P8q�P*TZs `[UWV q 5 (*� _a`[UNV` i V ( 2 VaU YE�*c[bÂihT�stP U Y _4V´X P*TaTZS V sKP*Y[\ U Y[� _ P _4` S�( 2 � ��\ U R UWVaU P*Y�sKP U Y _4V Phg � )

� »´` SJY[P*Y VaU YE�*c[bÂihT V c[TZgFi X S¾P*� _ i U YES6\�� n �[bNP�] U Y[��c[s _4` S�( 2 VaU Y[�*c[bNi8T%stP U Y _4VU Y _4` S¥s[TZS�R U P8c V \ES�k0Y Um_4U P*Y¿)

�¾Y[S�i8\ERhi8Y _ ih�*S¥P8g �'c[q�q�S�T V c[TpgFi X S VJU Y X P*q�s0i8T UWV P*Y�] UW_a` �8S6Y[S�T4i8b ��� V c[TpgFi X S VUWV�_4` i _ g¯P8T _4` S'g¯P*TZq�S�T UW_´UWV S6i VZn�_ P�\[S _ S6TZq U Y[S�½ UWX�y ���� )¦�P*q�S ��� V c[TZgFi X S V'X P*Y _ i U YeP*Y[S$P*T®q�P8TaS�sKS6Y X6U b V P8g X c[TZR8S V P8g%�*S�Yoc V (   i8YE\

_a` S V S�sKS6Y X�U b V q�i n S:R*S6Y�tS¥P8g _4` S ¨vU Y[\�\ UNVZX c VZV S�\ U Y _4` S�sETaS�R U P*c VÀV S X:_4U P*Y¿�0�[c _P*YES Va` P8c[bN\ Y[P _lX P*YEk[Y[S�P*Y[S º¹V i _Z_ S6Y _aU P*Y _ P � � V c[TZgFi X S V ] Um_4` _a`[UNV i8\E\ UW_aU P*Y0ihbs[TZP*sKS6T _�n ) ³ P8T _4` S _aU q�S��tS U Y[�  %_a` S6TZS UNV q�S6T Um_$U Y X P*Y X S�Y _ T4i _4U YE�ÅP*Y \ U i8�*P*Y[i8buvc0i8T _4UWX6V

� ©� z � � � {�z �{ � � ?az �? � � z � ���E| y ( � ��KS X i8c V S _a` S V S X P*Y _ i U Y�g¯S:] S�Y[P*c[� ` s0ihT4i8q�S _ S6T V�_ P�q�i ¨ S VZnoVZ_ S6q�i _4UWX YocEq�S�T UNX i8bS.µEsKS6T U q�S6Y _ i _4U P8Y�sKP VaVZU �[bWS8)�ÈÀP�]�S�R8S6T   _a` S¾Yvc[ql�KS6T _4` S�P*T n P8g V c XB`�V c[TZgFi X S V q�i n�KS�S:µ X S�s _4U P*Y0i8b   �KS X i8c V S _a` S��*S6P*q�S _ T n¡X S6T _ i U Y[b nUNV ) ¬ Y[\ES6S6\ý UNX�y ����¾` i V Tai8Y ¨� ��  ] `[UWXB`�UNV%_4` SJbÂihTa�*S VZ_ stP VaVaU �[bNS§Rhi8bNcESÀg¯P*T�i8Y n ��� V cETZgFi X S   i8YE\ UW_�UNV �*S�Y[S6Tai _ S�\� n�_4` S X bÂi VZV S V P8g _4` S ����b U Y[S V P*Y �� �Iq�P*TaS�P�R*S6T � UNV i �'c[q�q�S6T V cETZgFi X Slc[s _ PUWV P*�*S�Y n*  ihY[\ U Y[\ES6S6\ UWV®_a` S �'c[q�q�S�T V c[TZgFi X SlP8g�� ���¼] ` S6TZS�� UNV i8YeS�bNb U s _4UWXX c[TZR8S�] `[UWXB` ih\[q Um_4V�X P*q�s[bNS.µ�qlcEb _4U s[b UWX i _aU P*YI)��¾Y[S X P*Y V S�uvc[S6Y X S�P8g _4`EUNV´UNV�_a` i _� UNV T U � U \ U Y _a` S V S6Y V S�Phgji8bW�*S6�ET4i UWX �*S6P8q�S _ T n )»´` S�TaS UNV i8Y�P*��R U P*c V q�i8s�g¯TaP8q ��_ P _4` S¥uvc0i8\ET UNX'V c[TZgFi X S

� © � � ? � � { � ?{ � � ? � ?? � � �?� �E}

¬ g�� �� { � ? �UWV i V uvc0ihTaS   S~i XB` P8g _4` S _ ]�P$gFi8q U b U S V Phg�b U Y[S V P*Y � UWV \[S�k[Y[S6\�P�R*S6T

_a` S´�*TaP8c[Y[\$k0S6bW\   i8Y[\�S6i XB`�V c XB` b U YES�b U g _4V�_ P¥i X c[TpR*S§P8gK�*S6Yvc V (§P*Y � �vq�P*TaS�P�R*S6T_a` S��*i X P*� U i8Y V P8g _a` S V S X c[TZR8S V¾` i~R8S _4` S�g¯P*TZq y � �. 0V P _4` i _À_4` S�q�S _a` P�\ V P8g _4` Ss[TZS�R U P8c V§V S X:_4U P*Y X i8Y��tS�i8sEs[b U S6\¿)

�i8T _4U Y Æ T U � `v_ � ��� ` i V,X P*q�s[c _ S�\�ihY[\ _ i8�[c[bNi _ S�\ Æ T {6y ��� #hÆ T y ��� g¯P*T%i8bNb � Phg_a` S�g¯P8Taq y ( � � � UW_§UWV YES X S VaV i8T n�_ P�\[P _a`[UNV � n�X P*q�s[c _ S6T   �KS X i8c V S _4` S�TaS�i8TaS � �+2\ UWVZ_aU Y X:_�X i V S V ) ¬,` i8\�s[TaS:R U P*c V b n�Va` P�]§Y U Y�� � � � _4` i _ P�R*S6T � _a` S Æ Tai8c[S�T � ��i8Y U YP*� VZ_ TZc X�_aU P*Y UWVj_4` S§P*Y[b n P*� VZ_ TZc X�_aU P*Y _ P _a` SÀÈJi VZV S§s[T U Y X6U s[bNS U Y _4` S§q�P VZ_ �8S6Y[S�T4i8bX i V S U Y�] `[UWXB` � � { � ? �

UWV i V uvc0i8TaSh) y ��P Vp_ �*S�Y[S6Tai8b U Y _a`[UNV�X P*Y _ S.µ _ q�S6i8Y V�_a` i _

Y[P8Y[S�Phg _4` S�� � x � ¤¾UNV i V uoc[i8TaS�i8Y[\ � �� { � ? �UWV YEP _ i�g¯P*c[T _4` stP�]�S6T�) ��¬ _®V S6S�q V

TZS~i V P*Y0i8�[bWS _ P ` P*sKS _a` i _�_a` S V i8q�Ss[TZP*stS�T _�n ] U bWb VZ_4U bNb ` P*bW\ U Y�i8bNb _a` S X i V S Vg¯P*TÀ] `[UNXB` � � { � ? �

UNV i V uvc0i8TZS8�¢�[c _®_4` S�TaS�i8TZS _ PoP�q�i8Y n Phg _4` S�q _ P�S:µEi8q U Y[S

U Y[\ U R U \[c0i8bWb n )��'Y _4` S�P _4` S�T ` ihY[\  �_4` S6TaS UNV�VZ_ TZP*Y[� Yvc[q�S6T UNX i8b®S:R U \[S�Y X S _a` i _

� 2

Page 27: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

] ` S�Y � � { � ? �UNV Y[P _ i V uoc[i8TaS _4` S'P*� VZ_ Tac X:_4U P*Y X P*q U Y[��g¯TaP*q Æ T {6y ����UNV YEP _´U Y

�*S�Y[S6Tai8b _4` S�P8Y[b n P*� VZ_ Tac X:_4U P*Y _ P _4` S�È®i VaV S�s[T U Y X�U s[bWS8)��� ��������� � ��� CED8/,;N��/¯C0H�D�5�58;G/F;<3=-KD8M�38L:�:/F1.7 9:/F;<3=-¡;G-¡/GCo;N�$9aD=��H ��¾Y[S¥s0i8T _4UWX c[bNi8Tab n�U Y _ S6TZS Vp_4U YE��S.µEi8q�sEbNS UNV�_a` S V c[TpgFi X S

z � � � z �{ � z �? � � z �� y (�� �

_a`[UNV´` i V§_ ]�P�P*��R U P*c V T4i _4U P8Y0i8bIsKP U Y _aV6  �Ec _ i8s[sKS~i8T V§_ P ` i~R*S�Y[P�P _4` S�T V )

�E) ? H:-E�:;G/��3$#'1ZD8/2;<3=-KD8MoA03=;G-0/¯�:)¦�P�gFi8T ¬j` i~R*S U �*YEP*TaS�\ � c[S Vp_4U P*Y y ° � ) ¬ _ \ U�� S6T V g¯TZP*q _4` SJP _a` S6T V�U Y _4` i _,UW_,UNV Y[P _i�� U Tai _4U P*Y0i8b�uvc[S Vp_4U P8Y   �Ec _�UNV i VaV P X�U i _ S6\�] UW_a` i�s0i8T _4UWX c[bNi8T'S6ql�KS6\E\ U Y[��P8g _4` SRhi8T U S _�n � U Y�s[TZP=ÁZS X:_4U R8S V s0i X Sh) ³ P*T VZU q�s[b UWX6Um_�n ]�S�]�P*T ¨ P�R*S�T � ) > sKP U Y _�� U Y� � \[S:k0Y[S6\�P�R8S6T � ` i V i¥TaS�s[TaS V S�Y _ i _4U P*Y y �� |~}~}~}:| � �*� ] ` S6TZS _4` S � x i8TZS U Y _ S6�8S6T V] Um_4` Y[P X P*q�q�P*Y�gFi X�_ P*T6�´ihY[\ _4`[UWV TZS6s[TZS V S6Y _ i _aU P*Y UNV cEY U uvc[S�c[s _ P XB` i8YE� U Y[�_a` S VaU �*Y V P8g®i8bWb _a` S � x )��ÅS�\[S�k0YES _4` SC0H:; A=Co/�P8g � _ P¡�tS�q�i=µ 4 � x 4 ��ieb U Y[S6i8T_ T4i8Y V g¯P*Taq�i _aU P*YÅP*Y _4` S�ihql� U S�Y _lV s[i X S�qlc[b _4U s[b U S V¥` S U � `�_4V � n Yvc[q��tS�T V ] `[UWXB`b U S$�KS _ ]�S6S6Y _ ]�P�stP VZUW_aU R*S X P*Y VZ_ i8Y _4V \[S6sKS6YE\ U Y[��P*Y _4` S�b U Y[S~i8T _ T4i8Y V g¯P*TZq�i _aU P*YI)°¾S�Y[P _ S'� n . y���| �l��_4` S¥Yvc[ql�KS6T§P8g�sKP U Y _4V Phg ��y ��� ] ` P V S ` S U � `v_ÀUWV bNS VZV´_4` i8Y� � _a` S6Y Um_'UNV Y[i _ c[Tai8b _ P�i Va¨�` P�] . y���|���� �tS ` i~R*S V i V�� . � ) »´`[UNV'UWVJ_4` SX P*TaS'uvc[S VZ_4U P*Y�g¯P*T _4` S�È®i8Ta\ n � Um_a_ bNS�]�PoP�\�q�S _a` Po\   ] `[UWXB` ] ` S�Y Um_§UNV i8sEs[b UWX i8�[bWSUWVÀ_a` S��KS Vp_�y ihY[\eP8g _ S6Y _a` S�P*YEb nE� ]�i n Phg�s[TaP�R U YE� _4` i _ ��y ���´UNV YEP _ S�q�s _�n ) ¬ YR8S6T n �8S6Y[S�T4i8b X�U T X c[q VZ_ ihY X S V§_a` i _ q�S _4` P�\�s[TZP�R U \[S V S Vp_4U q�i _ S V P8g _4` S¥g¯P8Taq

. y���|��l�%� bNS6i8\ U Y[� _ S6TZq � S�TaTZP*T _ S�Taq }

»´` S�bNS6i8\ U YE� _ S6TZq UNV c V c0ihbNb n_4` S V i8q�S�i V P8Y[S�]�P*c[bW\ÃP*� _ i U YÅ� n s[TZP*�0i8� U b UWVZ_4UWXi8TZ�*c[q�S6Y _4V ) Æ c _lV c XB` TaS V c[b _aV i8TZS�P*Y[b n Rhi8bNc0ih�[bNS�] ` S6Y Um_�X i8Y �KS VZ` P�]§Y _a` i __a` S®S�TaTZP*T _ S6TZq UNV%V q�i8bWb X P*q�s0i8TZS6\ _ P _4` SÀbNS6i8\ U YE� _ S6TZq   i8Y[\ _ P�i XB`[U S�R8S _4`EUNV�_4` S\ U q�S�Y VaU P*Y�P8g � Y[S�S6\ VÀ_ P��tS�bÂihTa�*S X P*q�s[i8TaS�\ _ P Um_4V \[S6�8TaS6Sh) »´` S�S:µ _ TZS6q�S X i V SP8g _4`EUNV´UNV�_a` S'g¯P*bNbWP�] U Y[� _a` S6P8TaS6q   \[c[S _ P Æ�U T XB` � � �2) �� � ��� � � � � H:/ �={.|~}~}~}�| � � L:H'A03��:;G/2; !8H�365�5;G-0/�H�AoH:1B� 1 -I3=/J-IHB9BHB�B�6D81.;GM�eD8MGM,58; #��#~H:14H:-0/ , �¿C0H:-�/¯C0H:14H�H �*;N�:/¯� . y �h{:|~}6}~}6| � ��� @ ;G/GC�/¯C0H #~3=MGMN3 @ ;G-�A®AK143BA0H:1:/� , � 3=1JD8- . � . $MNH:/ w¢x y¯z |~}6}~}6|az��,� L:H'C03=O�3�AoH:-IHB3=7���A03=M�=-I3=O$;FD8M � @ ;G/¯C¡9B3�H���9:;<H:-0/G��;G-d D8-K5$\[S6� w�x�� ��x #~3=1¾� � ( |~}6}~}�| � , �¿C0H:-�/¯C0H w�x CED !8H$De9B3=O$O�3=-�-I3=-0/F1.; !�;FD8M� H:143�;G-d � ,

� �

Page 28: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

»´` S$sETaPoP8g�gFi8bNb V®U Y _ P _ ]�P�s0i8T _4V ) ³�U T Vp_~  _4` S�È®i8Ta\ n � Um_a_ bNS�]�PoP�\�q�S _4` Po\ UNV c V S�\_ P�s[TZP�R*S _4` SÀTaS V c[b _�U Y _a` S V sKS X6U i8b X i V S§] ` S6Y�� � (®i8Y[\ w%{jUNV \ U ih�*P*Y0i8b � _a` i _UWV6 E_ P VZ` P�] _4` i _§U g �lUNV Po\[\�i8Y[\ . � . {6y(�*��_a` S6Y

� z � ��}~}~} � � �,z �� � �

` i V i�Y[P8Y _ T U R U i8b U Y _ S6�*Tai8b V P*bNc _aU P*YI) »´` S6Y _4` S��*S�Y[S6Tai8b X i V S UWV TaS�\[c X S�\ _ P _a`[UNVV stS X6U ihb X i V S�� n sEc[TaS�b n S6bWS6q�S6Y _ ihT n q�S _4` P�\ V ) »´` S�TaS6uvc U TaS�q�S�Y _À_4` i _ ihbNb _4` S �6xVZ` P*c[bW\Å�KS�P�\[\Åi8T UNV S V g¯TZP*qQ\ U � X c[b _4U S V'X P*YEY[S X:_ S6\Å] UW_4`�_4` S�TZS~i8b%s[bNi X Sh��P�R*S�T�i_ P _ i8bWb n�X P*q�sEbNS:µei8bW�*S6�[Tai UNX Yvc[ql�KS6T¾k0S�bN\ _4` S6TaS UWV i VaU q U bÂi8T _a` S6P*TZS6q g¯P*TJ] `[UWXB`_a` S �6x¢X i8Y��KS�ihY n stP VZUW_aU R*S U Y _ S6�8S6T V )��� ��������� � � +:- �¿C0HB3=1BH:O � 1 9aD8- /GC0HÃ9B3=-K58;G/F;<3=- /GCED8/�D8MGMJ/GC0H �6x D81BH^365�5ÄL:H14H�AKMmDo9BHa5 L �/¯C0H14Ha�67�;G14H:O�H:-0/�3=- /¯C0H w�x /GCED8/$/¯C0H�AK143 �~HB9:/2; !8H !=D81.;<H:/� A8; !8H:- L w�{%� }6}~}v� w � � � CED=��D�-I3=-E�:;G-�A87�MmD81'14HaD8M�A03=;G-0/��»´` S�ÈJi8TZ\ n � UW_Z_ bWS�]�P�Po\'q�S _4` Po\']�i V \[S VaU �*Y[S6\¥g¯P*T¢i VaU Y[�*bNS�S�uvc0i _4U P*Y U Y'] `[UWXB`

_a` S'R8ihT U ih�[bNS V i8TaS V S6s0i8Tai _ S�\ � g¯P8T´S:µEi8q�s[bNS   i8Y�S6uvc0i _aU P*Y�P8g _4` S'g¯P*TZq

� {6y¯z�{Z� ��}~}~} � � �®y¯z����,� �

] ` S�TaS _4` S � x i8TaS$sKP*b n Y[P8q U i8b V� t_4` S z�x i8TZS U Y _ S��*S6T V6  i8Y[\�P*YES�] UWVa` S V'_ P�s[TaP�R8SV P*bNc[� U b Um_�n g¯P8T®ihbNb U Y _ S6�*S�T V®�=  P*T®i8bNbIbÂi8TZ�*S¥S6Y[P*cE� `e��  P*TJihbNq�P VZ_ i8bWb � ) Æ c _ÀUW_À` i Vi8b V P¥�KS6S6Y�i8s[s[b U S�\��KP _a`�_ P V S�R*S�T4i8b VaU qlc[b _ i8Y[S�P*c V S6uvc0i _4U P8Y V i8Y[\ _ P�S�uvc0i _4U P*Y V%U Y] `[UWXB`�_4` S¾Rhi8T U i8�[bWS V i8TaS'Y[P _§V S6s[i8T4i _ S6\¿) »´` S¾g¯P8bNbNP�] U Y[� _a` S6P*TZS6q Phg�ÈÀP�P8bNS n � ��( �UWV qlc XB`�_4` S¥q�P VZ_§U q�s[TZS VZVaU R8S¥TaS V c[b _§U Y _4`[UWV \ U TZS X�_aU P*YI) �� � ��� � � � �3=O�3�AoH:-IHB3=7v�,-I3=-KD81 �9:7EL6;<9B�§3)!8H:1 � �6D8/F;N�(# ¥L:3=/GCl/GC0H 'D=�B��H¢AK1.;G-3�9:; AKMNH�D8-K5 @ HaD���D:A�AK143��*;GO�D8/2;<3=- ,¬ _ i8s[sKS~ihT V´_4` i _´_4` S�ÈJihTa\ n � Um_a_ bWS�]�P�Po\�q�S _4` P�\ X i8Y�P8Y[b n ]�P*T ¨ g¯P*T�gFi8q U b U S V

g¯P*T´] `[UWXB` . y���|����§UWV i VZn q�s _ P _aUNX ihbNb n S6uvc0ihb _ P UW_aV s[TaP*�[i8� U b UNVp_4UNX R8ihbNc[Sh� U Y�s[i8T �_aUNX cEbÂi8T UW_JV S6S�q V cEY[b UW¨ S6b n�_a` i _'Um_¾X i8Y¡�tS�q�i8\ES _ P�]�P*T ¨ g¯P*T®gFihq U b U S V g¯P*TJ] `[UWXB`]�S~i ¨ ihs[s[TaP~µ U q�i _aU P*YÃgFi U b V ) �i8Y U Y ` i V s[c _ g¯P*Tp]�ihTa\ i X P*Y=ÁZS X�_ cETaS�i8�tP8c _$_4` Si Vpn q�s _ P _4UWX \[S�Y VaUm_�n P8g�T4i _aU P*Y0ihb V P8bNc _aU P*Y V g¯P*T X S�T _ i U Y��*S6P8q�S _ T UWX i8bWb n�U Y _ S6TZS VZ_aU Y[�gFi8q U b U S V P8g�Rhi8T U S _4U S V g¯P*TÀ] `EUNXB` ]�S~i ¨ i8sEs[TaP~µ U q�i _4U P*Y UNV c[Y[b UN¨ S6b n�_ P ` P*bN\¿©,q�P*TaSs[TZS X�UNV S�b n8  g¯P*T ³ ihY[P�Rhi8T U S _aU S V S6ql�KS6\E\[S6\ U Y � � � n q�S~i8Y V P8g _4` S U T�i8Y _4UWX i8Y[P8Y UNX i8b\ U R UNV P8T V ) ³ P*T VaU q�s[b UNX�UW_�n*  ]�S�\ES VaX T U �tS `[UNV�X P*Y�ÁZS X�_ c[TaS�P*Y[b n g¯P*T�°¾S6bo½�S " " P V c[TpgFi X S V� P8g�\[S��*TaS�S V �$i8Y[\ �[) » P�i VZ¨ i8�KP*c _ . y���|��l��UNV YEP�] _4` S¾]§TZP*Y[��uvc[S Vp_4U P*Y   g¯P*T� q�i n�X P*Y _ i U Y�b U Y[S V \[S�k[Y[S6\eP�R8S6T ��  i8Y[\�g¯P8T§i8Y n b U YES . y���| ,�

���� ? g¯P*T

� �

Page 29: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

V P*q�SÀY[P*Y " S6TZP X P8Y VZ_ i8Y _ � ) »´`EUNV,UNV qlc XB` �*TZS~i _ S6T _4` i8Y _4` S®P*Ta\ES6T � P8g ��q�i8�*Y UW_ cE\[SS VZ_4U q�i _ S¾g¯P*T". y���|���� � U R*S�Y�� n i�s[TaP*�[i8� U b UNVp_4UNX i8Ta�8c[q�S�Y _ ) ³ P8T _a` S¥bÂi _Z_ S�T V c[� ��*S VZ_aV i8Y�S VZ_aU q�i _ S � � �±y . y��K� #[y�� � ( �a�.  ] ` S6TZS _4` S�s[TZP�\[c X�_®UNV´_ i ¨ S6Y¡P�R*S�T¾i8bWbs[T U q�S V bWS VaV¥_4` i8YÃi X S�T _ i U Y^�tP8c[Y[\�] `[UNXB` \[S6sKS6Y[\ V P*Y � ) ¬ Y^R U S:] P8g�] ` i _�UNVV i U \ U Y �  �_4`[UWV s[TZP�\Ec X�_ P*c[� `�_�_ P��tS�TaS�s[bÂi X S6\Å� n^V P*q�S _4`[U Y[��] `[UNXB` \[S6sKS6Y[\ VP*Y _a` S$�tS ` i~R U P*c[T'Phg ?=y � |���� Y[S~ihT � � (*)���P*TaS�s[TZS X�UNV S�b n8 ¿_4` Sl]�i n�U Ye] `[UWXB`_a` S�bWS~i8\ U Y[� _ S�Taq U Y _4` S�ÈJihTa\ n � Um_a_ bWS�]�P�Po\�q�S _4` Po\ UWV P8� _ i U Y[S6\ V cE�*�*S Vp_4VJ_a` i _` S�TaS$]�S VZ` P*c[bW\ _ i ¨ S�� 5 ( _ P��tS X P*q�s0i8T4ih�[bNS�] Um_4` y bWP*� �¡� <0{ )$ǧS6q�S6ql�KS6T U Y[�_a` S » i _ S X P*Y=ÁZS X�_ cETaS  o_4`[UWV � U R8S V%_4` SÀT U � `�_%` i8Y[\ VaU \ES®P8g y ( � � i V i X P8Y=ÁZS X:_ c[Tai8b S V �_aU q�i _ SJg¯P8T . y���|��l� ) Æ c _ÀU g _4`[UWV i8Ta�*cEq�S�Y _ ]�S6TZS�Rhi8b U \  � ]�P8c[bN\ X P*Y _ i U Y�q�P*TaSTai _4U P*Y0i8b,stP U Y _4V�_a` i8Y ��  S:R*S6Y _4` P*c[� ` ��� ) �ihY U Y º V ]�i nÃ_ P¡TaS V P*bmR*S _a`[UNVi8� V c[TZ\ UW_�n�UNV,_ P VZ_ c[\ n Y[P _ . y���|��l� �[c _ . y���|��¾�:  ] ` S6TZS � UWV�_4` S¾P*sKS6Y V cE� V S _P8g � P*� _ i U Y[S�\�� n \[S6bWS _4U Y[� _4` S � �lP*T'( 2�b U Y[S V P*Y � ) �i8Y U Y X P*Y=ÁZS X�_ c[TaS�\ _4` i _

. y���|��'��� ��y bWP*� �¡� � <0{ ] ` S�TaS �$UNV�_a` S¥T4i8Y ¨ P8g§½ UNX8y ��� � y ( � �

i8YE\�½�S n TZS � � 2 � ` i V � U R*S�Y^i X P*Y�ÁZS X�_ c[T4ihb�g¯P*Taq�c[bÂi�g¯P8T � ) yFÆ c _ Y[P _ S _4` i _¾_4` S6TaSS.µ UWVZ_�³ i8Y[P�Rhi8T U S _4U S V P8g�\ U q�S6Y VaU P8Y^�8TaS~i _ S6T _4` i8Y ��g¯P*T'] `[UWXB`�y ( � �¾UNV'X S�T _ i U Y[b ngFi8b V Sh��ihY[\ UW_UNV YEP _¡X bWS~i8T ` P�] ��i8Y U Y º¹VeX P*Y=ÁZS X�_ cETaS Va` P*c[bN\��tS q�Po\ U k0S�\ _ PX P�R8S6T V c XB` X i V S V ) ��� i8T U P*c V stS�P*s[bWS ` i�R8S�s[TZP�R*S6\ _4`[UWV�X P*Y=ÁZS X�_ cETaS�g¯P8T _a` S X P*Y[Sz z�{ z�?���z

  ihY[\�ÇÀcE\[�*S ` i V�Va¨ S _aXB` S6\�iJs[TaPoP8g[g¯P8T _4` S VaU Y[�*c[bNi8T X c[� UWX�V c[TpgFi X S

z z�{ z ? � z z�{ z� z z�?az

��z�{�z�?az

����|

_ P�] `EUNXB` i _a_ S6Y _4U P*Y ` i8\��tS�S6Y�\[T4i~]§Y�� n�Æ�U T XB` )��� ��������� � � � 1BH�/GC0H:14H¥-I3=-E�:;G-�A87oMmD81 ? H:M �§H � � 3$�:7�1 #�Do9BHB�l3$#�5oH�A814HBH �¡3=1��&#~3=1@ Co;<9BC�/GC0H �¡D8-0;G-Å9B3=-��~HB9:/27o14H$9aD8-�L:H´AK143)!8Ha5 �¬ Y _a` S�k0T VZ_'U Y VZ_ i8Y X S  ¿UW_ ]�P*c[bN\¡�KS�] UWV S _ P�i8\[\[TZS VZV¾_4`[UWV s[TZP*�[bNS�q£c[Y[\[S�T'T4i _a` S6TTZS VZ_ T UWX�_4U R*S `�n stP _a` S V S V i8�tP8c _ ½ UWX�y ���:  Y[P _ bNS6i VZ_ �KS X i8c V S _4` S Æ Tai8c[S�T � ��i8Y U YP*� VZ_ TZc X�_aU P*Y _ P�]�S6i ¨ i8s[s[TZP~µ U q�i _4U P8YeP X�X c[T V'U Y _4` S X P*Y=ÁZS X�_ c[T4i8bjg¯P8Taqlc[bNi�g¯P*T �i8YE\ _4` S6TaS:g¯P*TaS _4` S�s[TZP*�[bNS�q UNV b UW¨ S�b ne_ P�KS�S6i VaU S6T¥g¯P*T�gFihq U b U S V P8g � g¯P*T'] `[UWXB`]�S~i ¨ i8sEs[TaP~µ U q�i _4U P*Y ` P*bW\ V ) > P*YES � VZU \[S�\�S VZ_aU q�i _ S´g¯P*T%P*Y[S V c XB` gFi8q U b n�UNV � U R*S�YU Y � � ���2)³ P*T®°¾S6b�½�S " " P V c[TZgFi X S V� I_4` S�R8ihbNc[S�P8g � g¯P*TÀ] `EUNXB` . y���|��'�

�� ��y bWP*� �¡���

UWV \[S�k0YES6\Ã� ne_a` S��*S6P8q�S _ T n T4i _a` S6T _4` i8Y�� n¡_a` S�Yvc[ql�KS6T _4` S�P*T n8 �_4` P*cE� `^_a` i _UWV YEP _�_ Tac[S�P8g � ) ³ P*T¥P _a` S6T¥Rhi8T U S _4U S V� �_a` S X P*TaTZS V stP*YE\ U Y[� VZ_ i _ S�q�S�Y _ Y[S�S6\ Y[PbWP*Y[�*S�TJ�KS _ TZc[S8)¾�ÅS VZ_ i8T _ ] UW_4`X c[TpR*S V ) ³ P*TJi X cETZR*S�P8g%�8S6Yvc V¥� i8YE\¡\[S6�*TZS6S �  

���

Page 30: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

]�S ` i~R*S . y���|��l����� ?$= � �[i8Y[\�g¯P*T�i X c[TZR8S¾P8g¢�*S�Yvc V �*TZS~i _ S6T _a` i8Ye( ³ ihb _4U Y[� V�º

_a` S6P*TZS6q UNV S6uvc U Rhi8bNS�Y _�_ P _4` S VZ_ i _ S�q�S�Y _�_4` i _ . y���| �l������y ( � ) Æ c _�U g �£UNVi8YfS6bNb U s _aUNX�X c[TZR8S _4` S�Y . y���|��l�

��$y bNP*� �¡� �$=�? ] ` S�TaS �^UNV�_4` ST4ihY ¨ Phg _4` S

��P8Ta\[S�bNb �<�ÃS U bt�*TaP*cEs¿) yF³ P*T§S6bNb U s _aUNX®X c[TpR*S VÀ_4` S�TaS UNV i�q�P*TZS X i8Y[P8Y UNX i8b�\[S�k[Y UW_aU P*YP8g ` S U � `v_6  ] `[UNXB` UWVlU Y�Rhi8T U i8Y _ cEY[\[S6T$� U b U Y[S~ihT _ T4i8Y V g¯P*Taq�i _aU P*Y¿� _a`[UNV�UNV c V S6\ _ Ps[TZP�R*S _4` S¥TaS V c[b _ i8�KP�R8S8) �³ P*T¢stS�Y X6U b V P8g X P*Y UWX6V�  �i8Y U Y º V uvc[S VZ_aU P*Y UNV s[TaP8�0i8�[b n Y[P _¢_4` S,�KS VZ_ P*Y[S _ PJi Va¨  

i8YE\ Um_ ]�P*c[bW\��KS´�KS _a_ S6T _ P's[TaP X S�S6\�i V g¯P*bNbWP�] V ) > stS�Y X6U b[P8g X P*Y UNX6V�UWV i V c[TpgFi X S �_ P*�*S _a` S6T§] UW_a` i�q�i8s � . � { ] ` P V S�k0�ETaS V i8TaS X P*Y UNX6V ) ¢S _ . � y���|��l� �tS _4` SYvc[ql�KS6T¥P8g�stP U Y _4V P*Y � { P8g ` S U � `�_ bNS VaV'_4` ihY � g¯P*T¾] `[UWXB`^_a` S X P*TaTZS V stP*YE\ U Y[�k0�ETaS X P*Y _ i U Y V Tai _4U P*Y0i8btstP U Y _4V )��� ��������� � � CED8/�;N��/¯C0HÀ9B3=-��~HB9:/F7�1ZD8M0HB�:/2;GO�D8/�H #~3=16. � y���|��l� D8-K5'7o-K5oH:1 @ CED8/9B3=-K58;G/F;<3=-E�l9aD8-Å3=-IH�AK143)!8H¥;G/��¬ _ q�i n �tS']�P*T _a` i Va¨vU Y[� _a` S V i8q�S'uvc[S Vp_4U P8Y V g¯P8T´stS�Y X6U b V P8g X c[TpR*S V P8g��*S6Yvc V (*)³ P*T V cETZgFi X S V P8g��8S6Y[S�T4i8b _�n sKS   ¢i8Y[� º V�X P*Y�ÁZS X�_ c[TaS U q�s[b U S V�_4` i _ uvc[S VZ_aU P*Y V

i8�KP*c _ . y���|��l� ihTaS TZS~i8bWb n uvc[S Vp_4U P8Y V i8�tP8c _¡X S�T _ i U Y X c[TZR8S V P*Y � ��i8Y[\ g¯P*T> �tS�b U ihY V c[TZgFi X S V^y i8Y[\ U Y[\[S6S�\ > �KS6b U i8YÄRhi8T U S _4U S V�U Yfi8Y n \ U q�S�Y VaU P*Y �l_4` S�P*���R U P8c V �8S6Y[S�T4i8b UNV i _4U P8Y$P8g _4` S _a` S6P*TZS6q g¯P*T,S6bWb U s _4UNX�X c[TpR*S V�` P*bN\ V ) Æ c _ ��� V c[TpgFi X S VsKP V S´YES�] s[TaP*�EbNS6q V � i8Y[\�Y[P _ P8Y[S V P*Y$] `[UWXB` i8Y n s[Tai X�_aUNX ih�[bNS�i8q�P8c[Y _ P8g X P*q �s[c _ i _aU P*Y UNV b UW¨ S6b n�_ P Va` S�\eb U � `�_ ) ¬ g �¼UWV i ��� V cETZgFi X S  t_a` S6Y�]�S ` i~R*S _ P VZ_ c[\ nY[P _ . y���|���� �[c _ . y���|��¾� ] ` S6TZS ��UNV P*� _ i U Y[S�\�g¯TaP8q � � n \[S6bWS _4U Y[� _a` S X c[TpR*S VP8g%�8S6Yvc V'� P8Y � \[S:k0Y[S6\�P�R*S�T ��  Phg�] `[UNXB`e_a` S6TZSlq�i n �KSli8Y U YEk0Y Um_ S�YocEql�tS�T6)�¾Y[S X ihY�S.µEsKS X:_J_a` i _ . y���|��¾�

��$y bWP*� �¡� � g¯P*T V P*q�S X P*Y VZ_ i8Y _4V � i8Y[\ � � i8YE\

Um_¥V S6S�q V TaS6i V P*Y[i8�[bNS _ P ` P*sKS _4` i _�� ] U bWbj�KS�i ` i8bmg � U Y _ S6�*S�T6) »´` S V c[TZgFi X S y (�� �V c[�*�*S VZ_4Vl_a` i _ ]�S X i8Y ` i~R*S ���£�E  i8YE\ UW_ qlc VZ_ �tS X S6T _ i U Y y¯_a` P*c[� ` sKS6T ` i8s V\ U � X c[b _�_ P�s[TaP�R8S ��_4` i _J�®X i8Y V P*q�S _aU q�S V �KS VZ_ T UNX:_ b n sKP VaUm_4U R8S8)��� ��������� � � D8-e/¯C0H�!=D8M 70Hl3$# � L:Hl38L6/<D8;G-IHa5 #.1B3=O@/¯C0H � �<��H:1.;<HB� ?=y � |���� ���� ��������� +(# � ;N��D ��7�O$O�H:1��:7�1 #�Do9BH�38L6/<D8;G-IHa5 #.1B3=O /¯C0H � L:H:M ;FD8-��:7o1 #�Do9BH�*1 ;N� � 14H:MmD8/�Ha5�/ 3�/GC0H�1ZD8- ��3$#¥/GC0H �Å3=1Z5oH:MGM � � H:;GM A8143=7�A�3$# � �

Ç �%³�� Ç � �J· � ¦

� �

Page 31: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

� ( � > ) Æ S6i8cER U bWbNS   � 3=OJAKMNH � � M AoH6L61ZD8;<9 � 7o1 #�Do9BHB� y �hY[\�S6\I)  K· i8q��[T U \[�*S   (�� � 2 � )� ��� > ) �l) Æ S�Y[\[S�T'i8Y[\�¦ U T®½�S _ S6T¥¦v] U Y[Y[S�T _ P*Y�� ° n S6T   ¦�P*bWc[� U b UW_�n P8g X S6T _ i U Y�stS�Y X6U b VP8g X cETZR*S V P8g®�*S�Yoc V (   i8Y[\ P8g _4` S U Y _ S6T V S X:_4U P8Y P8g _ ]�PÃuvc0i8\[T UNX�V�U Y � �   ½�TaP X )�P*Y[\EP*Y �i _4` )�¦oP X�y � � � � y � �*� ( �.  ��� � � ������)� � � Æ ) �E) Æ�U T XB`¿  È®P*q�P*�*S�Y[S6P*c V g¯P*Taq V P8g�P�\E\^\ES6�*TZS6S U Y^i�bÂi8TZ�*SlYvc[ql�KS6T¥P8g%Rhi8T U �i8�EbNS V�  �i _a` S6q�i _aUN¨ i � y ( � � � �.  ( � � �4( � ��)� � ��¦t) �E) Æ bNP XB`¿  ¾; A=C0H:1�14H�A87oMmD8/ 3=14� 1 D8M AoH6L61ZD8;<9 � ��/GC0HB3=1 ±D8-K5 � H:/ D � #.7o-I9:/2;<3=-E�Ã3$#H:MGM ; AK/F;<9l9:7o1 !8HB�  0· Ç � ��P*Y[P*�8T4i8s ` ¦�S�T U S V (*( y > �e¦   � �*�8��� )� ����¦t) �E) Æ bNP XB` i8YE\ �$) �¥i _ P   � g¯c[Y X:_4U P*Y V i8Y[\ » i8q�i8��i~]�i�Yvc[q��tS�T V P8g�q�P _4U R*S V6 U Y �¿C0H � 1B3=/GC0H:-K58;<HB9 � � HB�:/¯��9BCo1.; #./   R*P8b<) ¬B  s[s¿) ��� ��� � �*��y2Æ�U T ¨v` i8c V S6T   (�� � �*� )� 2 � > ) Æ P8TaS6b  ¾· P ` P*q�P*bNP*� U S�\[S�� �? S _ Rhi8bWS6c[T V \[Seg¯P*Y X:_4U P*Y V " S _ iÄi8c�µ±stP U Y _4VS�Y _aU S�T V6  > Y[Y¿)�¦ X ) � P*Taq�)%½ UNV i y (���� 2 �:  2�(���� 2�� 2E)� ��� �Å) Æ T U � `�_~  ½ ` )¹°l)[\ UWVaV S6T _ i _4U P*Y  ¢y2· i8ql�[T U \[�8S   � �8� � � )��� � �^) Æ T U � `�_ i8Y[\^¦ U T¾½�S _ S�T�¦o] U Y[YES6T _ P*Y���° n S6T  ¢· P*q�s[c _aU Y[� _a` S Æ Tai8c[S�T � ��i8Y U YP*� VZ_ TZc X�_aU P*Y V6  �i _a` )%½�TaP X ) · i8ql�ET U \E�*S¥½ `[U b2)%¦�P X ) y¯_ P�ihs[stS6i8T � )� � � J;GMmL:H:1./ � � �tH:-0/GC ��1438L6MNH:O�� �§H:MmD8/2;<3=-E� @ ;G/¯C � 1:;G/GCoO�H:/F;<9�D8-K5 � M AoH6L61ZD8;<9 � HB3=O �H:/F1 y S�\¿) �*i8Y�°¾S6Y[S:gjS _ i8b �: [· P*Y _ S6q�sKP*T4i8T n �i _a` S6q�i _aUNX6V�  R*P8b<)!� � � )� ( � � �¡D8/GC0H:O�D8/2;<9aD8M ? H !8H:MN3BAKO�H:-0/¯��D81:;N�:;G-�A #.143=O ¾;GMmL:H:1./B��1B38L6MNH:O��   > �e¦^¦ n q �sKP VaU i U YĽ�c[TaS �i _a` S6q�i _4UWX6V�  � P*b���� �¾¬Z¬a¬�y S6\I) ³ ) � ) Æ TZP�]§\[S6T �: ¥y ½%TZP�R U \[S6Y X S  (�� � 2 � )� (*( � �E) � ) ¦t) · i VaV S6b V�  ¦�S X P*YE\�\[S VaX S6Y _4V g¯P*T¿S�bNb U s _4UWX�X c[TZR8S V�  �E)�ǧS U Y[S > Y[�8S�]¥) �i _4` )��� � y (���� � �:  ( � ( �4( � ��)� ( ��� ����) · P8bNb U P _ � »´` �S6b��S6Y[S   �����E) ¦�i8Y V c X ihY[\Ħ U T�½�S _ S6T�¦v] U Y[Y[S�T _ P*Y�� ° n S6T  ,¬ Y _ S6T �V S X�_aU P*Y V P8g _ ]�P±uvc0i8\[T UNX�V i8Y[\ ·�` i _ S�bNS _eV cETZgFi X S V6  �E)JǧS U Y[S > Y[�*S:]¥) �i _4` )��� � y (�� ��� �:  ��� �4( � �li8Y[\ ����� y (�� ��� �:  � � �4( 2 �E)� (�� � ����) · P*bNb U P _ � »´` �S�b��S�Y[S   > ) � ) ¦ ¨ P*TZP*�tP8��i _ P�R�i8Y[\e¦ U T®½�S _ S�T'¦o] U Y[Y[S6T _ P*Y � ° n S�T  È®i VaV S$s[T U Y X6U s[bNS�g¯P*TJsKS6Y X�U b V P8g X c[TpR*S V Phg��8S6Yvc V P*Y[Sl] ` P V S �*i X P*� U i8Y V¾` i~R8S$T4i��_aU P*Y0ihb � � \ U R UNVZU P*Y�stP U Y _4V6 0¬ Y�R8S6Y _ )��i _a` )§(�� � y (�� � � �.  � � ��� 2 � � )� ( � � ����) · P*bNb U P _ � »´` �S�b��S6YES®ihY[\�¦ U T�½�S _ S�TÀ¦o] U YEY[S6T _ P*Y���° n S�T   È®i VaV S's[T U Y X6U s[bNSÀi8YE\]�S~i ¨ i8sEs[TaP~µ U q�i _4U P*Y�g¯P*T�stS�Y X6U b V P8g�¦�S:R*S�T U � Æ T4ihc[S6T�i8Y[\ VaU q U bNi8T�Rhi8T U S _aU S V6  �E)ǧS U Y[S > Y[�8S�]¥) �i _a` )   � ��� y ( � � � � � ���B(*( ��)� ( ��� ��D8/F;<3=-KD8M �À3=;G-0/G�   S6\¿) � ) ³ i8b _aU Y[� V i8Y[\ � )6� �c VZ_a` P*b "Ày �8Ta\�S6\I)   �¾U S�]�S6�   (�� ��� � )� ( 2 � �E) � S6�KS6bti8Y[\�È�) � ) �IU q�q�S�T  �· P*q�s[c _aU Y[� _4` S ��P*Ta\ES6bNb � �ÅS U b � TZP*c[s�P8g�ihY � b �b U s _aUNX®· c[TZR8S'P�R*S6T �� EU Y�MGM ; AK/2;<9 � 7�1�!8HB�¥D8-K5 �§H:MmD8/�Ha5 �K3BAK;<9B� y S6\¿),È�) � UNVZU bNS:R Va¨�ni8YE\ �Å)�Ç®ihq ��c[T _a`�n[�. 0· Ç�� ½�TaP X S�S6\ U Y[� V i8Y[\ �S X�_ cETaS � P _ S V6  R*P*b �   s[s 2E( �� �y > q�S6T6)���i _4` )�¦�P X )   (���� � � )� ( ��� Æ ) � TZP VaV�  �¥P8b n Rhi8� U Y º¹V ]�P*T ¨ P*Y�q�Po\[c[bNi8T�S�bNb U s _aUNX%X c[TpR*S V6 oU Y � � #.7o-I9:/2;<3=-E�§D8-K5� 1.;G/GCoO�H:/2;<9¾S6\¿) �E) · P�i _ S V i8Y[\ �Å) �E) » i n bWP*T y<· i8ql�ET U \E�*S   ( � �E( � )

�E(

Page 32: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

� ( � � Æ ) � TZP VaV�  � ) �'P ` Y[S�Y ihY[\�°l) � ih� U S�T   ÈÀS6S6�8Y[S6T�sKP U Y _aV i8Y[\�\[S�T U R8i _4U R8S V Phg � V S6T U S V§¬Z¬B  �i _4` ) > Y[Y¿) � � � y (�� ��� �:  ����� � � 2���)� (�� � Æ ) � TZP VaV i8Y[\�°l) � i8� U S�T   È®S�S6�*Y[S�TjsKP U Y _4V i8Y[\l\ES6T U Rhi _4U R*S V P8g � V S6T U S V� *¬ Y�R*S6Y _ )�i _4` )!� � y (�� � 2 �:  � � � � � � � )� � � ��°l)¹ÈJi8Tai8T U2  �¾� VZ_ Tac X:_4U P8Y V \[S �i8Y U Y�� _ T4i8Y VZX S�Y[\0i8Y _ S V �  �U Y ���H:O$;G-KD8;G14H�5oH�¿C��H�3=1.;<H�5oHB����3=O�L614HB��5oH ��D81.;N������ ������   S6\I)�¦K) °'i~R U \  �y<· i8ql�ET U \E�*S   ( � � 2 � )� ��( � · )¹È®PoP*bWS n*  �¾Y�Y[P*Y0ihT n¡X c[� UNX g¯P*TZq V�  �E)¢ÇÀS U Y[S > Y[�*S:]¥) �i _4` )   � � 2 y (�� � � �. ��� � � �$i8YE\ �[( � y (����E( �:  � � �4( 2 ��i8YE\ � � 2 y (���� � �:  ����� 2 �E)� � ���h� ) � ) �E) ÈÀc[b V �tS�Ta�*S�Y   � 3=-��~HB9:/27o14HB�,;G- � 1.;G/¯CoO�H:/F;<9 � M AoH6L61ZD8;<9 � HB3=O�H:/21  �y �¾U S�]�S6�  (������ � )� ��� �j¦t) %) �'bNS U q�i8Y   > bW�*S6�[Tai UNX�X�noX bWS V ihY[\ _4` S��ÃS U b X P*Y=ÁZS X�_ c[TaS V6 �U Y ? ; �¡H ��A03����HB��:7o1�MmD�9B3�C03=O�3=MN3�A8;<H�5oHB�%��9BC��H:O�D=�   S6\¿) > ) � TZP _4` S6Y[\ U S XB¨ i8Y[\ � )¹È�) �'c U sKS6T y � P*T _4` �ÈÀP*bNbNi8Y[\   (�� 2 � � )� � � � � ) > ) �'P*b n Rhi8� U Y  ´³�U Y UW_ S6Y[S VaV P8g � y ��� ihY[\ � y � #v��� g¯P8T�i X bÂi VZV P8gJ�ÃS U bX c[TZR8S V6  ¬ " Rt) > ¨ i8\I) � i8c ¨ ¦E¦[¦EÇ � � y (�� � � � )� � ��� Æ ) ��i " c[T   ÇÀi _4U P*Y0i8b UNV P*�*S6Y U S V P8g�s[T U q�S®\[S6�*TZS6S  0¬ YvRK) �i _a` ) � � y (���� � �.  ( �����( 2 ��)� � 2 � � )¹½�S n TZS   È®i8c _ S�c[T V S _ q�S V cETaS V \[S » i8q�ih��i~]�i V cET�bWS V Rhi8T U �S _ �S V \[S ³ ihY[P  °¾c ¨ S��i _4` ) �E) � � y (���� � �:  ( � ( ���( �E)� � ���'¦t) ÇÀi8� ` i�R8S6Y  JÆ P*c[YE\ V g¯P*T�q U Y U q�i8b V P*bWc _4U P*Y V Phg�° U P*s ` i8Y _4U YESS6uvc0i _4U P8Y V6 ���P _a_aU Y[�8S6T � i XB` T6) y (���� � �.  ( � � �B(*( �E)� � � ���¾H:;GM ;G-E��3=- � � � 3=-��~HB9:/F7�14HB�´3=- �*A0HB9:;FD8M��0D8M 7 HB��3$# � � 7�-I9:/F;<3=-E�   S6\I)��Å)¹ÇÀi8s0i8sKP*T _6 � ) ¦ XB` i8s[s0i XB` S6T¾ihY[\�½�) ¦ XB` Y[S U \[S�T y > X ih\[S6q UNX ½�TaS VaV�  (�� � � � )� ��� � �$)¹ÇÀc[� U Y   � bNb U s _aUNX¥· c[TpR*S V ] UW_a`¡· P*q�s[bNS.µ ��c[b _4U sEb UNX i _4U P*Y�i8YE\ _4` S · P*Y�ÁZS X �_ c[TaS¥P8g Æ�U T XB` i8YE\e¦o] U Y[Y[S�T _ P8Y�� ° n S6T   U Y � 1.;G/GCoO�H:/2;<9 �¿C0HB3=1 �3$# �MGM ; AK/2;<9 � 7o1�!8HB�y S6\¿) · ) �¾U P8bÂi �.  s[s¿)§( 2 � � ��� � y ¦�sET U YE�*S6T��S X:_ c[TZS � P _ S V ( ��( 2 y ( � � � �Z� )� � � �¥½�) ¦�i8bN�KS6TZ�*S6T�i8YE\ > ) � ) ¦ ¨ P*TaP8�tP*�*i _ P�R   �ÅS~i ¨ ihs[s[TaP~µ U q�i _aU P*Yfg¯P8T V c[TpgFi X S V\[S:k0Y[S�\� n�_ ]�P�uvc0i8\[Tai _4UWX g¯P*Taq V6  °¾c ¨ S �E) �i _a` ) 2 � y (�� �E( �.  ��( � � ��� 2E)� �E( � ��� ½�) ¦�S�TaTZS  ®³ i X�_ S�c[T V bNP X i8c�µÄ\ES V g¯P*Y X:_4U P*Y V " �S _ i \[S V Rhi8T U �S _ �S V i8bN� �S��[T U uvc[S Vy \ �S�k0Y UW_4U P*Y V S _�X P*Y=ÁZS X�_ c[TaS VB�:  ¦ �S�q U Y0i U TaS�°¾S6bNi8Y[�*S�� ½ UNV P _ ��½�P Um_ P*c�(�� 2 � � � ��  S:µ�s¿)(���)� ����� · ) �) ¦ U S6�8S6b   � P*TZq�S�Y�ihbN�*S��[T4i UWVaXB` S�T � i ` bWS6Y  ¢y �ÃS�T ¨ S  KÆ i8Y[\ ¬ �l  � � � � � 2 � � )� � � � > ) � ) ¦ ¨ P*TZP*�KP*��i _ P�R  hÆ S n P*Y[\ _4` S �ihY U Y'P*� VZ_ c X�_4U P*Y  �¬ Y�RK) �i _a` ),(�� � y (�� � � �. � ����� � ���[)� � � � > ) � ) ¦ ¨ P*TZP*�tP8��i _ P�R   �t3=14��3=14��D8-K5 ��D8/F;<3=-KD8M �§3=;G-0/¯�  ¢y2· i8ql�[T U \[�8S   � �8� � � )� � ��� �E) Æ ) ¦�bÂi _ S6T�i8YE\ ¦ U T�½�S _ S6Te¦o] U YEY[S6T _ P*Y���° n S�T  ¥· P*cEY _aU Y[�ÄsKP U Y _4V P*Y X c[� UNXV c[TZgFi X S VÀ¬B  > Vp_ �S�T UNV uvc[S � ��( y ( � � � �.  ( �4(*(*)

���

Page 33: math.la.asu.edukaz/colloquium/02f/SD-DE.pdfmath.la.asu.edu

� � 2 �[½�) ¦o] U Y[YES6T _ P*Y���° n S6T  v»´` S · P*Y=ÁZS X:_ c[TZS V P8g Æ�U T XB` i8Y[\�¦o] U Y[Y[S6T _ P*Y � ° n S�T   i8YE\P8g » i _ S  tU Y ��143�9BHBHa58;G-�A=�$3$#lD � 3=- #~H:14H:-I9BH$3=- � 3�9aD8M � ;<H:Mm5=� y S6\¿) » ) > ) ¦os[T U Y[�*S6T �: °¾T U S6�KS6TZ�*S6Y�(�� 2 2   sEs¿)§(���� �4( � � y ¦os[T U Y[�*S6T   (�� 2 � � )� ������¦ U TJ½�S _ S�T¥¦o] U Y[Y[S6T _ P*Y � ° n S�T   Ç®i _aU P*Y[i8b�stP U Y _4V P*YesKS6Y X�U b V P8g X P*Y UWX6V i8Y[\eP*YsKS6Y X�U b V P8g�uvc0ih\[T UWX6V6  �E)��P*YE\[P*Y �i _a` )�¦�P X ) y � � � �[y (�� � � �:  ���E( �� � ��)� � � ��¦ U T�½�S _ S�T�¦o] U Y[Y[S6T _ P*Y � ° n S�T   > T Um_4` q�S _aUNX P8g´\ U i8�*P*Y[i8b%uoc[i8T _aUNX�V c[TpgFi X S Vl¬a¬B ½�TaP X ) �P*YE\[P*Y �i _4` )�¦oP X ) y � � � �[y � �*�8���:  �o(���� ��� �[)� � � ��¦ U TJ½�S _ S6T�¦o] U Y[Y[S6T _ P*Y � ° n S�T   �ÅS~i ¨ i8s[sETaP~µ U q�i _4U P*Yei8YE\ � � S6uvc U R8ihbNS6Y X SlP*Y· c[� UNX ¦oc[TZgFi X S V� jU Y �´D8/F;<3=-KD8MIA03=;G-0/G��3=- D8M AoH6L61ZD8;<9�!=D81.;<H:/2;<HB� y S�\¿) � )¹½�S n TaS�i8YE\

� ) »´VZXB`[U Y ¨ S6b � s[sI) � � � � � � � y2Æ�U T ¨v` �ihc V S�T   � �*� ( � )� � � �¾¦ U T�½�S _ S6T�¦v] U Y[Y[S�T _ P*Y�� ° n S6T  ®»´` S V P*bNcE� U b UW_�n P8g�\ U i8�*P*Y[i8b X c[� UWXeV cETZgFi X S V6 > Y[Y¿)�¦ X6U S6Y _ ) ���X ) � P*Taq�)%¦�c[sI) y � � � � y � �*� ( �.  � ��( � �E( ��)� �[( � �E) » ) » i _ S   �¾Y _4` S X P*Y=ÁZS X�_ c[TaS V P8g Æ�U T XB` i8Y[\¡¦o] U Y[YES6T _ P*Y���° n S6T¾i8Y[\¡i��8S6P �q�S _ T UNX i8Y0i8bWP*�   ¦ �S6q�) Æ P*c[Ta�[i ¨vU � � 2 y (�� 2 2 � )� � ���*Ç�) · ) � ihc[� ` i8Y   �¿C0H 'D81Z5� � � ;G/F/2MNH @ 3�365JO�H:/GC0365  oy �=Y[\¥S6\¿)  h· i8ql�[T U \[�*S   (�� ��� � )

� �