From bcdd21ae6cc71eb57a9464bc0ef5aa97ef719a49 Mon Sep 17 00:00:00 2001 From: JosuaKugler Date: Tue, 16 Jun 2020 16:20:20 +0200 Subject: [PATCH] =?UTF-8?q?nein=20spass=20ich=20w=C3=BCrde=20niemals=20all?= =?UTF-8?q?es=20in=20eine=20zeile=20schreiben?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ana14.pdf | Bin 254956 -> 254831 bytes ana14.tex | 207 ++++++++++++++++++++++++++++++++++++++++++++++++- analysisII.pdf | Bin 785695 -> 785690 bytes 3 files changed, 206 insertions(+), 1 deletion(-) diff --git a/ana14.pdf b/ana14.pdf index 2468cdf90a08043c4b2ffa3fc15fd120d657333a..bb7dbab2c7ed50c63aba31e37238d4e76468b1df 100644 GIT binary patch delta 44228 zcmV*DKy1J4#}Dtu50FrQ>vP-25&!PL0{KQxjnlm!an)&D+ewpgCUul)X6$^RBS?ZJ zQ7J{oO8)xp9suGFJUmF#vfb&2gfw(o?Ct*cd3!oKdGQ0w&Tk*k>^!5{*AE__JUDtL zMdpapTxTa|nPEf$zaFcXL5!#cI8Hdv?BfJ)^1LTfo z3`L-Y$dpsW2$TJPdi~(#FEl%apM9HABAm_M`IXnWni)d5$gZ*e8qOAF2 zAvsMH2&Z22q+TJz6`b(akYLxmS|p-Hl}_tLFKRM0)5IBJy&yc-f@t8-FIpy~

8P z<-w{Oc%dkNmJbuGb3=sPuzpsp2t@^2R;Ky+mkA1*aKoqX{fpGbfmdm}8;=kG8&|Em zyvm=J%j(MyZqOXnazTWqI@Nc zSp$23$QY~epvI@I@JR#cHgJty8`GA6W5~3X#8NS2WS33=8+>Ey-V7r?xmJXMzO=Oj z{;T-?9p8SlOd;D62r3Ic;brl=YdVtN~_5Pe{8(K1C7ZxqCulP&`x6uP)8Z`UP6Nn8y-(Ucglpa|&K zWs_qAC@kDC+l!k=iQPng$|fjaxbAY%aYeNCG^4KQ^D!3MMwwTD!NhwYcU5}G$L5o_ z13-Uy&W%lR1KG0F`cuHOB$C@t0MD$#vz&%!c@Iu&Pds;WAKW5s(wdxezR4h2TpuuF zIxI(|BA(BakwHU5FtH(gdV!(WSpN2Lj79Ys3$A%sxu4U;63Z+h+o~pECyto5fVxW- zrIc;R->z5X<+`}tE_FBs>_MeN8RWWhsM3>m1Q!O?euIjvX_J!#H-8O=9~bLC=eeSC zMhQ=~KQ&=8vc2zb)`*pN_`d>ss>{>z9KJ|lW{->0(my)KD%U?c1ACDx{9bnfRE<1* zP=KW@+^%ub8?*8!3U#h!i^w_r9>OzdYF>qh!n1_FQ^bnakNVyyV( zgey6%_|^TzZ!R3<$A6%Q8~~~eo6^#vRJ~tIi%B`SeFmKHk(L%^Lme#$kJ^gXBNpot zP$LwbotA!Yt77>lRQRtKtLpLF@1@eL~%K zz{_nJfJwOx#_FA?(ju^N*%EpL4pke_Fns9h!!#d{s)s3;!LeWJg03VtX7(CBz-zBp zi(e6x7Z~5yMSp??X^Cs?;XL8DBrz(kK@&TM=)}yf1?W0OCpcSZ{C|@^;%Qsp$1=bT zm02DI;dHRk6PeR4@BW1)72fq^n^^7Ajh18*@$TrZR4cVHDyT(nHvc(f*kFHJrj(fTzZd*zw&>CV# z2~Zx^Srt~3y0xwwXPieB6O|X1*wdiI!AGe3@Ku5^EQ8xMf$nL%j){&gVmg#{H`O_Y z#H<8rCutl3&pl9^aS7>*i)-^pJOT^w`9Yw?ybOC=KqXRmkW*WWli7uIK>^H01^doc(R$K zTW(7%him~vnsW#tHL;%)#HcTMu~*J(ulXjmRUH&;NuPn82ZevJZUfN2&!Nstwz0AO zCJtUlZ}*P@d>B%u;8=m=Ri}X*r!O45k&1y_?eH!kRqP3&CW9lIH|c38p)9r?T2#}N zczqraWvZzFDSeNleP#J(TW*dRFzsMbvL>vCHCw;-(RC>#I1(ccqD!Gbms;+mOA`w= zp-T`f6`?9WV1R$d;1)D369Kx8+RJT8O3Mhv-BZZ8jT&yN0|GW%?R;t`wZ9Wi;a&Ggn>EKkm(`37;Y zYO-~+KIH?kaOk66$r#)>m^adi6? zRd&yuA9rZpAd;uI&PyoZ&{LIP(W049eV6AJi@ZmI!ie21HANJesX*g2sC$gp=* zNPx*n$HGw|ZXFZ@GI>*ySRK{G-;~Q$QC3II#q!kG1k4W!)dNEK&*s1izemRz_#m97 z=$reYCY&M}tWwa`sFArJD5KlVrf2YlyGgfR%zF?1po&FQB@7LJ{XjyoJx$UXgK3!> z9o79uDkZTZDs9q%wo}iPXAn*rMMIh*+r!K?L`@g?Y-;lQi08Q=iC zWIdnAw(Plc_+|Ki=4Icc=O=8O8YTtBw3;|qzRx|0ZW+B<^P30#&270Zf3Ht@n(M%c zCK&{eHYGhComo%MT*t>c)5nxOckyvE<`41cupAn86F7@(_Ddo%#9OV(%86IDcQ|yY z3FuJwi;iV+zL*3I$j%(!tf94psXIf%I2rs74u-Azr(+2m1RmWt}_c?Sv#i7d&c)zfh3h7l{@&BTCxgSpY`~e|CVSBEz==u^(i+fB!@TFLTOY|HRm; zv}^)+YO;d1(-$?<5g|!*JG*+hEkq>1Gf1|yjcesd9o$h+I@KCV|bMvyqZ1& z$1E9kmBo~Q!^ae1f3lTlFUNost34Z2akB`AM|=%gEYFM7Rd_JcZr`haZ0sW0 zz*FrWQtjKQfw$A&EQ(*l;`8J;RqA{>e|jJ3BKa2pOc%Vi@3u*2ic0q0E=&F;0bwEI zqMcr8b%JD7RqNec|F*bHRG)YYe!Tj$BcsR0ry2BMDc{HQp7)UQ4V#$hS{P_>Szh}p zyjEy=fn~6j;pLI)0p6ra9f07iG&yF0d5bS|#_TDDw2!*_$V}nAAFSHHf zkfnVO8?+Fo!ciQ2QPKuA5z-CWARgXs1i0{jN)9wtU6C&O&R72nS86uwlfVxr12!=* zmv6@dDt}pX+eQ+8*RR07QRM)h`$(kJ=1!$5RoPfo`(WpRiD(%yC1piAv9p{1zTJas z255jFMajw5zSt5un4_nU@9S>p>BaZo@$71I!m_K3WnZ0qdU5jfORh6tcq2@9@iODg z7F_epYF9WTvWv^?`9hrjaPdw1P$pF2naOft;eRn%4f*oa$;F$S*W2>;<}_E%ES5I~ zKKArW7fo+$p_GGZ!b9iP0IuQ@1#|6y7fkbd43EYLR%p+=L0k@3;=HgzW~`9PRO3Fx zL76BxQ&}E;etw#3DHqq}O}&(aO?WF^Sjrd1*er+HysyT;u7|N`n1Wr4W<#%tc@#{e zntuWH40ddSb55KG&g^s+?E{3v2;s=WS>4R^HL+J$2*>e6%ByPB%i(gm;09M~$4so& zp6e|&kMU~9btZGX`sg(0#x_DsM4&8g>^DM-N(dgvS6mYlIHBO zt$-PHclH5lya;0Ddw92-xJaV7v?pH!SH&S z8)6w-7zu`*jLL-&l;k;h4Qu+kpvH9JiWI11t;_l`2z0eBS3j0lAT#T>qoE4K|cR?0eiaum)yRPCO#Y$i_qtC)AQQNzLRUGlvx*c+UuTn$>jQ3mBGxlXGr(JL?O>=?k(5uqxH$RO3EC6OcqS!) z6|go~v5>-MKV6?Z|AA$f@Z~p%0Drv8ehWjdQGqN2Sdgu=XD9!t0Z_cBb}ok!7t>eI z_q_UK0=QR;y5MAevX25l0i`oc?>MmGg#`UQ%7L4?<{WrmU3XLG9XDS^M({Y8I1zwk z4xX8b+8h3d5EO%$qZHsy@V~HG5EOnP70fO`%P~&et+JP`WO|xlP%!ngVSkN(|4uBT zAeMlP4_X8c!r=R4kv|9knJa*h7ETP|(V0eoLKAjw3L(r89jEcCQVSvCVA9UUN1fov zbgp1_m!I5eEc^gRnXH0N-6pQ`V;V?vGllDN$vl_@4>E6L{m&OtsaNY2$8F&T zoA1wXYq~n}@F?N1ZH(wpO&ViCpJg-VX9_$Q*0e$5Z$r%hZD)K}xCc`-e($I@i9=jR z4o1RT5|DpxM3dzcEdfiD6BICi;|#ad-kk8!Tx^#TzAp${nP2bjDB79WAC3t*B=E=p zPPo!3dHj5NjpQ?>Olgz@=b|IGI6S=u@1rB)L6QL8IiLqYf)>8_kxv!YwLUdK9Shf- z3j-D~f!n<%RAC>;4$!QP3{QyX9e5|*JEt2tW~JG95K^;k0QKy?r`eBxc7pOaI9T{^ z#9Cluu3h+!l!0X8u{S=1zXB!!DhC?j#=)U|pjZ496fVxh@QUMD{fOO5H zqu5sf_z`fNL)MU}1v*{hkD-IfIk*oNkZ(QcoDwS>)_GVHt0Ope z@(52F5(g-$<8YjioQ0=<;8_D>J?PX#FmD`86wDJ)*~|oj)Ye6NjX0P5AIKd%WX9;g zo1Su!e_qHzrDBYV#>`-XkLF+-LHU9~%cEDh#wL@G=5K4p1-P;wyv9I9Rbr#I#y za6kiv!52%IpaF!3R#L%+l()T2^zspmx4YPE@5)Y3-S}&IC=7Vp`oAMe7 z*_&(pKrMd7Uv6)xg(YSkZO@q`E63iXNGs4ecYseqK!Wk2PSJ?osOPz6Z1Kh2X0`pD z%G)RadwgmK9-vKs^Zn(UyPxnhlzL$y+(zD$G_eHa?>g@Xl|V1+nvh!X<^m%jcvo1> zYZ=s>NTxFA!&?!|w_e2Wo>9fPCpLg=!`^@V825A0K+ z4qEBQ>5ufFPA*ek0zA79`)MX}*kfbH1Nfl+w56;?LpW3*U^$MekaYuHh(?XnE)ylE z+mu{kx^(}4n5KyxjG`uHdPe$N?@uWx_2!s>{!{0AT6OGi2VjwIFJ&JU=$GU%zuZ9A zuxXeyhtujtzdmg~gcCLclh17LGgHXsv%k}-0LyZWiIXT@Iu0ZC4AYth=~$aI!V{K- zMW9flZf)twWa+-~1Jn9@obt_q`a9%0Occ(G8B&yg;BfoJmedb`R#@M2Q6sA7GSR6O zRrwCF6{4Umm^GuhKgB>taYY6Pbxm3hZv=ju&dcF$rRts?&HX6>3X6Jn$RZ##MCcE*0AB- z03I`c5=ePeROggq_>6E2pAnAXGr}=^LI4SnIH(q;K838Od-+0Ibw?ZlOxo!F8Car( z-nRbnYDzqWmk8Xt81!+WDBF!Y|8Y42|m&vAs&N&HED3mb667-;)t|K&!A2EacNVyh+LYc z2+&31`a0@?N5z$l> zU9Zh?GDiK8=$#DyODPwE_9UWoVOJx4Cy0PF7@Ptf)Y_(b-}GiBXczXK zn4~OzpJeNPgq92+7Jzlygb)YXo@cRs06BJ5{kPSBXM{(PnD#aOs5_cdDJV;DT_hW` zd;dswf8Q4rs(9Gu^g|a=-9P_r;{@M-Vm8_?-dsP8GSkISzrn5n9wKOlm6@B;Z?rBf zR#*C+PJsv?`|G5b?#%MxRf4e@U=MD352G^(353h+e~jdHFyV;T!H^1aUz_5mt8(*8 zxq=WQom_xhOmogr?m4+I8?~44L7JM6rm2siV>JkPN7|`0!!iqadxp=?d3}t3s%{j% zv+cJ-Oi{U;2&W0kT$xT^PlNmp~Q|AVG@UZTG6O^ZX)MGX|i8N zvY2l?`c>vei1rL=Zy*Ep2gbydZNol^>m=ax91A?-nZ?BM68;QmrPaOWHMMJ=cf(!l zK$YWi4{TbJudX52%;V#oSF)ggkR%90cH|=tX#KP@=p!w%USR9V>}|zkLtA?qFAX*dBng3+(}>ZV$i@lJ)@n09A2w zd$qi)e!W`X{H+QBw%Di{D}kn4H}P{fU&r8-JreNM3xSfk^AFV$W58b=VSla*h5!^Sqm%e;@9=VLd2cEl#2zu4{ey5zCS z!>Bi@hc6(Re#em-y8`M}Y9vZs>rmR06zi2rJWj8HNW+FP6D$y*3bKJ{4l7C9$gpq| zCkpxfO0jxwJRW2&8XcwRi`-3L61O!%U(^mVQCGq!N^yesDkTRy@kUAhB>?X`;uIt?nrm(&?nk7r={j`9u9q7}CTJI+EH4*c zfQ43u1SF+KK;^13zpq!D%?i`a`6;jS&f7cqv3y$rkaAOhzo3{>%h2t(x-KJn8HV?* zI;$E40Ih`1zhQock5(M^!6&^cLeGDHi0M7hobT|@dsX{m)qmv%Ph5W%((hqli^zse ze_9s|W*1=eelkHVE7eVVYLcXtM;{NSB_Tf8daDWL3;=NXwgG@2RPFOzi;(R~2=k~& zMM&IDi_}to5plOjH8Z402jZM;vmY0+z`u;4r(|rgFlZ-aEQJfs4c`V-BI(s#)_=n2 zuKM++{0ZsXE8JCmfJzI9?lf%+E#~YG-`-_>Vt?4=#02l{FI=-Q;1``4uhrexbmW7W z&a>rqyITK(!|9?5i1NQm+vNocWo~41baG{3Z3<w9gja54$}PR7OqD@eiM33F`>n9h0DxeCP=iFPD z+ilNuTg0u$*|~rM5-Wj3=8wMk+rQD<*MIm#ZC`I{`^hIi{^FA#edg?zA32z?eevrp z>W8y#i#|P4FxwZu+WzjHe|nh5r!OIdcYpEUKYh{f{^qa#;{W=KFaG?~7i;ai&wlpT z|Lu#v_&@*p&;Pnz|Lm{-3;yBW{p`R0w?F@{fBR=Y{_$V@)qnZfU;pIs>ED0xmp}SU z`$?G7`VqAzC{>QdD+ji{{PQn9!5eDRTQUzD{TArYe2Uwb<&)oSfAro@w)*lX>S1&8 z+rMnMP5<}P>A&icHuX^bf8*a@mi?ulE`KxscZcwz)gwEU<`Go}hBtnC|4VHjA%@NQ zM~wMR?7!=%_q*yr6{iM4L2W}n{}Lb65E}^z)QU-kygBvI(UFjEw@<8nXk(Hgr1Y<( z=`OvR-6JdO&1T&re<+6+LwFeHgQpi0{kriMFKUw>$^_~PCO?$5#^T+aA2uYHwin|b zbkF4aE&$n;FaKX3=KwI;mo0bsJPo(TrMW6|SiDuK0?&+xU>gU_>wT}<+-nH*? zb}6;)^ck|EucWDNi6l0Z_FaSL3OSh@YM#on_ zlJ$?|U5s=)J`7b)kH(ri8of)J!fE=THa>i`745Z`q2tkzb&CE?0+lkZekZ1r)viZ> zMJ+(xiQ2;Ie-P`tYbV3IJ`yBPH|dj8ThuqNJ*<9F8n%k6v}sTEQQCj(7j^WKwbb_M zK@g1>`yDl~_IFa&!dEW~yNhZu&w;_wHlU62P%nm*f0`#W`7`?*r8 zO@3*9^ATO%XFgqLzGLITxcUp zo15)Mlf3n{-Du2{+TYu4uFC$>{?2HVQa44zOI6Qiej?UPu#}9vMX>;b7#wp|z?p`avo+kwP25vQDg06IX~nn{=j?TGPBh^5ThCl3E3Q@+V?RddGlD zpNXXk#9(5)$t)91T{`$Qb4$9Xw6+(Ue?~>Het#jCB%L0&N*(O7x2wx1da3ERXvi6! z-tix0Vp_fSVIprcc;@eCPWFCA-RVa~X1z9vNUhfv`AGg>uc+Jkl*8KysV91Cb9@`1 zBmHkeuIBc&X>lE48#wP@=@a8F@J0VhmpJai%=5~*yXH8}OVm@(9l*gNbT;o*fANY& z;I)@z8tvZLyY|>5cek+(;p_(9RqFnVw+|DNwr8Vuhdb#Y`WiG)k~zJ2K%QRAFqoUJ zdc$D4c5TU8dSe_7xn_YrD~W(YDmqv0pL_R9VSKjod@InNG&cp&7ax>^Zh zy4j_Vs|0D|p!{o<=x@|Y7@-p5e^)M*co`Vqql0n=>v79Jg46jm!+qS3@XI0PHWbFY)M7edGB~{(EzzFHj|RqOe|3KO7@f?o zupM_3`ed}PEU=Bcj>e#JudJ|5cd?2>jQ!OGws9Lh!6^Ol3h-ExqR&7Sf75araDT^P zU}$S@==%@QR@_41!sK3A2wa(*yb{O|IdL)Y!s9-;yWvbuZX;^QV+YBBrxDi=R)W93?n4>Z%m!uf=H(}6fi4Tnr&e{2G@u(e>F8Br*2j|T+OlYDfmPH>I5=f1QN+TjQPV_VZ z(M)8hTSXb-0Ar1JGnoB)fH9<^e1Bt=OJM540I4yUpotV;14Cwtqo$*euW7_a5#R(R z$25PY{dz#LI+-2He<~GFGU*=cVfq+Sm`%5(KF73wrpdnX2BId}Q&&zj2{=@d`tIAla`65~r}~*4|iz){Q?) z!zUBdq`D?Ddg?$@s6w(9Tk1ynXheJy7MJw=efH}Ki%oTTf4{!5$ORZ^aGEmcCbj`x z->@U=)jCbqD}5IFN60e92~PB-^O>Ni}*E76SdlDwl!`a)y)42~EA6 z!K6dQFmfwJewJd5pIkXO&#>BUP!QFjhu*yWd=;0ts1xq!Ns-cU@ni)DGR0s31XcF$NLf?Gd$#tt*wJRjqLbcWKsa z7Nx_cuc}gMYVWfr2k@?{S;h9fW~Y2G=tz-Foz`VMf4mbMsW$2y#3DM2o*g>Ud`}9A;E*|L zZSo?>^Q`SJv$kJm?PGkef&nMhzcY0bw4qtd8rmdU8k~t}kcvjj81FKgRd7WEKNFbG zAg(I3f2mt1N6IZ={LyD|&b|V2zyZHA7rti}zBk}3k}s?kJ9BZT$C@k`SSE>Df;hLA zVNRdRq6|BmJ{anQsiE7fnjYT6+JBGgQ%L1H~e~yNYGAW&YJt-XxJEGIiMQjn6CXMDC z@s$iQqP{P{!JCkzsOE39U(eG5&5b*ftI)zTIkIJF@X&J_I4Y}FeGc6K5?ng<-u-$2 z@j4x|=_0fUJT7A#SZX_(6;)1cb%=#GL7`zWf1Uk$f6GOJAanyx^^jal(^yp5e61361f7P|rnfky$473rin!qxmR?+%~Xyj5C_H;XmNC z>lsfzGz=rT$UL+9Pf|;gUh%ngCLZL?f5utG7J+6GNhN(q%^2{H)HxOh@OHnsiY+3` zq{+w=$FEqN=|&!l@a~9OGe1&`POmQGTFQn5Mp!ObDOJ7F_&i%c@@eY#Zu~x)*m&`DNi%2pPoFqKi!WeuCyM7|byG)OfTx5<3+LDOoQze{MZM~F7IU9BG z6rGK_c0C()=Agn0aV!$UBw#r!S3Oj0S98>hmGRB!F*rPOdqHBAStNRi#cpHjTt*f~ zP9nN#a3?)CNxz(iw)2PSv3c14`1~PDuLY1jL)M^128u61@y{i#kECg{+Mc;Y~<_pH0br zIU&hrtamO#i@+@{eyBP32AWYta03VRA3J}X{dypY=2#bBWEP3Dfte;te}lwBpJfr< z1jL$o;PlH0hV$`ID67mOb4q$eYm!UDUS(4=$O zF7a^EF+Ru`Cp@0i!;Gh2Pk4eI3VD@WBuzgKKA>JHLjNBp;OCqKrQahj(l(&7Du^8`Z z8Afgqc_mGpUJ?JH4ni2(i7X!CJ>|QKEh4K7WZDpC+PrHQ$SEUM{S{GT!cVh$^Oj;E-rP`Gf5jG=QAT`6qDvs| z#QUNQ@yzS;h1X?~*D;cd%qSg^`am-PHKbBztP8Iw?;1=CQ!Emt^l)Nu)uAl_VE>oW)cfU0yAwUG+WbDVnp}{R1Hs-QIa||d56kA>rlPYZB}k2O{s)$U+295DizC)!PMb-T7iL9WP+U^&iL#B zEymYsF>BoIC`hN;N!?x^3|$+yQ9{?cV>$W)s>t*^e?WKWxH(%rZi85XyVDdSxJV2f zV~r?b!WgFxo|BC6-t@hSEfN6_SReKD`5kE%Tx13w;YLv>7Pl`^a|q9D z&tKS{U)WxlmZL9Xi_E|s0$<>R(%8{X0C;=PU=>p}KeSA_wmYe%Up zPKoy=>Ih3f52hGH9PEdCv9v@}s}qio})VdbAStVD)rw2ji#{nx(Cea2M;=1w{x zKq=5XlB{mpgRQ`hHTXa`Mo9!*`;L`=)|0R=D+2$llie>Z0nd{QFd%<)Ya69=*18=o z`>extBz_yEC90K(3 z?V_(;mGnT+ay6ber7<7aipcUk%tLt5Sa^ixz7faP<9URE!B%8jfOu%O^b3>8Ftq`n zlYcQR0X~zgF)INNlh`pU0)N(%1u`lE|E!ZSGA038lUOn>0X~z4GGPJPljt&~0ke~i zGb{mTle;rM0iTolGb;jr){`eRDFHu|Q8X+8S(AP=DFL&Ss5B=6eXx_!G*f^4dSCkF zYVkZpyCN&Hb%97s7)b;%rtYE0R$`vQ^fpRNQ+b-g)@Nf1i`yv26vjX+qIKaAtqH4H z2EhxX)?ziDxVg4ME244XL4*`O24S=z2DutD@zUEUL9X&dyw+!fc-d`~AYSR%O1WSw zvT=csPnJwC>IaTAS&-}TJXU|pU@Ni-g1Aso*V;uF_XE{jJ??M*ERu_0ghWJzLa40T zE0cZ& zAkYVJ;vbg<+|dzm(+od9d-U$*Q(W#f0Rz8?$mq1RCv0NBLD%DA@l-O%S+&& zw_3WuDm>lCnN@44*brpOslUBz{IsdE^EUzoSY6WQ;pZC3;3#=lUDz0X3vzRtb zbnP%STY&?swQXegk{%zUpMTb|DkMi4Wym;g+jn;^z>2{15frTMaUp9`T~w8-aG+I6 zS46!`8R>jFTJ~8(tDwNPQAUlC*Y|fWz=~w7AUYO;)e_k?Hnx8wscaQ?v?|rmaYvc8 zI@U({XB}+^ zvz03Gyq?P-E27DQI9x~!1h~4dbnumUmReYW71>|`O6p*QE1TRyf3tO%bggb-Ls|)? zYKFRHo(+xbw^4sW(aP1BEe5xZ5>`Q&EQa;jVKIcXjZ)QEsryFI1zLZRt(gS35p23qV@5=#k*aY3 zHdy`{rK;gQ&_+HB{va~OR0of}Ofeo#ApEn=6mx7I!M%Ts%5=}*6<850fuzZ(OrfTkrHQ;B zx96&GuyD9eZ==lC9wVQB*3owNyS*d?KJL3o7hpv)W)gKSP27k$izTBdh*dbyT3boV zUgi)EBEvuHXiaEmSZfgv+jxalBvS@4*gTRNQuVq=2#f!KYT%qZN}3?zKC<{{9cuTj z&^>>gL04}psKOfnHQRwh&}&U&op`#D;6JNT7gR+yU=YVm(@>ztN%uGmd<~x0VHs>i zwqX$N4Mvcz$#6TxW$W=gg}`7dk{Ls$u%;O?p)pPMvKOoIl)c3TR1pjqi--bb6r+pU zka~_B7XA}AzpluNXux0|rVou|vAU~Yz7BsgDe&7UG4cC}6s*q%DRA2;K??SrnJcm) z8!$L`7Jb#L2EJxuEuMI)Y=u@t1IFu0el3!8=1kOAuErD3J+0V^Xuu$hoyEQ$yrNx? z%Ene>wi4VnN>~D6vJ%#3!%DE*D2J7>LMyWUfJ1g z=0|CR?Jc|DD!d5*U()2-#%}h#1F(M;c%DP{f~!b245=y~ZNQQU(n~wB2FE-WYp92# zq%IhaJrVXZpleuzmp43JRtRjCKC!UNT1isR$=BJUX@Vx%VKr5ncqTwhBRtdn~YPl9q z-0NDQ71=gHaKT`Z;)kqC?d59Bs?%(vgnCSq>Rg{4s?+6dl!x+UkQLcP36GQzFGJUp zxKXae^Cl(+S`m#DL>V+S#V73aAXegur?3`aMKncfjwc*;J^0;x#WNkGJO zU%h>N==*al_$ThgEWnCno>=(P$GURgzPnU~XY9tTkcvR;;e8qtHgA#RM9bKpu|&8c zDw0t`y2+z&BTZ}W2T3{X^U`bFMv3KBJ1xDg&&JYg+C~|Vu$X@L{0P?4RYNH3Jd;eds_>pA`yQWKvAQHSk_Q2Aj}H? ziCZu$tRk8&2&{jIwZw#=LH*n?SAl2lzg&S8*=|8>MHk$Ll(TmCK4j|fl--vVQ;{qd zGI;_{-r!Y7-7ARjpLzD^imXTm3(nMOt*Qx3-(x>O#Z}^o8!!vBB3mqoujuFk78*?f zR<6b~cVHH5MK)OwVllRD-0a7`*IJDyZosV2ie$Rf@YsLE<4kJ}CsyK#+b;{SBHJwp zv1oHMe5|%&9iDY;W;I5NWVU2{HjjIxyBcGF<7dEuH$>X=;k+XA!ylVjaTUpG36C08 z*-lQVTm>fg&9EH>95jdT=JmmF+)UdjhvVi7s0f4~E~(V94Q^W1EOHL{^SokLSVds{ z9!q16dMJO60eI8md2tn(9FE;~6k!jDz&?;&jgJkY3!)+zEb$QxLeLqkm1^rm=+8JY zvmh#xy+S=hA*!3kX1MLKsX6S=a*bbM6@lx+8#bb*bkl{tQ@~c>dFKvRU_~-lNTE7J zQNdgqa#Nmc6`r@!y#gze(SrDm_-S2LeH3djADDlM+bD4;#!d%j)@S3uOxQ+=12b;l z9J=5t9PiHv@@PQ;P0!uHVSnN&q%)!-87zqKNONmK&WB`qwvtTNswGyb+;3AO_ z;i)HrU(rA9Xv_ktNQMe%twu=GjL-$yi}0VJQddYtv{MX@+6U7NNEJF+<={W@P|OOg z$d-Q!(urkk;la{@WhqwUi3ej=Y(+Lzz*DjDq!B_(hga(H#FH^Awj!A-4)Gp6Ht_k6%irb>F4Nckn0lA~US+Wo_d=4b21XGq zkpRmu{%6Jee+sj}tcYEj3;4DXJOt&kISmfGRwZcUTFb z-^I2vOjil~6m`UGM}fE^Bjm{S!IQf{Y5}v8CP66yXR}a20RaIclTt!90cVqmLQn$w zY?IqUDFH8&1w$tgGcJXS>j(B$OhvLA5Refsvpho$0RhjGN<JpzB$ldMK60kf0WMty&NyD~TDdIlY~4tFP_Rc;X(I708iXUHI+ zK?tuvyf;CwVv9tt4FcX}eI7^thHAfvp?bQifrWZ1}=;5_{re4i8oXNVXF8m|zfbimoK2OdX*ZQq+%$a8-Y zXe?av5Fc;zIAPt_?C|3>IB9CmA=wMD^Uy#Oj&?c0plXJ(!haE21PWV63-t-SN%Nz2 z61eI6KxItdr(I6p$6>eqSmVFQEE2(vh-65;$zcbn$5dv~O+B}|x=lT|Urz(6I=mh0 z{1>rBqS(|wNL{&hbh`ydVew64x_W=&jv?JHCm`OKz0p9zy{K7w(b0K z*zVcTSBXVp&WZXEcz4ZU?PX|_?#JTlto!@*q;JikQC6Xa=f|EXHG)@RP$%6O2M2D# zf#Ed4XqOX;x_YVhrS+427;F=a z8j5+s(XZzT+4_B!TxAxCwIXJlw>4K)in@mrX3*VhkyU2l8C|ju>dwA?L6Y4RIScMy z2U%nmfwE-~aFA61p)!c*ZjP8$YLO@Q>%>FBGN^TuxJ{g=17092dAP8@$P)XNG>vEMBpnePu?s0^}u85WZI{ID-A3>Bg5ecgZQJ+`X;nbM_L9>Wg#I$%=LdTtFwCkM*VW2 zupnDw4f!mwNMtc$afIp)&g+hVVGOqE18ID-=>zra=>w4wwtvD^Y?1gPjo+xr8X!)A z@bVJ82}z2lA-!KuNJ2i$ja6vj85tQ#+R^{pS}OWZaaVNrte!<;k;q~~6h-dXmY zo`Fw=z9mSb6b6O`cRfzvPNtsryQ>#NYhlg3P)Iac+8(EL`!Ho-`GrcM2F zf?@oD^D40jj1eKVl&|^kTr)d3U^50RYtA6r^^77P&T)&#B2a&_h-g{#d8u>S*bYiD z(C#(GDzHdw5z(-)^oZ?{&=1Wp(C+n&Rb-LKBDM=bHGzW^sTRTA9WJZbBC$n;vx1`v z)!#VPR+?CVXC0z1N<=SOWY>4A*dlQ;@XZu4ds45k#u>tr3-CKB5h;j7MU<2LKMzf&>XB&?rpw$(o(=l~ld0-JZ8I5B^QXxH^uwaQ>G;M99GD&~+UM)i!=ckq8{d$6s?O?i# zydtqh>X;58s}NZ`1{(xq)2A`s%}Dm^`O~PeX1xe4EE_`#0-b}|Fa_2?jXipL7VDzyl7jL2rP7;lb@#$??HXUqc z6=n6T68V$H3l_t+>Lya`E7)-XZo*R$G3B(zQ<7qLZPjLR0Vy&7CcHqcC#_A zGK)-Ibx_n_v=;&CSP+ngrCYkCyFt1`IwhsP(k(3A-6@iiigb6kNOw2y`+GBQ=KZ&G zXBT$&+;i_apE%q74pQHrAyC6tlzwhDv2sKXMlX&7nnfAthJ$}#7!Cb(QTh`z!+m+H zmem`gk0761YZ8WRP0iDDj4`~?TA`>_Cbbof%(_(?VTZ6xrjZHnHCy|fZ~l~TepJr| zzQSS@IV?bu8z{(o?YYe?E;J@|&vCW;={QYWh7IH=2scPbx#MM^?Qr>q;S^3|0~K4) z1nJ<{`P;p||1?zWdjs2)v_N9SEGFAB{T=5wm4E5eih*7!Z7gQsi^Y8R5S=DcX9r#p zCyzC$__w^Qd0usj2Do{#5y7+DvvFZ;I(0H%XcSrixdAbM!)<}TE{y9duJuAniL z2kFX~2jTqzib~)o)+z!fRT6LXz6n z*Kx!HGK{H40(&W?yQ8Z8dn`}#2Tl0bOv*9+MF#CFTp!S&6l7}F@LtBVGs2Lc%;(c# zvPiRoQ#Kp_Tw*c3E@2qS}l`f48ryFIahoFVKn z8o4R`9)B#F$CA1LR7-_~GWV}LAG5I2QmUC(X2T~AF73!8xg^RtccRMZ-nkRSaNjgG zRvmNseMF#%5e7CE=f>fSw~I@{5@P zN6pJ?FaZm(GS_A4)OrvlEXM!~+lQdpcxB%5d;`WE)lbE12EdMGX`B zP2K7F1CcBJ9_KCLv6pEXmt*BB+uIs(j3`Z;?SAYbP0jhQ63frL>Ij!8PS<~#R6gOx z6d`C)Bv5o~LYZw1znO5%QNVyc?w+s$wj@X!W~9={rONv{X%tFueT97};cQ zE*p6xLIQ9t86-RvJ!MT$yT(*I8QT7zG{}}prx0m!B-^BdrwoVldiR#`D&^WU`!cD< zjMZVXLeO>y6H}2xl@3!Z@7S?X4pRqTDx(<*^GB`2r19d~M%cqum0jPFHP(0gB<9!1 zLEcn+n$6mDNKVIJs})|pKJW@|-ZWjOI9B{Tz)FBqBj*JF_$|AS-+^8krf2RarX+Js zEm?RxCoP8V>h^ZAbJh-d@LiH_2pX=oXq5%|@sD>PC1|K=frxXtnh4Nd6XSWW3p1;U zfw~+v$bDGEDV^(XYj^4$Jix{XH19d36_vsLdPKgn*yygnGN-Dym!!;otQX9Fah;^> zvK%r}vi2j+=+2}6)w;fHzSss8bg_x_O+M zZ$MdgO#G96nkLrRx$dkRPv4mMRomo25E2pmLvU%pr-<%TsBoU2rd!qBCDd0C+wkUc zv-C+$q|2tw0N@ADHZfQe(0(Io{6@W&L(TkPgNg~)F7(rb?Wb$cU^dq;QDwuT_hj(d#`JuS z-b6EHHO?0Yw12BZ|8|VjRAn4`#{9O;?99vUotB@s|K@Gf0%ey)O7PoZ>T5-aS9~EG z{pZEIekiu)oL*v&kuA! zRzAVq=NK;<%HA<4v@$WIXnPVWvsRxNy}LsKvge5NTxuKp@4bml3g&jQMQ&qwa=#e@ zhPG=3i0keOvteH<;L@*apB0HMUyHyZ*hw-|85vNHMwk-3TB74W2;~Z^N^hl(K^shvxHt3;>TL-AvU?N5UhL zYp8~*~A#lIL?lD*Vz zfpFIhRI!u(YdX8ivJWH-+{iD%W%`0X!kn2id!pO^!ZI}P!&Gh<8)d}vq(gudi;;qb z<(c1Bllbg&TdM_pUNY7ZL>|70gE+Io^#~7@Vad6@6{bVp5wfk%;XxdK_SID1mIaxR zJ*85ko5|Ffg->5(XkbS9b?eSnlVZ*YGpU?!ht~}YFa2fC(P-JFS{Xvri1NpgB|Y%o zXIN|&Y!%G>ihA{a%SC-1)gKio@274N*<|DxleU%uF`zuWDYB=n!16E=m(%hd!k781 zYR+0tBBv`Q0`Z}}?hO&M)W7Jz)Vyxdu3l{)1d+8b&o;+4hM-}|tS4|L3!kF+NntSw zq-A4B{Z>E?fC=>mJJcTN%#RH$x%Et&XW!bpUy04MI4S()DwFs{x%ClH^O=8x|J=WT zS5czI1Xa&FL}^V%5=PUiXAv@G8+dg9ane#1rijEk6$|A)Y>~hjO|vDbrd*VW^EtG{ ze}U!k29vR}U!F!>BbW7y9C)#%h5mtsc!e|~Ddn*d+u47rn1RG~7s__8YL4-d{<9_d zDwg>z|INal^&Apuxf(YhzGpOPfb%E3^s{tB*vAQEG}8#|yWHduX|u(ID>n@cfXd`%O+n6k51 z$V)8w)IBw^2-_pRsAh};Zzb0WE^t+RO2`*l+@n} zI=gK&SKwPfa(H#RN3`M^VKj<%@?z#-jVmM;k)#~r?x?QI-<&NPTh*(G+6A&{kB(5p zgE}$PO`bt9)OUdW)iUa=F#3Y9W8$y?gd~ z0y&VS8oHOumI5xb;Y^G`sIzNk+BvH3dw6{Aj5p%`Kf}LkEl|NlnWk^U+h<7G!=l5Y z3qMaIz}#BVR>NQIpcC?hYJ^662!}xi{_6BdS#MOq2Z#PK9Vo3JNJ4OkERQc%rJ1^# zc25v_Ee04!;b#lbRk|~lJAWN;Y?)~=6AYxB6b*@7P$0_n;H|l}GL)h*l)EnS#}B!N8#a} zh7old%r^Od$>KFhtXA%d)u1%sCjN48+gH-rAP#SWO!Sya-HwY1h9ke1Bp>~#rL#j; zn$7A05iQMhY&=4Sk-F|I_$>4IN}wol=GQp1*CVs zr3xQ6VBh;=iQ?2ppxvAj@&uuY)#!4h&R`>BadE|*TAne8(zhZN+@yO2Z3?%~9u7!t zcOiyz9=#q&=pt=2Wvjc*6z8%E%5*H$loXehn@Yk7M6uxH+;F21n@;S~ zA9-=0ojTTl%B^JHKB8YJ#>j$k41ADPuGB~V(!u{EE?cp=jM{lp&?~~7SqY4S4fm5G z<~CPr1-}S~vYz$i|9WH(BA}B{KXhEam4ECt$%3_;EuocOnZYC#F~*Jkm~?^R$g61u zsVseRv7k|qVO?TD$2EJ}?Gir5*AL+DQ30_^t`Cm#XLTQzG#{G~~xJ>wYof ztW+U2u()e%IiC6rrC0pMoy!E_(89wrSm_g&(FlVK+w6vJn^}M+V9|AqjY_xg{g!{9 zer~q{kUjmm2A&=!HTppikt4zh{(x%)*7S|Py|Qy?^Eqo%vRfln{98LKrE8EEAD;}&G*A){+RCjFWy>hdZ zDleq6!_gn9IL`@8`gS7^?ggoX8%D@{VJP}j3gJQOj`o05;a|AR;PmuF=kdmxv^t6t z<$o3H&#n_7Dvag{tx&HF0=Cal{e4uUQGcW`9C@eF^*{mJ1AAbUVm}qsJOtlq z=*!WkI4h$=wzhe;Ha-aB14vH!%o9ixBhmoWqV46o$4N>mMwKBJ1n%kU*8 zeIkg6r8<6tGyLh|JI@JRu644Q)`y}!1*(8q9E~wUv;Chm#YSYUufa`7(S~gKacZf= zd+|Fr8j|xqC(lj-!%WT2Y1mZqFDXmdF5i#iiM|ld_!(gVgs9_V9X(L|7>88)Me}J> zb7YbZ4%9wIQ(*`)rGeVVG{}Hek{*HpS_*ebxTmT@XlZO!0Nz@m|H0r?p+AV|`5%Oy zKGtx9MO-J&`gUedRqjx2-9j5=4FsT}y8ZJ(u8iPd0L(R9 z-v0>G?MY0s*FUA3T*U-yF_8==wcRaGXu6)sfVvN${G=g#kzHO&x|VbzeY8JPvIK%O zpm$h1?rS@FY}ND8Z14--A0Bz?rq5JIco+5-9gE*yBNa^6{s4M4Ef|&BL);`A+$wKq zhupo)E~@7vWlJ{7uwK|>uL|G&1H~&hto@Oq#Tx_%V!gEw$G*nV+xyO*^PogXG0|lU zn6T>998IlErd*u~i{66bZhRd)ft^kg_klD&yDcbyn^VpA_XH7~SHu#Hc=v;7DT(SPm zKx8{v+tz(Wf9w+>#rk{knB4!fU>kqy&?|7SYY9VkmhQLz;fgcNy2jD=eRq_Sb%mz8 zs5P|t(j4yzjKkY^$uX4fqM0x+Emt?IGbM@sxwrq!k*^MHkO_f9kxH{DQFgG}408G^ z(Io2YOZB>KWCGk&>#5GD3KVDWxXYAMm^=6t@-a%>zxoT)y~xS*85QA2z~ct0B2|lh zxbTpU`ok+Gr>cdU=r61!w=c_AhsuJ3=rLW>UOu$9NYC~s!%`;5zHI7nzGi3PY#z`l z!Ve%c`>r1nITdV*4HHH1@~i-5;`$cn>DaIP|7xbU4KrcyTv}RXwHlA@jXWarx1q9f z2wqXg(VC^BX(9`I5Qm*z=Pxog$xcsEqzfpScUYbrF*%TQWn@_<=~)=1kxK_8duPg_ zWpOwoJ7LmFJHjGpMi;uu;y=iZ`+W1k|EoJv7?my;l}>mk{a-33j{>s=|J+lA(&l-) zbrYNJRv3SV4^Phh!RDGWAK^L*W7kP6SzySfD`_59>XUwCZ0hgkhe z4k|7cv9`!2^vj9uR)FYD3T2Z-k&vR*niAE%-=yqw+8sL7mJ#Ms)vH4ma`lHfZ@w*a z8!7j1cz57fzO)H}VD1P`EQ%?pJF_HohW2h4DF;iw=5B-s@fKQ<1= z4`?#Qc+Ye%1Sv7mCwqgrlIbg*Q$GYY0BJWe9g&*W5NqN#AOSnuJZjyv7SUjRT`7`A z&u4LXcP_y;Pj?@=y;GLubwxCm!$9>PRd!Zx5o`Qa!>m?W7M!x~+tSRpD^t9@R`_dN zq-D-ON^TBp1j>o3By_v+0~&r4GE&OFT^TMT{cLqMf<{g?vE(bv=37``T%#T(%OqD3 z7lV6{hCUUT4LILAJ`7fYtBk@|_KF`pB-#Yyz`NTbvH!j7xSg%y zWT4%j*WS+bw34W09Z)tCZJM2ye%he`NJA@emvX_Lj5K6 z(*zTS5z-;qaq&oBJO?ci2w<9_>3#1ixuzarS?K4qNv5C@S{G_!nQtoV>3YcEsT=G` zxC-u%;Ys1~hVbZP7g;{voqD3@c&$HdHr(Id=4m(TAZSJ8nsGTzXK-mZsh&v&xJi#D zNSf&bM%J!)K@RG!i&-6Dcv^>}h1e_lq3rTeB&SD4yX^Jv>|Ny*iFY{y%#h68Ea1Z7fYwz>K2P%ivqq8y9hrj$YuzO+av z6ZG7yZW`==chWe**c|SLqZVjT>c%x~p$1w3{6Gn&5jKO(Q$*+}En(|I>YGiF$UAdd8SK-cxMwrWtY)iZ4G z2Sv@@5o9IihHla9k*}Oumy9g%vp+)NBnnT0@cO1Lp_q$DbzcZzA)g#_1>j7B*W(X8 zJdxfkW_Da~$y^V~)!b_EOeJ)8aW;mR4XbCUt$9>yC}J+~gH|WdHky7Q6wN(THx3;i zAo8m18Box*W6H}Ymii%Lb=_&KA{yv96}hi#4cjf0kY=S;t%gwh;cu)Iw?@rFPq_|$0P#_fm+hIw*>3;b<2lccZC{zok2K3-*yw`Toi&CsOwxewqpZ{lQnpM?bjbX8h;x`&Bn7(4 z1kH>RTQ?zrzk0?C_q;=+`WYU~L7|8C79XF;{N}fbSaw;~PiH|_8c%;cZ7@C@C5qvb ziju4w;}%*Ry#dlR?&<-c5GC3DrmSmdz5-$*({Cnlf(6A6cKyP-%UJ_3tZo`8)PRjE} zXcJm5J3aXuY1GC`m8c-YlK{y{JU*@g@Drb`DOo5Lh$XG405dVyk*lyuPRzXgv=!F1 znL%p{e==s1e>{hqochgh2|VqZKfQL4cR!+UcHFCbbrPQnE|t=lW#+cxYNKd1k=o<0 z2lr4ltp*nvt01V{*+g7{)j%S~{urO48Ml|X?FD<(DNn0DB0-uqbQDEP9ze!)$HcRU z^AyAY%^__#6UvO7Qd;3}tD?`3znW1KL}(M){>H8~a3DuzjpFEEqRS!pEtQ{y&!nI? z)8jhHDvZq%byiKV3HCs3IP418n-R_MnEV6CCnNF|(@>CZrw+)LQY@OkXsbm&IaaR$ zYi7ocND@Z2p2dh*v0g7A_SbuGjRB(4y}chL9n`x@WDb3H1lIxqxEd3y5N!1NG~-;|;bPZ#)vW>19t@m%tAhv-v(mix;iJ*lud zGws3gD?&&W$jPJ&c=D#jl0MPVs+EGKkX4Tt68K_3RzBbVcx7c)bZL`TZamgKxZ7?8 z3Jye9I9_|`-1FX-AG;9fer;Mffi{M3;Xnk%OZ)oj4qOg&f<1{tL4%0n9y$M4LhQ_3 z$dYWASJT*k&Kj16vcs-9WU<+69Zlx`rRQ*vxdL1j4zCdsPE8Fkg{i2b&aZo~7=QL3 z{=g9no{%unA}|*FqJjikb!6EfJ@7_f_#6*>G6~5xR|agr^FJ^`^w(umQvMqb><0An z)xr@l6@N?WUCOH&Vi+?^v?*AE=SguYavdB{@CvJRS>xJ;;si$+EUOu4e-uEZ-1+i= z<7m6=K%xFtAupP5Qn}8j7INW5DX-NN>k(b|d*!KearQh8HaAaC;ofvH*2|(N7@ejO zQ9zIFGb(i5JX83v{h!Nbou2t>@feZpo0o z1{q9rOn)ZGs;e`}ShzIhMk}eSCj`)^j+CRC%kc1;5?Ff#$>Jt$0k9CxPdyg1hnL~y zo-7fBtgUcYXxc)MNv;l3A^N2~`Qy-zrHsd@X(&XILUbIvaP}d5WO!mD;Yh`1d|eH2fT}?Cj-Hr@O`@- z7P%Py)`;e)rXwtkM%;ozlS5g2YrMb|eEsn>PoL!zq*6A(X2|{|vzMSQ{$&#`nhAW8&J`ilp1;TLzORGfvrXIQS&hSN)|Q+Tt@x5!DTK=JEQjn!|A= z%B3_S*5xs0=L&Y0quVa;;6^)kv+s1Q4L4R?N%cPAM*wQl`*E^7vCVc14XQCDEv=&& z;JE>j-)fhc>zjN=y;FC4+~QcRjy5#=JtyAg9rUMKbQQVl`lwPon^OEI zU^c2BfO5!5{zB#+!a2L#e%lq$+L;Abz8o-g*;*+LTsc~YWiR-!)5eNz#-?yCbt=p^ zZH9Y378TbczLyw_FB?+g&2Tdlce=Z+jSf( z`)?e*t)uB+-5W1!To+=&Oa|+1Gz#^)*}WUYiV~%^p{DvQ7jqoWm&MJ=5+H4au^DtL zOYU;+3(@r|mZp8XpwmKj3|-d^^tfSOg4Q7mjZju=RcCwChVTeGuOE{M8HeBl0aDQ5 zd@)hSd*tu()NYhLH#JUgAHzzA*ftT0EUB-q}E*1(jsF^cPtttZ&%kS1K zjzcgXsX2rz*;_vCE4HLffq|ln0E>=%*mn%K7FRI1hZgDNU6_n_2uv$7@I0B}Ljrp`g1q-aA14Q z@@?2JD<#NNQ1$nJIvyg-2;1hc1f5QBWG38dNelS>Ov=Wb5;*(jp1{nG>#)Mk3iU2^ z>vpAT(Jg<1Ems)AotsW*3}KoK4+iRYls<{A3ORIyvp_@Y zkwfw{KkatcyTwzkesJiJXXS{I6pbE~;KmUv)mB{$G`uX;7 zW+B76(l}RuJ0?b3&csl-lt^)Ic zR#<#lql&r3Ye$>;S1SYpMu@@=_BOJBcA0sU2)Mz10GDA&Y*(+e2zVImMhg!EM z=htm8!1g$95#R0cGZI+hg~E;V=Kf@G;5ILHfCce?y}{Z5VV7ZQ$kS^dIZ3y0$1-a1wv$4j1(_9^ zz!e?_3CYfsET)t!ePMcFC}ae>f1j3hD0G&Cp~x7i0+CoM+!)kM#C%Uq z_1A;+PBPJ*INeZeISXJ22#*HcCE~`2bklIKRU^Uu`*KN zj;Dv@t{}qEyhg4MU z&K%44iM6GjS%KQQ^IbBsjoZeTg(FC9tE=#Y-Zr_>K_Pd4Fw#(Lzp7>w*)F|(94NjAoT zA)eYj{~=qHRIiI9lzifXDY{zIR6Sw3)Nl6QD;4sD>@Opeft$E^L(|rjpXgEfM*%;u zoa@=h3CG;rijrCb2Av$1k`?W2?#xq;VTQSUF{xGaJyMcgc+bPvi0E4}?IbqiXMfEh;c}PzPE9;s&BqiBD&{&t)59TEA;H3r>RhxEZfU zc`QxQ@JjlhL=HWFc#}p|O<+{NXRTvWgxrL+?C&d~a^~LFv{|rEE^I~a5W9qr3?|~W zCEWCr_s27!?zO}AhZl?==X~IoAp3jx+voFI5>6%C>nKHc+^cZKa`@PwePH61jC)aZCPhaPsZI`pKmpG;yxPy#FQp z#hkPZNz3v~>ApwCeUN??sG)ljZ#=B;_5mZ@q9DOLwH`+MgMlmM9p>uFOrDfA zg>^A{#%~U(8+P*z(hEA0mnOAxTa%l+1i#rG6+SJ80%|Fx=@K@0r+I2 z^W3eORlhNLbdk_*=-q`c7k-_O5zS-Ah4>{dn(3V=^jH~4<&Hc9rgpSi`-c5a~^D+Tvk)~*g3JdSZiN) z!3u*xX{|iV!ltN}A;YiXF0foSlW;=Vx>}KI=0+0=PGC!HVO8!VEvh=s#b9`;q;f}I zVwcy*l77&;MEnTA=sEL&Uo^bX=r67brsE-FRQ2?+p1pt(1%)q{E>?)5V@acVOC5K@=^)+$Ri*9S;Xe+)x!8Ie_|Rg(T7$% zEyw_uS@e1n1a+u^5LWT4Qh$XlJw+i$lbl=~a7C&OpHEuN%klgd& zmJ`a{wcv8%J@A1kA{p4yETw)xntHcGVln2i7{ja;QqlO5+E-Ps^p3BLI5an>^mG?U zPlAk)C(LPF;>aa>L&J}@qIXsprD(|IJo0unfwnpXIw3pt&T*)it@zUi@k2O|@a0|E z$Ml9rFQkkQl@gjdPDDi`3vS8&Qkq`k**02q9OV$(mM=iJ>%3>?Xu<1(LPCQRr9lxp zK8LSO^YnK2VdbZ+aj0WG5S$RNNwnu5NKNK(r|ZbGLb*e zE|`kDI2Xdb7^csR*LydD!k=tLetuATL8;_uk%Z zyg3GRP_hBN|7rU}K+>J>F{-+NEBtoHg}hO1vu2s=c+SdvISh1REVDI7igr0*i0q!$ zCY_*d+9PIL<*+nIN;-*c;_fM{w_nn&Set{^nlF)JS{CujBPSf}%(2x8|Ku~RynNC& zs`&o=*Ii9OHhL%~uzl0~@FA=Jgh77p%rywWW}~=0?K`{23nousyJc<)zs!6lE#2qF z4f94^kDw0Yl)*%kQr0M52hk%9*X}QrLnYPaQYi%pr;VOnAua>~4{r`@2oFCHuUT`( zRJQ<~UN5xj*(D7d{=!!}4!v-{rSu^5{4sU};T~G%s=R&;UEZr6+po({Gv`fiuzY~J zX}_2wrX}Y49TpeR=qI$%rreJ*=&d6@&?UBcDxZiIiUoE>UV91vpJ}+%3)Q_)ZJlsOz3q!2gw)dI)#R z9V*ZehF=r{-$~UV&z-zF^h$Uf27no>{`cbvr0TNpXeGPiHo6=+ID0;qhl|fxh@^4R zgBA%0vvo7%78p#$-mU?Ymb+9yD`mU@or`5jIOB{^YlES^k4px4h>zp_ZsoVVexmA* z%=gzlaiD9c?tgQEAWil~)biWz`DCR_WBW1s~spCR@3{I6*ak61fOtR7v zz045EK1rTJ4tj}y75Pha^A09?8RZz};RR49L|{V|5hx1AR>v}bGbFVIqtp1R!z1{g^pmtrlaGu`QmSssk~E4X9xXUG znYePuu>j#fzE7*0P!D#5h;ZR@iK-R-StVe2c5_(kBBTn~4yoFz8Hf$OLQVDFDI?{#w0ds#BYuKP2`S?LbcJC zQa<_1i;(*k$P_9q)vY=F?K~JG9M(vE3*Ui*@$tTZ95@?@{V?_YnKPaiI6Lr`oaibE zy?nQFJ97bUtn`89!Zk8Uy#@JX3;Cee_4Io%h?Vz~dis0Gw|0Nb?k&?Vm5r z6)30mazA$z!_~e{eibb5_vrc$?}FHw#O*=w<(@95V;6hvY3!<`l+fOJD%R#J+R1a` zol;PN*xv2Lwe)hiZUPajujy99F*k38=wWk0V+LS!uaeV|T6p8Bf`ej+_Vm&CF(*^< zmZ2I)C5L(Eg2si+t%k|1E3Uaoyuz{Z>0tReHZp`*)qd}>QJt6uKI)cfia zzh@xor1NEeDTnC*aLjtZAwT53dQzFb=Kg&@p!U4WiEGNPCzDJy@bNOLzcPIhYvR)F zzV7Hm_M7_2bN=I0)7qoRn(xa@2^&waXO3U`RV(m(Oy2pj-ITNMx#o98>L}B4QE?rc zGqEC0e!e2&Z+q2R<1f)(!4!z3aS{rb_6uMP9s8hky>bGaU1=AxguAyr=-$bEw0=%n zBoMg+2O8e3jN3*|t1Qa;VatUC-nVvYFV*Z$fQsPt`X>dy+I7jM8r+_yrJbC|#0hf& z%F>H%yi8j~ZHt#>BHmz;=|P;z7h?0jO>6A!0nFGZtbS}+9d&4ZsdpNAf_oAe{=i5K zqoHT=8j;rXB+Xdo1W8t(wO;42jrFry#Z3)a=Yq)Q<#tqKcplo1!@d&5hA~DRjqbF? z_80R`Bs*x6&&!w25~|XcM`%T`wdi_xh{R+7?;D{~ZoDg+k8#|WLgYC(?O$091qD=G=iYFNWghsI%)&#n z{DDSDp()Fo0{rzHm2#ztJ_?+!Ix^ecQ`)_a``%@;E9B8TyV}B%=L3;U{AiJnHBBO^ zIF-uZ_WstdGv5}$$`WuM#~m;H7tgzrgJMn=oDN@du8N-Eu0Gv1O{BO2Rca5)_c%%~ zoht8CIDcqN>%$)sAuu8DJpTn$p1QgVh@ZEXYC3E@(5-pYo@+WDsIqM{em4{u$2_2k zJYb3}Ft5}+pJ|Y@HG?U+8eR`p~%+d%+8NGPrGIu`Atb*BT{UpGgu$2y)^Y~H>c z6Rb7rzZj$rrk#Rl>W5~)<8_Rn<;~pmH<2<@=~z>nh0h{hx*heV&+Whvspog$i#{lH1*QB+re!ZGnK@Wj;~;9-lSi9-A_PSD@c`Z!%(UWtS8mmwSAuFX$PDoyGLRz z67dc*jI1BhJIKChvToob27z61sE%3H7i_<)YIA z?}#gBSCRQkEh-&QcRKx(CElsT5^q28+p&XrBG2XG_49K3SQV~_waivkOmCAzVCuKM zix%O;qt+Aui4vRM7IB$2%@MvpsBpBqz4sljnS)a_}`cqT{qemR*$iajP$u4FLKW_eBSiJeZAdG73QF8NVHqL zSJxK{-Y{6e%jJmn8jnvT{R}v7k1^N%vS(F}z`fu}RTi&xo7BnRLn0@4j;r!K$*_b> z=T*?zc~DNR&lrCQ-Th#FZ11aXvFc0OlnkxtcRR(6He75>PbE!_(MIkGXFs$@8Eu(^ zY2;E@g{4C~!(}p6_A44OE7{PN*Glai8e;v_v!ow^Yql9C%rI+{W2O~X_&E*k>y0Hy zz*uYwmr*zH@x>U}8w%83SB;m2JPuuZF0IY=bwa5Dxt9l|$0XY9nkM{Qq%gQsiG?|d zw4Yk#FV@nt+1`pZQVfe7l`nn^Ug3Op z>qOuu$3>@pOz|7y(Xhx=iZ;X;s@}Xm8!TS?x&JBqU%U=X4^gKUhel_Fe9SuPd;JNk zR+iahv?UapQk%J7h9zQ-J)YiA3BI^^6E6&xv371cU5z|j`nhNLvCkCEV4h4TBp-ky zJfBQBuSjWxEJro0eu}oaC9i4~l<8S55Cf&tnM0IjCzswzBD*nYGGDyUns4L1SI>Kg z&Z*c+I(x}uQer)CQ;aQj510;445Xg6Kdy}}&79!~RoKP4J_<+2rB7gQIA$1!L@YKr z`WfU(4-vBs#kMi$+ddreZhZaLANaQ<4z9oPR{HPe&_0L1zv&%@@}$#j%ZnG0UMCz&}k~PsfZ`^At)CwKRzPz|Kuwfjv#2k}$lUewUd-W=34T&3aV-2(^ z2-C-WaHP#&;-G)FW{~fvx7#RPjI`7lSnPo0ld8nggV2}I z2SOdJKkxfs)dMh+zD!MSk#7&cU~>a6Kze9!ga24^Wu4@1EvA?&d;Pq>!6*q=xdT`7 zM{}=G^;Jo61??y?frOnxJ$4b*Y6^|KdUFoCXkl7HCntXcGG)g2M&H zx&8U!jfi0Hy2+X#B>}fzjM-H!W@HfJy^p?Z&D0ncL1nNOpSwK55yjt>;*VL5&MH4+ z2n%{;5US~TOl`YTBCmMYd~dF_jZb8pJD;~x)_6HLD@5uAj|!dFGCWTF!AYw7DI)<* zS@TMd5r*nWb)7)h!L#B`XBokrxQM<2!dtS`&ifwn=TnayH@fzHo5o3k_VhI(&(7`r zah&+bJ_i+{J*#D`LfqJUS%NjJl?=-pv_9#hmsz9l%Fupb$d8HJiEOBZboB?$WhX#> z=br8@LG@0mm~vAmR5&xd5Z9LHrgY6J`g$EMnbX zEj_K*RL7p;wrr{{#s{ zjXUn|=y-k_h~y^>p=1dGms>LhY&q=QF5NcObTzBW@3?~*c-RMVD(<#sdbZ}ugzk=J zqV^S*<~ln7@5>w+^wmiev^8AY{8XCJy@8Aana*cvI)c-MC={Bmg#q%70lV8ie)vV( z&)i?`cF9?OFeuTibj!-u;~PrSXmkspoM_~K*P{rA#8~c&%;TK8CM+syT?{O(4t$aR zBG8+5JYbVgYNu4K^wCAE{%x3v`MjRv->OgFKljpw0{UI0tc}ImYHrFYU^$&oIh%FB{&UKU(yTxOFi4*EK&oH{90G4@Z&bWa;p?^QYDBov)x| zIO3nAT%6*LoA#LOp+8x_ulvjuA8u%FMv*<}kmJj5oak!B&M)DmbYydfkY&9Y{T&g~ zGlSrT2^9X3ARh{|knDxPlaJ+>7-{^DLXvu<_$w*jN_t{tA%sns65S-JgLy))((8i- zA9;7q{zKYPRMmcs=#$VkON+G43T)OZrV`SxY9q}uX^-TmC1e`U9upO2v5S+YoYD`= zv+NRp@+ro!Z+N+qwGo*AmmK8LMbjJ0o3PEgQFS4xouN(_uF!9&6=&zWoo6m z9GQe|oArYuse6kn9#A3g*5SzM*#uxIsW6k~dIyep%Fl=vG*MME-kfX9tSXj)i+erL z@Oj&g$NJQo(J`$6+1p*qFB4=*?5cjSIKea<{wL2hO#eqzO6iEA8QgafgkJpmX^Trd z0E$%e)`a~#M)($g+Pa%*Kg|5%_@^dO8nR>W+TY9_G>JSIQjD^_D|`NtY7ZzK)5@LK zZ0oU*56;S|&J6(ij|VsQKKo*dP_0R~m_y1+(enO=n-g-&f!`*98UHG|Qj2V!&g|>- ze7ilu555Nzw#o0bYy}48q`?uvf1EWB0`kK`55K%Pj-~RnVGQ$>L)C7ingO5} z&AzLV`~GXAY|2c(D=IVvq?xG|8E-Kg%CixcvZRQbF=)Ra$BJ?UT3UYjIAMDl?(k(x z;7jm+_U0)GJ#M$rtW@s8P{J#-@)OTU984I>I#b=BIbwxO2KY>*CKJ;s?B`bHXJd0H zs#pZ#bdebPN2_SWP3T)Fb?&_50%Eda7)Mrc(JPU{#CN#A z22`1d4V<6OeZqA%oeh4$b=YnJI*xJg<7gJ+ruZp|TG!%K2P&$h{(Mk2RhYLPajfjK znIodnEQXG~|CqKNRRzVPbhwj%nZesNdByJG(epMVVL{B?p7E#H^!`3-1joOb5X=l5 zgr~QrNjJ`QdBZmbn{t1;wvb`UX1;SVji1{JG`W72>_jy2k+#$nKA&g-h_;PuM=}t_ zAeXZLRn7fcPam=V>;o#INB(`f)KLKv3Rh0#tB~10oMfISdY@4a3ANY1lK%P>zDH8` z@3kIg>L=1ueXcJCONl++8sS9q-=gl6U`X&fs@sQiWUTi1b8}=W_VQ(I6rH#qvtBvQ z++9ncTCff$|K*@U9d{ZbGzQXj@`7XP`ZE4?+%+S;^0gMyqovuh4H+7qU%`_7vb=*5 zC+0i&J^gsD9eam`N2x@bCiS!(62ubrYG~sk?-eBpn_n_p*_Dz( z>w{|1Zu^~0@h)1FCRM40n!o0c{EOFcIgD>R#qrhL56P!&afDq>ZsUPMWEcr0X(HlL z4COuk-t^Ni&dTUiopv=%Ntn<_x(6zAys3%cxzFvAVvR6pgizZh6*!=nq8c2gza*?+ zp>FvJr&(Ujm~G`9n?SVE3X1=gbEH6luT~+n?9iJ`zAM%G$KuvjN*+2Y_jdR=+^5gj zqZu}}GN?!6H1*kEGV_5dMdJ?%TmjpoV)J?8$uNhsGmGaw-&$J2*AZ%JiQgCDdE#vo zjIAOk_tpk*!(Vltl?nV4N3ClRI5OGaq?}_yEFnum8|}J$FY(Ij_bhi^%{XgBi^Nw; zsP&p>eDdbpNeW!?s|#OI^OB2^{ll2pPySeN7+P<+O}eg1?@uRyKk(lOmuR1(gZJv{ zK%PZ1nYMi~s2zC-x-U;OiItd}KStZ&@sj?dm{x zijY5eGj`SjC*rthj&^6aLRg?-6YdwqOTsW^PM@IU8(_k!REU@Gk%%KzftOudeK(Z! z^%VQX7mbWL`*E5o>hGeqE;VFy8IjVsP|Jd(f1*pLvakISk zkCgRcyq^V(Ln>D^bxZBfCUL|Z!sHl;IIx_kOx*=6$VjulyiMd|U2GtmrDs6#cWQ>$ z545LNu?tU@%jYldb$%H))G|f1MwOb0V5xkY@(M>Cc_j*$N7(1$*>^)@o%431&tXq! z{oAP`ecfLvfLn|*x7s^+#*ZEG1iv&q3utMBrj3fyA=MqeRSMO~XQ1cXF>xKCKrRdw z%@y4G^no0zcNaH!;hQp$)@lSA7KSgXZBzwJa_!WmV_lRyGxPa>q$s@2b7rT^YX8v3 zB_1ek?fE~2ol|h&!IQ^h+qS*2ZDXU2lZ|cX7aJRFY}+x_iBcnVPPe zx0$Nx?&)uTf`2Z4$&lh))o?)-y+eEOl^;f7YfsP(yG;PW3d&w9XwWIhAh>nTbL~g9GlXqd$$Y{D^+RNWU~l@nsZR zZuRA8!lr-TuMX~+#pKJfkas^Mzmke?f}*U_ehK8 z=Bompy(YQFY2;7Umx+;019q=i3xz6ToKqyP2O_}ZjP;ml29CHaSfjGA@<;ggCqGRMI}b^Ehsh?mg%C!x@v#+j)&quP3POC z4wnHvbx!^0-8GIpbwP*X;84lFb}^tdJyILLn$dBkN>$7DVUfn673v0Lhi+gx*SsQ& zOA9yygs^kHVp<5eF>!CJv@b!x=o=G*+9QhuK4+GReW0o8bP-%4y*=u?>A3d^D#k~2 z#Z#Y+2=sYI!kwjb z#xqo|)@QC7i(C;#FIHH6n5YV|bKUXvW%opd5yledVr``A@>@A6@=F|K-NHIs!M`69 zHHtEIy0~Ymxj$^YXw^ zW;?sMvY+9&6Y=GTsx&*}wbI^_`MCRa8sX}~Q3<+=^>@93seoag0A9(kx2_A=&QhD- z2H~xzX-Sq0=j%@Rx5tSZnhF*pzRfQ$eL1}Co$1VHO7_VqGzJpfcj<-0NVNC+Z1046 ztu3{}T~$RUiXm#o=WGzjGi5(Bwv_BN+_dvvDQ6}OcF3RMk~VBS&oJ@=DT z?rHAAQREog)e))>74+k_ao{HH+E27_efm_lbjJ;wIxStA`Y`pLH0a2CT@15K zKByI4#9*P$w@Zch^XL3hAtYN~Ml)6SeINF*&winEr;$}*(Tl<<~kj2_<@^vmg*083zd?O?*TlvlTH zb=Jh1^j$7fY(LWAa-G&tcEdnEuRS>U{Epw+XI_~%Edfg$W1;?2dDH+rqzOO zt7SGnPWw=~OV=nz?>8`jeG&q3yh{;W;X^#!TJ#fhcZsO)8)tq}X^1=VYXajdTpm>J zrrnSX>JN3~-d0A+Xh(rV(ueTaTJIFRzRf^T1~#bmd&IO1wRb{9DOxg4X6KknMgPgm zDXv~OGx>cRp8)x^7hj$pUIY87scCFYY{5;PjWren%rBGq7&zym-jD}+yn6JfQ||9R z+pg`IYL4i|<=^|j;5=3gB+Xy-Q}gqSKJxU~AkW3Y{?EKuw zqg2DrBcbT`1PX1=)4=~AIvbBwq^y@b+sMiM{Uzlh@%SlSb}gU=4f}ZTOIeACZXpp| zv}-}F1egmP@zUdSx9xaIXaFzDjyo1-Sh;%$&i$@KO@{fLg)l^1bq6`PsO{)(^pabd z{5DbAQyX5=G(wC;yd+WQSog|KAK}mT!)*F3tRtEJ3i#*WuUh3{;*EEinS3gkmGEh% z=TFm*Mti9Lk*Om~z!EMap~Uf*M`2@H#y==2kX#$vzBduWK0A|f9L#Q%|2pvldL8eo z{UwO$CZpxLhKb99IL4j_%Uf^$IDD?XWLfQ-XKHe5*ST)pwi^u$O8IJLKHl)}f($;n zY2v>ZcmU+nCDYDuf)A<3iyr&~Q}w`RgLs(}rI2@Ug25*Ob;GWBuUNpL+eDe|gLtvg zE>mM+bWhmGg-gbQyc%cM4_yz1*HSD&iXx zOnKV0tS?JTO5V<|?7fa0uP^b@5K9ON5XlL;P6!@&Whujlgx zIopCl+}xh79)3??u0jv`f`U}}oVe5ww4%blZHSD*lG-I;=g}+~@`LGKV4Z{Pq}aG- zJG24TH&$YUX{XUbDs8+!nDh{UJNb8tcoq3|P%srm@=%I+y9r1voB`s!p|ftS z?cr7my4X~jUVq}Rtx5uj$l;qY>x^t56VA76IUvDT0kw}o)@obmJdoBU8oZbS%&`D| zb|me@;!OA3MS{mq0r$jpI~G~zYgdk_`HBpexqsXr3q`!$?S`4teiOq3K3MuWA+&!A z>56AT(P1c3EmC_E^H5WY#w~@xT7haGAiZapV;NVxw~532XM>g)t!RNbS!J~MH$+ci z6~y90$ZR~n=NMTP$b9No|26L$D2XKh#NK!D8Bwy)UkF0-5EfNBzJ#3Q+>0JKPs_&N zqL;~3Z6ZfPhkNsXXqvDf@gim~4dF`#*FyT9PnojhFC2TRAW0E4Ncc9j`~)l=QL@jc zX3xlbBmCKiLfSK)eND8F!4r61rlJztbujWLJ|eOEX^_pBLF4COxQ|vBEcV`@!hFb4 zzuH;jGoO-y;`k%i;xvd;9@SfVi_MCpYhJ z{*e~b$r$5;-gW+dcG8&Q>?ATr1B2C;c`v=gzSNq2*qRM-BHa%mF##-bj>R*b4BbMi z`JnxhLC80ztmeqaOdDS`-Y79%!2q8UMHszbcW_~Yrw?octAb?}Zy9m-iNgrEP9^(! zdZIl72S<~?CE~Z!^!^K_Otd+8W0=C;Qhw3;DSyT7mfC78pB~J5`rXf3?cX}IO)=;1dC>&VQgF{G*UaPk>P@Vw`=Mwo7 zZjm6FB9ZsNzDUsdJ%Tl8z&SG zD;{SMBN-$8SDDU)YJsDLWG*sh9BfNc;ZcQ-Y&{*=xR(DJm#-EMVbO# zDdsUWjL+Bwg;g5%ZCNZ8yI{eUA6$r=HSm z-q%pooWOJHDcAx{XXr1&>xev^l9j%&jL3sF5+U&1DhAFTP_+3^1!Vm()aYvXp!V8XSe;>Id4Z9j*VzD=ng_9@+=Z=uoCmC@U0Kf3c&>wpJpMqWX8O4RwXo$o6OY{=`9WG-uKN{WlvbF0kSg zT%Raa7!>r8iN)0Qg9e1xJ3*D~lfRYlHnVlD(zt?NoWdt0oU0lIvhf+o@u}z7Hc5^F zuVZ9zpF(VTWsTj&c1f%i^PRJ^x(QupdmBjJycT>-xegD#X;PS(*cgoPF#!Z z`g&`~1o2{^PYKXAU6W5P6$IWMmpj2ADn$&l=*MeC)B;KnG5}Tdz?$umjw_L zQY)?E@f@i122)8lL7|}yz4<91l-&cENtMk9=e~RbIKsuD3&B%H_TMrbX6%f1yk>l^ zLvr3;1`RI~GL4mw6(1eIUSYt%_VH}hlXaVb-P)e~t2at={#ta2gim`D0MW;*Skf?i zS*7=Z09h>xLf-rIFfPoRJA{bS4x$qE?A&*$bD3epI^Y#0c&$~j+|P!;NlyY%`O(+S&>c$CiLuuIY8dI}Ac-vH$^wg&*9F01 zNN8_Fuyp$k2KgMK;7K=dc_%n*Uyhq8bApCjz-G48Fa+8KXbe<82))K zAn(cwVfD;0EADNUWq`6*&oXtl$;l>ti?31sP2PQ+T|Tju-%S5n(|`LMDUz?DtpZJ9WMnr!O1k zBKqxC5HVrYnM30XiQnfJmARbDz|LBwi%5Ubd$G@@zLYwuEqtU!lW_U|bn~J!JE%r^ zSjG1n-j$M2)H52dt`j(-Bw7{Tp9dpkjkG=&7OcG~Fy6Y7e}2?~1i3Q7AZj2HqnjS@ zM&Lk+u_8S*>c&oogCqw`ZfEX8FKD~Lu?uOh;3d`^qJR9Akc$4ig;4Lxt*av8KyOiV)~s|tJXpTECjmO+r$0d>ZoyR zDd~veu)Zsd8RZNH@~nqxvxp1M?{$mXW=o&|orC{?sV!zHYrVB&;X$m_W{CoOkLnL& z7MBOkOk8W)%;lZ>DdIl%&8nf^%JuGu5|YGY@n-Lchm77)I4yQjz(`?G$&H#_BIPjX5gC<9iJO_Gxddy1e<~UL5dy6V zFa0YqP3_>DhqHt065`KJK;62xWQWCp+y2_Li8SDIk+V{wy}l1H zRfjJP9N-lQ=e297j5PxpoCL_tlcx0k`s^Nu2&Lc_-yF}qjh8A@@@? zX}hgG&i}fbt#Goaz;6$^MTSd`OZuv`{-ZJKA7#~_7OA1uW5>c`QBz{AG|;EzrQ1;y+Z0u>mYx~r%?|W zk+`roazp$lFWDQlg8)Q((J(iKe?ufF)EWtN7>rp#R!>Ez{Ueh5xZ6$4;UuaIM@CUN zriI8n%B{sD;uk?CTbfr&9f3ibC!2Y&+a4WPl}aU8J6?YijrClH_>gq0Ck78ecr)zU zs2uayUS#9DfQ)ro5UhZWQN@mvunYaNgQnUXG!vT<82}nekqxY??DKTRuV_H=JxO&!VKhFMX+@jbw&5z|0sYJkW zk+^2da(+D=ZTvYF_S>CYKAwunY}iwIl{bMS%? z22r289_tsOaPOEU2GA|4s6v_ju1AvtR>b>S1Qd6wz1@?16$S*D9x!VI?zS{VXf55s z6C^thAPnL~b*_0PSwkF*5XyRu9txx~5fwlYLoHZvRDm>Jg#vli+f=+jqh=Br%0?~8OrM&+o*M|xlS6qd*EVUgEqb30DItRcjURdat=*rk zD0P7FVpy)6iA50_edeIoRrTl(%5v*>gyg-1Lp*@Q3`;k;BoU;1wo_&DZo*_WGV7vv zm1?|$2&&?;9vXYdFf=y$u^}EurEjX`b01eesN`=|?nwOhz$eR_s$ghA?C#H3ahenr0vs)^DVi5U z43q#>@uoM?wU-=!t0tc+!D4@B15`(6zQzcA6O|v1;{K4@k|M?&!(u!Sf^=Gt?6UFo zfuoqwFIuOEq3{kJ^K~J;%DyB`#sS1+X)#4^z-NXbV zmopBWF4e48B?NMXbxG^rw{fx#60pHQ3{Xx~b1536qgiI>KTt5j zz)lK5pw%rhQUi$p>kxJldD-xeIZ%aBYS zFhS7L&)ZTf;l@6b`p3_OXu#CXxx*1Su#QDkP-6%wtE2`RY??QOPHb5E>Aad|CNm2eIs=d#4Mjwb%GE2gu^$s(YjK&&U zUC}7gBX95_K6u}OldYCBE`vhcs4VWB{ZxGtI*LXiNL~-6WLD-;OGLQM?UX7!HPC`D z(=zjutpy%Ua||iSx7V_+bPyW2OR6nRP=D}_)ddR$MiVaC^ncN!5iZ_V})egNVW=8Qwpqy^EGLu zCK4fT2lZ;CBD1F}Skp~{n4?Ugdw^L`32t#vZDyyF zilAygV{r{&B1x7Nu}b;ah=r}{RbHy+oM93jSBp)zSmTWIgP*iUGo_^r>7191%O(n- zZ4Xd}i-@l_j2*LV=l`nD)l3im9mbIjhHRltmLN!~AzP1*B3D?U7aq&XsCQ4p!5SRk zNtT8Ws~K6I(}|uR;L{);w7y}mccgo-OQsy3rGxiAljnMGb2VA2+>`QPk#r6_r=G2S z!-ni<7k9{`GN;comg<;6dl6Wz;gV-`{sJ|2&Jt6YnBYX2=8XXS@G-oM56v$k%VZ)T z>KQq;^|KjMhZ?!T&=U{qbSK)edMw}XFb5m?dY2eQSrn)zHN=gq}8^lCt)!|QDL(3v_SLg zOb{GOdn;t*-{!XTL~mANahdkkLH>1XhB(~kPPAvXdWHbOGDn`!lhEnAVTA`&>D7=( zTB9A#Rc%o0xZC{H3x%4=BjqnN)#@J?Xrv0-X~vf{XlB4Eo5_iazM5cgz#)ouT%QAe zqvNg{Jfln>oO?jhpRJMda#GRf_5_5HFkHZNEobQv;8^yKIah&Pv-Ht6|y91bcdDUt2(_3+@pPUd+!)e(+s#Ef&fd#T*s zbz}FFbMlMb2CHW|H1v3r6KHSOhw%s!b=W~o>IA$O?*|?Dc>(@5yk%~W`cnH(kd3GE zdHw-BNO)aj^B0aZK;67y*HaHTT0S2k-8BG372krE} z+gptyXYnsM&Oqq3U;+8ejm>s(`};ha47F20D=)ljXbBdVWE)L#5+muLEKr72VZO-x zRRpg{sKHu1K9fUkZG?BQN8axAe2VmUwsvGA9e`S{ zzlP9Bz&FD0i(k%$tM6qR=3N0k_9ppwHMH!J=!oSYAG2aH7l%v;0}*RKqL^;|nfZha>I9$s%})q3|m61}xnTf^R|1X@t%N@5%`$jUB0lE zkN>t1ht)q`efT@en52N^#lr_XsGVrs-!gibyA|P_i{ zmotgiOxNktaPG%ODvc5BZN`-u+-=%k-zm z^ZxAU=PZ?`vw!g73q6Rn_?$#;ogeK?AyPz)tH-J7RuTz+jlTNu$5$x}tAT)@m&40j z9Cg@R$qT=)*iP?=(GAyV-ww8o8ZOscq{8|km}d6K+MW2@_wRsUjUrtoKC_fWe&=3( zT^SCKo}2n!=VwRu_xssntJ&$KiaPa644p6Kq4)Ef3DoGC0!dk_NT4-^F6yQb?3`L4 z%-Q7`0=j~ZsjawqeBBEK#}Xd@gW~MvpnaG<;0ftLKs;JIYmw{W9sCbYco%sqfiWKx z_XCP}1q+#6jvw%=itRPr`VQ33{wj)wg=S*7W|BlfJ}P)s4JdPu@(12u51w0Gt|q7Q z$#~&EJGXg}r_eq}orS2}J=5)trtbwVU;_K?7tha9Sv=uOJHd5B8KK0aa~o#g$#JGV znG~Hz6?a5~lDxfEgUt4EH@c7DCoDJbu`b@Zbv93D9VZf(UFGBL^AWLn*GrVq742_Rj_pp~P$C58a{r!$Y zm20xM9X1!if2{!Sn$r?b`e}Uc;*ft;A$~y~+(7yycBz{3s{W(XN_Rl}^YrZCo>%NY zF|(??Eoz@vZA}C2&#+X}({gIPrn$r|y}xtd1*W@R60I)yg;FgY4@Jkk<8zlwQgN2< zyK=}X$sup1yF*F+>L&yt2h>TvuB`m*8+T6XjK{$h!wq8EA_uF z-R)AkeB2F^yN&Gyc5$rnb%5T!lYq5C1wwJ=zr0H}7dic%8(Kas94W zhKOZ)+2y)pi}r!6@exju-JbBZ%@<8O+;RLLb>Qb*wj+LFh*in2U*@-1h)Y=mF-J%6X6_foH0>Ws@*ucYob93ydKmpiGc~- z#f1bX^RBbHl%e|1-b8b3+_oaoxJw#M>-$Vb+uXFF+H6{osLs4@@t zg!wfU6{9?ITn@S~``951#ns)&>zyLbQrPqDH9%}c!|S$oQfi|ny-Do;_v1Q zK*@ZTB)8I6iRlOGBdi z(4b+keTJrqwLJyf>~WHJLBm70s)H{9X-~uBz-V1+&!{r8kZ7p9RMMa@icC+St4T$? z#4*Fgb^xh@l_OlioCJo3hK%C$B=>t7uAK*1XQEY zx0Xn+-;d>I9DbLw7csH>h%pz})oe9TWimPU5<5`XwO$rBP`})tE5J~QY$XRohu?w!R!p6cOA<4xq!6PavAt5Qk!pgHm z&dScsEhfn&E=a=v|8CLyKbZ#`7e|WY)|bkIaE?dmy9!R&?K4ml6yA?O()tw%Mv?Hs zR1b8IQ0)l%2Nn21>X=N(6xPM`uZ3?N87i`T>86Rfq0aXL+ zQY7`$(ge-l5WqQvLch>Yi-s_~j!!29V|?%bN=#anhB(h4aMTY~Zo3gN^-7cmd0F^N zS_CRek1x&p>A*LN;_C*wo%j=-B=~+FWH5_y>Fz&#(zDGBC-hJ_68<#;vFlZ9g>Bj4 z+z9CS6(WKYXoM4>f)nJJ`pb4K?i($zetMUl@WFS}PR-%_ke+7$MoA6YG3X(n^S38bjdb58R3 delta 44382 zcmV*NKw`h|#t-br50FrQS#ul55q{UNSib=kVL10;g{fGPWXCSK9GR}TO7a6_F3A;1 z&;cQf_ODO(>@~BPJpe(Jl*)q*XgTcd>Hhkh9v+>%_=Yq}N%>|w1>&^1*+4=JIe1ZR3CExt?jy7?q1h???AweI;cWKKue`?9%n-^&c9s43;9py0Bbrul_~1lo zHbhf0MXZzXJ*EkI13C$s0iXE1{(Z(Y(OhNgJUbg>4KKh#4jrVNTZ+Nd&LX%Po;k)GL3?i;6!M zlG8+iaH=&=sueO^!3l2-33koPMIu^M>a<$)q9Q{xO`H+d3&L|Phz1V*qE$j#j&MNI zB3N|;FBJKI>S2O)ZiuiA>u2SPP*k90W$K@QnV_HvH+=fuzesHyc$K!h@dyF1an-8J ztNdxPD!=^jc6GT)aJCRq5J`CkE`~T&3F9uoBIsB*9P$Jy2XtI_lPcDoX%<}M#VcXV z2G|2c##n_16+Uf=PZ~hChHLEFn6?BQL#C}HmWm-`yL1BB;Ek<&GmQA;S`h~N($*6A zujKc4eEZERg=|Y8sC3{F5mcmA%)@ejXUnTAzg08Ki&b7e{OS5UfBklKo)WGhoHyeC zE`fX052_JNQPn2I^fp)_`oP|zWr`-=D2O$aF9RVII=?G!Hw8pVTn#_blo+s}2E zlVk%ZENqzV#m%F{ZX!Qr6BIC9ce&`eBHDVIQCIZ&7z=Hq%qzfP;ysYNDm~<57n8UH zK!5py8=K+=vSq3Dr+{ZkB)6Xco>_@!IStS99-P*mc<$srxJ6p0H4DyplR>h$K48Xl zSdK_VJf9~cgNBG;Vng`!0zd*9z|5G(KSe>wJ4SEt1}e38P;9_Ocpe{_yju77j}_9EB#z3Kv}3VHaT z>UdVr!zUhk*wny(+@MRCo#xfA10)fOJ^d_h!EB_M*v~T7jr6k&1OPYa+`*>ASnaA`e}f(t08nMvl$I8y>it?;Ov=ISGvI`ew6rK2>S#fD)K;_}G2aw` z8lmXywD5ad=c_-V!hgM7myh4BPE)Fa7jZclJwf4eq`9T-aC3{pW}5BhzV8McMUch^ zF{JUyKxYY@f@PCiOQfXtq>M)J?Yy|!l;6Hg?0F%;`#aw0e~7G;d5oa^{nLTnN(dkA zkIN65*g1!kNT#iM>gUljNkf4sO$5io7)2PZjOcOxE-!8ey@fDdd=)FtN{5#06Y91D zUTn(%Ov*JFt9G7Bi@?TZOXv|eRBb@R@S)2OQ-3@vAEsOe$9|~`x{} ze??GUV0_=?e+d?(C9buH^Mv1$#HhFiP3#z=6EnLOpqmt(;B2Ax|8@F^r)_~B%K$f2 zW{cqY@ds=;41r!lyluSBU(yAHY+lkRA})=%F#CvkG`}S!0G}*V;e!%u`ADyy+s5Tq zk*EaOZ2?}|(_YTm?CJ^=f)vIy3yY(jFcz@_Vc*zrP4~;UXC)`tdA}}M`R57&zsc9X zm$~oNwiF@tg~Z6J*k1;!m;I#zO~C`w)vzNA6)m&1Ij0L;2xQXAktzj3cN^?ZtK&m# zgD(Wouk(`>2Rj0mm6K5iB7YX;;RtscvNacXjq5CBizmn)U7P$J_jE_xwvk5KpFs{~3I`w1ajTu}*Y=8o*7>^6uoz*00!VKl(KFKWj{lK z^Y^mip4A64Zpk9o=B=9{$f~`6mv1)tYMr0wSIPb^7aHHo>b~`HiaS67Bnsg0WIai@ z+LqV?vIP+7fec8%{Qs7>Y!jt`V8zmD1VG~4M6`shdMLa#>V!W zICves-9HBKVMv*RV+E2|od$B8zHsnHDh6`7!@Go3u_uI@4322tq^F^Tve2UThO#j1n4SiueK#AEh7|nPa)$rYPhWq2x!JSV-g$^776&ptaX3{ z`r0-=BuG%q$JKr$43Wa^zLEZef$A7TSb>k~zSwoXzVvN;L2)Qm40qJIpKq=^21P!{ zNtM}8*Oyp~$(tjW171aUv^X1*q^YR~J;N-#qEp9$XlT@)usQP;$F(cBR0`H zV)9~}>8-g~o}5SX4dP(cWb0;q);-{f1YvR?#cnGjOf1s5pEO~@R)zKiJw|;w8WQBG z;h~t`K@c@z69B1T(IiSd>3;FZI#^Wp9l=Wb`hSjs@8C7IE6?(L1VRs$O_I)e3g+N9 zvIWi_v7k?k6=&$;==Lkh?4CJ4?ohu$Bu{Uhmr%gV2MhvZQ%z$3W6WxvXwcnTU%U+~ zj!$*>Cgtu;?e4`+?tX`J>Zv!?ILE7Aa^#j3UYCMJ6UT%E&OQK44mlKkEZ4Zd81nfe*rIioUrYYKhj<=6(#(?J!f%+>e4d!AAyPxSMqA#k}|6k0ngo zs7e?b`q93&>{w}M45npjbX4~rsg%TusDHFc2ii_$GbK-W;$7mSbnfC*H+ivrxl7NH z-9k7{sO#k2J@Z5uk%p&<0vEjRRk`5gzrCf@sd~7+hoVRAS@f6*MNc)q|D+k9J)g6g z9}&FTd>LPYUN9VZHGp;qfs$AAiEPWBJBMF}H!u4pJwIXN)G#S1rq#r`@_p_}bbrg} zX3cLN^f$M~rue-&<*Ba&Cz@msJld4>cywkxJ#!r&>r5Y0_T0tC&6q#Lqr-A&*iGOp zve_?*$PjOpD$zRYg6MGQP!rIh?iU@a{CqhH7?7PgzQ=L>hXh7@p?4e@c%x114GhRt zv>y`+3JLK)?KXhJ6DseH=71MqKKa+e78Grg1w`W?SA%quGXxQt-k!gc_!m4Ak==&TxpW@Kv z2fSa{%kjm4rd4RZ{hF5hX*%HwMCX{uL^C^_?$c=2~Cf`F6Wz%G}}Iql-ifn-L}L@GO9%1b;igQjy_X zf!GhS-M@b#f|ohvuYY1}Ra({oJT+NC+v$sn>4=adx}9CU+!i7d;29)a+QzkVqz>+_ z8y;vi{3>y6z!=ksL*$A|80|%P|%T3KHL(!MP-sELXFa z3q@&9rwPhj)26*6PD@R2T7T+;IIV6L$|n@BRoVYIylG;a3l8xh4bI%iwiB{SC?9*G z)H^s+n|2>w`50bh2d}1&z%fgPU1c%lZ}^xZ>`%7x?B)1y@ksTbXtif!DsC3x@QAM= z^VNBtx(W|Q+U!)>))1_iRu$?!H-v;c4YL}_%wqaEam%n z-t!((zF`wHT?+#Zu8M14g;xqKF0c%?GQ2!eKERt)sRIzal_tk5kPm^`5ux0W1%f?r z&EDGt9ifmHbZk&)kuJ7D9I~|UVS^UpR5*%*FG|{=CPKO)8^pugMu1EIr{q9WBY&@FS*Wq;f*la#mkH{TX4-Yt6kxY$Sy9k=L>QA!^Jo4Lzz&8XC}*qg@4CnwdBiF zCl_ySUT@3Wo6}r5vsm5~_}J4gU9`Qig;EZ-2@joD3%H6`6wI{?UNFt;H9T4)SfM@d z7I8T|iSxn=nXy7DQ?2_D7iFU0Ol5iW`T1$CrCeN>H}z2xF5#_o;V54iW3wD~^S)aD zx?aYjWeRRBnhU)m_EE5nYJV2gJJ_)e&N)F3(Cl;;-2<4z25dPHE~y02*;5k z>3U1;W4ziS&SZ~QAD!mh*oKJ-3pD5*p&KzDE9zt71aYD- ztdlWy9}}m^f|*t%4ehb5fEjc$`vA4xE-sgpz6!XX{nN`NrpR*1-lE-a*PvqtL$aVnVu#X zWK8{RIOE^H6Mu&&h$X<|gAM^g7<```@&^GRa|IC60>uy>ooN^-G-2na5W)=6aT>2G zwGbi>Chcx~)CrDk=L%+b`N^He!Vhqi$tmd6ZQ?3Frhzm!Q@Add%!5hr!1IRJ|9l~p zdbM6rt`J1%Y{Vf0I7mzCaL8PooCK8~Fj8kl>s5J!$Xb${#pX7EhRfw5AkbDS_BFMs%nqjj^E5vRU&p1)d9Q+92__p=N-#GrlX_ zgDD!ncT}6iA+AdfM#5VXkbiDilk5{M0Y{S>6fl3w8E&b&IpL$d*e)e}Ul6u3zn<)< zv@@|kToYwT2lOCF(8BjV z@~OhQ)~5!jV?oThFkk@_xZP_)750Jb0L|LS@Pv5Yfp^k9I^77Fm1ZL$q-NUy>e+p% z*)MRkhF>mx9*- zb)AN#0uGP>>6%AJv9AE|BY>Pk&XA}DIz8i$p@YdexDOVPZ@uW8Vk;2qJe-Nv=$eFT z)D0&_y=NlQ6P8Nq2v3?M4p35;!*POh7M_2CXAP|Npi>jUym2s5Fi${bvlB2jEsiN3epe{IBz*|31EMQn5rg_#9~nN?3`bq{wjMvIn<8N4*;s@)NIX|R4HQeLX}$;*F( ziVFeYQ2hd+-k68L0Sy!eUo2sR1`r2-eR-8D_?I{n+kNVZ(pBk zy;!cB556sL%4-dy7cYVj-na(hD^EV1fnd(I>|Irb(++JVlw1AGz!608?>iAMBR zJV=JP8+lLC#1fFdi{1|^hF;b^ zA+_L*03#-NS6Iz!9@IIJOnK0Uw<4HteTd(g4?S*ouGfE5GU?7?t~WF^z$Ip%wC69= z4byY6jXhE~%u?O39fjZ=7)RDvJy0ok_0D-u+lLj-B}Ct%Qo-#NF{x{FgU7Ryt~7w^`$cchq{!K|-+VkEhC$H-B4 zA#~Zx^@V7XAGoJN9kkMqr$5qzI=M`F3GnPb*-x{P!x53vr!8eI8p5Fh2Fr0& zg;_VG3(=~P-escX^q7(>Ojm#IAJa6kBcrGjGd-jHt@lqUDD~!8fc{h0dRletUk6~3 zZiTXs3iMm@SYK|SYq&IQngg}E(XUUN4}rpFVe*;neP#;TeD-%*72sHoF>w;5OUGfQ zo?%+kART*?MtH)qun81;)U7Q&nQYxReqdUEkJG+6P=ANH4ikm*Vg`Q~B{AD==R;upV(b}I9 zps=WCha3V-Q%372JnBFk&u)}O?6H=hR+Dc@EPG4J|i5%Cj^iHj)Q7p=~KvhyO%Gd zRd)mlVA4kSX<&&GdfWQPv+dd0a1+5n1)C+Dob}`2Wd(9oHWR{_ErSTD2mqOE(m6P0 z%`k23qi3`zXUKC{!_hauE4#Y8TwPNwBwHA*W~WRqnBXJx5#oO_vL+2~c}~{Egg9c_ zq-Uf}`f<~yAc$O=rU}pqrk6o6X=Yc4^^6Qk?~y?%Onms_j1Evr&7MBS8B*pKpM;6r z& zRb$m}=3Syol+%BZ+Jvc+G<3$1Q9@wFe;Rivh(PaT@Lx)~5VR)|r3<$jnL9xQq`}}6 z=%Cg%t^1~um7rZXcVd#Z_}AZySFn_!g_tcJb!=X;hgmhWZV5 z4e$^_Gn~xalzyXiVY9l@?{o`9_&8rD#q?yB53dr8?ErUh({~u%K}blr%>HB~uY(Ck zybgw`AosN?e!40*zmzKoG1AEexW%;Q9Mzr^f!S!hgb&Kpe6&n`3>~XMz&p}Tr5Uza zz}qu?e$IdEV^nve@SSbH9b$^g-9$J|Q0B^X`hv<61T>5+ZgEV-TK2x9P83S)I1x5+ zIHwh@n&&2RE|ezcWh9I9jYk({ZiHxQNM{2Xs6S*(OxZTFDN7l%#dC35eKw>SsC<^W?3(= z^?3HS%45S=dcuUG(>T0!Wr_0yE#pgfCB5jvMABd3aQec%h!swVuToICvvk;Vb)od( z8&r0$nH+s+>1?Kv#iieV<+s)tH`B;?V`VvKjExDQ`ce@`R$_>|!Xwhai3iJTaX9Kd zN8x{T^PXe5rQT)6-lI4qj~c=$nnz<+#qVWSX40&HEv(0u^Gd>$>);MIyXx1g_3e+#bvyMttSO1`W>k)oe!{;G06H}>r5K(O( zP`mP5908NF$DrTQ1*|Sa(63ZS1+34h>;!+#=<=6J*1f?Igz9AH7vHe4P2y$VhvfN~ z8)Qt|MGfA;tsl?;-28c9V7%Ra7 z0jeMeh~{J^=@=O{ZsI~Azh5g>&yB}}%0;82RQe)!r!R@y8cAQ&j%1?lk|*iQj`n}{ zoV10;&8nBS0Kb${l#uX+Q7dOO%56`;8XASn_-TPN?IZMIi zJ_=5!4%Veo`eHNr;uGLE2EkB;KNsPe!p+zVEwVqB$a-GQCRs*F6emOVp_gT_I6Fun zr0O;Q%-+2*J7|sJ#DCysl*CFxiwT~<&(m>a~f66H~b|TgWW>`8;$T~ zSeh_l;&xuvB-7Fal{A?&X<{QulNn2zQk>wuN|S?~c%vl$5`gzz;uJVBT5E12?nk7r z=_Yi=u9q7}HfRr^EH4*cfQ43u1W8JbfXY>6eqXOPn-!Lu^HW~eows-JWBGr!0wCq4 zenBx!EyJ|q>bi{5%dos})mb$lfP%!o9y6@Z@X?OLKKP{9Md;f91VxvnIK5CH8%OMfqEgt@{w~ zFVxBrOG~7s)yhlFFgC)X96=77IMCK2^{M-Mcx8MBnGqwF@Q`=8I`_Y%5{ou#WZuyae3EP*y-lBdu>$d3ABL%a4 z`K#^kf8MEwd3^p7LU{L=|M~M5{qDc~^~m!IJYwdpOHhmC#<^k+WB?W^+HfA2SX?k8J)_%rpeIr;6MHvCNg_tWXW z>XA0}Q2l@7-(Qvep`R{)Gyivo@S@ctJCx=TRR)GPUcLXNwvQ0Q=KLeZ{3iC_chvi@ z>OmE!20=k>L$7~{4{C^wgam5Eq(a`Ddg$m#$T!<();_c`$q-WdSJLz=J(}GkE9=c> zf88S}hX+G=80Ukh2NS*Bc#8+MNe^WLbp?|j%35RbY|alGl1tl*aS!^<{Lad34)1c(~*ADQRC4sw&QCbAL4^+-{tI5YTfBGWJO;| zQ{56tY$)x+CXH_w9!hEDYwc^>CD+?7f2Dy@)2|I>%CDl-Ihzi%pq@Ux% zQ1$d^thuAnv!p4UrVnc4!$(`uUV9lj9t~Ni=-(tzDdXyQVmevvdh}P+0@R(Te=V#I zu|B(YGCb=eLGtt?eR67x`sB5T)h|lJR#BBU?WsOW`%nF%jvlg>+CDu9qVZzCqo!2J z0*|K#3L%T#F8D`|j-c60HO#?HU+C!GeQcT>L##jQl%j7fv~=*=LiG`q(DzLCacZMe z5AUs{s5v+4WIjjj*hYGD_?s7fGed#}n`S(@bNB|#H#8nbMjfA^>Mhd#8@ z{XTv4i4i<KF%Tt#sb+iL4SEZDp(6$%SL?*eLRiYkgNxzarnqQ*jtJELb zpLy12zKFTBNw4mxEjv6gf44xHkdnb{daF?k5z(OPXK)Xv9NQ@SL;ExQfWJO~Zq%mq~e)S>oU$Z~7Kf{mO>r>ZCb+`j9Td5qYzM`2;f889l#4fu6_cV6~ z%~z>Ev_B(!r^{2bV0rScJ~zf4wSR(FnZul1!uB8++Ft zo8;~`wjrF|z`IJ_AMy5KLelnZ)b4O69YkM)21+ug2M@^8gBb>M(^YR6OxLb0Sxaw> zqd|A5FW6@?i4s|#%7XY>T4^rtJpV@ZlWe*;lW%W1&<8Ha(Pt+}D^KSEn^ z3xNxhdu1VTWpeULAVcKD#lQ=X`{Niyp-Dz$+k-qV3qX(DO-4hodKa!;uvyZQG@#AN#T&wgd6`y_vi%QDKWPJaP z(1d8>e<$_si4HKjQ6dD4YW#%mHDPG#Js0G0`94Gv?5j zn?=puo~2(?bm7xAJFhUV_@Vzui9juiIzec3y!0|F=Kc|MiE)5UBS=tNWe# z`VYU^_XWu$q@LNHm^!@$kxp$WU{i)UnQ31`VXdj0A5WaXu?QA=$4dB&1#Jj${!EeMn5w z(a#u~Ci})4h?-YB*t zsRK!&3dve*sT<{^5%En}T+;LR*{>%oe>T9f#3LY6U3aH2mAqy2h<6I47t<0`r0Aq7iJDWLhIYHA&`k0l*JaN>jPXs2b66P%1b zIKk=H6P)O@IpACc_a!QCF?FU`=dZp=s?n>k5a=IPxfEoOGn`~jXzJw*CLJn>7l!p=B;)5-Twg++c#-L6_3#>xn9 z_nWI!AR#M|RD#j>0Do9XVj6no=dqk~b>q;eQRcoBVuQY2mi_&4!S5>Jr zwfEVR19(@}tYZ6Kvr|48bfie8e@^Q%9^MI#R2y{;ViBFiauDyTU`B9dj^SZrN*066 zrcNnfS5xPrza=gcQ3Da<-MNVo3^);{DQU6|3)eX`&kh}Fz9)r5aL63BHhB@`dDixq zS=%qO_A$O!!GM$M-NvV@;L|ER#enL7dymFsILD(FLZ@ z)vxEzMH;a`!SKf-5xHPsmr-}8x!bweRx~~7oEVA55zpEUkZELD($@3H>eo|Ob}q-A zey(DRL^bK(#-b6Z9YbIDe`NtqN=L&+nUqeyo|KM;9ntCMBDM%jlSXrn_)3NtQQsHf z;7v$URP#65ujgrj=Efb#RcPUv9N987c<4S29FfbP-wv z9+xo=EVUiYiYlkJI>f@8pwO_Gzs`O=L7`^amGCOIh)grWt%bzWe>-De)?on-EYU_g zf1mw&!jkeJpNrHY^Gt_b*lf0dy~~h9gcF$HPh)7m9$*3us(7GTFTV#@nrIVDPRBTRtaB~RnPS}jxe?fhXnf2~3b&)kEL@z_#qE66jA!}nqcoUM~XH&9YPDrvD>z#|x zB5+HK7d7YJKr^ZcZs4H)W9N^vUk@bF9P8qX%p!3%e=yTzX^?p6vn-;UfLJpRoPIgM za6TRiWtCZEPDzhwO>$}2t4u8*;hV5HYO)zjzn;+eJS}oAa*NC<9WgV^B_2*X#s?YW zgvXP5nDO-M2~V&?A+M5)#3^w|#>Y(U0RO+-Pff5OZ#T~(xX83J;q;Dno5#_&zO~I_ zyxDP9f3Zd8m7XR~OjZD6ebdA&#GAv3ky}J!NyIcnY6rA}^0p5%7UMlF!^kZnucV37 zE8;)YK?p-Tk;Oy2r+injMP!wMOdH}%n|ICV5dq$vl2*AzVwFIPE>56m82cb1JkQhS zGEtjlrcTqE>sf3OIc3DEzanZ(_-R&e-cl^Yf14XBtJoql%82hsbP2?rcwdwuo_Srq z@VYGWI!1Dl8KomqA7}=khE&Rob>S7|U4v<1ibbN79!?CdI+R7Ur&Q6;B?oq(t43v zeO3OcYY1bqv7+py(%4Vw zOR*PNh39r~FV&!}o zCmdO&L8o=sSRDKvYQp@ujgqFaw6`jMuR#q&$F`%ODV7MxzL0wYRb=8ydJFRz&DJ!P z7!m#fRl}2Il%x(#-l4MhI#jQ8o0VHhQ!3%x*Lg3%ipXpc!@#1N4wT*W?IorXe+OQr zVY(ls&dJ#wFMF@!bySIMl-k2ZWnT%tAS*KUO`!0k)Ex*&?kh-aEuPrIE4CuCV^4E3 zO?m)q_dR;H7EkNE3$7xvXLz=G@NcZMdcQkcFm-sIR$!nNnPBIKGd_Dji}AHu%o=w) z3eu@|Qn!~IL)XS_l+d;ASdPAce=0Kl4$vJsZq8PZ+aOlp?li>+E)oOBSR+c9Fvh8a z=OkmiH+`>Si$uT!)<->keg~gn9gK|dF0EMw7ny-axKY%J#qCSf9Kti(^B1<~7q%Cs z<>-spA~SG@z!&(SGSj_kc4jw)5i5 zWo<@WnptHQiFSvFqlU8>YxI){?ofgr>+7SW&PadM^&oq%tHOZOwWCxPr^Ndb^#xYp znJ;$kG`xGgs?PjT!hg(%XqG14ZIn8|!JyBF?7co8hM1cgXeIVh&Ax_xhE;gx3lE`` zLiKhhL$M7t7XOYlTAC=gQMU2Yu=4jBRw6?*+D2*W{%c?6KI19^b0?h;pcH5xNme)Q z!B$|$8hoH1Mo9!*`;L{r*OR_4D*}J7lj|=n0r!&`Fd%>Q(>6-ytaUqF_FjkUNc=WR zOH?ZzYwj0lMP~1wI;V>`oJ3Tr?@)8Kcw+Oe$cjYX4UtMiBbeZ7m-<>BSBK~I-4$4o z$$Nen1@;fT%J%b{TqT~T7#L_pB<}%d*+_ZdtQrvtC)eV6T6cw3L9Uj6EZ3Sf3K52GA03DlV&n40p638GGPJUllC&D0lkx)Gb{mnlgTqa z0q>ItG%EtX*ON6gDFNS;Uop zxQ}%B3B#ch;h3S*!Z(YkPm)`Zn8gW!cxYq1(n z++16s716lxAVLZsgD~0rChKT*|96o#x&~f$KLDhm#eF6FKD8%Z82!EK#IWiv3coWkP*+N*S0vnF#SV})uTiQ?H2dm zyp2-V1MtJWcfBXO_nLj*<^rrp=1NAyP;6ns;orzLc;0^k$%?D6=1M>|2=oD*xNN_1 z=nt)$a$WJlC5rBZn5PE^CT5|8hWybGctSt^KdkA;6?4cew= ze6!ehm^y#ux(dFRNvmU}oWIuzme-YEwSqK_&->oa1y+$v6<0I2SxlQIx^@_vt-yiR z+BUL#Nw<&D&)@4<6_TTjGGrXL?Ylb{U`1g12ntsBxR5oeE~-jZIM6DkE23VejC4L7 zEqkw_RZw8tD5J*6>-#$wU_~-k5FHD_YKiO`8{2=8RJIB`T9s<(xTDNk9c!cfy^c0& zQ@tvBNpnjK#|F)bt&K7pw5Rwq<}7(PYsGv1Ji_EIgd|*-DjoUe9Ha z713lt94;gV0$kl!I`~RFOD(Lxifpg|C3P^ul}+xUzu7uWx>mQaA+3Z`HACGp_lCyx z+bDmbal^6GbAeVQiv?O490&*JwU+Df#67b?M1epoaMbZoq|bF2U4a$Bpb1Ep zfE`$D=(MdjwgQv0bN6T7-I<4{^YzAXb9UP(ZH=(eHZ~_+e;U8cOY0$6~P(^Rq^GvmaE~GA>f}@5W{v9SR)+E zhu0fp?J#Yl#M+_R_mM7$ie$+|ER7LeGRQz=U%~#274H>NkxUuVN*?)KNsYQ3+Q?_Y zA4JBO>fo`LDaOMIgumCBVvfxtxR-xXneG|90xN ztqJW6Yc1ko8?Ug6WXd20n@3VZs$Ta9VeubO4V-gFNfSieM;3ptL+!p5x`%%==<00+ zRd@rSW;<{QdaY@!6Hhl1{AV@lf~v>{4C1(H8VYnf=^lrHufg*=EQ77cHVneO!3fed z8E&VzY(1W*5EyJlGGoXT)-)p~G^VLu_F^@jvbVT^DuMxH5mA7QVsud(QqPgY!hhoC z*A-b24H(SB^r4X~R(JKw*I|Dq1%4YPCVoGWg7w}Y1#TN9NWs1{b46BU0|w{LqOW?@ zz}HNy#S>4JtzSG4O<+1P5#R)X6` z2}>YMR>FF3SP6C;<**V~XhpVP5a^EhFIbR4Y$LfElY3zI0Nfn_hi`x2^~P`vblWIZ z&EH!Mn#Ce1lI;>O=TTi;jWg2JD?DR^qp8d#cmtd}f;P>cpqHkGy6zo;z1KTJRLX3l zG=}D}eT(UWtMDcO9Fyv5C4pAawT)-C0>?O`Imd3JbTwLHA7|{nKF;dG{3va(y=50% zg*O4r)P&Aj_ceb;wh9Mh#BuCxl+;g- z@x|ZkXh|jNZIlQ!+)bPXSdpw5Qn5(X0s4|^1QDqU2U@LbB%{5|DIL*{_id?;47_aej2$7&+C5-v?AIj8jg}+l>qFmmTU3Ey{;8nk!=$M z7Yqg|e#olSUarQhI?XmpsK+#^&h_4*I$hpIc_>c?S&>bY@JI>qGITwO8|6woZ(?Gg z712mRltEKde8Nr-VkMq<3TpvYL=(l~=)Xs%O&rn5d;WiBrV=xuGTSIYn$kq5)_a3c znQfFHRDHW^g;rz}1u?#%TQqXi^~9TeHD_k9 z(zND&kd(tdFTKWXlvrN1)6(mDZ!Ep0ZItl{i@AT|Dw0itqjU&_j&QCfH-~Pv2G3jC zUV#;X)FbmwhEm32xXEqqWvei48@TN#urV<946ZlEhJo2eIkpT|NJX+p5E4T+K5?D70HA_)I`70AlIFsu;8Dzx3yp@67h!t6g6syWewE=!mRM0xCOJqDx&Fv zz>0rZOH2qF)Xxob6?o?U%N1CW?H0sVbir*%IcsC6zk1!H-roi=0FLJg?XlRuNdg$I_Ui9*Tcs z0N%8CUR(txhhw)LMc4x(us38^<70#9f~W`vOMC=_5OfA>rP?|X`ZG?_1-uz6Sh&}z>M2Bhc37Z$NMva zJX%md({nd)*q?X`>5QmI1`8rQ(%hPm^C4Lt`4i8_ESQRDtpsW{xJV>Kc zc(8O}S&G$o;=z~|Tair_@KkI(X@t3$h}bDhT^%O#PMqUS+Wo_d=4b21XGqkpRmuzA17?)TWY%8Q&cnPpBIvB0abV+@30a=@5Qzj(B$OhvLA5RefMvr0n^0Ri`uSVSWMdy{-bJpy`9v%5r~0Rdu@Tt*%N zUXy!9Jp#Yile$JK0lkyuMty(YU74G6J%bKghr1KeDz}IX9HIB%Gh`6ZAcR*S-kYFT zu|=ZS27&cL3tDK%uyiuQyToM`Tx5cs2t{@^(WoK{X=RYNDcvf$$Q&Cn^-QQ94AfOp zkWt=cx{KT*GHm2baGu@<-zSB@86w7k#w!FV9dP#RfydEc+xI3G@?3ue8Vi>^#K+q_ zPFVLfJG`6*Cr!;cBzqxt9vW!E(Jm(#RLw9}_%9-hKw;}>p+12pX@1mB0yljhsEq0R zw9D!HIPA6`Yy20PMIzV{kqoIfIqV?yn93}=spnQ#x2fm$>uDfWhqq&$|01?X6r1`7 zsVmoxZnxklEWT+>S5JT3F{In&1jHM&ciL5EVfiT>-U1>X*g(3?ww+%N+dUilDzQk+ zIZ+=1@2(lFy$o&A{a9R`b$`E}^sPBG$||(*{MZwvM(`>O>ZBXv;J{5dFq|eB?Q%j< zS1;ARw0@FU1ojw6i=fzTf}uU8j4Rj*OY%5gKdIQLorV{`t>{^ zTffhetIQ&?R>W-cw&tozQTK4d47z(QvdSzxqf7Qd-PzYKNV1zEXTja;AdAc*P__&L z4zda$R0a{<%@MOoEfS?fL^UKcLa(+y%lK}t!!1IKOcx#Tc|>h69;Rp0pe(?9YLyXN zM7pRE7EL469O-}c;8c_$-kon4$wj7&2z&;omYP)$<4#0*Zzg8s7MU@68nv-1RT{in zMtGB+tWt}}7A@kJQ7d4%EGE~Fo-zpUayCY8k=Y`Glp#nFk%*9v7(*P!frp0b$$MqL z9(YWhO#2jYrGbTKWH=mQ5PuX}--Pz*NXx*fEF@%zxgLLJbym;cs9z2g7G!IzA)h4{ zi7ZAej!@mfdEF5(jKMa2AdPP}eV~3leIPQz_D{HqEfQa(@f$T+1H>s1US5JXAxZHx zr1$FyNyvw}u?j6bBO@b8JNkcHOGV!)?uzc7)w4(}5?M@$qDX&7ojD=QlZb5s0+Vh6 z(k>?;b+v!1Q^-YV5%?kv*Ni0wq^0nxXUX8uP1@GR)0DVhPf$`mc*G*L2)qn#S5=h~ zW}2|tax9B)f)e~ZaQfxE6=V*3eHB`G(wM0;$Escsn%|1pCKyT2w5eZCFpNKNUL_WR zF(Rav@-_dRYi0)rY{r0P%^5_yo>An(Ic^bI1WJDv5iN^8FLh2E+d(M?+P$V&1r~`d zA{rK!9i=?yad{XsYh1uUUw9 zXB>YIFr|XNsz^OrYQQtQS?Gm4m4M} zMJ9}i22FM~acau1AUxBybfIlYq-~7cA`?bLh{8DwnuB%9y=@qb0}Xu^JGab!JI;KO& zDn!ST7L zSCK^`iZo&~ayQ`pN%P&EpN`sqs7molOUL|+mvQ=PjSm@Gak~T!d;w(aq^=&8Wq%QmP z?atwaMVW$hEowjnWd&Iw_{b;0;uLnJ5cvATI9bmb<=gxVldYFUY}xp z-f;Dxul1P%R3omD^<{lD86-56c+dB2zcdwj*l^myFo&Xl6;k%l{Gx2$5xS#U3p{Ym zFj|wCO0?}CYT4g1m!YIwp%xu(+~tooeroEZfZoCk<^FnSVX%rFKMGOu53gTEZil|` zB37H^a!!>k|M?U?|*XhKyhfFQ_^~81rKhaM(i}biRp)A+RrJ`W6 zyb>z$2-O-+{--|t@7A&^2xvAdtwsRIT)aGt+U=>5quGo2-E5lLn$*m*RCrG7uTHDRZZskyMR$5Yd z=Fb&=l|L$}vKhH9pS)q@aZImB`*p)9Aufh`evyMMB~ogZW?Y*o+svZ&Pe8)9)gs`; z>L%Y5-rZAStdX?yxjybN=S=pXXb7#} z=dN)m{CHVM{;W-fV<_bn%L25YD4n4HFWk2Fl38*A?w7G+1R`)YKAfg9v#m!|M0dhZ zQ7q`*i}*}ebo88u9G?H!JR!zf8i*?E74CVS-#72DoMD6Aag7du*SXiJFwK^A+6-fX zcUm;dmZG-=+|*p%Rmj_eYI|E zruh%#$(P+FGRSV3E6FD{Zl*6pnU|`OK`+Pdxs)-x*i~tD0_$_=y-Q76pyZj;B z9&ygjbZw2tY{w%UuDjOM3p*ztvFkR@&0(dbFvX6p@>;H#pi?)5?dmpJ(d8K_L#fk( zvi!m}~)PEKhA#aHa9Y+v{4K=l3#Ma$_yx5`suK*+56 zWZ7j==@(MPgyh?S{hzxv@X!|V7T)Vwzt9=6+F;pz6_Vtm{%ca&KWcmUiBqZ+0Qlfp z#`@WxwBSUH;8bhbRm_eyu^F)KzF9okTHIQNu()|gmX3)$LcwGi(eVacbC#(YZSF2A z3{(@vK5=(8I{tdIMbX@TMt#p!q}gNXHg`3H8zK5B^_bs?FN^xS`Qx24^I#p675d}1 zSpf^Xl7cL;YcF0aRXLptj?I=|z$HEMeSJit5w>7}0)l(a4({{?)%Z_75}oN(>gfY5 z*!OY1GEOFv?$7$I<9DntVBn9-Vkz~u2+WcQ z(xq4P7_Rv+szSKUY*r6Dt7@xnb)QIw{#Ngm!0w(O81~=F(9HoQ=Lr?woEfd3?&X&u zyXk*@ap2Dom{fXW<5YEeTEbF{&u*r$=DfJrb= z*vh$N-vktq^t%BiQQS0%J86B%-8FqZqwz3Fm(w!ZSsIW3xD1MfC@>#PUO7pd(^ zp#w2%zMs6$(kO13P`pU!KrUIAU;A|aj}@Z3=|<61{yEo3GcVZt^+ zc(Wp%3rj85J3~myETuZx6kBE2J4k}!R-s&$Q)Ot~} z%`I8W6$g|vd6xTHyK7aZ4xeKdKC4Ff+ScRh+!45#mn?|WAbP{hh8ThT*l(25KJ1@z#nZ zks`S+Blt_9&en{!D{a;~&tXCuMB)#0sCnLJk4f!|;=xp8M=At=N?$|<-fLhwysQ7D z@ja_RmTL4qZkWC~MYHmU#Fw}^yXSTx9bo?CRz){m>BPl+8@ggB<=scPzKP1PhqgwY zvc5pA(!Z+v4Kq6xvBSyjix-9ohD(`o2(e&{XBlS5U&1!Vl3G>lW4mfZ{5sOn-~4Md zm9*Pu&A;t}oyoSPC+eg2KTg8K+y*qv&2pH_XMc3LAhFN1Ylp!wwX-|rUxzz1=K~gJ z9`znC9!art4Z;&4`Z>5ukgwIe)`jKBE(4?UUYbu}Sk@XtyYFByfd0#GyI74{ET!X4 zkU}7W<4pduuibzQv0}T2uzwe;wk@JB1!*C_XlvUuXr7y{eG*mI<@sj`$ zci7tbG&Y;QE)`Kasx-Uo<0Nlq34rO{hE+{%Fa=9b;RbEOfY3bs;{4Glrq%r`roO}o zIT>Qqa|zVPc}6h-jskZRnS+t{J$SJG-SL4+iY-PrwgAFS)^F4(U$=0RBegR}5@j1Rd2 zGQH}fyrRk@c?!4QRMAW%;4*IWUQ=R(j67mlX?r@;_1cs)tDgZ6(3Ye*M$2{1tCEh- zk99?j70?}Spx!amJvoVg1^Zk#xR?|>E4(POim}YWjj!2mD~xMiVA@?>2`(R~)Aj7v zFz?86R>Yyb#N3nCYerH!Ze^&4TzSN8jJS*bd0Y_Ei`YAO52uZJ@QG~?qm=EaqiE5; zGp4lf%AGm5VkiU;7Dm7)BciH8|xjAtS|A2Ohl=qR2C85;VGRD z`SMh~1Bn@nbkLL~(@}8dE3tz`c$4=9q$%7_V5^N+e{SQ8_;`+Fv}ZFp)$gGY0SLRM z^+jXmq9L63ZvSgf)Jsp)RXvAuyuO6kY00ZLv&8X0YHKnu5|j!3UBV5if1bJcoXR^y zTu0|i{5|iMBSp79H@e*w*}F(fLh~K9^bSreYbLgvztlxpdZvdvwG7UfdaylF#!_7E z{$<6edUek4iiOPFgwj-Be9!=12Sj~yyogw;BQk$|UMii{eDRw7)7U}{`*;NJujyLz zEQa%p{#adr>@e?4OYE)k#m1$~dv>

tLNy> zxu8p;JD9j;1_jLD?cDkC>#>iVyQuv&tZoe<$G{oeQQh4q{iqVse06i270k+m=n5Hk zBA4$K{bb1F#WW`Yn)1{|nObc2`4X-SJTr!DcT&6MCv?uN^rQ?=R{D}fGjn<}((0cF zeVl&jhtYDMMOKEp6Jmp?QYAKpKZ6 z2mxs_|69GC=wVyJEd&1{kd4Gd`{Os|pdu3HZxtx+?N(+Ciyx2RiySM)LWNo;e(H$8 zIJ6S&o~-|Yo}_F&R%xD;`-79k_q+Xmdhq`d%yJhsE1T(zygN%^1_OWItg4m-tm?(} zcUV2X>PhiB5=io66&C+uBX&8~d*Rvdfg%wv*`F|*H%EvuE@5?W;tt4@Z2p1~8T&kU z%7&zY*GE0b6Yw9(t6a|a`|F@)S!jI&i>x^orYG5>B))QpStkBWH2(6CvKcDriRN;^ zOv${;)mKPfzK;)Li(NJLD5Wr4(S3F4`TS8%L}DF9bmZbCMoQvC_hxSkzo=5mPb2xo#G3{lgDju7FzV>v+1F{D2b;1I0xtZ^AWBej@YFKP?9X1Sew$ zUwi2r5XLZqSLxL=X;6a0VIf_ufF5c2Ur6F&2!;q*+{`}2M7e4UJJwlUU@|HJ*RMxI%b>gTrQ zb@!73#l75E*<7RK8C@pu6qB!IgVx;s))7qF(iqIi>&}f0@wCV=Luvu{Mlz=A)_=o& z9h4J!ZR2N}!;IPfme`OhC2Zvq5EKcSt5H+Nk2L)bo(Fu2|{$)`w+zS?XI z8Eo%_kW}q4nA}c|&OY7DF) z0ol`Y?g}VN$!yv!9Q)DUuH1~o$^FDHOVxXHSFb}_U^E!o0SNZQ>T$Ej6ku%picHcb zTPtCwltK97qsqkjnKbtf>_x`O7r9JJ9t89Qg}d}33=~(G-1xpj;lmv;fad(w=R=APj&zS>t3h*M7Z6$OYxl()^i=0yH?j3o? zKzkd)F6b&TB6&zOa#Eg!($o3^hI`G$4kCoLq$2<64-8|Rp?#2Y#w4U>N5Q&ICracD zjbR^JX>&~0HQ6U^(?tbj%Qt`26G=x-G3YTumrE$KrSlxdacVUwUq}SHCn}j5 z08LZ7^l&Q$yY>A3KVE#kq!D_XYQ{^NCgJgItT^QkPMOw>e`HJkRC#%O#7AAEGN{rg zX^DBS8{`Di!a3fLYr^oS+$#nlc!3I1>>KW4boPhnwQRU(t05oCXA@Qf1jnZrNPO+U z{-7)?o!G9`@w#g-TtA%4zn8}IL*hyb7~|}eVZu^_!HvPptvxTlkd^ZCm9@pym*7ro zaia~m(Ih0?UTp+<%XmeI+;`;U*l+2Ka(!M+lNCWxL$nd!zeO)WcYb0-jiPXxZP9Rz zN2Jbs-T(4sp**J|`m!qB1gv9a$WUA_5m(Skb3jFsr1wJSX&$he2lUpr`NRX@lz4zK zLi>2QMgSJH<{KC4+*c4Q}ge4bSP9!;qF+QUvgY8s9b)~i3_N;dgXpmDfm*p(K9m~o{Dq6-sLp3#{iR= zLnmBabvp_rLG5`>_>BfH5UAVKE>dfL4S%P^q~BKkK=Kr1&FwgBi?3u1R(QFA>g?fv zze1Z@@tT+PD9N+%$Hc<_fPAuy0nN*Z!h2jM& zE_6jPDH@LG>LIRjG9|q5)xEv=Kx4IKXsSRx-Ir;WNi{cZ|2tQuT8g`1w8d`q?KFN(>s0VDMoBlYpN4j<5amW#l+PYtP^ zOLhIuwx-q@4UT--FkWKLXf~JMWToHUdp)U-(KihNepvDSvi+heG{sLuWMLOY3|rt_ zYLB(4j{*(9O;B=IEz&k>1ZLnqBtcE<=G4?o5wxPiAz#X$jD@bGF*IMzw*1#Vd=_F! zmJnCOFq*mqHZ)H9(nO@c)wG;*E}pz78jM)w1q)Q%XPa^dnFnbRC?xU?2p6`24xw2OxV_HWp>it9-p({d;RFE0P$!h%x&1!pTFg5T274X z-qA9WHQ&%MzW6}AMwC8R{O1kWD+2>2SmoW=su^)QAx%rG^uLGNuv`>t097&P61NhH z;IU~Q4)36De7EYmiljQlNp$M#9tStm(!P3WNwWV9>mwRL97 zn6nbU90Xc}tDSZX{AS62$>BOd7)pJl&O@LTanjT+#7k3$KoGAPP3ily_F^QxJb(Bk z>q%%LYW-98{83Y<)e}htI=hd-W zZNnWAMQ2DJkCOA_4eVgBkVjdwYCoiQ&7R&brN%?|niIkPdY?2Xr-?r?)P zKgVb?TaFxAul!Gs8qco^lmk9=Q$_ps7!thoxzH0Eg#x1qQUV~%m4C5qgb~^!xDLt3 ztATjGoB(r`vuZO~82vavFvdPR7WWc#oPSft7{`^+hDS;@8e%{xK2Bummp<-PoP3w9 zyc!3yq`#>>?4#I3de>p+Qy^eg$}z-=V=Z>9;Dn}+cWh#2Mnfkzp{Ni8g+2#R z*`-jBt%=6wbV*Oo2scdK50Qi$L6VhW)V5Z7*6-2Q#w8~tE#vPC3*iMmH4g@e^J7os zwc40hgTs4cqX}zM5Pt9?+8PI|V};EV0GlK)mD9D|CX5Gc$?AyrUcQBHn0g-~56Jqs zo!!-_{9CXuRt@UF&K%gK(iuxw8>-8Q#I)7gKK$Rr*$oQ~iz;eMO0Mf<3EsA+5Be>J z5Rz|BDhv*@U5qU&*Ph@HT!3?pjIM>V9(w1v$4KzCX^{tHM`%*Q!$pDFPahJ48b!Z( z0se0B`sSt)B?P>08ykJ3@(C(yh8z0S`0?6UT!m1H^j{8-Cx+?Hm|d!ZqGXtuesf|W z%+}qWW15Su>F?<8;HBw6B;Zu3w?9ar2@=snUEF@!gMe+0MajOekxQ0#5HF?^ZiQvW z%m{+enIM6J^*Kbp3m_90K`A|sLAbc3-vjM&8+iLhYgk=WV;KK71?V>Gmzu`QSV&3m z#kJ{h?{fj0T(cGWP6U*3HNqgA1{|E5+w z*7OTO-w$#o*dmnTsJ~)sy19D8rB{fTRlRvNFW_7dQe#Nj(@OgNkiK5Vo}Cg-C3c8& z8&gR)#B)*91sp-2A}|QFk=@N*pYV9I`#l#u+>&**8^66I$@gLm$C%o`q{4VQCivMT zFh1280up%fvJhyg5}lOrVxD3Qw7)MW&yu$B5#}P1OzG(|>^9p2#JnL*23(-~Ul32^ z!n$!Hw@#(k;I!*T;Phl$l@KLEh05t668k}El1$W)FNvr9zgsQi3pV|w$Uk%DpAu&B zg*UBoX?05$#fL|jh@mQ0{~BQ*$~a92wNF5;Of>P1p8cb_fh&8@%33;@Vg4mk0c)iJ z^MlvSEs1T-V?iI({7F2fGkq|-20?JV6KO2$4dbp3Iv_#kuP^` zu0TTJV}T?cN+rEGZBbz&%EhhC73k2~e1JApCo79b5Ni**+;zFqaQxpQi3%yTD5({_ zuzs(nwg>v=aCt1$67`|^X${FYq>&ZT6ud}nuO6udoQCq!Q*XtywAu|#IW<-2>`thqUyhy#JrC!S6Tf1aAR5Cp8sPu`z-E*%1>~kd_P<2e-qb+Ofb)b*rwG!Vs zf#ZSw0COwL)?&J!zta?IEhZe~grIez3Y1B>HZ5I%h2LR+tiWQN|3ZQp5?#n4Qj`wl zNimR)3@A>pflU*0K8IM;{mHbC|HtR`_euE!Zuqh`LcZebxgK7g108Bk|nuvY;Kc69a;_jcRF}M5y z+IdN5%(MbglxSmm_iZ0mXT0r`)U@&qh}&@Z*Pg0$wdLk4_%Im0$*E$&$0*^@m_*H3e9Cbz$+*AT1eWET9QoRjKkR2 zI&w!}CnJGXwxm33GSz*fBj)KTYn!b5f&SpT6NBTtw)GdGs8bpbP&^h`1Yn{n5DmN4 z2%Xtq30Y08${&D08&-tGCkc|L5;B9b%I>=T)1nrge!M^lNZWh7XX|CTZq7wzy!BNQ zk*K+zx_Wi0B(*COhJ%{B<^0OaBDa%7GUBEF(Zg%q@=BQfk<6}?tZR<9%M4<19(-ZXO z;Ubjrs~YpPS{X>;FDwVLC9UI!^L@Rv6YyfgiYAA_WxQbtxc0x4j#9=)AXb87R?uoK zXN0-1?8pY4r(vof171e)*=P%YaI1h7oc~tim*Lj<)AI#@p7~6gmhzg1{%_YFNUbzP z296mnye8X6(Qt-+4jZNI94jx9jEa^kw z(G6?UEb>RhTQk6|1ABL9G(r*@*NG@r5W<Rt?gbqu;e)nIFxF>3Lop-n_(UiZUM1=0IXF$< zDC5r4^NwmK#-o5tJFTDR0H1vRV?F@FZP&<4zh<${z!u(TMqQN;CQlzKn$kDr_43pT z!^@R4PI6AjO=Gu*8^u{2j1Ki2%XWTVmntm_T-{T;HOv;?)NY7Lz~NHK?m+Q_(u7sa zBwGknvb#1}Ja%n_0}R>d~Trkw>h+XXj}8+ZzU6)js%0&))%J;mp#q{j9gP5?STWawKaqH@gF9cLV;gxXfLibshXHlS zQ}0gBG_@r^%!acCc~aq|*l|F#0(F{Nz`tah%v2t==jv^9%k86Tt`Ob_l@FN{6BeCo zCSrT!LM}_`?WPP&iq|OwX0`+zvPQ9#(_6uz^#Q zs{*;A$4!@4=6#p}@}bAD1tKV1Hc|3Ss=9qc12aFl^gsnq%*?L+r_Y^v^a7NVoaNYr z^9Wq40$Y5?{EB-jfhsd7Rp1`=XIz=wU3&IOTrc7pUidPhds_$*Vgd*6fu-OTwjuPS z1r5mWALeq!Fx0-)`R?ccDa!l*-ap`~ZKl~vwBa1yvp8>`i}tFc)t3LzCw4Q z#P5j+)kd z-lz(+*9!k;&+n)%f0)}vlrPyaPW5S0TC1H#Z2Qo{3<#5Tj1R@k@(ML@u-=e!1Z|DLZZ)S7GKHsWgdJ{OI^zD z!Zm$nZXG8z{G>Es#gy!oiHR~<$k6m0_!EOW#kpFlh)6Bu5v~?A9_*c%_WWOvGvB04=T6HHi3Tl4xE%41HA@|`P+o&6=os?VuDWyRBKNTsG zFf4(9Ndx_T+Um?oaY+{tP$I1WixWBgk}%ux$;n38MjBFa9sWuoTG-M`wyqIJa)Sal z4dL6E5HR=c;vcdosvU`+{;ovWq8Upmqf66R()wqzA>;hGP3s=W-~PRiNW3m4(n_c3i|zntBz+@C#u3id`yH0{#L3zP${^I3$S zfbKZ1inr*3VWZa;GL`!CQc6K7;b+^YR?oMKADz#S8AdPLO`Q+Vq4q_dk70Btt(~ur zE5Q9Wy71c*%Uh&jZn4%}xD@N~q0w85V0;n$bAFa`X%kS@X1WG7DB2`^H3>XEM4?@* zc2-p9S=Mwbu*|;0Uwa|EQVh-)J@B#td?AkW6|$)K9juqScj*}#6kqB?YHJ9WetwVtv=0`N5!?RaIht51-K;uX7?5)zx za1hmM#{|y0y^-IonpFHjNXbM-bu^`i!fjlohxol&CDZ0(CmV0T9TPB1Is|)hI`c4d zAl~YXk}iz>m?(6;;q=?2@Mz&uZ+pb(@!tdK_^dZd@y%@Jz;1TE&)ea}$_s;{QipXE zMT_w$7mxWC0{BX=iul&`Hx<&07)G@0MqZ@=$AujiFHSg+LxDf!|!X zH!t@{oUe7Fsr-K>ls)ThOq~;^&5=$g|(%LSkHePuS}LN#eM|GkpFQye#^co zd?CBBxL=(9;RaL!CQpiwSPE~#N(4$AztraRU{HuZNGj~TZY#a?^yL%3{>!*C;t_}A z%;0@{a(W9XUbb|0IH+_1HmugSlE>%#-_Wz4&QI~z1CXKC+xHAl$wvZ=*DuItJfH6t zJlpg)r@hfoMcWKIpN5}=U$?EN&jS7gyu@`rQ^W94P_20Yl)K9nr`8LXH=ZPHa&e7; zj5-v&npwhs(ONp&HjLN?a#oIsEioX92XQ(B_gq>_CEL=({^7C0r6!LsjOh2}0YCXI zya?RdzGb_Z=~43xdg~L>hW&K$M(-TyzFJE zO)P~`b4bjH^6B>S*Jt1E$%853P|>i}K@6v{D6?h2_o-nqV!A1UZ5TUW=`AH@swcpb zUoFE0BfDhiUY{jv)2Um@_IKnBT0ZIOMrQ@%Yesxqv>I@S%X&?U{g%>|o396JhK5F~w??>Je-rjc_DsjHS z`I%7A4ft=sePq+9V^f6bemiFOGM@cJppLi4wt>x%Xy)MKUaHHz)uuLIwcKkhIl z7O8}3W)t^|_AO}{9;LzOMjkPfj~cVgaWQ@=0NtVVXXBUl6B6Jn?)ieJ!0{h^B0UGIOG+xe3w5AONIk93zqB&H<>3ny(NgNB`&PCWiDzw1xKB)EgiS} zZX6TQ>PT<41B%bidV_y|k%f~C_MM3kg%rHHpIo1nlD<~pMjm*E|1r7p*l0IZinq}* z0EiIOXg>-2yxow$JuRMf+AQE;mJQlzs*7yzM>QleFH1e;#?iZvKdt`u+qbk8+{>)t zN)3w)I10VE>>;n?pX3XrZ5gR|8hF>oS9ML3=)2lf>OlC@KV$6=>mvaR;ayotElSD< z6CY0k!cz#Di4c$YHH>s+C;fRqepSP~5OAyGL3ptutWnIHnHyMJ?XHBXk3Rn=%!wAZ zgO_Z^3*$}Y1$fD=Mj{jao+9A`3Vl0QUOU!aaVZ}d9@l|hBb%94AG-^0XS{bTBOO`8 zV-2b=g@m%chacPV3ap>XUhgbAJ)T!bU&97Rjc%KO2LIX@_m!Wo0m6PS_Va%R07_^o zB8#fA=QGXx((Z3(xezi0E79QktP%2wpklOTa5K+`fEdFYO!b%G3gsJeyywH z@P`?qCyRRUC=(I`RHjw2t%Yk2bgq__n_l_J=cfY)bGG7^`^e7rmGz2h!#Wl?Q6gD2 z(uq=9rp4@K^2arrLw-3g`OM-8z%|h8fU&(&O08ky7msr3#~gz1a^A{K-<#usy&rSrCT)C8a ztK8K{*bPprkH1A?QfLiRezY;>**={J_q>q=z8@_A{%)x8Txxr({E#gm05JB3r#)!0 zTLm~#D}_m@3ixiw9U43Bn(ZlWT?6dj?*6hqIk7C_w~YaOO+y#OEXVmQiAg(wJEZuO zdyW_R2Gh}S1Tm23=h;Nvvf_8O8=IH#`yWpI#rfw^F_l=T-jmjC8|?kJ|LQ-)oy!eL zY0eG))9e6O@iq$W*SlE^pm{gZE1&dHe9=-{eP^fHjX{0vGjYA|O}NyH)}#ucnOY~t zYjnrf68CB$qqyox+GMuR(US8b*_zcLxkw&W;9bDFyg#K~kzyCTqkBgFlen}jI=?w2 zdy=%j>*%r97wN9YMyyGO?0Vt*fl|AmIWxLp@%%UaeHNZ{Xz8I`psnLaQI}{I&e|N= zNhwd*_)Yg`HTQpeMaF3O9(JSc@YpU%LYl!nd(X>k zVa87oSN`1^o}#~Fz!6V}5pYG?89il+4A9l>61DKvm(J4k_U{uR?>+V| zXs|qU^!t`PC{~&G%}dyTe0E>djwJ$AVNB1Dh`PL-C?JTF|?N z?XR=bW$vnUFB*SpP^*9EP<@Mf`^s#-uHVGDjq>9G1J%yOnFz`_TyN*T?3<)%XXHkO z387%5GVH+5a^U4Cdu9K%D~4Q&XlGyUEoQ~vw{NrgYI2N0QE;cTmVa}N?Bt1@K*8O0 z+sBTv?UUSu5Eh>>2LA1PQ36k+jXR*k;R2Pvn&3ImJLuJacQ@03dv75*yUFtLr|`#X z_7`1Y3CCkP_g-w*^wYLp2LGE=T=c$7sq8tZSv}IJSK!E_%ffKzV4r<_1rGI1W-6SM zTz8Q+h@9Nax#sNW_o(y^xZ0LFvhg|*PbcI9kH?3LpyVl}V#SD5QuAj^4Rk3$;30`P zt=860V&f6joatB(w)5q2WxeffJOKKR$LQ^{)l$0c&Frc0C(fzw?CI*{h|}8JuZOcn znAQm48sPJz@vYW9g=Y~l`Fs=zF;r5+A;FIsy==t+eywUPiKaJ?CLU@PQc8}()iJF` zZW`7PGT?~@v{~02r|%E?@`a|m?*`z;T8wzPZQqJ-zj|PazUrN-m3m)k3mu)OBpZ_aZEEl^=E;?`e#_a0Y0je2V<;WD~}*8qOFRz6=9PTEE9(;wQb@3y?VkG~R2KWzvoJkRg*jQ0@9wv03cPe`4Btoo2xx#5dD=8a5?wH*N=D3jOJ}g_ zbYC3H-F3P^HtPg__F^eX8=={?^A&cyVz&QPiN_x|?EBceyfzUQhjqtNp(M51nws^f zR@9q`_hNX)@`}2CfX;g(Rqat6@ff$>{&s4|DkIrNC?pD*zNV86I~ zAJ251)zK>4+biqND9HB?@c$Y=k|rTbs^nblReF%)7~+TdkZ^u7SUL`j8k`*OMkVj) zWDaf5Ni&%VzxXc4J!S=W^`kqF`_*?m?HRd+wb@#Vc?yc{PH;H~q_3{O285!@Nt&O$ zK^^=+-AA%;Gq`?~269)E0@Gor0C(zY75YBo=uKM=N)03QkT|w*Nbq%s?j* z1irzuun9U{7-*uOl()*XT?!)YX=ACi%1rq*T6Jy5XP9@s|J5!x%7)veV4Y139;`|l zi5jBrCd--LTSVQD`XTH$8CgC~rK^uy?{&LBJSuG@U{U4_9)=zD!wE?W)g%&EO2?Fo z#n2E>c>LeeBuh4raGBBD^W`+qNzUf=85u8tRz0@>Qsm`-wO^cVHk~j|Vw7H`{P%%v0e3p5c!z>GhOLh`)%X4pR|AKVDxB*V+12Zv zm0OX*OIs?=aqbj5WMBe3qHO60?zmUUx}e^>&X480cv&dIS-!CmhrP5nVd)6@qm!@U z=fP|}o4ym_s^c}yyEl;l*LjZ)&9k<2^i+Q))?QgIXDF+F$IS{ z_l!oQhyah0TyI6RQED`;=i<*yYPfI@n#C0CoQdbj0^^;M!JVoQd{h<#JQvwi^>bp| zc}FaJ0L{j@F*#owHi%N6Jjzy=39mL?vg(H<&%CQye+C2Ewn#G26|f|iD8r-x9bTF{ zi9<~?_AOB7^x|p(X>a7+wj^6qjTOR^0n3vUD^y|5YSFtoK4*~M z;**aLtWHDT$SDO-dM#@OGLR`PM~UC)t=8A8)*M?0^!R#y%L#Prk1lme-3M1uy)%;b z5QL_J%A9V@4ZO!P{`D_{;eeUIFWkZZ6v8HDI>?+kr~Q4U8Kj=E*o5ooBe7fHte1uo(+d3?V{%bTyB)g>5F z^F6*Yd`}mw{<;1c{ro8-N4{WcF#w9#Egoi${XI3PfD`M!Tg!@${i9ydjBQwAx-imA00~Ma{US@KEr zcVRj$xf(!0OTBb%N+cS&IF1|N`q1u89j%`q5>r>4vGX31XE5B+ovnWd@s#y$INF6dp8QT=7UBVvYi}bdF+x zz9=7YefS0m^;Zmz?CxeEIi_vVdlv`Km*G<#NFqB!VwXnbqYodVGWlRpfR|BB*xjJ_ z5J?G;AB%h_L7y;EwU_%LI)b}gK{0@EgaR>bR{-KJkw${MH{QVePK__hFS=3AsCB$i)Zu5L+K-V~jc#9=jvMbhRjy{6PJW1KFGzj`24)jO^HpB|J%8!^t@ZR*}-gv&Gc+Kh?Ez zh#j}ejX+{0sU4XuX(Dt7f3(?zYmJ-cuOWJzOji15tkAg!cVlM8e?1*uxZ<3A`A1UT z05AXizG;wYgvOz%#Xnv31q>7$u{)um`Dmd~iMM;oTgJ z2d?~I1>t#?+q-VmJ8`sQcz=)X*6WY9E8C~#{h{x_6c#sK|3LW@=jqqA&a$aTu$9;9 z*KEAo67kocr}ERKoc$f*o-=eD#vch- zMJrkL&?q@RpTQD!h?8aOLvn#8Ml7F&@^l84p zNBN&Z-YG~ICR*BT+qP}n?%r+Nw%z@=KjAE=Z~3~tGTF%s)%*BuCg+p zjQG!K82Ukeki3HdYXd%Ut$pH4;wi6nc0%#HPX3&XXd!5 z7$l+<-@E}!MklVT9I|&*%&IJi=rR|DhFm*U^iyM-Q;4T-IBP!`tH6H#-yIr0wrB;T&UnI{=rh3XJve52N&Ii$q26EjED%DmKqn4=mC7$%lDx>qfx8 z9qhdJ3IX0}3GV^wa%KV=nlx@>hi-vNUEb@o1P;5uQiJ%_e4Yz%-r}9tD8aJQ0T@vU zXs2`NK(vE?T2OrZ8!d0Tz(_2$d3MP;o)uJ1u+DRCyc~FHIrKI6Rea zSaN@MRRygmZWHWR5dg7*+jBBdIT5jD-b#t2S)7aamz@s(KD_T2=%I^Dwe8q4*xRkEQ|ya!JY}!M zz*9MW<0{X0814-QzwFA0<3_)A;BFvnRGD<|7aZ%2gHaNW(mzE|?BFunGKo)8)e1zq z1XD4zgfKD|Gm=dw%-U{>)cZ`Pm|KjrNzeTFl@$B@0pv`bz_8Sij5zlkkPTAma$H#3c30hwM|LowPgwr#bWKZD#X(Wg>s1RVQy zd|*w2S<8w|G`iBO8j8r!S*#Vz#hPYxUH$E7#d(3Pw{yKL#N{SNW9F;A@a6oFM(40FC1OOCpD7IF62Q{>BvO07D?te7+& z0Y4B>5yc?x+k|vYjsq?WVbe`dA;GQHjI>jc$I=Dae$$rFjik8A32gnd>`CkKk?_e| z83IaTd+u;LY^NR*$w}h*O>>11okIe>JmDpLrlGp)0aJwg*fuf(*X5^Rd5x*%VtPH{ zP#M=yno=A@+xYuAXoY$&P&w7qMme|@0Hnfk*AV`WZ7xg@SwnCfRU`S^S2Tdaury4h z6++T_E)thV_g=)Jxy+NTZ=`JKPoJi)f$e`K;h57`f8yWMI63K7PqLq)t4B{?usNFz z>h|n7qF0K#f^W08s;@?IkWm30KqQ#A@ty$<_$X>@Efu7`va&|b%;1Dna4#chfFC`q z-}iuiVAaxb$i6N=@))feMoi9pld0>c>92-^;znv0S=&qE@7ItO{0KRCaEqRnA^uql zqS=}Mgpur1CKP&M;3?}F6#FlMHDIuUr{tri5UjeSSF2Z06~`_AC%)X9BXs7(UeD*v zuKPW!nbc^$+=aN5JYFwCwG}oI5cQ=FafGPx5q0?4)6>i3tCTv;&vsc)cWA}@xOGyW z-L%=U?Vq6`Hh^B+B>9UE-p9864+H*B>tXGU6}zi;86+;x&rVmD;c9>icSU|P#WB49 zpGVc|LRKYe5V#Vw?};*B(6|!cr(<6=bVu4s3Y(7Xr@Re@kmb)b!k!m-$cpi@3h*Kl^Hu5Pjp%HR#u4*vS}7MCKJ9E zvQHPDNcO^|euyC=u z!Yqk$6}eVQ7Otd*5kyHI{o6w=oK?-r0lS8)>u^=gj(L(@Zh;~4U{Hs7^%PA*^33^V zD|p@Ywqz0tal8jN>Ti7>cUt%!nA{&hX92hTKnuZ{6s%Wd<=~L}R>~qO5Y?(yHZPg!gXZmpQmB!!uqBXL?@L0lx>ES4#DAjg4#sPKJDesD2|Lxm zsoMRU?SuMIOjUqgdv^S?rlcMfzdh@|!9{PPd`5}(JCjNqOFrY2kqXtg2YfOWm6Y5) zH~g?7b1hi?&G=k(XPxYyK>^0h$;$b^|0^q6lg?WbNWD+A`?#9uF%||nkApPyiDtBI zqI$+WYE96KbdfE?;)yxOd&|MZ0l-ZAB+1Sv#zUsn<>hBDS8EyNy&euyY&mX@Y%z|m zOWfyfc?2#P*rsHuN_RBMU`}JC*@`ZLM1+kcp6;je+?nDz5<{$6i~`}@x;2csX*m;UFyzTUDS?O2px)FMPF zw)jSJQR_t^R*`HNN+Rgr&@K|=er4~j7cri55yGk!+iNdNK|UBbN?KTo76ZPZ!CV}j zgI+|2L5V~x?O1X4rw2kz8c6KIVE`%%@4IVNvP30W9o9gqa%9O6I*-+H5XK#&`M5yv zdb{bjVPea)e+U8*tc&_$aMB{hWIY&_Sk`JaZfmY&32Z%07&)4fxPc$i5JvI%eYYDzXh-uz2xBPZEAmye%qftMH| z9m7SLIE1D&T%>Rm_Y)|BwgeCZLnbYYC`c??!5a6f=LLEfxc9ifqoFUH-7dIuC$eL9>F-M|=0%*pa6U9)0n3g^kUx ze!rZaW%`RrE?Rc2O1~}EWKK?4NgB4c34Rj+6M#+6bHLK9RVI}Hg}$)xi+&BiF74GCse>X+r93)FlpZ|X z5|5o9vt)&{4h=bL4l~sUQ|mCIKGL38VT9EliKC^w={J#XSMSM~VEqEDI~O<&xT6`uRZt8;$)(-MJG&b zo}onZ>rP!JTF_{<`5&9GSc_Qi(s$*1S$Hpyj??K6t1**W=O| z?6xx`x9j<$TVEmni?PpLh(}$SdlEegHvkQdRig5Ce|R(_+&Ryc573bfllw3YAQjFD z$e8$fW}cMri69DeQEjca85F^0;e8^t@5?q{8w|YhHnhy`preeGRAo12q^~$Iu2!kD z#+uCpO!>)r1puN0Fhgq6wcI9{vQ(F&MK5Y+On5WZgFF`t)T+wC(2^Xj|4eZ> znGER$2Y#qQAmznG~*}kI8J(J^fO|ZA- zmMmm8up@B@3>fM_3aPHILJ8r#*4v55pIN*t{`KA{qn9re&u}~q;4FNYIg!olbj9nF zY8pD(|0Ttv({X7)sboRmJT4N7R2rFb-ZNj0jls+&nIH)rg}kuv;dR;|e;5+*&DHTs z`f)wc!vVY>SPm3aT~3KTzQ@D=b!k>JDsfuG7hwSM+1>w7EuaT+e@!r9EzKn~C4GV< z0!OCY5jy%91%ZM6?HQxx<@#T^cu0}>CjHL*61^&-eK;}=8d40KQ0tIUlHq58pY7myq z;^9}<8z8P4_a|PJrKo6oJY4O4nB~Z4?;9QYl@z(l>vVv;Sm(#nJ!KTVCaOG~zZm&( zn4G{oYwc=3 z<0%!A&YQoLta(<-%49Fv-s+QfO6NY1G5X_p>;eWBf&6`~ zxRJuz`dOBpOqq+QqV0?JcpPjPsO}Sy;Dm(LI~z=z$j8+&kwSYjy%JzUxmdRx$SAwN zp9D>X#s`A$3c~LSfd4xn?RN#BP!u2>?ElwI(Ea0r+upQ$r7`nZqI?RsE2KuFmS)TC zUNh2oJu)7LnIbmbQzoUCBy;_Ab+@*YK^%sd*_7Md1lZORDoz@)XIEydB(3ir4|lpWwT*Rmbi3{WE$Fyvc^lRaS)#OX2dhwF1L zQWN9JoG1*(sTisOESKBm3QfUQcMJV(uripjs z<2Y%Ii?4jtq{MRD$L)c9K6-~gJ~RbD;MO&W7x&chEd=hv>|jV0t{F#%?E*pdf{M&= z%8!Fy?|A}YGfkRM1~SF=k?5K@tAW`fEEf;*Qb>eT^e~pzPkcd&i z8TSXi2bi$QRiS9YyXUZVp`jU79XEEHN@Eu!l;Sun+=B@=CLvpzAgY4$E(8MZ!?RMX z0s^d=7S9Ko7o7Q`uqi+^&QZHe1Xu`cB1BH6AaIm2jFQU)?s+U9gn*?^;MyuH_Pp>^ zt$K8W2vWzUZIYzRE_$U~5L-|_Nit~~ku(~}Qe{WQS+v=Llp$MuBNhahRDdPJ!H0cu z_9c{y%KXGR=rk$dwbmmGi6H^kR^!YA73WCIL(goR%%zU^!-#5r+AHzqjOJ=T1>Q#o zlJnJ{o;|)gKkuYrh?ki8b&_g7ww~NP74hcFCzf5hs)_FimuJPJ*XI5L+5;LLN2adK z1O#&e>JB)r`|qIeq><|I#@;>wYNr_pCPue(hUBj+{RSF^*qYaPX9NH-BIe#q+1=ZH z)gh~FVKB125Y{T;0A^ugP)O8IzH(ULuNEIEP9V}9|Dv9~)lefSw^2z=h7ZHI2$sR|hC@B97HdONgLb|1h44m#vyhJ%$xDLp!e zVwUWD3QzCrYs%2Ep3=n4FnR^}t-X1vYn&G;B&L;mNlE6=dALxq0fjVAYS zmg6Z;g@VIe+V{tW=jTCzK+Y_qgbT)UPc5K|a81$y4N%~hd&D;pADn^`Rc>yGl4@y` zP(iii9UubY`v$0Pm3^-NqAUk}!9RGEYcmYSwv4Kc#RfaykN^UyiE#!cbiW-);SK2A zZ1E_g(Mk<;MPp+aYBGv$e>mO0q~mBKhkHwFH5YPAfXA5L)bX;DY5;1*bVzSZ&^xqH zZw)|axN6Wze$`5K#<)z@(Q=^?G2?I?{KttRiw!*mqyzv99r~DdRpkO;(@Y^Lo7bpq zzl)^5dGDO~J-LmhZwi_{d!o;P2F~*GHq70QZz*~|_&g*v`4f$jOxua8tW}-o;V^CJ7JceSRawLVs8(1)ZNecIqH<8oStpG-=->48_|9Vij z*FdzDegft%!1MD2RRq5k^ATy)2zToL#tS|M#Hhc#oOEA(oM^q2@w)yPKCiv|K7TJ7 z^+mk6EQ3hf@ns`q<&ge+DTgMgzqeRsLyYt#Ya(NzWFc`DKfb#?>M_Aa=_G!Pn=o7R zSY=k|?>0mAc0rBj4O|nW5E~QwU2aX=IRtRD6$0Ll>sID;<+`Z4 z4cxom7iN9-a{b&!n&l@-bWB>wj+ozX_hVV~;s0(0b0TxZ;I>JFsz~EvzSiFORVY)q zFW>Tb)t5eP-6kr#14=ks_b`!W<;vP+S$N=WImgR|XH?lM`jo#U^1`Dae?&qlhmDYU zwE^0ABBn>$`~~I85XA(oWp&1~pc-DGwqeF3%7RUi)yf4qB3DOAgzQcvG@&$PMN^uw z?IBIW8kJ5+gt#ek+oJO+30SyU$4-)E7HT5NcT{b?czQ9FsQl3sM>W(W^xZ)u;VDRh|QKY*;B z@g14uQ&`gzzDya!wVl(!)7>+x2efVhK6 z@2yTwY7j^UA?yled~$R{j!LS==Wso=z8k9`*PX87Ya5*O_63ar1R9?`u+owZVi}y9 zEL2|!7>C|7wkx!QAuM%Zy_4fkbHE=b#J`ykqQq(y%X~C!kK6D#y30P5W6HKO4k+%S zY0Z4)QF1uAn_CoWS)5sWMatI=vcX5`Z+Rg9YS|>x8X;%Dy>8?o{-EJt|0#VBhEHJw zTi*_~f$iUUtvrB+mg?*Ysp*QQnXbVO?d#ClT}sn}|5*`B+qzEdJr+&nPy{TXXk>QF zKQ^7{KwYWO%a(Z#O63x#mv@<)i9EIX{zl59aF1H zPP^Bzmyc1jmRXXyC_gu)DmvpTI~txfU;9$Gn_b&j>NgjVMIIXdxnLE?}i;HB661!8VqXky6)cZ@xawNA$*pjE%lT4CEV_gWh^ zY}R{S7($zwNUZedfEQO?|JY^BBxduh+k=t8`(S;Q+(>_7jtgq$g-H|muNS?1W6siI z$chAi&1xDuw_0@8vdoxWUrT*Lxdlu^jt7Z)Xp*flZ;GH^Q{BVbN*YjK(}B~RlMzz) zUps$Q`-rG_tI4PnJsq!2<20?ccuxiU1>-)SJ)gzR2=S~le&(>y@^L|vb)?5s#{m;n z3YWLZOs}@ZUOK}n5>0j5frK|#rqsy{0v?~1f1Mvbub?j*6XD)^0EpA^dyx~OqCota zKwh>+`Y{Ev^y*zw`YmAOh&<8XZQ(^77?{zV`z+ONc*ABi5jaJ^SDQ2~v-1c8VK{J_ zOkV*_q*&2xe=BBN&H16U!xvVdofJ~J#@q@g`5^fBKzzs7;-2@BrNKL zXUHB!8e>*{CWTH%IcDhPm*1RmWVF={1#?4cVTQa2{78H)zlc6Z?hI#9i3e?a+Je5= zIf^m5SkM^uISKHJN!hr$8EdTPW@el1#Nu9VD!Pc*MKkd&h&Tt?TfKClF+ZcHgV(^mU> zg>reIHGB(c41uW}>=(lffzR=`pz>2b#4*BUx!_+Z1p(v~b<@e(%%~AFy|u#og~TYz z&q3fZF&!~AYsR(zl{#n$9Diww%@ z{fm-5shB#<)uYwc@r)}hcfC9NA!KhKJ&$ghKa>q1$V~C?aj)|_E;Md>Yqc$J+n7d zP$fColL0{_vEuY6)Z-+bM3VW0XV`Nfrg?ah^m}sL@RW7Sq%vGIcki+h8pKWot~2AI z$%X3?%$kAQ*|lo3zyHoa-@wnTU9N+t__oKDC= z^$^5NUU7yU@?~Rq1|eOO$71U9l|>WG=|m1p@<2E0QW!xHH@pLSd78Ab6RPDA4g5le zd+-rw>WP~sgH7f{bgVh;Rft{jbI&OyRJpaZ#ri@oxiXT{k_q12=kpKbj6+2%+z{%96Z~~HInGkrFAC?`<{3(0^!4`Ay{P0WeRC;i@z77-fn>3=LPOo-Opt-+IWMeg z9;?|+YH?6zOQK zrILkc*v0kb`+JhJMtZABOdjGt9)FvHmAGBw+cp;^V%qp)&}V`$a!89g@1aYYs~uVE|55NG(EA_G89A>CKE>(#0Ue7oZT; zm9>sSP9!nrz@!%rXU8BXlwZ5<^PBBqgdAj0^VC#1=7#69o z13d1ttv9RL2ZpeS%wXF%?}j~fJjh31`V(9$aR5;1#2t2?Hb0f0{jNY{hDlu(BD`s9 z0j|qJ5v|JUkBbrPLEgGE)wZ1JSx|0%W?Bx5LCWL8kG`xsQj;xG<5PK`6o@R9{_U}0DJMV zfj|SJF2k> zGe5pIy`kGj@icu8LmDqnAn5b)^d?D(Xiy0OntX6l)HLKYVF21yZs(>z`H2qVS4s=^ z>wJ2c(+6S7d}8VOvux@60r1y40EI3L^r7-Xyd0lCF!l8ZcynVf?q4{&zVJHoRv4UM zuTs(7`*W4oPkvVmWFyW?7RNvS`g8a7W&H5x#_ZC6@w;SoUv_ogHgD@`YTLHuhFi0*(M{mq-nEv(Gj)C626Ehh%A<7l`*r}Pv9=L*3 zAmHZ?1kT~xH3O0u20BJ``z~=Q1?1aW0SROcfuX^tQ8+4WtU?9tT7gNv1b=q;tNv3J ztOV5MuQFa89Dn{a00eO9!(Tq&GX7!~LvI#mdGD^Iznu88{imB=zPxn-6_mxJMSWG} zFEx33u?*p0_s)Ddx-omb)%Aso8{+)+m@iGgnt)5-^jnba>7)Ni{0$9s)^=nj7!9{x z;|``}+njldW-C9X z_}@2ku1ZP5dCKAH&z<*rr5}5DR!A;XVKbhp=zBG>>`!m!W}=YY|5oSy@mAAU`HwYg z{?o9tJKzOy`TlX&&rjO0jj(0a$P)Vlr#`E%Il7~{IC-I(ZPBF?;Oph>>g+0Jv3Y>l z_2Tkby4St{SobSa->ETHzu_3IW9#yI?5w!JRv6?q{5%uxLK{x(&w55?F zp_iEcG>4hbVGK*ouenrWBoq%ZMO>91jh3l#c)G63h;qJvo zi+NFto9Hmhbm^)^MlTu^TlHcAYSMpXJF#_Ea*eO5!|M(Hxf=*IxnoMvm?t>Kj-W68 zBopzQMkqS0S)^EY(b!P(OvjAwRW!Ek7v7=mmk?-FG3BF&G z=6koaRLLv~_da#b4(JZ@`QH58)5AY7Ub*&$<&n_*Yx7s;v6It-Lc>jh0cyf zA+hGY801|h=k^Qn?zZihymwD5{n-IIBvTBAmUn`B{o&Ow*R?u7Z7gk$>hhm{l6qDb zj2wRzbmU&-8$Ul5hba{dM9s<-9F(w^)YP?H4T_t5E?`|(*D}tp(AmbgU<`GxME%Nz z{yi~c)+B0l>u~@AR1EtsA`X=vTAI8raq}{UZmnK zro+IF`*mBy8jsX)HO5g z-1_=fYSi(_<1!^YdL0O zs|HpYvqjqayRTomzq+2`eXgL(HMcfDG4XtFs=64qcB>pe6U@RHnc7-uPr7UYSDo2I zt6AOCyFDK@k!!dzYmH}?-l5$oiF!dr|6tubj`ljtY8)(w5o-2YiND}ETC`5<|9f(Y zU_&NGtK)3OF8l+g0R{0hOLy1?%>ZX-VPa-VhuZ-q_}^r8^$8gidAs61RbrGmpT$`IUX*NbQ3+}HSJw(^{?QCQB>5$MN`#b5 zcwOO-9UGE=M>nB=7J=-X{TNwP(Etye8#ir}S}JCS&S99nVjj8~yQd6n0S%Po1q2jw z$lFGbHw`iJ+_Nk9^x1|pFmr3r(NRB4Uw#qkmJ`x{p!w0$ zzaK5}zV?E*XTwE;kHlnFP)7a#au+@vKYwoz4EZih#=Bgl%K4(aT#dWv75>Wk=Q6lr zhXnh^RyH(aZuErAx5W+EnhqpeU`THpFPiX9t#oXG2oULoL7Gv zPDp)RECwJW7Uz9<$E{nCdJjx8WKtNcjhPGLM)vvJhGzJ1NnD@FB>2{@?X6m+o=gv) zCv1|eJ=Sj+>shAYVDC)BHh-GqThsE@jqTt|Lpss&JvLmBJ}{|^vmlzNsTMOXi6_?O z?PgIGEqBOuwHbx0WvZv^?StY2$hwc)wA#(}ti9FShZ?G<8_=`#n&l1m%FVVzJWPhZ zM`VSbP+h7kG_>lg^pDo+Mt0A+b{8hTZ+V`q6VxwU`k=P=LDUgrXMrAX`{;oV7R}k{ zp1Ja`9JMcRTDr6^P9n*HW&=Lpn;t;F|G(_P{}2bv=^y)``pRM);#}M!BCO18!W`^k zVj`^E9Ne7jOw3#y9PF&Ztil3BeE+{AdjAi*z{<%=^uKNc{?eo}suEiG!_yrH5eYPc z5)%tok(3y)iCCsmj-yzlB}fLOqnVkck{jkd+=gadOB6Y;0TC|--S%kGn-ulO+^NRi zP1lZJvVghyKhkC~4ErIbj6>~RD&g=M<-V9KusU#qF-I@~QIOR_AoACbH1u~Udfg-H zND|)wGb&L4EnX-G7sSmy^y)qr;=M**SP5P0orSb8Wln$^7j*ePTuvx#LfQ+$r=8vc z{23AGSs|o%A>!u&j&6$G+RyGEN9?)*HLL*Q4uQpUXhwi^MvzrTpjJk(*T!2E;ZY%==5_We@p=!l9=y^(KfZLdW)0Pf z^yWpi>i6E8;%D>0o8f2YNjnASk3J)s8IC?9I{f#mcj>v!##L+4PR<(c=wjNM&gf{q z{+}pLSp7S+f7s<9!Dt!519)KnaUPI^{r-?-fer*b=xepyjgaWPV9&|iVgK)z(-n_E R(P3CwIAO@i#T6uA{tp*PR5Jho diff --git a/ana14.tex b/ana14.tex index 518775e..26a5a18 100644 --- a/ana14.tex +++ b/ana14.tex @@ -1 +1,206 @@ -\documentclass{lecture} \begin{document} \newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\dv}[2]{\frac{\d #1}{\d #2}} \def\mathunderline#1#2{\color{#1}\underline{{\color{black}#2}}\color{black}} \section{Extremwertaufgaben} \begin{definition}[lokales Maximum/Minimum] Sei $D \subset \R^n$ offen, $f\colon D\to \R$. Ein Punkt $x\in D$ heißt lokales \underline{Minimum(Maximum)} von $f$, falls eine Umgebung $K_\delta(x)\subset \R^n$ existiert mit \[f(x)\leq f(y),\; \forall y\in K_\delta(x)\cap D\] ($f(x)\geq f(y),\; \forall y\in K_\delta(x)\cap D$). Falls \[f(x) < f(y), \forall y \in K_\delta(x)\cap D\setminus\{x\}\] ($f(x) > f(y)$), dann heißt $x$ \underline{striktes} lokales Minimum (Maximum). \end{definition} \begin{satz}[Notwendige Bedingung für lokales Extremum (Min oder Max)] Sei $D\subset \R^n$ offen, $f\colon D\to \R$ stetig differenzierbar und $x\in D$ ein lokales Extremum von $f$. Dann gilt : $\nabla f(x) = 0$. \end{satz} \begin{proof} Für $i = 1,\dots,n$, betrachte $g_i(t)\coloneqq f(x + te_i)$. Da $D$ offen sind alle $g_i$ auf einem Intervall $(-\delta, \delta)$ definiert für ein $\delta > 0$, und differenzierbar. $g_i(t)$ hat in $t = 0$ ein lokales Minimum/Maximum $\implies \dv{g_i(t)}{t}\bigg|_{t=0} = 0 \forall i = 1, \dots, n$. $f$ total differenzierbar \[\implies 0 = \dv{g_i(t)}{t}\bigg|_{t=0} \oldstackrel{\text{Kettenregel}}{=} \sum_{j = 1}^{n}\pdv{f(x)}{x_j} \cdot \delta_{ij} = \pdv{f(x)}{x_i} \forall i = 1\dots, n \] \end{proof} \begin{bem} Die Umkehrung ist falsch, z.B. $f(x) = x^3,\; f\colon \R \to \R$, hat in $x=0$ $\nabla f(x) = 0$, aber $x = 0$ ist kein Max/Min von $f(x) = x^3$.\\ $f\colon \R^2 \to \R,\; f(x_1, x_2) = x_1x_2$ hat in $x = \begin{pmatrix} 0\\0 \end{pmatrix} \nabla f(x) = 0$, aber $x = 0$ ist kein Max/Min von $f$. \end{bem} \begin{satz}[Hinreichende Bedingung für lokales Extremum] Sei $D\subset \R^n$ offen, $f\in C^2(D,\R)$ und $x\in D$ mit $\nabla f(x) = 0$. Dann gilt: \begin{enumerate} \item $H_f(x)$ positiv definit $\implies x$ striktes lokales Minimum von $f$. \item $H_f(x)$ negativ definit $\implies x$ striktes lokales Maximum von $f$. \item $H_f(x)$ indefinit $\implies x$ kein lokales Extremum. \end{enumerate} \end{satz} \begin{proof} Nach Taylor gilt lokal um $x$: \[f(x+h) = f(x) + (\nabla f(x), h)_2 + \frac{1}{2}(H_f(x)h, h)_2 + \omega_2(x,h)\] mit $\frac{\omega_2(x,h)}{\norm{h}^2} \oldstackrel{h \to 0}{\to} 0$ \begin{enumerate} \item Sei $H_f(x)$ positiv definit. Betrachte $\min_{\norm{h}=1} (H_f(x)h, h)_2$. Die Menge $\{h\in \R^n\mid\norm{h} = 1\}$ ist kompakt $\implies (H_f(x)h, h)_2$. nimmt ihr Minimum auf $\{h\in \R^n\mid \norm{h} =1\}$ als stetige Funktion an. $\implies \alpha \coloneqq \min_{\norm{h} = 1} (H_f(x)h, h)_2 > 0$, da $H_f(x)$ positiv definit ist. Sei $h\in \R^n\setminus\{0\}$ beliebig. Dann gilt \[(H_f(x)h, h)_2 = \norm{h}^2 \underbrace{(H_f(x)\cdot \frac{h}{\norm h}, \frac{h}{\norm h})_2}_{\geq \alpha}\geq \alpha \norm h^2 > 0\] Wähle $\delta > 0$ klein, sodass $\forall \norm h < \delta$ gilt $|\omega_2(x,h)| \leq \frac{\alpha}{4}\norm h^2$ (weil $\omega_2(x,h) = o(\norm h^2)$). Damit gilt $\forall h, \norm h < \delta$ \[f(x+h) = f(x) + (\underbrace{\nabla f(x)}_{=0}, h)_2 + \frac{1}{2}(H_f(x)h, h)_2 + \omega_2(x,h) \geq f(x) + \frac{\alpha}{2}\norm h^2 - \frac{\alpha}{4}\norm h^2 > f(x)\] $\implies x$ striktes lokales Minimum von $f$. \item Ersetze $f$ durch $-f$, dann 1) \item $\exists h \in \R^n$ mit $(H_f(x)h, h)_2 = \alpha > 0$, sodass \[ f(x + th) \oldstackrel{\text{Taylor}}{=} f(x) + \frac{1}{2}t^2\cdot \alpha + \omega_2 (x , th) = f(x) + t^2\left(\frac{\alpha}{2}+ \underbrace{\frac{\omega_2(x,th)}{t^2}}_{\oldstackrel{t\to 0}{\to}0}\right) \oldstackrel{\text{für } 0 < t << 1}{>} f(x) \] Außerdem $\exists \eta \in \R^n$ mit $(H_f(x)\eta, \eta)_2 = \beta < 0$. Analog $\implies f(x + t\eta) \leq f(x) + \beta\frac{t^2}{4}< f(x)$ für $0 < t << 1$. $\implies f(x)$ kein Maximum/Minimum. \end{enumerate} \end{proof} \begin{bsp} \begin{enumerate} \item $f(x,y) \coloneqq x^2 + 2y^2 \implies \nabla f(x,y) = \begin{pmatrix} 2x\\4y \end{pmatrix} = 0$ für $\begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} 0\\0 \end{pmatrix}$. $H_f(x,y) = \begin{pmatrix} 2 & 0\\ 0 & 4 \end{pmatrix}$ positiv definit. $\implies \begin{pmatrix} x\\y \end{pmatrix} = 0$ striktes lokales Minimum (sogar global). \begin{tabular}{cll} $f(x)\leq f(y)$ &$\forall y\in D$ & globales Minimum\\ $f(x) < f(y)$ &$\forall y\in D\setminus\{x\}$ &striktes globales Minimum \end{tabular} \item $f_1(x,y) \coloneqq x^2 + y^4,\; f_2(x,y) \coloneqq x^2,\; f_2(x,y)\coloneqq x^2 + y^3$ Es gilt \[ \nabla f_i(0) = 0\in \R^2,\; H_{f_i}(0) = \begin{pmatrix} 2 & 0\\ 0 & 0 \end{pmatrix} \forall i =1,2,3 \] Die Hesse-Matrix ist positiv semidefinit. Es gilt\\ \begin{tabular}{lcl} für $f_1$:& Punkt 0 ist ein& \underline{striktes} lokales Maximum,\\ für $f_2$:& Punkt 0 ist ein& \underline{lokales Minimum}, aber \underline{nicht strikt},\\ für $f_3$:& Punkt 0 ist ein& Sattelpunkt. \end{tabular} \end{enumerate} \begin{figure} \begin{tikzpicture}[scale=0.8] \begin{axis} [ domain=-1:1, samples=20, grid = major ] \addplot3 [surf] {x^2 + 2*y^2}; \end{axis} \end{tikzpicture} \begin{tikzpicture}[scale=0.8] \begin{axis} [ domain=-1:1, samples=20, grid = major ] \addplot3 [surf] {x^2 - y^2}; \end{axis} \end{tikzpicture} \caption{Links: $f(x,y) = x^2 +2y^2$ ist ein Paraboloid, Rechts: $f(x,y) = x^2-y^2$ ist eine Sattelfläche} \end{figure} \end{bsp} \section{Implizite Funktionen und Umkehrabbildung.} Frage: \underline{Umkehrabbildung:} Auflösen von $x = g(y)$, d.h. \begin{equation} \left\{\begin{array}{rl} x_1&=g_1(y_1,\dots,y_n)\\ \vdots&\\ x_n&=g_n(y_1,\dots,y_n)\\ \end{array}\right\} \Leftrightarrow \left\{\begin{array}{rl} 0&=x_1-g_1(y_1,\dots,y_n)\\ \vdots&\\ 0&=x_n-g_n(y_1,\dots,y_n)\\ \end{array}\right\}\tag{*}\label{star} \end{equation} $n$ Gleichungen, $n$ Unbekannte $y_1,\dots, y_n$. Gesucht: Abb $f$ mit $y = f(x)$ ($f=g^{-1})$, s.d. $(x,f(x))$ Gleichung \eqref{star} löst (lokal um $(x_0,y_0=f(x_0))$). \paragraph{Implizite Funktion} $m$ Gleichungen, $m$ Unbekannte $y_1,\dots, y_m$. \begin{equation} \left.\begin{array}{rl} 0&=F_1(x_1,\dots,x_n,y_1,\dots,y_m)\\ \vdots&\\ 0&=F_m(x_1,\dots, x_n,y_1,\dots,y_m) \end{array}\right\}\tag{**}\label{doublestar} \end{equation} $0 = F(x,y)$ Auflösen nach $y$, d.h. Gesucht: Abbildung $f$, s.d. $y = f(x)$ mit $(x,f(x))$ löst \eqref{doublestar} (lokal um eine Lösung $(x_0,y_0):F(x_0,y_0) = 0$) \begin{bsp} $m=1,\;n=1,\; F(x,y) = x^2 + y^2-1$\\\begin{minipage}[c]{0.35\textwidth} \begin{tikzpicture}[scale=0.8] \draw[->] (0,-1.5) -- node[right,pos=.9] {$(x_0,y_0)$} (0,1.5); \draw[->] (-1.5,0) -- node[below,pos=1.3] {$F(x,y) = 0$} (1.5,0); \draw (0,0) circle (1cm); \draw[fill] (0,1) circle (2pt); \draw[fill=red,draw=red] (0,0) circle (2pt); \node[color=red] at (.3,0) {)}; \node[color=red] at (-.3,0) {(}; \draw[fill=blue,draw=blue] (1,0) circle (2pt); \node[color=blue] at (1.3,0) {)}; \node[color=blue] at (.7,0) {(}; \end{tikzpicture} \end{minipage}\begin{minipage}[c]{0.65\textwidth} An der Stelle $\mathunderline{red}{x_0 = 0,\; y_0 = 1}$ gilt $F(0,1)=0$,\\ also $F(x,y) = 0 \Leftrightarrow y^2 = 1-x^2$. \[\implies \mathunderline{red}{f(x)\coloneqq \sqrt{1-x^2}}\text{ für } |x| < 1\] $\mathunderline{red}{\text{ erfüllt }F(x,f(x)) = 0},\; |x| < 1$. \end{minipage}\\Für $\mathunderline{blue}{x_0 =1,\; y_0 = 0}$ hingegen gibt es keine Umgebung von $x_0 = 1$, sodass \[\mathunderline{blue}{\exists f\colon U(x_0) \to \R\text{ mit }F(x,f(x)) = 0}.\] \end{bsp} \begin{satz}[Satz über implizite Funktionen] Sei $D^x \subset \R^n$ offen, $D^y \subset \R^m$ offen, $F^1 \in C^1 (D^x\times D^y,\R^m)$ (stetig differenzierbar) und $(\hat{x}, \hat{y})\in D^x\times D^y$ mit $F(\hat x, \hat y) = 0$. Die $m\times m$ Matrix \[ D_yF(x,y) =\begin{pmatrix} \pdv{F_1}{y_1} &\dots &\pdv{F_1}{y_m}\\ \vdots & & \vdots\\ \pdv{F_m}{y_1}&\dots&\pdv{F_m}{y_m} \end{pmatrix} \] sei im Punkt $(\hat x, \hat y)$ invertierbar. Dann gilt: \begin{enumerate} \item $\exists$ offene Umgebungen $U(\hat x) \subset D^x,\; U(\hat y)\subset D^y$ um $\hat x$ und $\hat y$ und $\exists$ eine stetige Funktion $f\colon U(\hat x) \to U(\hat y)$, s.d. \[F(x,f(x)) = 0\quad\forall x\in U(\hat x)\] \item $f$ ist eindeutig bestimmt, d.h. $F(x,y) = 0$ für $(x,y) \in U(\hat x)\times U(\hat y)\Leftrightarrow y = f(x)$ \item $f$ ist in $\hat x$ stetig differenzierbar und $J_f(\hat x) = D_xf(\hat x)\in \R^{m \times n}$ ist \[D_xf(\hat x) = -(D_yF(\hat x, \hat y))^{-1}D_xF(\hat x, \hat y).\] \end{enumerate} \end{satz} \begin{proof} \begin{enumerate} \item O.B.d.A. sei $(\hat x, \hat y) = (0,0)$ (sonst betrachte $F(x,y) - F(\hat x, \hat y)$). Die Matrix $J_y \coloneqq D_yF(0,0)$ ist regulär. Definiere $G\colon D^x\times D^y \to \R^m,\; G(x,y) \coloneqq y - J^{-1}_yF(x,y)$. $G$ ist stetig differenzierbar und erfüllt: $G(0,0) = 0$ und $F(x,y) = 0 \Leftrightarrow G(x,y) = y$. Jacobi-Matrix von $G(x,y)$ bzgl. $y$: \[D_yG(x,y) = \mathbb{I} - J_y^{-1} D_yF(x,y) \text{ und insb. } D_yG(0,0) = \mathbb{I} - J_y^{-1}J_y = 0.\] $F$ stetig differenzierbar $\implies D_yG(x,y)$ stetig $\implies \exists K_r^x(0) \times K_r^y(0) \subset D^x \times D^y$ mit Radius $r$, sodass $\norm{D_yG(x,y)}_2\leq \frac{1}{2},\; (x,y)\in K_r^x(0)\times K_r^y(0)$. $G(0,0) = 0\implies \exists K_s^x(0)\subset K_r^x(0)$ mit Radius $0< s\le r$ sodass \[\norm{G(x,0)}_2 \le \frac{1}{2} r,\; x\in K_s^x(0)\] Ziel: Konstruiere $f\colon K_s^x(0)\to K_r^y(0)$ stetig mit $G(x,f(x)) = f(x)\; (\Leftrightarrow F(x,f(x)) = 0)$. Betrachte Fixpunktgleichung \[G(x,y) = y,\; x\in K_s^x(0).\] Für $(x,y_1),\; (x,y_2)\in K_s^x(0)\times K_r^y(0)$ gilt \[\norm{G(x,y_1) - G(x,y_2)}_2 \oldstackrel{\text{MWS}}{\le} \sup_{(x,y)\in K_s^x(0)\times K_r^y(0)}\norm{D_yG(x,y)}_2 \cdot \norm{y_1-y_2}_2 \le \frac{1}{2}\norm{y_1-y_2}_2.\] Sei $y\in \overline{K_r^y(0)}$ \[\norm{G(x,y)}_2 \le \norm{G(x,y)-G(x,0)}_2 + \norm{G(x,0)}_2 \le \frac{1}{2}\norm{y}_2 + \frac{1}{2}r\le r\] d.h. $G(x,\cdot)$ ist eine Selbstabbildung der abgeschlossenen Kugler $\overline{K_r^y(0)}$.\\ Außerdem, $\norm{G(x,y_1) -G(x,y_2)}_2 \le \frac{1}{2}\norm{y_1-y_2}\implies G(x,\cdot)$ ist eine Kontraktion mit Lipschitz-Konstante $L=\frac{1}{2}$. Aus dem Banachschen Fixpunktsatz folgt $\forall x \in K_s^x(0) \exists!$ ein Fixpunkt $y(x)\in K_r^y(0)$ von $G(x,\cdot)$: \[y(x) = \lim\limits_{k\to \infty} y^{(k)}(x),\; y^{(k)}(x)=G(x,y^{(k-1)}(x)),\; k\in \N\] mit Startpunkt $y^{(0)}(x)\coloneqq 0$. Es gilt die Fehlerabschätzung $\forall x \in K_s^x(0)$: \[\norm{y(x)-y^{(k)}(x)}_2 \le 2^{-k}\norm{y^{1}(x)-y^{(0)}(x)}_2 = 2^{-k}\norm{G(x,0)-0}_2 \le 2^{-k} \frac{1}{2}r\] Für $k\in \N$ gilt \[y^{(k)}(x) = G(x,y^{(k-1)}(x)) = y^{(k-1)}(x)-J_y^{-1}\underbrace{F(x,y^{(k-1)}(x))}_{\text{stetig}}\] Induktiv folgt $y^{(k)}(x)$ stetig in $x\in K_2^x(0)$. Definiere $f(x)=y(x),\; f\colon K_s^x(0) \to K_r^y(0)$ und nach Konstruktion gilt \[G(x,f(x)) = f(x),\quad x\in K_s^x(0).\] Aus der Abschätzung $\norm{y(x)-y^{(k)}(x)}\leq 2^{-k-1}\cdot r,\; x\in K_s^x(0)$ folgt, dass $y^{(k)}(x)\oldstackrel{k\to \infty}{\to y(x)}$ gleichmäßig auf $K_s^x(0)$ konvergiert $\implies y(x)$ stetig. $\implies$ 1. Behauptung für $(\hat x, \hat y) = (0,0)$ mit $U(\hat x)\coloneqq K_s^x(0)$ und $U(\hat y)\coloneqq K_r^y(0)$. \item Die Eindeutigkeit von $y = f(x)$ folgt nun aus dem Banachschen Fixpunktsatz. Für $x\in K_s(\hat x)$ ist der Fixpunkt der Gleichung $G(x,y) = y$ eindeutig bestimmt. \end{enumerate} \end{proof} \end{document} \ No newline at end of file +\documentclass{lecture} +\begin{document} +\newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\dv}[2]{\frac{\d #1}{\d #2}} +\def\mathunderline#1#2{\color{#1}\underline{{\color{black}#2}}\color{black}} +\section{Extremwertaufgaben} + +\begin{definition}[lokales Maximum/Minimum] + Sei $D \subset \R^n$ offen, $f\colon D\to \R$. Ein Punkt $x\in D$ heißt lokales \underline{Minimum(Maximum)} von $f$, falls eine Umgebung $K_\delta(x)\subset \R^n$ existiert mit \[f(x)\leq f(y),\; \forall y\in K_\delta(x)\cap D\] ($f(x)\geq f(y),\; \forall y\in K_\delta(x)\cap D$). Falls \[f(x) < f(y), \forall y \in K_\delta(x)\cap D\setminus\{x\}\] ($f(x) > f(y)$), dann heißt $x$ \underline{striktes} lokales Minimum (Maximum). +\end{definition} +\begin{satz}[Notwendige Bedingung für lokales Extremum (Min oder Max)] +Sei $D\subset \R^n$ offen, $f\colon D\to \R$ stetig differenzierbar und $x\in D$ ein lokales Extremum von $f$. Dann gilt : $\nabla f(x) = 0$. +\end{satz} +\begin{proof} + Für $i = 1,\dots,n$, betrachte $g_i(t)\coloneqq f(x + te_i)$. Da $D$ offen sind alle $g_i$ auf einem Intervall $(-\delta, \delta)$ definiert für ein $\delta > 0$, und differenzierbar. $g_i(t)$ hat in $t = 0$ ein lokales Minimum/Maximum $\implies \dv{g_i(t)}{t}\bigg|_{t=0} = 0 \forall i = 1, \dots, n$. + $f$ total differenzierbar + \[\implies 0 = \dv{g_i(t)}{t}\bigg|_{t=0} \oldstackrel{\text{Kettenregel}}{=} \sum_{j = 1}^{n}\pdv{f(x)}{x_j} \cdot \delta_{ij} = \pdv{f(x)}{x_i} \forall i = 1\dots, n + \] +\end{proof} +\begin{bem} + Die Umkehrung ist falsch, z.B. $f(x) = x^3,\; f\colon \R \to \R$, hat in $x=0$ $\nabla f(x) = 0$, aber $x = 0$ ist kein Max/Min von $f(x) = x^3$.\\ + $f\colon \R^2 \to \R,\; f(x_1, x_2) = x_1x_2$ hat in $x = \begin{pmatrix} + 0\\0 + \end{pmatrix} \nabla f(x) = 0$, aber $x = 0$ ist kein Max/Min von $f$. +\end{bem} +\begin{satz}[Hinreichende Bedingung für lokales Extremum]\ + Sei $D\subset \R^n$ offen, $f\in C^2(D,\R)$ und $x\in D$ mit $\nabla f(x) = 0$. Dann gilt: + \begin{enumerate} + \item $H_f(x)$ positiv definit $\implies x$ striktes lokales Minimum von $f$. + \item $H_f(x)$ negativ definit $\implies x$ striktes lokales Maximum von $f$. + \item $H_f(x)$ indefinit $\implies x$ kein lokales Extremum. + \end{enumerate} +\end{satz} +\begin{proof} + Nach Taylor gilt lokal um $x$: + \[f(x+h) = f(x) + (\nabla f(x), h)_2 + \frac{1}{2}(H_f(x)h, h)_2 + \omega_2(x,h)\] mit $\frac{\omega_2(x,h)}{\norm{h}^2} \oldstackrel{h \to 0}{\to} 0$ + \begin{enumerate} + \item Sei $H_f(x)$ positiv definit. Betrachte $\min_{\norm{h}=1} (H_f(x)h, h)_2$. Die Menge $\{h\in \R^n\mid\norm{h} = 1\}$ ist kompakt $\implies (H_f(x)h, h)_2$. nimmt ihr Minimum auf $\{h\in \R^n\mid \norm{h} =1\}$ als stetige Funktion an. $\implies \alpha \coloneqq \min_{\norm{h} = 1} (H_f(x)h, h)_2 > 0$, da $H_f(x)$ positiv definit ist. Sei $h\in \R^n\setminus\{0\}$ beliebig. Dann gilt + \[(H_f(x)h, h)_2 = \norm{h}^2 \underbrace{(H_f(x)\cdot \frac{h}{\norm h}, \frac{h}{\norm h})_2}_{\geq \alpha}\geq \alpha \norm h^2 > 0\] + Wähle $\delta > 0$ klein, sodass $\forall \norm h < \delta$ gilt $|\omega_2(x,h)| \leq \frac{\alpha}{4}\norm h^2$ (weil $\omega_2(x,h) = o(\norm h^2)$). Damit gilt $\forall h, \norm h < \delta$ + \[f(x+h) = f(x) + (\underbrace{\nabla f(x)}_{=0}, h)_2 + \frac{1}{2}(H_f(x)h, h)_2 + \omega_2(x,h) \geq f(x) + \frac{\alpha}{2}\norm h^2 - \frac{\alpha}{4}\norm h^2 > f(x)\] + $\implies x$ striktes lokales Minimum von $f$. + \item Ersetze $f$ durch $-f$, dann 1) + \item $\exists h \in \R^n$ mit $(H_f(x)h, h)_2 = \alpha > 0$, sodass + \[ + f(x + th) \oldstackrel{\text{Taylor}}{=} f(x) + \frac{1}{2}t^2\cdot \alpha + \omega_2 (x , th) = f(x) + t^2\left(\frac{\alpha}{2}+ \underbrace{\frac{\omega_2(x,th)}{t^2}}_{\oldstackrel{t\to 0}{\to}0}\right) \oldstackrel{\text{für } 0 < t << 1}{>} f(x) + \] + Außerdem $\exists \eta \in \R^n$ mit $(H_f(x)\eta, \eta)_2 = \beta < 0$. Analog $\implies f(x + t\eta) \leq f(x) + \beta\frac{t^2}{4}< f(x)$ für $0 < t << 1$. $\implies f(x)$ kein Maximum/Minimum. + \end{enumerate} +\end{proof} +\begin{bsp} + \begin{enumerate} + \item $f(x,y) \coloneqq x^2 + 2y^2 \implies \nabla f(x,y) = \begin{pmatrix} + 2x\\4y + \end{pmatrix} = 0$ für $\begin{pmatrix} + x\\y + \end{pmatrix} = \begin{pmatrix} + 0\\0 + \end{pmatrix}$. + $H_f(x,y) = \begin{pmatrix} + 2 & 0\\ 0 & 4 + \end{pmatrix}$ positiv definit. $\implies \begin{pmatrix} + x\\y + \end{pmatrix} = 0$ striktes lokales Minimum (sogar global). + \begin{tabular}{cll} + $f(x)\leq f(y)$ &$\forall y\in D$ & globales Minimum\\ + $f(x) < f(y)$ &$\forall y\in D\setminus\{x\}$ &striktes globales Minimum + \end{tabular} + \item $f_1(x,y) \coloneqq x^2 + y^4,\; f_2(x,y) \coloneqq x^2,\; f_2(x,y)\coloneqq x^2 + y^3$ + Es gilt + \[ + \nabla f_i(0) = 0\in \R^2,\; H_{f_i}(0) = \begin{pmatrix} + 2 & 0\\ + 0 & 0 + \end{pmatrix} + \forall i =1,2,3 + \] + Die Hesse-Matrix ist positiv semidefinit. Es gilt + + \begin{tabular}{lcl} + für $f_1$:& Punkt 0 ist ein& \underline{striktes} lokales Maximum,\\ + ür $f_2$:& Punkt 0 ist ein& \underline{lokales Minimum}, aber \underline{nicht strikt},\\ + für $f_3$:& Punkt 0 ist ein& Sattelpunkt. + \end{tabular} + \end{enumerate} + \begin{figure} + \begin{tikzpicture}[scale=0.8] + \begin{axis} + [ + domain=-1:1, + samples=20, + grid = major + ] + \addplot3 [surf] {x^2 + 2*y^2}; + \end{axis} + \end{tikzpicture} + \begin{tikzpicture}[scale=0.8] + \begin{axis} + [ + domain=-1:1, + samples=20, + grid = major + ] + \addplot3 [surf] {x^2 - y^2}; + \end{axis} + \end{tikzpicture} + \caption{Links: $f(x,y) = x^2 +2y^2$ ist ein Paraboloid, Rechts: $f(x,y) = x^2-y^2$ ist eine Sattelfläche} + \end{figure} +\end{bsp} +\section{Implizite Funktionen und Umkehrabbildung.} +Frage: \underline{Umkehrabbildung:} Auflösen von $x = g(y)$, d.h. +\begin{equation} + \left\{\begin{array}{rl} + x_1&=g_1(y_1,\dots,y_n)\\ + \vdots&\\ + x_n&=g_n(y_1,\dots,y_n)\\ + \end{array}\right\} \Leftrightarrow + \left\{\begin{array}{rl} + 0&=x_1-g_1(y_1,\dots,y_n)\\ + \vdots&\\ + 0&=x_n-g_n(y_1,\dots,y_n)\\ + \end{array}\right\}\tag{*}\label{star} +\end{equation} +$n$ Gleichungen, $n$ Unbekannte $y_1,\dots, y_n$. +Gesucht: Abb $f$ mit $y = f(x)$ ($f=g^{-1})$, s.d. $(x,f(x))$ Gleichung \eqref{star} löst (lokal um $(x_0,y_0=f(x_0))$). + +\paragraph{Implizite Funktion} $m$ Gleichungen, $m$ Unbekannte $y_1,\dots, y_m$. +\begin{equation} + \left.\begin{array}{rl} + 0&=F_1(x_1,\dots,x_n,y_1,\dots,y_m)\\ + \vdots&\\ + 0&=F_m(x_1,\dots, x_n,y_1,\dots,y_m) + \end{array}\right\}\tag{**}\label{doublestar} +\end{equation} +$0 = F(x,y)$ Auflösen nach $y$, d.h. +Gesucht: Abbildung $f$, s.d. $y = f(x)$ mit $(x,f(x))$ löst \eqref{doublestar} (lokal um eine Lösung $(x_0,y_0):F(x_0,y_0) = 0$) +\begin{bsp} + $m=1,\;n=1,\; F(x,y) = x^2 + y^2-1$ + + \begin{minipage}[c]{0.35\textwidth} + \begin{tikzpicture}[scale=0.8] + \draw[->] (0,-1.5) -- node[right,pos=.9] {$(x_0,y_0)$} (0,1.5); + \draw[->] (-1.5,0) -- node[below,pos=1.3] {$F(x,y) = 0$} (1.5,0); + \draw (0,0) circle (1cm); + \draw[fill] (0,1) circle (2pt); + \draw[fill=red,draw=red] (0,0) circle (2pt); + \node[color=red] at (.3,0) {)}; + \node[color=red] at (-.3,0) {(}; + \draw[fill=blue,draw=blue] (1,0) circle (2pt); + \node[color=blue] at (1.3,0) {)}; + \node[color=blue] at (.7,0) {(}; + \end{tikzpicture} + \end{minipage}% + \begin{minipage}[c]{0.65\textwidth} + An der Stelle $\mathunderline{red}{x_0 = 0,\; y_0 = 1}$ gilt $F(0,1)=0$,\\ also + $F(x,y) = 0 \Leftrightarrow y^2 = 1-x^2$. + \[\implies \mathunderline{red}{f(x)\coloneqq \sqrt{1-x^2}}\text{ für } |x| < 1\] $\mathunderline{red}{\text{ erfüllt }F(x,f(x)) = 0},\; |x| < 1$. + \end{minipage} + +Für $\mathunderline{blue}{x_0 =1,\; y_0 = 0}$ hingegen gibt es keine Umgebung von $x_0 = 1$, sodass \[\mathunderline{blue}{\exists f\colon U(x_0) \to \R\text{ mit }F(x,f(x)) = 0}.\] +\end{bsp} +\begin{satz}[Satz über implizite Funktionen] + Sei $D^x \subset \R^n$ offen, $D^y \subset \R^m$ offen, $F^1 \in C^1 (D^x\times D^y,\R^m)$ (stetig differenzierbar) und $(\hat{x}, \hat{y})\in D^x\times D^y$ mit $F(\hat x, \hat y) = 0$. Die $m\times m$ Matrix + \[ + D_yF(x,y) =\begin{pmatrix} + \pdv{F_1}{y_1} &\dots &\pdv{F_1}{y_m}\\ + \vdots & & \vdots\\ + \pdv{F_m}{y_1}&\dots&\pdv{F_m}{y_m} + \end{pmatrix} + \] + sei im Punkt $(\hat x, \hat y)$ invertierbar. Dann gilt: + \begin{enumerate} + \item $\exists$ offene Umgebungen $U(\hat x) \subset D^x,\; U(\hat y)\subset D^y$ um $\hat x$ und $\hat y$ und $\exists$ eine stetige Funktion $f\colon U(\hat x) \to U(\hat y)$, s.d. + \[F(x,f(x)) = 0\quad\forall x\in U(\hat x)\] + \item $f$ ist eindeutig bestimmt, d.h. $F(x,y) = 0$ für $(x,y) \in U(\hat x)\times U(\hat y)\Leftrightarrow y = f(x)$ + \item $f$ ist in $\hat x$ stetig differenzierbar und $J_f(\hat x) = D_xf(\hat x)\in \R^{m \times n}$ ist \[D_xf(\hat x) = -(D_yF(\hat x, \hat y))^{-1}D_xF(\hat x, \hat y).\] + \end{enumerate} +\end{satz} +\begin{proof} + \begin{enumerate} + \item O.B.d.A. sei $(\hat x, \hat y) = (0,0)$ (sonst betrachte $F(x,y) - F(\hat x, \hat y)$). Die Matrix $J_y \coloneqq D_yF(0,0)$ ist regulär. Definiere $G\colon D^x\times D^y \to \R^m,\; G(x,y) \coloneqq y - J^{-1}_yF(x,y)$. $G$ ist stetig differenzierbar und erfüllt: $G(0,0) = 0$ und $F(x,y) = 0 \Leftrightarrow G(x,y) = y$. Jacobi-Matrix von $G(x,y)$ bzgl. $y$: + \[D_yG(x,y) = \mathbb{I} - J_y^{-1} D_yF(x,y) \text{ und insb. } D_yG(0,0) = \mathbb{I} - J_y^{-1}J_y = 0.\] + $F$ stetig differenzierbar $\implies D_yG(x,y)$ stetig $\implies \exists K_r^x(0) \times K_r^y(0) \subset D^x \times D^y$ mit Radius $r$, sodass $\norm{D_yG(x,y)}_2\leq \frac{1}{2},\; (x,y)\in K_r^x(0)\times K_r^y(0)$. $G(0,0) = 0\implies \exists K_s^x(0)\subset K_r^x(0)$ mit Radius $0< s\le r$ sodass + \[\norm{G(x,0)}_2 \le \frac{1}{2} r,\; x\in K_s^x(0)\] + Ziel: Konstruiere $f\colon K_s^x(0)\to K_r^y(0)$ stetig mit $G(x,f(x)) = f(x)\; (\Leftrightarrow F(x,f(x)) = 0)$. Betrachte Fixpunktgleichung + \[G(x,y) = y,\; x\in K_s^x(0).\] + Für $(x,y_1),\; (x,y_2)\in K_s^x(0)\times K_r^y(0)$ gilt + \[\norm{G(x,y_1) - G(x,y_2)}_2 \oldstackrel{\text{MWS}}{\le} \sup_{(x,y)\in K_s^x(0)\times K_r^y(0)}\norm{D_yG(x,y)}_2 \cdot \norm{y_1-y_2}_2 \le \frac{1}{2}\norm{y_1-y_2}_2.\] + Sei $y\in \overline{K_r^y(0)}$ + \[\norm{G(x,y)}_2 \le \norm{G(x,y)-G(x,0)}_2 + \norm{G(x,0)}_2 \le \frac{1}{2}\norm{y}_2 + \frac{1}{2}r\le r\] + d.h. $G(x,\cdot)$ ist eine Selbstabbildung der abgeschlossenen Kugler $\overline{K_r^y(0)}$. + + Außerdem, $\norm{G(x,y_1) -G(x,y_2)}_2 \le \frac{1}{2}\norm{y_1-y_2}\implies G(x,\cdot)$ ist eine Kontraktion mit Lipschitz-Konstante $L=\frac{1}{2}$. Aus dem Banachschen Fixpunktsatz %add reference? + folgt $\forall x \in K_s^x(0) \exists!$ ein Fixpunkt $y(x)\in K_r^y(0)$ von $G(x,\cdot)$: + \[y(x) = \lim\limits_{k\to \infty} y^{(k)}(x),\; y^{(k)}(x)=G(x,y^{(k-1)}(x)),\; k\in \N\] mit Startpunkt $y^{(0)}(x)\coloneqq 0$. + Es gilt die Fehlerabschätzung $\forall x \in K_s^x(0)$: + \[\norm{y(x)-y^{(k)}(x)}_2 \le 2^{-k}\norm{y^{1}(x)-y^{(0)}(x)}_2 = 2^{-k}\norm{G(x,0)-0}_2 \le 2^{-k} \frac{1}{2}r\] + Für $k\in \N$ gilt \[y^{(k)}(x) = G(x,y^{(k-1)}(x)) = y^{(k-1)}(x)-J_y^{-1}\underbrace{F(x,y^{(k-1)}(x))}_{\text{stetig}}\] + Induktiv folgt $y^{(k)}(x)$ stetig in $x\in K_2^x(0)$. Definiere $f(x)=y(x),\; f\colon K_s^x(0) \to K_r^y(0)$ und nach Konstruktion gilt + \[G(x,f(x)) = f(x),\quad x\in K_s^x(0).\] + Aus der Abschätzung $\norm{y(x)-y^{(k)}(x)}\leq 2^{-k-1}\cdot r,\; x\in K_s^x(0)$ folgt, dass $y^{(k)}(x)\oldstackrel{k\to \infty}{\to y(x)}$ gleichmäßig auf $K_s^x(0)$ konvergiert $\implies y(x)$ stetig. + $\implies$ 1. Behauptung für $(\hat x, \hat y) = (0,0)$ mit $U(\hat x)\coloneqq K_s^x(0)$ und $U(\hat y)\coloneqq K_r^y(0)$. + \item Die Eindeutigkeit von $y = f(x)$ folgt nun aus dem Banachschen Fixpunktsatz. Für $x\in K_s(\hat x)$ ist der Fixpunkt der Gleichung $G(x,y) = y$ eindeutig bestimmt. + \end{enumerate} +\end{proof} +\end{document} \ No newline at end of file diff --git a/analysisII.pdf b/analysisII.pdf index 124bf23b10e73f6751723eb188cbfe9754b4751f..d0866f4ae29c659df9349b123ebdb342f248e9d2 100644 GIT binary patch delta 8856 zcmaiWWl$VIw=BB2JHb6TEbfcD1$PS`EO>xj+}$O(1$TFMcXuavAn40??|t&`{phNm zt|K*7J?Bi$Gh4-v3Be*gd8l`H`HFw@`zcwYbiy~7Uuy_)T(&<&e4h=vMJ^GB*}xWq ztIuRCQ^*(@IuWMPSV){R&i9@g*rQ}AQ3{xAGYE$12ib7saMPz6mJ$ncVoQKL zUS9X$Jy9995-sJJ5-r7S#nzv%_da8j2buILv3_aq-mja1$BI%Gw2K3XfJE(TK2s;A4nEw=8tJ-Duys$5AveV=P;ww)Z{K<|pw&DmDB zplyvh#hsG;95Pwrz;VK7v!C|i6=Qo``Ey-hIt^7x*}*W5E}D5s!B^d}6*@T^Q-`>P zNrH0^POR3i1ALl=08XVV;X?*v#;!IP z(@sEsIaZ9NF+Ix)k+FKDfqID3F2FaTL-o2_Nyp0U)0J_fg$37g6b@sguH-Ryk!wVj zZ{sLYNiHd=RV%mrp1l>YMUrXYKmf?$A=&0mF|8VKWy8C(>I$&EW;ZX6Vlp$C?QAHe z$;Z>)5iETsVMC2o=m>(VQ_Aa}(MWr}T7_(~MOakI4p$d%nSF$OS)vSf zf21q!bXP37S)v3}AZtVhT5{sh85(y^+hf^7U^>(N@Af9qJ0|UU>BTes;uS4qKldrp z*EUyw*3MkB4|g%E@#l$eQ>zL~J8g|wg`TfnpjsB^R@Efio930i0mw`Zl;}+AYEDv z^&Ijz-r(PF>ky88ox-w(;aaO@3+(p?WoxikTwrf?6 z+(NKlyNI$=PLq0pVoHr(7D@+Kf68QLV(0sU5SYn^AFbJyQ+-wz6gz(XcQjK6C*S2gqwg%hUpmAmTz}e3cYDez0^4=s}aiQ`0BD7WXBE_#^vtaiesi% z@4r=at6Zkb=Yhv2fOtsS_boUp7`UPg)u)D6RjqT^fL^MgoA?^tq4F}YkfX)p*=Cvd zw(R|=%T)h-YHH;C(WXypR%mik`HSfWQ=G%ycj~K9-s;^*UW(g?*Ry;3tuP87T0{c> zR+`TVn2Wzc4q!=Y8U(T7Wm`Qn=?iK2N|TNfAiFW~Q2ShM3dWo?5ES95L3ivLs9khp z*@SC`rt~((k=TmbOIBlWQn`oWg}+Wgi9KiA`I5Naa`2&*X84x4c`?dS;E(6wJr}&} zPUiCOAC2O69A0+LF!KXg`n5aV0xUn4bx2>YSMUT6}yfPP- zR|rS#1~Z8wHp_oWfX_@dos8|oap;$KPv6E*4K7C)8CO6QMrLK;84GNa$T%6g)1z3w z8@JZi7UTq}eo#GoFA%b?qxsdGK6?9EwUYQMVAs4$$Czij^FC$$l|W!j_-)yT-}qHc z``n#UM5UM0+R(g#h%s>q9#%L}M>y}UHVz&VnS+EGwPYxj{95(|e;{^xaMEU!Oz@=_ z;tMjpJ6{6slpKP$q!QuKH_!{rOuStCE9wr>@s;Fzm6^pYnIXVF4@SYhGXJu=grN3n{vaoI| zC5OZrnU!jr-J1Cs8s;V#847u-vD|d}Kl;iS&IMr1DbIp+ds|fCzMot`Q|XF)+v|(@ znqzXBQR@evKWa+ZsDe$!B=8ttElPx_230v^$6T*G0-dbOb%srcdwA=|x;9KTG{E}E7t32>kX|(Zc#54b&?iF7 zZTU(6rxp>|CYpjjwIPsU|y8<;UC8>nBY^!91q1hAu2Yd;`N zRI3uz-qnGA)QS}66wOWM$%9OXkvDQDnDQXbZH0kY*V6Sxw&hJ-Da`kMSB1cew%C_{ zm$=W1{{0Gg7W>jKaYNaDtKfLX+!n9uHv;RFl&J~Uh89`1Sy3wjI;?n4)NrQ=%$DA^ z=`D-MAcqGR4?rks!w01_9sN597urT=e4^J1m~v`l7s+(bhYrfW;1&?wXd(b#WMTlR z{3G_|TS8+MbpHlokL3+CZHNJ?0anpVfgYIDJg+HsKR2+IIa8--*ykdOY1iB(U?AFw zWeBV(?F)F&T3^uLrmX&0{@tDHuAlGOF&-6EHojael5fdL5(fx+fx=#&&O5iytOAo> z525nLaVb=G!}+F2y1c-j0Nc%dwQ>}45E!aY?WiRU_@>Q7VAIiM7Q_m;W zL%+-j8_J^N)i#+Y#~{*f$`(-*VZqoq$F4u8&+DP$<1LyA=Pqeju*9&Vj~wSR zNTt*ZDSp7JC8!&${De`%O60%}F1bm%M3bb%>x`_I@Go5-b7VrLWoF2?FU!`xdj{{j z(j|kQMvrXuG20Y(O2}F5G*ST4(x`id0{zI#=?5drtM4NOUGj_QUH0tLT31^+zIp9JnJC7ace#qLfc^OizWYI%uDcht=&ZrQuhQ*O~vK z6c=abv1c=gEYsi9Q~%x5n6%sO!~*64*nylDI$f9CbYFlVDNxSqn%3bh^}?g$QTg?g ztyjW2Q(D;g?21izykIVi&?IqS6CZQ2zQd^-F5j&{r2tI|x*KtJ)Q)`a*OW^%=6JCy z-7u-^-xYcbyPN$63K3A-S5HtFx?k%FL4QkB0Fu}~z7=2+BWue;5|!}VBe2i>yh%~` z&ICCi@)H@nD8@T;B$n;+x37(#q1hRhZFK5?8^{mTOb-oJimvs%_eG+!75jSOk{yu% zWGWVDhy99$t*f;sp}B|tv`ui6>+X5m@r`@3$VJJYPftWpa)R)^O>T6LPZ+F1!}T)- z_MG((29alo?FS;VMc;!=`oTJr6>qZ|>(?o4WEQ+~%~&?N1HrA~X&y&SGz>fSUaX#C zYc_w;vvMA1Yr`c#qvGQTbJ;4Z7PUPW6eZ>vgTb>Pfs`7U^t#%cvt#WNe;mn8HW;Rl z+mbW4)@NZFib$o;;%g}BIi#*2X??clRHP)!Oo(FCkU00az1`H`JurS>T7!#BFpPgY zh8z7PR=y(+W`$g{>OuK6(Y{6q_t|uj`IbpT2#kFqed*;7?YO&nxsts%_RD0)7gGmd zOrlR_P`!65V@#u-#-#+bNcGLB{nz>^9k# zT5>h1Jk7$>0JxOO^gnWV2!PCiFC4R5^R5Ql`+CPGsyqe@JA?_L%A6p74i0bBQCbUj zR+cH8zlg{VxVQr#cC&r#I@n@_nRtjtP3=(8| z^Of{ITplqozJfCo*e3l!(0bBliEb4849tkLMOjOr8#2ey^6v4PuU5ARyMuWT)z@C` zg+Mc0FX-l}Coq(k0xbOiB1JFvYfOf4a8kVkQnILo|MwT#PtD4PnavLNr4C0gT@~jp zt&G!SQuy14;`Z7i|` zI617EGMf5JRuoUU>?>}_&#|AAGIU8k-S9e(w^2owV}PM7R7D0K=gr~swPE4-Cyn_$ zsgoI9bavgOD-cn3wWR+%yMOYw#PT|6%&0RYFn#Xu!lWn)<)`Ak<@rGit0+jE#DZ!X zs&la{@?wOH6f+PexT=tMjVnghkh=N|Tvfni#=Ft->w}8!W$9x9LbrvljbD=9@7RXz zU+|FL8-T?t0b1Zlbnns&%7Bex|MwlY3;Wz-3gcd$m*f6j$R~tFN(yO@IG-=Az{DI@ zALAchUAKkTSX=kEj>^Z8#f{H?p4ah0w7LZ46nizIFD-Q{9@351-auP8J*-l;x@@J4 zLKQqNZTa*E?gXbn1Sj8sM*7f2wtDHadp^DC0`NGpV5Vv&+Feog_PoI@sdAe zW1viB32t16ppsoXD71`ZZws*Hh9!3&fxt&Ozi*YM4l@CwA~T!hPU@XkZrsPYulR!M zI`DG8Hk#$aY)&8ESu=yYS@+2((?|+${0@$hk@}QMNd=G7yqgU7k_643x;qKYkP6QP&kJrK zVYRY4zMY(?>PUzJR^!eCFeYKP3a!Lv%gq^egka};fz|v#y}eD&eGV0cte!7*m^ItICPk?W;>c`z4VUk=BQy}LMh?7eIx#~o@UO9 zE~jFim<@qQipbC~eMLR;j__fagy8T5?72=&MdGZ*9|J{PS26ebW( zpl%67@mYpW{;Kv@q9TBALZ6CcQm43(mTWKyS<<>VYJvJn=nFo;lOW9{3;4Vzn|(|% z0ybhl#LL>FMf7>BQG-{Xv2zg~1uA}zsjLy}&xn7_wvnoOM2yV$+%4gT2c!O8Ab$Bv zfu(}v?+=5L*bz#?WDSrsS8~GDH2X8*1V0K2gvbRD!3Wcl&9Mg~B%!8pAUhyuiNH}^ zkM@K?Jc%PQ0m>s{vM>g=xl;_+J$35;qMXffe`i8Q8OWG zV{!@b5hho!q=))t{j9ld-u^HNXcvdfGwSzzwf6>p|1RSF?jr~UHwOn%BXHQABa6F^ zO^l`1XRf~9yZh);_DyE+?zdGPeW4?$7WMlTt8(sa2#vwT!5C{fpFv7eb1RD(hHo!( z-LW&yEhDR1pAQd3t;MyVQ|2z|q+eO&U4HeP!i76e8R+xUiB^Q1(NC7wCB zi*{H1&WKo6?5wJy+b5`JyNFug>BXH>>%c zcC5AFfzXg9p)6&qIL{?lQFWVd$44)vLY>{EnUBVF^RWfDgZ6%!lbNJXUVpo9OKuO- zt&L<$6bPw1n7GQqj@ot@vJ(>0wo)Tm;aS9Sw*tKPC@gJyg;Z;tJZ&GoF{ww4!QCP` z;oQ@YC{u3emzosR8Kx(iyd%jct9|n1F*6n_FBz!UAb}JB3<*i-3R7QL;mN=yGcf2L zh02KQUX-y1dJJ;(`;;93l~Ipt*2q@1ndK2sM@A&*MS%#}VuvD~fj*dcj0p|qYnS)F z9I%m|I+1H)3f<;$WG>o)lT?aweEom|tNu^zsLg}4i(cz7e_t8G!hMvNuJ=!A7#;pN z-F&3psLP3R;Ab}m>QD~5gFG;z14aFgl=hVSjeJRuT_{H+1eb^3D8wAA5ij;;}PPNNv+Ew+Gq+KE` za4Mf>{TQvqH1eDFVxmcUzL5l=<% zczSqRePk82e3glvjKF-xYlOht$7 z{82*mM}`-rOvOJ|a^Bzx`JI-#J>Y?HdWGgCI0!lO@!`7W#T)UQ5Y^IicEVFTSru2@ z_W{;f#dS`>{*jzR*I)MID-j7v$IlKd1cbxGwD8+QN}TB&UUGb6U9TT+@9uuSy;J#% zhvjE?2XzAce$G=VoO2HDe@%T*y2QEAF55lEbYC|Fr~YE00g1s5)-A<(`KI#yf>r#& zSNjv10>0R-^jtnCMz8!1@k-=}0ituNGP*1u;yd+d1;1xDPS+T{CyAM}LLOiE_5`Zz zI-Q9OZ-1F;>ak^yNoBjICKu*>e-n`j@e0)fJKjZX+>ezTtnE`_rN-{psr+h4<;s!@ zxw2`=-Sx?Avi$;H&)IW#+jtY$?o=2j%@c^h#ujCT*>_!Vn>;MvQCg?42TFfLR((*? z8O466kR5{SE4emgI_X=9qB%v6oS%WykXR0GJz#atLrUpx>9X8$}NrG|jiHY;52~>HEjo25KnByFT~%-P3Fr z{?)W_fx^VAmR;upI*n#8?(9KfcY)l=t40{7Sv5(z+S_gwIHKA~5NYxmA8Ynu*DpbA zD~My$w{10jztKTWESS4!C5!@k)>}`b%CFWQLi#41DkarakE$J&bB4lF=SG*y&uoI&9GO+6w0~s?6AIyhn?Jag3xX%&Evmn562SQ zqj}}3Zlq>Q&v%3POzlJZ`8s5dI$k|A6ohi2s1}56I2pK$MVXcpeUZAV=!t84B5d)8^|llvjWn zuQ*78OPUV|l;Y+S=i%Yw=H~|TNK10@b4mh1;=CdhLcIUK5JO}(#jg%luGSQsyzKn{ z{jK^AoUr%-*J_VBj6Sz(P;d9@W?O|!P{rFh1O*DO1%;Ed_P9cE!GW+@DUcWGDo|<~=1Ch9QWGAq1Ntn3EwC zxim;g6@IaX$Tk=MESHSf1cgf+co2C^e_z!k{^hkX-;EEJzyu$Em(Zik*iNva*C@>& zha)61;|?>2324YsMZDK~8#++0M=>#X+$0OHJS98I0@9Pk;mGs@yqACF=^bpegTtAEs-hcFNN9$-a79!&?mk zz$6Z~ZG#du(&NEP*gR-DEpXrU-xayI_SvCdOwrN99y5pUH{|#bLwDE}d$H0&bafzJ zH(Ns0C8US*N4s2xhLxz5Uv+S=DiK_>wr0TC`zh8xgTD0GNGG;X?RCl^`{gOS3mM zN+=D10lw^UdVZ%qQn|KjLFe6JTL>kJ)dw%*w{pJh|5l3Wf_Ll}-NApwap>>f*}g`9 zMR-T~2o$Wrq9#KGKVgc&B*e{4qwQ;DH~?-bQQ%b(rBqRARgn+9fBUy+f|{{;KzI#j zYK%Qm;X+-7i4wkqXZ@<=A$sgzsJ3U$IB>)BWG7I^)1s^AerczNnzPg2b{@N4&pw^p zdAnjUphF*6%ieNa-qsM7nGsE7>u;fSH*drD{h4|2dZwRHL77i@SBsRFGtfV4VBPyzPeRB}f<;!) z-fF=A422ofQ}a|Q_%KFYLm>U{we<2S_V^+0LMirv_~1fKBd&^pU5N90k6i_+IsUq7 zfnI#b!zR#6J1tK*Z{9=LS&&30WaWO&iXI{z_(_1NQ`^z`-ZWUf+J)?C`t(FVeNQ%= znwGWq5j=$`frzvCMcL)o-K7kE76MUYHByfsgBR?`ey< zcsUt5QfJqJSu7bDlK&ygh?3J-!ZIxeS)-vp?!9Sd<0WNu3_qQO>p7Ux>&!yPnU zORCQ?`_4vwQJ2I7wW_`;8_&!m=l=6X+9`8!Ba%FADdDX1 zS5>6N8oVa4WVi=z0SjPFCrMSXKxrso1w9(uRrpt6W-gH)I2W%8T7Q0*2T7WI!TCQ1 znSmEHp}z=lIPo_+3EZ5)_+f(R`dx^jSMisT7ooHXTT~yRVqXg!Q>TCfj5uGCw9xsky++|%J$;el4nu0MUNk`pGE}WJ& zMRrx1!n6r3gZ8Q4QaJiQHNzul)l=+ss7%&rTfkmQ!&rC<8q8|7l7rZI3e5JgG#%6D ze79z!kzXIab<;$!o&9T#;D7L6iJRjjC(S~L2qucy+@dq{vT39U;^tnjbvJ&KRv*Xp zA9t%Ce=k0K{(wbZM04J9rr+62J_u?)P*y(RFFs)YfT3TyDKOM~{;v%FKRV>i_`mfe z5dZ&^0peb`mN3N_tki3BQ^i2LP!SfEPr59mSre(+PL)wmIAF19F5FD`B1j)D@}^d! z5h0BH@93_u_FuX>-XEH{9||2TH%-Kj*#?(&G<3f$qswaSdSG2k)kN6B!GGIP#tK|d z{2^?kG1Ydw@JMnSFLmyTal?8k^J8L2xF9`j3>sGh5sN%1u`r+^wF&)?|H61)xaeLl);Gl17+G_M)-b`yk) zAVYRU;RNPZmFfQ;K;ZvuG&WSZI-5{*GQit$nIqzm&8ZZhU1?aCz*a{T1=eU2Ne6eFHbeuT-v*+1$eS zr`)CxnxBSqOEra`-pIw3;efGFAQLj@XpIHKpcD+=u3DB%w880oFj=u9OlxqBs z6*gyJJ98a--6cdX{Gn%>O(fV+^|GEntZ|1-$gL}tAt^S9dU=VtgIfIKKjpsgwQ`2z}MjNr%(;6ALvAo zD8!fl?-ki2;qAc01p+oeIpWs2tedo3RAUSqA03r+e87$dssBv<^NK==$c{`)E2Sch G{J#MHpWi9~ delta 8847 zcmaiVRZtwz5-jfSBtUSNMS|<%E(z|oI0SbVcXxMp!UDlvg1bX-cMtk<@BMinuj+kt zovxWPH6PQ{lljb+_sr%W3>^#;3=4r%2!-JUq-{??(}VZ*w4Fcm;P^e(rFSGW(MG@2 z@74_1@U0YQkFLtuRqhB*&&YhrD(s-fXL`S1Lm!7mV;5ITjH~gjmqi!LpLtxeyz0G< z%F}ji?vjpb3?%ro;qBye*~=-8DxH1M#w*N%Mx{2=M=xJ}FnBF{y(WPpS;|>+zjw5> zdmOE3Kmr!}c$|r$VmUT?;A_=6|05KRdO z$GktRErN3Txc-!U)9Kx1xNbFz)S}Ch)i8JbGy$INplu|N`awW7bd;!odwylZV7&HC z4%G&>%)U#T?RW5gD)#PoW4Mrw)}F^l`Us#S7h14CG?S6rdTIepJl2q zudFUOfysw03H~)~!7B;7081jk+HqO|)K_>~Wv>*>y+NBX zqELdw$uhjzQK)BU(?~NKkqc$l=USwPjXkGIngm;FS|M2(QtRoJQ-HJ%X!NC+3kI7N zP*FHkAu}5#5UWGiEwUeGopETOzZpv;-E+sI$ppW&-dwA|vn^GFw;q@EF=v;=JpHc6a-B z(pRBtS$UAtflaD^jfbMhBRki`W`n|=f?M+Hs`h2Ic5P=o!lE>Ra>Z4ZoqaZ!Nu&2) zex3Y95i#i6h_}tGKfUke1O$bPjq^DKazaNZx~kn4`xt7Rx&&;}245%CZ-R%$Dh2C2 zA8*bpy)>Qg*7mc*%Gg*5%kJC%I`Lr&FlavZ|4QrGcs`Zd#B?y|MXU?hKmEIKywU*@ zdy>Iog>jTsqo%0*hW;Co*GTf-qY#RL(BUT71qFk#q?Ou*3k2b-dxkGJIglWIoItojk{h@p&YuC1B& zqg-98ho6tI*LI9YWx`euf6;KCnE;)P?VqYjJofZcq-PwU}uWMTQ;<)Qz z?C(yUVv{}h&K#bE*eN<0S*<~}c zu>Mg>3C%Y!sn;~Q@$u8w&(Af`7xXY-x$X>j)K@E38$paHvjn3Xud@V7LTI5InDclY zjfZcP0d5moEwHK<4e{%A3<>x(h`*^+5ESqLnmK38jeaA&%#D`wG|cp)`)_)aQ6^{J zDp}Od-<#IJ=^0|Ui!?FqkMHXCyD-y5=MI81P|F_FtZX+_NS2$V1>)a*ZaDcb(E>O} z%8|C(=l2aPRKbDsXHbBQfQ0gc34eNM_(27A0!8e8nV`=#iuxpTMhN6;aIt#P zv|ot4gnBRzTJ7`tu0!WKF)K|{)wnT?byfF6 z7P1sg)5%2jMG{kNl@p>9QP9hZp$B5tng)Nv>D6Cmbp&WJl@nq>1L~v-XNIkOzs48F z5$+FEK|EsF_fT0ZSoAYxg@+OJ8{O4S{aW*WLxsWt4JvCPj?=w*xr)`ZrmSv5DOVJH z41+N$EZ`w^DPM$p4e^tj%lO3)-1GN9U@xg9k4cHvXf)jV{!B$&c?KNfFX$QJ?-A#R z)2`gbr-osHrlFrkGD;Q=1BK-&Ti;;rGBz57tENh(Iyvf&^RXB8c&7+rh{J;C?_`up z%-|2fp~BE;_~H5cRdCYZpRBPh`Lw2VvFR8h`W9945CIuI z2aSa_#nw$J{_MbuV_+aj1(&HPqCBTu95m_ga4cRzI<-aha=ggiQcC*8lzD5>#boRu zrkhriWR=CB>KR}pp4p{baxogm7r(NU^6-&gI3dH=$>;jvI4CTavEDHDW8J-s$@Q_} z2;lpUYzJWl3pL_u;T4kR5`U(|lR4$I&)-wq;-hK8lTOgEYdL>s)HQBou=ylclKN!SYNPrPlhC~>(2rqdR@@|uBytxp-3I9&#uGS_oeGkoMMt1mYWR@9X#*I*CchsihqTpI^%L* zD%i`a^pxG|J>?JRbQMibwO?sT7v`H49K#;wDzBI|^xTn`nPv`-9FddC%LII1S@w9m zt(p@>PTgRGVfeU1Jk4^tw-^y6SknvcAXF1s6QGTdU<-4EDOuqr$4xVZo&t z0LhayF8t|-Oal+nJB5LFIWU0>e+hp2;J**aq)@~>7agZ_6V*Tm*v-LhbveWHDd*fV zZeorZ{2lso&`ibm+g*;g%Dn=nUEaOO?bm1LaJv%vm#S@RPKQ4cLge6-nH$QX=?9J_9<03C#YxsA~Z@>8hLyNL?=QnY^Y=VB??w-cr7Yrh6_ z64>_9t7A|+Ps)liODR;4sy*?KX{M_iOme@-NM+aAp#sZXfiGRdZjvbwtCXb@~^x4dYUr^FRQBXektiGGoS~yvb*O7ica1 zU9!O^)ou`N5!s_jQ#NNBWUx$k0o!(rAH7a zu*}-&_GyrypQL5En^f>?z1le6|2wK25_i>>26pB*oJU`L><_}+L;lUL4bL9tdqOqF zx=Xy!e|r84`yjGSMRKu4HVIN0cp=BoxG9BG>rn^F!vn7;!JZOobGmwUm`xQip$fdH z*aUdqotkHgcLTj_ImT~`k6qAUdU|TnUP=}kga%UaOmuWaR;!F)GJKs8>08)xxdur_ z`V62c_#Lij!)*|E2MSfy|I~O0cl%v+{55@o>feT!zIBW$56dj>jF2szO+b<@RONH$ z+QoBLNd>2Vdl0l*vuXlWALdwi&-&!>erjs*Y%Ex8#@3`pO5$NJ=>kY3{%oK?B#x;t z^#%z`LC2Nso8;P%1CJkU_FL2|gxnHp+m;jrPyJ&QK`2eq9T{9I-gxU5m;BAW7QA!; zTjK3m=h=sMdRsN9fBfe?g@d!1@5-q!evEzQI0WmGWx`85U@dtfNEW|39Kqp-(3_KoLoqqu$_@&$^fsd?*Z*?4)`cP{mo zk|B)LT3w+C36Y5A4CvNzWQF*j%J!T-PB`!`AMlXyD*N^Kwy-X~2|fuQANKxp(tedJ z4Ddo1f4y>6?6oG>(3*?rq|?$l(}nO1I=;U%WoH4$Vo}k~pFzF}_Zr)evwkDpa}lSC zph51A*z8_^qPY&`LIOXTJyqf1AC-A#%M?iLKiar)F*vN2+VeQ4--8lNJN`<$l^>S2 ztQH16Ur4E9=DGk5IfrQYu+qkj^l(@&0N_BRD~O$&kWDQSD>vqx_ZX*Cujgmf2*6d^ z(*9hHz|0$g#oR{ItMH!-BkhaFue16bHNrbFIl`NE!yUJB;MQ91g}i%D1?M21Zxgdn zm}j-H-`k5s_=Wsex=WV7rRAcTW-G7xNP~~XjsDcPy4HWN88ApyqnO*9IlDNS8@J4) zz`>`a8G~RLQ<>no!4ep0jg8KK4}hLD#1;k&rResV@Pe{a!G+Sp-|K%NmIhfEID;Sq zjpjq9InBHQ?Pe6^2}`la6vyyz;RlLFK$L1Cj6PuVV*EtIj0~*;IoV)ORehl5rqLh~ zR)7aHu2@I{to+r$PWs+#z!f>kUPe!G{stB+{H8ZkdUEyv7)y*_XRsI>(DRA5W`Na3 zOpEh|0466NAIT*&onqR#TC`0sgT51!oH_|w_$C!enkkjRDngpQM8>=5KTT_FamTQ z!Wl#d4hsXWj{u9=220@vK3fK^S%6=k5hRpWMBX3-jwe~Rg8d!vQ%P>5cS=@G-0=WU z1<%O;+m#;FY)(TDpNkVMmT`tNdK}{R8&bGwZ2o5m6f_tCDCB@7VJO;OFGEO4LW6>x z3xXbG6gG!|P8_;gkBQmS6Kf7X_Lsg%89Q7B3I@9-dLQ0lt6e^cY3Kgj z0m)Ww)>uLrBO~;7|ECzD(hdW1mmkd#$*T(n@$?Xu7z^ccdRNorifUKM));YRvlmMB z^c`AcW3qPwf;jY==FbKCGCr1;WBtc22oV;Fk5nm(O7@#^qjN7&ZJzJu9B;gghT{jC z3mtp`d1)U^U`E5-h=Tx^wqW+Yc3_dPRV`}eOJZVrZdy1iJd60Q;jNGfo4MFkcUvDH{^JJO;L;Ea3TUXO3_AcA#k7?xR~1!*14c`xDhoX z{RB+D5nAjT$k46CKjbl!wyAWKNH%HTXq5}JK)YHzoJrVk`#hKWjUvnAq+Xqm`xaa? z&=oTs-Kv`Ew@#+IJUv6O1nneWD)OOKBUUSU_o2ED*)3)LrI2OI&z6i$hn(F!1jaoG z75l+fx*kH|3;E8@-8t%(>RQO&MtK_)8tqUd;bI-~_}}w2%-nx3T-D@*jZvQIY{HF> z@ewwzT)p%X7pN z>+QiUCK<7nbWi$yuFFN-zfES|ZI$%|v0s zj~^k3*no?3Q=JAcM8(jT{31f>nRx9@Q$d4#_Cp&15mA}WVW3yclMhtr<2f>LdpHp_ zsH?GQ(A}M5XjtE9>H4?wS+KQQDb_H%u_PLjrSC`dC$J7nQT*j1$B97XG1l2nA`MKe zZ-3;Zm=?+Ut*v7>bM8&JRPWa>OPoE2_0& zHH3FqMKI1{e>QY^!(_kqdIjIme`MK}YzvvUWm*G@XOB7rV# zcN$aWu~W9L=}wwUa#+!p_#o}`NMW%Y@tF4+$h&lDu%ce*HD0;HP;*GzngVwB^sL}6 z?2aoXh4#oIY|g?^eoP+wJtQ|zq3<*N&l$fKv`ti9Gv_wULEHJ&;rmC$QgTG3eE3q1 zgxg`9lz$&~yk&k=+(_Vs^=UkRie0^tXRVFIN_QafA`)^;7{0Y;5pUmJfjRYU^TBTo z+p$ZzIAPFvJn2}x{Ni+axB;$su(>Fg_36-Jf_}SJ(a-SBe5~eoc>Ho}TiJqqDKlia z`>@E_*g<}IsdXCda5)ttXtOJ zRNNYs)=jB?_*eD*Q2fu{w0e14XnNDTPs3p4@Zkz*pMOQa)kV|i&mQUXLMt-QM$I!a zDTg4TtAlB_5=yf6Vy_dwgxGl_O}${3Y3$~C%~bQkw;2S1pJ}<@VV;} z7@q97yj=0M5S+?tz4R6X!ATi+UwS$!7n7gP;9ODo6Ae%xs_l4&v>C!Ab`jcu@vq2# zTUd<*`Oy0V=>tvZ1Jvk)oR(qmRgiTHP$~-$tP3!j#e}$kwH|&7b=Cr05-s@lJ&f5w zd|H7;dKH_Ctd^m^08~0kG`g=A6#N>XH7Y$C3DbSd6d*B)Kf`kH5z-$GF%c4w%UJ%x zZAZ(6^E0buAjUF<0aq-<&B+RU`XkgHwdPxW-~I_4WHtwym19%NCP_o!EL3P3bpwEu zvy(!7FW|+SqGb2wEX0q+Wi8%Q3MkC|~ zL%WTo(91;S13CmkZSOeL6a-?MFCzVxPNqCK;LxMFF5C8OxxlvMcx(z$gIsQVjwqdT43Xt`*M@ zVA9y`Pzk`v5nO(9%A)W@j6EvQ{|dhQQyi5_nK914(Zh-p6OHzq1)o#Oi!9cl*TZHK z@po2oFVp;daNP(Q$s(|X=dfCp6u$emAs3G(#vw890>q4G;)L3u#QaclrR|=wOmTS!oQ>NY!kf3Jc9wIs;6I8DK!s7=Q}4B-M93g=+*EUcW|XHCX9eoXh?DJ!L*oIl@x&|0yaCoJ zGb~{&D$t5;DJXybV;z>MH18RZv~)(NFl~$-p)G)8t)X+3pT(H&NWPU@=RA?z8?jc~l7S z^M7Onoih!;F%8=}jp$wQ1z`(-7V!rnJtc_tV)So0M523nN&L}Bf|1jM#R2eh>shf+ zIOE}e-g0yO2o8+M2)2+&LL{}qOn2fQkFkfh;j)9j{rHq>46k8~&hN>PQDAk>F&8_{3*2=I@aKiuK?Fq+N;{c=8gn;;>LLu%|g zk&hbKSdXMoL={}C_*!Q$-|vzEpR8W=02De5(Y39w<1TSD~NQ1ThbqHQigf-s+`G@oC zXR*LmID0ur%wwxczpn>A)%h7tDGY6}lUK*q^9!Bf$pd0mem;S#mvUsQsTyJj_;F6d zJJz5kSdCo|c^6?5wge34HE>|AFT092h?aQdnK1a`!*LuRg=_u^7Q2{QMzA()UrZQJ zl%ii-#^#@g%Byf!zodxreeZKyM!?6J*EnX?4cOmT`8qGkGT+Nuo|gX>d9e-^^Kbi% z=<2yhty7KmoZAflqYp^G{W2`+G_w9Q`ai$%ttUhfBOHds87DZ+__u5iy-3l0aM7~$ zvo$X<;s_WVy!?Ks0EXVbWiDyr(Qlt>ta#Bl^TD_5h_^prVr8ymO32*s2#k?qdmX3C zvjdWF1fPR1eEzGJfI0}!Mn7=u(WM)moQ+#Yz_w%3BK2#>Jx#soDmpoW#2^m-*x6b! z+|>LLD+4)K>a>=c$!+VKiH@M&G7#B7;dl})VgbE);O5mhBCZDN#w5dDm$lbM`4k&0 zF_?%WFUQQ#P&|Ocpo>X8&MjU92VSiV!^W5Y&aB!XF`9@j4`&+%*Wz&G_n$`piw@$T zuFl0GcBD?`aj9L_}^36CLR*^@$OsXe}t;0dPRW{?l4y z-L?>eTkVg()gL~KVBu4dJn((q)bo_Z+hOB7s@4a5^#{x%82Yr^|K70YnQbHhuKxcK zHhLrcUzkP(<^L*BK~{fNSr%8V%CtwH2a=wCQ*OaA7occ3P(FRTv9yZlpgB*4=sH;+ z(kvE_M~~7ep^)(RWLa;oUtcyN*`0xVW1C$k&!zCDm;#baP5f+e-|qlAUmDY zk1T+@`@UZf&zO(X+=o$~y1dVGU;n8*gnt>~zMj-WJRWQXm)p+So{yVJ4Pre;&I|x= zpkc0NxGkf5=ffi8%kM~jS)ghwK=dlWd;p4^1&KxQ+#OAd=YwmXLye+Sb6rRo z{oes6Y05p+`kmx{M4aDT?kUT*0s5QxW*(aHGsB3TnC~FYTbKJykD*DDf~poU9>MWR z{8c5jGdS8FmxT<%Nq2O@n-~1e#T84rpF^K?8DFcUDyS2|;^+kKDv_#DOq0$WcXDRFZ+vx#us-N3ny6;N0%Sy|T$_rILKclU9DpprMh&81M~%>y83!g3 z-9*wd!_5*WHvKCwV=)5(VF+=4#rK53u~Pp47x-lv1Q?_kFtlJf=~A?Oa(XG~iZMwC zhO(ORiuURF<%2IzDlO`13lh0V`k znro-JW(4Z%!(qr*`rmhz&7suVniVz1vdt}4HhI+Dcllo_f7^Ur{gm?kIKTNp-ujMe z^;6RPFi+E0)`}J~y+6>u^&1v*Sd`vGJ_V>QW9n4x=9_0i*GwZ% z?+C)_q&@#ArQw6siE+*k`30pD^qea{=70>ji47vyke&7tK-aN;Hc_dvMThh(irLuU zLaGZRBh8B<+wO@&8OGLDnFo{6{rN|S0&qrY{ae~r@b-Y8Zv64(nxV9YK~X~aL~{Og z_qN0UeC<-ovF~OXk<8x`Y67jx19ZxR?0Uq#+yuBpsm`D_&CK2VSY`i^#MvJ_5O#dP XLd&alr(M6IQXui5(9lRKOQHN9WM|LK