From e7b2d4b76852841114e8e76af4883b1df099ed3d Mon Sep 17 00:00:00 2001 From: flavis Date: Sat, 18 Jan 2020 14:18:45 +0100 Subject: [PATCH] update ana11 --- ws2019/ana/uebungen/ana11.pdf | Bin 142237 -> 142168 bytes ws2019/ana/uebungen/ana11.tex | 12 ++++++------ 2 files changed, 6 insertions(+), 6 deletions(-) diff --git a/ws2019/ana/uebungen/ana11.pdf b/ws2019/ana/uebungen/ana11.pdf index 9c6e92498b37b7c35f39210eadba7050f4e31351..30a6c2531968c9a06aaae3545a9f091fd709eae1 100644 GIT binary patch delta 13190 zcmai(LwleNu(V^_wr$(CoryEC{lvz^n%MTlwrx)|v90et-}wP&cQ3jZwdt#>I%gGT zZ3X5R1sDr^va=!$aAY9wirxhZB zXNQ6tCX;ZfrvF|~B4^uf$CO=uK0ZAqEPDdpYqcH;Rp)Q%-#QM&>DiOUbpc&3UuS=J zu@Tw}A%C{zrv)AjZ$4w!0VTtXUuUDie-jjL=b8PCVfrZXgll88vA^wRm;ElU%Hv3| zzI`pF9d7>t9=#ULVU~a97EnrD_2+`=2iht;Wnk7(%rpy9DGfvzkIFjV&$B0!vWFX) zQ6^zn{NXlY<;WnOhYBcX-Y9UL#FQn}_pW_8yn{^XjXf9)e$H@KfW z>sogK2xjpb(~X{waH!QHK;eeJS4-fg2`HM_c^&qL#x=?)PD+EFmPw!T4JmC_Gp$@a znR*N3_VT%PLFBSYmwBWpi5eSOb*vXx!OI-+Et$@JxLg(qTwA8~%iGefG@8Zc?g3Gx zRU21~)$Doq4_-Z{`5qX^R7zWzVPs0U;n!2(mz&#MG#AQn_-sX_Mi)(pWalzfKYlK{ zGdkJ|qE+*;0yG9@BU20-qWaP~8cX2MnDCg6sJ5V!?zTRC?wUsJD|P|Xrqo6zjRqZz zc}8m}NHd9rCq7#FQ#Q53a)S4e0J*a!mFjklWo$b8fak6bd*+Hl4UCehn;i%?{aq*E z%!iT>Epxsi(od$)jDu2}#uQ#=o=4$)b(d*Z1tlBv@8R0ZE%6pC;hfQ9ZTEmwj8S^h z!f?zHxOdu~0Z5`AclGuoCPlpuL&Dns<5WL(zYE9cj?j=Du45z5K2%8|ROHT~sska9KV4WX633q;2uJ$v^S z&7ro{kQf1Fr$BPsU3+X^LbIleIgL2I%5cRKe5cqh_1}^Ecb|gKn544pqvhNgF35cO zBiwe=a29AsEvG#@n$ zuf_9OqXa`SM(cvsf#o4%rR!%)hj9%t`hyDXgHTsS3NtBMN00ehdfdpd@fjgW8&X`P zM7Sx`?DTu9D%}#*)rE>#hjn(z6gB(IYrc?e@>fG{a|d=-5V-*PfVly07A$9fSOW#d z+U{t5P9QvyMsu}G?%mc%eu*^z#(_?Z$yNfgk1d@%t0x!2BqkqaVAhlxWal?jJ-G(P zxQMfIcg?Pkff9IQKI>*d_BR9%-3Ygllur0VpH2^aU+S=#KiIf}ktJQVh8jpiC>$tE zWF;nODGwQ}I-(m+xPzOsMPJ0c29&&vQ|KMjOM?-@?IiL!Tx5Q#y@vps=Zc0mZ~qKZ zTvd?IErKZ-BTE@fXipQ19cWv|+;rx?OlW&0B|F14wz}&s z+Q`GLUq9*G>5kcmxOA`M7q(=WfJzK|pqF)bp>rTUEn-QobPif&zh%qtkz)iJC0CZT zYpnzhlT;vr%O-@Ya{#~)qAxXsL2Iz+&f^F3@>p9itW`?7e?-Q-yw$f&*z-^yFO-i= zO^!pcjfeiCGo!8hQGpv%QW;$`Se}e8j>D838G)Ih+t9ooU1VkX!-#?Nq`k4d0b}N0 zqQV6+p)28|(mziW=iJ9u+IGplX@hBLRK!~cL*2Dd$;LHVIA0)qi!-(Hhf?87`izdO zW-3d0!icxsY5EJ~cLztkPAyS3#|pIL_o+iUT=_4n+Oiy3%h}?AM*R~7Ld^Z&(w^UJ zs*tYJu{A1=D#HB^OM`WMciU3S>;wD9Dv*7_hEAFXGWGgG=3-LlGJ{ z3&C-rY?vD^R06=gP5J)ZVhd+w+g4yMC^vJcm}5<298u_)!N1=p&<}Ctn_?UOHe93( zphI#;U?7pPR%Q~NNfy5A$fqiF!=?pGupx+B&U-)6{fbC!b~=v&NH}SpVec?zs5fH; zKw &L<3dPecb7dKVVc%1f4zfvf+`co1q2)YSSy(j_nnR7b|&kf76Ase$T6IP#6 zC-J&B$8K^&q$~vkP#V)0@+ksU|DW|Yi|RQE1qhB+xG~7JMBd|AlhY2G+J9I|{0z8D zCPRCi9<)(trfB2>Wd2aQ5g>EXev|XKDdbWB#oP((<1;vy!m!HcH~Q0XfYFKv zJiQX19|M5vY`oKA<1ek`tcdvJ_8_eEPQ953)%rdJ>{K0?C77}JUY-|4i|QHDZV!5q zy7d%I?>nh#BtJNl7x3jBIf2Q2{gg-z{Btj{<%^#=KNA|nQyg+kV&2PdfP8mp;+=}^ zj1}jh=%sjTLK;yDvNKq{2oxny{FM%mDqTJ7D{H`Cr2>0pZcK|6vl0XWrvDtFk%+o9AY$GKC~2SIEswP9Q!u<1dOa~m@G|W9bQ6@(XPMFE;DyG zmIj@v%QT+xk*d3bVciZjgl@->s-*XExdV`_P_+e;%n_rjPtEA zr3S)!I-sI!#_@(Vz|@IYa+n?uYWJK)JY zmZ9^>rbnu#mAnP+JX4tB7NR~Vpr+wqrk%n4tRD*P)47sY;J}2wIh%Acl-of?nT^KJ z!FKag;^s72I0-UoGLq(-nd8V%zpu8^Bm!Ke9H@iW)V%qzF<7cSk#K|56~mJ2&<)60 z)8L3UQq9B?U$eQvxsB7kdwyr@K5xeo|K>|gpTAGqxT2kJF3=xKAP~Sgqn?txYTTGr zL1kB``*9pg*a)i7JkDs-x^YLPomWTPD z88KV}#4&bocSuVmN1)be%CYR|+1b#$cAGM({??gO`l15`N7w2(&iQA8YDh;8T+oAH zwrJw@X_NNtv&$|=77tVZNc;BHsG!?jPP@&MCnl?njaIhL)c2UYY?*Kf6#y%Smnlrk zuz%XhXcZ#$)TEnL!@oicJQRvWqVD3-yJU)t4z=}amCM_vz$BxImgDR@ycI@}q@x?z zUzXHd>)wU7w1w`T%bu4bw08u1U6Fj-eYTe8ugv~a;j&-V!Lc1rwAwJQ$-VH%l)fDq z0z&sa0b=We*QVcsI)dLa2~-)7Ozfnd?xvmwzj;iHXoYWp^R2VTHL+za~;2 zk8;4%GjSw`D3=d1F32!H$I-W{%0XZnX zl4a*bXiICZo6Cr85^1lG?jb1NBgXX}1fL6^n z%c_glu?g5x<6_nRgkSZoION-*?)PfWE*8Q>Dk~Cuk0~-%!dTHGOdK^rPDsSdR zpFfXl69^P`P8QT^5AB9l@nN!6Zu=uEqnwXGYXu_l68KnXfwS^t2|`v?bfnAr2%M<0 zY-JKs;sBI#7LB(2h@=V53SUvEW$j%5rnri@7R$Df1-r0Bvu`O&hwL$HnB)b%=(~9u zzKx5;vZ&H7bj66i(A4{lUs0nV|6~tgBF3>jepvTYR74ipc=F>gtWKNw38;JJoiRAq}0^jOx2}G#?rK&rQ<_xj|6~J zaun`OY8|d3)o2(IzV{S&CX$d$={STDD~=cnI#=cn(Jfbwi?d2H?3Z-50=^ZJ^5z6l z8vW|NAJCmU_hmQXRB=MO^sO2(>s?y@DX_4PdX)eGQ@9zwcOCRIf148;>e?j46#IK9kR1D)1AFVa&N%!Iy4Mjj*h z*u6FZSg2eBq%u7TQYD3t!(TT-zt@N$+f|SH9X+qUXnFL``f89&s`;g z@x7|{jn?QAMs1QM12jt>KGFM%=ub0+3oj=*i=Y0_vXNGqZbjq|P_;~h9U zDsU$%T#){hlRkiSJ^I2Q-=6^A^qT3O)Quh0=8vPv%z1f!U!3c7(f?Aaa(k;|(X=#! znn|v!POz7s%Z#^v8}{~gcK9|f>g9c%+WfZ+Uo-u>nLXKgf8O-&`!)3&O>kY61Yha9 z3vgYme!siCJX*gTwwm4z_&$3&yZSn-{yg%0-+cd^PA1L;eD9X;lYu-GB})1;CDNif zRZZa@q1P{HUBf-?T`e@?uruMufz{M`oxisS?vDw{y_i==#VT|fJNgC9HcU^;%hSNo zmjm!U>F=6V^`yaEoT~jo%-OoJYh)x?u;{vV*=*?T`FMbisbtIvbn^K6yU(WgnX%wMHSdZ?3)el5i&MRM%3^c`9sj3}g7NtAba44n768l1SMA5+zw*(qZv{h7O;Mxrq%xhMmJ7JnvSmF>5bQB+ojtpbxE3VkTy*W1sj z^e{=YSv{S4-QT6S#tj7rhZ|ib5omb1br2-2XINX>Q*`mDULCOJ?<8eBT~~I+(lJ$s zQn6%CYpnB73th_^nZ-r&iU7!IWl+kZ{|tdyH+ay(tCgI}P@*kyXnSOIhoLonHksLa z-!k|@;|}CcaUD2|OzfqYhjIvtr(|8@p*;L(34t-!i3P)eO?OI-kWcIx`_EW2)43_b z_V?+_xs$b%iGzlwpuFLfx~)Wz<272P&Jfsr^{-S2U>OK$*9zXe5+y}J)~4kd zZ+1xwN=gpN4tFRKmARAeLjQ)nx>CO_K1>+*s&V+s&P+y zPuFZ9$cuf&ImOKQil*UiweDgt-6_%hO65?1)WB?tya!d|4SghfHIo(_eiXy|c5X`O zvIi4=nfR0hQmbh>_Ynx%efR2cE-gF8pK+cosDnteasY^Qjve!JJ0QtQXAXB)X7A`w zN7@@~KnsTdgE>xo7VT^(V5DLQwiZcZV3scGDJ{EBam?zJOpj*L`*#8@9~1W92!A^# zXgMdSf+V3OScbSu4gpu+jg(blrd|}ZF^UXhaIHRTFV%tmAK_@23N5n&tc}nVSw2uK zzZcbU0br1nk(7B7q1E9+eXl%Le-dtw2zJkRumpuH4q38nVFAzBR@Gn1lW!DMWzh2T)TJxh&ZkG-W{)8K#i`OeS=l(D?8kVTHMiVa7iTl>{K zz+J*`5uQ$);jaXb7#2T{WD+vI+BC9U#T9xSBd{@{es$LOo9?Mh^=HxqpPAG?#u_ep z(e=dMu6ymvkKGUN&rZjktxrF9z(deu-ksTL=}dmpbBqHN>g>HE?doy)+)rs@b&Zs= z@oB7D*^uUW5kX;w2y*MIgRTcluykMFapccVrG++ry->#lTcAdS$Yw4I8uc|6II)Yw zI6z5HB9N?AO(R*9wlzel*la~sk7$wBCE0H~N|t>bfnI$;da!4E-)K5^F_X^5PsHfu zx~{X%>=<$NzCPzayP0i%-!J53l4DF8Lf#5XuwE2cbWr9%j1+&t#=~|Krb2!I!dK0Y zK;Vsr#RG;bcmpBG#zS|gA9?|;w(eBs4%jWtjJ@7Iev2$pyLH4Y(Uw(?rYJC2vz+mL;#57IS1tRjCLx#$y)AVp z$7s%wUnr*1`(58!bkxR{V|&X(Kz0}kM{A|+X>*O8Oeqd~XjnUP0ZP}pL4>P$21K?L zkCaMp^7d3>(_%bL#8M`8W`p)LnS>{UfSs{KxivFfkarZxnKxplm;YEKP|0qvjGyIa zreM+>d}#pK5~)}M`tDzrsa>7R$CCYBiqywLylQNJZqa7gY#FFH*PE*BfX!!jusGm% zGdD3#t))pwn#R@KP4|I|yhz`b0%r%988IXR6M8X-8WHec(D^sHNrZCFqmujC5L}{b zOjKaHA!}M7_nf#FDm(Reqm;3q)5~8TNuy-T-5uNCDGg|&A*~42C6PX%dzS+s5riRA zW(9wbM3-JIY1z_HWIEO(2F;=fpw3D`?Ky{k+^fMV|A9#GW`WGJ9{t)%1NIw?f<~j2 zm0i7J`wqO}ibh8j#muDzSZHxAwrD(<9$?Bb__H8JDBoGkmCI>>H4#F)`?Pc);xu=ni(h(jql*TFEDG z6{R)v{6|IR!d3T{?g2+Okd>_~L8u^cu zIyS7wfJkuYln7bjFmuqrA*Am54E>&HYf)h+%!fZO_P{(G@Hyy`Au1K`7EBZfCNZ(U zY`3y1a*SI0stzlx2yG~> zE2}}GPWS+j8`^gPR0Vyv2$&p;+EQ#JtJdSV!C%}FC{fHQ;~G9b#^u@Y8?qb9KepphKx`HjJvTdv*#KxGr6z+a;INZ;6%eL zD=1(P_}$KfB=~C~xC1rdjpGC+Yu#7(Nl570UrvUxjBcxh`H=@hLlO+hQ3dmchDSTl z1d8DYQY`0D2#L;T$CiD#3G!kxlDj0S;d=YvU)m-Alp{R{!dukAlrRlV8Xt2JO3jS2 zYnq@qfI|9;p-w`tJYpothCAGfWUa2O&T8Sf;IFpWfE%-&kUfF<4zn}i?1q}Sq6;UG z+{mllT9O7@&h_{nR^3P$2F~+N#X*Q5@~8@GZNm~vO!@E5Brs9MB%BQ&q4guBvaH#% zwKkV86HCYv2M1JKuJ*)L_nA8^=-h*^H)xR#$aI|1XW6QFr02TO9{tfa+4y<14kzG_ zEI8xF$GgHvAQ`~e-`r=^6RK1KwgVPgV2RWas^kxqqcq}p78~GX#(N~!Y3ywj-a*KH zR(-+(30#6{IDw;U8R;OVNC~LmW;Z>WHKQ73h+4q8Tci{;u5t{er7L?(5B`e~Q2#3n zOy#OxyOJ0$Jftp|n}==|S#Xg_#E_6~SciJNI5a9c{_ zLx-au6HQtXRn>H0_@yN^jyUxc!8`3&Ahl*FAnW z_y>L!#VHZXb#wxOTrP7|K;}fkamkB!uS&~)YH~;Ys!N9XXv!&boN)U=H;nif_^pV( zEc_=gbBjNyos%?nqX42CspdjlHJ0r_1I}{$CXZ)Lp)$g}t z9Rnhw+u@^N=C9lNnAL=@n-G?mzTi}k+qXe6dCS8A_~B63m{sf;?n&Xx742T)KM|w! zB}>3vUT(>^W9q`Sqeptgv-bG`YVV>vDN$W0_6YO0_q|Q}NO#-1aqTm<@?shA(zZ(! zzv=#XH#+^*wk{a({5G4=1AP7yCWZ}+W|lyNL{@-oe>r-a&-V0l2Ce(*+Pg_L{=kAZ zVr~v5MIx%h5*B_-w=^F9O1Ze4$O3NgMWlUR9vdFyU$i=oc5j^T zUu}Gx`-RxNlKi|?mG6Jt>-3dvQ=)*Zf{#f+Wnu*JV%yF=Eu*y5Gw>Xd6PW}W&;XWU ziXmQjhxaxa@Rkh|Q2SI-j~Mq#t5t^2)OS;-1gc=I@bn(0s#_0S2ZoCKAHS16Yp z@vDrn_B>bHj(pCG;fi13Q9b4I)F3`BP7`d96vY39_AYC4!3mF`Fdr&d^Ch+#8u~eJ zC6RuFIqXo0w`i`qN>g6QpleVf67w6(mv1G75l{;>DOf1A^lExNHKkQf=QeO|{moBpE1_Nm_!tsr^HD-~{4&qyyrm{A0;wFiBR2 zA00lLR^&P?8rFRry+lY2#FEQ8G7pg#b9&ZCNErEE+w{TKB`TP^Sjyk_i)egKwSQrN zrPqnT3jeVc;v|+|n{I#LktIKmf#x%CLvwa8u1k391&q5k0&ln(4`9UvK|slhK*jQy zB0m6){;B09#lt~wW7vohJsNf}M%*e5_tEENy&V5HrICdMefgJFN})<%Kz*13zqc=x z`?F}M{uwN{wDO#KrkFzwU1`U<;T#7{V@VK&&0bALaZtc7j(>2$6PP=GqAZYnB!2_{ z(Us>WZ{Hd>Lu}F`-UBVakaZP>?T^{3Dx|h9%(>W=n=z{fq61IL--Ish2{1MSBXc8? z;(R5`kM}Z;@|6(V7}Vk_jwxhIelT&kvSZSaq;AH}Bxwa7z#nih!2YqAA|1_?T75vF z?}tI*9jT!$u_5I=paMDeBaB|mQU{mXAH4y&4b7613n>tNeRg&OvFS7Eu#6Z(;5M_QolkX1OIrd-x^T3hhY~br zhsm%)%mV}p&@U!3^gBW!TSy#kDLHU>Yc+mN#!M3tFS&He8#fQfOpMYu4AITHXT2(S4!*sFRX$onYVE|T8%Mm80Uy#@P9k}$ z!d`lpHvT-t+ts_;%ec&$Fga&a!^eD@(5dF9`v7zH0V|pO>5&7x%(_7x`PKgY^lU@Te{o<+(r**NLR!YgmnA;d z7%*y=L1OCDje-`%(O_o3On%RO&{J6qGcj}Z+W&#(I_A!AceLM_oX*Wh|0BsEjoGrgsd8#&E==j^hYv+~`VIJyBG8jTz3`~j{ z%tNsE4Q}(I0(k1x9{&>i`r!+52NKH5z&S$D?gCiYdW`H1<0Ro+bG?8!AsW02+EphYo{lMv}Wkj zNW9v2E*I1vGFXgWyjlt%^iL2P$UEVKnei6MmO|@&e%|ji2if*C;BoTSsh?=4ltRlR zGqky&K(DcCGFwF2_{PbKaApk-C9)b3%sozP;9;F$O0K>e%k}o|$pCAyiI9xGYK0j| zp_|4|8DNjGpFj)XridrAmt=i}Kjbeh*ZOl3HtFzW?u}5;2)9`^e^ zFJ|&jeR^YRV8tUQts-1iSS`I`$3sua^`2-rs6-dp9w^=>S4uJXd+M>y2wERY{!}|L z6tzX-CKe(4y0L#@G(Zz_mGL_34nYV9lQ7Ll^rd}T+N&nyHiIheEey;HSm=z63Y){wgr)F!8_ z*P}@)eJ=mB-&%I{^VMvJsBg&qoHq#$4|5d`K7riEit)#z5&%E`p`ovE@j}EvyTa~_ zsc9<@sN4^%cOBTw{fzIl{Y9bq=onaVBFF#}>*AXgG>20j)l2?Iamf~9Vslq#~(@+K=u;Zg6_l}L?<{(B11x~e`BwUjpwuB5>Mvmq=AIq~Be>e_5wOSVjq$QeG+O!3G`YtS zLwuciFKzi6>J75bK^MKpCwWWu5=?m!Y2|VK>A_i>C}6bCx;s;^l7dQ?PqT&Af^V*% zuWAJXm|O|XJJtIW=kN-CGVvq4l7dSMT~ja436N?iI*DSW2|&qx{p0B}GT)r}Gito0 z{zi^59MSc8sgUZ88v#S6Ww;=*iD}xg;US}O58MX((49TikIPJUm26U$+K}DUOVXv$ zI;h501TNdD9s=Tg)p?i1^MDuLwa23+vMwkFamp@}&{`t6UY3Wye`#YF3yL9de8VSq z2r#xFmpwKRKy}+*$bO%=cYlK>;ESTl<@*&l-EgDYkP$;HunD|!)ZZappEZIF={c!DV{KM8Fk|TJSt277d6RoyQmk zA@piNgoIWZHtx3CAE&QM8d8`SA~_s85n=^3Yg*dm z1c`r6x@xbqoXV#Y+4>JuS@uS95bz>KmnhnaCAuH{g#3*zP|hMlh-1+`wX7!Hjh$XQ zJhO&0f^jMZ+m*veye`2_Z-$V$0})4kDA6OCU9N5VT$=l$FMw=bp3LN>rTk-hY)H?5 z;4`ce_chYK0sHFKoTWOEv&4S zA-pZw&XWqeK~0!Lpy`8@CH%cLwl9B5O!YmELx~M+;`I?UfszGgp?%L=4z9M2xKBw@ z=h^x@a4Ag)cvd18C_oUz1MkyUZ1!(&i&u-RJG!=qcY|+d4LU~l9@q^2w=OwSHI6LH zf7TCBCFvT%a4~Q9-o^N!J=yX+4K45vEFIOfICJ&F5;WEBeuZDm`wl_a`*}JyeM3S6 z#Y_VTd`4k`p9W$TtjMGXTVR5BGVf z;;6al?s=8?z_jwavg)?c)`wpAn|@f*%bU{{L2J4yyRWVRogt7eReVwz8~Ca;NDdY%a*j%yOjP@(ET9Zb^}=i3I0`Sl}xunQlQ& zpIul#<&f(YwAGv;Zqp~(pN86dN{~(cP~|kn+ey^rM${O9Z@A?H0fHR}HEM0;+6Tl| z1VD~9^j-;Z0s>&yH`hHwy zEhAh~D+{ie2>%W$1E%&F`dqsApmq_e#Xaf9P2Y6Jb&r?e>$^}9Ff)>5zw3^ z2v%+u_H-dkFyjADt)`e@zUZoKT%tTI9Gp^8tgMm}QoK@}Vxl~(l9FPgV&be4Qer$p zBm)2Eh#?%af`x;Xn>7h12YWKA8ZAIMFRv13iUWOf@`{qf)9$u))+Q}`wLdpJf*8Tl-_z5l>D(6xxD3v5iC+|A99If5=4P)*Rlp1D&xoMaq6A%Q#y%O5D z0ZmecZ{8Z>9%tfDxuycoWf}CF=8nkD2Z8H8h7r@`C+k|84Pl!s2d-Rr;=VaVl&JacEJS0C-abeqU)!Ecf^wjlCy7r8LQ9nLF zZ2~axkR0Ojl<%qV(cEp?ppQwgMJZ^(<_%+Ds}3z28FgkXCV65&7Ec-XQ2DSd{orInScP0^t-+zOO;99 z)ou{&oxnH8{C#d~KWMS8IjS$MMc28h!MZ=eKkfH?IISzdYjXCaFs%>Vcg+u4;Y55< z|G|W8GK-!NJFSvddnTj~+U$LO2Ms4zm}kjpx7SdXKYTT(l;b5j+H~0Z^{FLQem9li zR$2%%P*#QeFidM)(#kb($Uh#i$v0`f2d5pEAHeBn$+Im*^?&?^$ol}tE($@0vv4qX ob#t*Wwf`S-{9y~{>SpTV=H+5x3CGI8%Fe|LM@cEEECu)f0A40kt^fc4 delta 13250 zcmV;zGd;}M)(D-~2#_QKF*TDREhvBGTv>A*w-J8luNc2AD_I!m#;J-^WyN+XuEgco zs>*}q2TEL#Ypqt5C0SDZ>(js-3(R0JvzL$LMO=~~y3y$V`h-4x@!hwgx_h|d)m_D_ zzg|6marNm}LRF4A4Z3>qrjo!ER8}Ap##AqEs~;AhZ*MmLd00POFA;I^w`YI#*O!R$ z)zkLw;nn8m@zMWGE`+#V3M-Ud{C)L)Td#Zmw7a|6-Ttxu?l1Sd^~2-(rupSRR=da5 z_WH*cUtfLs;_Ba5g1F+9Xu05yp@Zu6yQ?35g$Trk8JfJEPPi@cmUs%wu^I}*35LR)gSOy`NeO{CcErVyq(ET&Of)6^en1m4G>w<>d1zYA?0_Z? zX#(lGJP%C(!_}|6`Mk@$Ubhi)G3&2GSh!9zSv*4R%ht$WAj8(gW z9DarmNl;0idkKOm-N4qHW*x6Ia|Uf#$E>l{QZVUM!n2JxBRqmCTJw@Co*{~~nQZy` zINV1u?YiI79u&`jv5RM)HG8FUbfCz-7)M321|dUUXtOe_CPruAP2@#6f`fPAJgcW)-w2eBrPD#GhEbvaJzcRQEt~1 zPNx__|Ik(x2t{^r(@J!|>W8J`eDQpB&z|K>6`3-oW$sPxzAXJd*r<&3sK+|->#Ast z$AypubUGXp1;Xh)%4C$Jw$+#@ABxqGtV~R_AX;IAcWHkpK0K}-*LO|Y(gx;IQ8cm}~Lq zQLDT4<}pK$l)~)Ykj=pf`A*fdjd3`e^x@w%J79Dj5BUeM` zhT&~2HE9#b07rg(;Gk$zB6COE`Rz2+4eT>@) zdLP4O#}hyfyR8jw6#2NNfF`EMdn?BSZWk=2jY48eNuaUa%2JMD#w~@iGwUkg^aVQp zLk3*=%aRS*i(HvZ_~6#YS4M2e8i#+%d2D20Wl6h}FfF$Qc9N?imyF6eNu9By&htzk8HUvzwqAebT@*X}ofV0r|>dfAFUVjNb# zTjx$Cd}btO>)lbBo5wXWkD%61H5xxg!G49$Ti*)md`G;h~Z}e#-3;-EJRS6fX!!HibveTu-|*>K!?pXTyJO zme5oODAlE}BHgYp441LUUivTH%-mJY$X%N?*_4%X5)Jb*Ry8AIRYzs4tS(GgQb_9& zTZLRAEcn;8#l>p<*ru_;d52GL#;S#ycm{KhGwH&adO89C&E+>?>lc}NneTd}9_c=- zy+T6_{ul6V;V3zEC#MS(qG(DDFF}8t2m>tskRvQ8ckeXuKIeyw6cfHo$JSi%&v4nW zp;vEkzGjJ)^nr~lmhUM>25CyzkA#k-+)`1ftnH|
P8cL691*d~}EecfO7ecqQU zd6+Nt!Vn{O9HDK7Z?Yr^28K z60m3SBC}UbEVEgX=gH8BNi%;%Oj$x*>{pQuC>8Uw&;LX}|2y^UbZHc_Mr+#CJjPAo z2VZ>d>Hx>1VCj=OLHJ~uu|-|>;Wt;QHple#TLFo}B%yAFWSGy^#14UZmCbSNKR(+L%4oF zFs8TP>J+VucXYw0Gz*)^$ki?tYyXSnqk-vtQii!RTHjR4!riy_!(uQP6mV&H7zNkV~IA z>Lk&e)a_5QX_X1R;AnrJi5Oias2Sf0iCB}ladLk;Who8h(Mq~G>BO99Q_YAr)kq+} zq>N`oex%WqmmeoTTNIy5^%X<+wSZ{XI)q z9w5XlVaZ8h>AtXvQxCR3^^p1jy830jE+_9K4ljzE@(k=@PCtJ$5GtgB=!&SLJZEGY z^asAqTtHSGhE5XPNg`~egPlt*DEjOHYZ99pF1>uc;Ce$8sq-+t9utc!+`JvR`KHeE z>%q)s2s2^8>Rc#ePC%K=Dao|4U&v1o9xV@(XiG9w0l!2^%+4u^*$dXhisiTc0ES&O zv**nK24y@u0@8nl|B#Miqyy%Zy;-1G$CCR|4NrQh=LKK&-^$w)i#WWeCj!N32 zqg3GI4Ck6zDtF{8)t3*0hIZm9DIxdMs($FsPE@7cg{smpvvfMMw|Vd^U8av(D1Ou3 zqQcq5Z#CnL>fp*$zvewr45b)b6D_2#0*b?Iw8HiyuwHSQfSC{iy>oYFlq;XzD>19M zEQZBp(=UHMMQ{PR`C%~_AHHyXS$pp|OQe1vTw$?SONk9dwNoOcWLt-}OIf5zFY-59 ztk0;pD#nSN5Yf|Xo5FNAc-*)~U);!7yL@gV59%pFF*=p!i5q|9jObL3gaa&Bw2?O0w9l%%A1dBe zW*FY~Bl>R5NKQt|gC^^m(wxj!W7?c-C=oIW3H<>#(Z`sG$Z5S>oZ%(S3@>?^0id+3 zLSIZ4YS34n;bNiQ&{;;2UO(?t{k(HO+85>qgzt7*nGO!jd`vVp#s`PKt0)HBHF5+H z@dbbL@Zs}U8$uxe8hN+6TWz-OO#W{5_A!^6IFhv%;qj4Zr8#;p{^6ROaWBf4>3h*Q zse|fB9W;xPx}iWwYUq5f+}S*{VXXRS!V2M!}qarEprp5h4yIW0fDUmT+KUKb=oD zOw#=vN8Q!zMm3Z*Jl$L^4Cl@3KN$HG=54ltm>U@5-&A)${ znRYuk%kAj1xE*;RsELCl$oHO5VcGvsq#{V+q83Un*W~^I8HTSy6^o&i+c;;(8z}8o z!uhCKoTp1<4aP}n<3?f9jZXIZ%z4}?^xWxKG~v`~UC7K^074Qq!D04+og7;*E7BYd z*hA-XGG4!!ffoBj#q=KtQ0p#{Y8`*e(OsfA{V(P$x-0{nQ=YDK7MLgm7oH1i;^Zu^ zh4tLm!b#3uXfX*xDY1Z4dQ8Fs)2L7BF?lYq%*9lX>EMGDnoJlJLnuhI^4CW^P3IsF zdLe{!o~EC*SahBT>056OPrKS4R%_qie0Q^%(fn-daJlCpoI*JBi=HWIK2Lvib|fcI za3{W!Kdg9h`3tp~LdIx}zC^OG`jskgi$lNkYY058?=%%`^pJJoK%!n z`B3=8z7u!EF`1Ev{a>T)gxPpQ+w_6$S<6gg?G%_0JNo}IX_nX(HX-!*lYGiK{PX0p z*pbZHn4?-#1DQuK7E~C%J-xhbC2ObR>csd8y$a*y`ZFT>EbHH&sjDzW_YClh44~SB-4`aPrlQlP|s@+e8taa?Oj^jZwm?CXg27ee>e7`Qh}-oAc|BcbmJj zmDc+7kAKg)uP?PW7w>Pb?k=v+@9+6%=9K7cMS*NM{nPs0O}Fl=_qSK)x0j!H|NGsW z+s)nm=6w6h@7K5Y>zlJ5Uwn7+^^22#oe)kX8|qIXqtH<`uil>g@S|)l@vH9|i64dL z-|fP;jnP=*@$Gf<_mjU??&YLVDm30(p|t5P_J8%}=4ySnzW?REJI!W$s+ZEr4gR6- zR_oiV%^96u;BKeem3Nz0XDXb&>Hc?l{&sWEm%yGT*^{&q8mCxk65dd^=}RePw>G3p zoiNcf4(qyQw9gbJNtoapZ>)$x*Vv+@kbwrawCqT?8%SvU;S7h=jQSQL2&#dON|gml4X zw6a1wT4FQS3u(i&%?zYqGo;3+r;JqrA`e3P+8wR%PBkkq%WT)_!ujFfB<~r70|P3@ z*C0Hy_;j~{5Lgy$Ztx~MJJxGqwB5TQP3hTB`z)}+<_w_1UMywbtfUBWYZ^ZnIDg{+EV2b;gIMT;m%D%`f`=YM}utcb^n753K~E3(~4M8l+mRMRekK_Eeql@k6S z2hkIP$!cY)frT9e*tf(Cv8}1Zm7WsJQ1xy}l_LuQiMhBaw9tHL(ZWaG7hk15D7JKz zT!i5JtYIp+(8B1V>d^(UtQ21|Cr%NDONcPq3MU`YiH|SIUNB!-vXTJRwto!yb*y53 z8;}?vbdmE1c>95pb0Qf1Xx<*AIAH$YH~-K3W{SdCKkJ*JZeXno*f`Q*-nl0Q<${-9 z;GLJ44b8y*o)qjxOY>B?UqJ*22F?Yr%^u>M2TEP+cnarShW`1YmI)Fzyj}RLWkO%T z`QjU`Qe)R}ry;T{=PY1bmw$Ty`s)1RY^Bxd8vh1;szpX*M=UD^S=QUs-OKad42C~x zepp#4Pro1PI+-HEXUBoSmqllx2KV;tc_Er;p^%N1R3+QV26AfxiL+dO1j-*Tyk6R+Q0U)>k52f~HaZTjfs)>U2K z%q7wjs-y>h-Kr9!KoegkoW)zunmCy{BtJCKsF+TLfCbcmVAQ9kS{T_s$` zd>?diV9K6{IihhSfq(1}<4DLnxp5Hf7fyETo)vU;r5r+=V{cYdZ)??U*?K~C z2^c4Wa0{7sx|=|bQAWaeQ7&*#t}X$Zk$~>JlbaKOniIfS)L~Bm`t@Wa%QdI%k0L^<)mW&7HX&z{&X5HkJ}K50IDe!i@>X zWCZS!fcY0(3L(TIr=G)hD^rAqJd~c_af4kQ670QC1zr&UN*OOwTFnAf1}30Y6I3s*H{LS>JVr<5JKYQ-DKbk!hcBaR0H9eyii6t8nxs`E@`Aa zBY`cb?vyPGR-1}!oF9&`j)#ncYt74sU1oPNmD%x7!@H}_%6@sj>-(SY`u=5mlWacN z2M(7J(ys6mWF>)a}#3k(~xd2EGt#WE?<;b%%i?Pr?F6AfpGKphpHrm3)hk z{Xi-U?te6oje1(?oesf93$*7cp^;ezjeH#$(IRAIjuJM^gE_Nz*tu>6m(xvp?`^J+ zI4TwYR#ASH{G><{>h*Nyyk}n%l}?U2uGR+OfF_JLZ6~Lq#asy|HPYgYy*(|*)#O}@ z({PB86B?wPecfeyznCrAl|Vj}s=Xeda;fc+QGc~+@0L_*dbFJ5vwD$3VXsUok<#4^ z@SO!SoCFY1he@QTWnmxD9aSQut++%5{*Z3_Dhsh@iO(x9Bres$_|!j{NFx}(!}Ivn zZ}!EZBF_}cg-)dm;^ z!hfI0o2sWPFjql%l&uyUs16tW4?y7qCA%xd9T$Z!64(e~yO;@N4q{sY%-3bGyYhyW zuck2ex106tD33*|ySZKeRIn1qBfku+rHDE~@OGj-X;YASEL1!_0=vIij*KM)l5h6u z@K_>;fx2lxTTQXryi~%It-)g+iNQ~_V}E~c>+o35z+=Uyr+s*gz~i~YIN;+hA+iY% zOio4_m&DGA9s$*2XP1i4tH8sXTHWClAHJjxZI9jT?K~2p1CxSZBEmqz=<3{tYR7^y z36=_LCIW70j=Uz&(%4Zh@2L68nyFUS^zJ96NDTcpA?NpEWqE-3x|Q0b(zET1C4W#L zZI9lT-hxd_Z~GCPCpM_9oY8>JSIS5~5yz288toR9+QI`%ZJEymR`u;rSg#oN0S@d_ zy``G-mTJa2OgoCCmRN`83JO&moZ-O$Rq$^-!bB>h?Z|q3(>yH}7~BCUx@5-ZakK+kZZK`R7Y*%$w^j7@E`fH<#Vd|Gd7r=_BOb`sa?u z?fI6#`Nh@xdc6TbQ7kPX3RCFfrJ|Z(xtX`ze(pAt)huYUbPuf$jU6M8nZSKAG{xfT z)1DT?R$^N)?dYYJNLKrA>{z)w{bXsll>$Ghv@5HKk*l4K9;fY*WLjaAqCySd$A)Erj{G!~)V()5SaW z#3FlKBhk1MJF^As=6u52+M^Y2uc|V;Ggt!*7i`AOM??=LVNCSp%IG1E_jq6;&m87* z8+?6VpV;rgJu5_9sJHXkIDbxImN@Pg$-{V-cpy?n5Cp8PJ@@QPG)I<4KJD8=MxHO| zhka#)>U|++H~I(d68B$x??3}yMMxPH+2O-7PuT9 zKq)E}B0LYy^W%p8dFXcZSQhLlSLl2QlMxEnwpH6Ri@SGMR<8Ge)#wn&d`ve=&u@1V zc+b<&a1);uj2{-Qwf%%HR^>s!63|qN1+d|$DurtXas1u{nFRi*swQ3{yQ`uh3%}?B zVl}nZpM%g47mEF2G=F3s7+SOOWNK$Txn5lBt|9AIrkiRcs*XcLS$H^eavpj|N(<^= z4$x;MN6?M@%ewdRW$#l%Y9&?SE`6=Xp`a=faCm{LfJec(3IsD?5py8bluv?S?Ynlj z7=k@aPkPi9g>-f2&X2Z2+eewM<+c$YF-%!VRfVcY22@g&fq(j*64*#Gac5;E(n38n zmW;bGJOjagJ%6;5HxCs*P7v-1I8fucO`HPgzxwb7e4M4SNA21Y^ zEh`+r*>|DoU;wI*3P5XvVTkhl?yWW$U?dcnq09oqt-s(JD9zceYo|`eg^xW9jDR5} z*Ymlo`>VphZGStRpWbPe%{P}V81B^i85{PqV=>%iS^nT{2A;Zy9%nqA9NQL-B0ra4 zT3|0kbM!$1oCFRPyUu9l~wjGyr)=QXzKkxx6xPS>W zPpo#75+106&Dr)bT7nn$pfZKFGRjcdYud$@*JrEl*?-?do+<(gH}l|E!|V zt8)#lUw?M(FvKIgaVqx|=Mq2jP|1u@HX$1aubhSISLdHJbkx>56ACNHL6X)%QB`)V zaUqi*mOe!xQ?stj3^E2-m!~Bx;2GFTn$Z(V?_ZH0+%WB)WqvG$KH#3%Ld0lueva^f zic}ve0tZ6Zo}-kHN|{jylYdg4B~nozR)b{z7=O>NQru3Qd(%Yix7(kY(mR2DYeE>c zyAt#~w6YInW!{8=yWsR>pfl%aw|wb8ilcj@qmr~=Rdtd^i9LF)q|;*Js=N?!u{Sgh}O<+6x4 zF+~SL#>T|4u1iTZ;H^J~uzN}cVdrHh#c!H-vL01%Y0~oQjq+_^SOvBFAQ7GY1@8SH z2N*ed3T19&b98cLVQmU!Ze(v_Y6>$lAd?I&5(7CgG?O7MD1X&G+j1L6cJKTO_zf%x zy?wuEO;@cVJC&_i<;1E=Y9&2@CPZOPLJ0t_WhMFdoYOsn>7F@r0SRikFXCX(=stb! zw{HF6&AtNG9zbRC?=eny}6pbo&RaG zy!m~%-kmR$Qh)P*d6E9UQ%YanZLW8hH_QDV-|4v!=L_M5@$-LOecGgPFYmV3%k9<7&HKZO-4n`SaQLZ_fVvOaN4TCTb{nU@FMjFSloJ-}BiOeD&uU zhgW{~TO4>h(+Y+-c)XeYeD-gLeHHstva>(u43kP%`+vLpaJ{?)faH7y|9UmoEP~XY zt%WKRm^L21iL<=phA-hCQT%lFc40W5zgol0T7>xz=P>=<=GXli@q=#w9qHlMbLr>s zR=WA$@NxC|dHUjtePr+?`TVM$=H=>Kz^BbM5CGQ$$V3L26oEMxYS6aT5H+(bIOQWA zV1S${W`BiEgLoR`^J$n1X12%%;U=mDT<^PJ z3hfvdvjr@!xhl}&GStD08t`J6)s=*N)17Mp$bW;c7ul!_0&_5z5q6wVnhBxC2$c;w z9igQ22gr2}xnm;de$J`sg4VDXr$QPgx}q+f0O-}sDa#a)d`Q??@ZuorQML;#)H6vS zqn2q!S%J=T0t#k;z91yhSG0Lu4a+GAd=KydA%m~L|4(8v4k(i_N-&VRG;H1F8Ogjj z$baL7Ab3v#Qz2xoW>VQhL(jZ9aLn%nM7m=5$fY?z1^o;&O3boB4GD|o;?sd;K2zw_ zOd|>=EI4sJRj+qR3WRiaR`5DkU4&`|+}RQdk#vIaz-l1?g%wlYl+u}lY#H~ZI1mzS zU`xIwQw6|2Y665r4*~3$LWPlyyXZ0{GJlAWh;pSXo%|C;lB8CEcePa+hXa@k#{$56 z6r+|KmQ2FhnOIOv0eUY~OsGqqDD{y7Ftb5@{A|dnXnxky`MICJ3Zqw(OBB2zHR@CI zk4T1=3YHdz`I+N5+n~QCIA`3sQVAEXhXMU$LFot7u6i17yo8VJT4qZ{?dkiAnQ%|@ldV-$H6mn}-G1cAXil#x34Gkno18>DB4fKPb50$|S9p*XEMHDDBe3=a@bh{>9Bm^NktSAS*WF1m;k z9CbcO62N7(9Hx!basW?f#+N#-!vgyr8zFFsVl408fDYhnU^NR1hqB+0PLH=2dpuk|TnDn13o;^7jEIv;pk+ zpmTxtdvHk%`?#4IgMPDV^I(fWiu(r>$ixJ(v4DT^F#1$8<&Y1Ai@whCNdjyTRe`QF zW|ta=rmuhih7QD$yO?n{QZ{zsvagUpyFEaiM#FueG+;1ovqNNt4`U`N(S>rBmec@} z?C)hOdOc**3z%mIK7V;eC)kxS_jCyhYC+uMOK321T&N)DSF0^PsQG%6UNps^`0;x6 zWwpM>N>}_Ci$I^Yx2v1&rcBNnBC=TL_3!LHR7eU#+t>-SVF~b@}h*>LYtW@i&mpxfbc# zISm+)=QK7HakNxr>hn}Y#_G?D3;gTmJ7h`3{-&GiP^cIk$Ow0By`C#k%OqHDH1%~+ zjH(EM7X~6YGk+YkVsIlBA!%C5Jx5 zQ}6IV}IS-S9S00^=eZUdZU2Tk-oB^ zkJZ;#d#N581%gz2pwQG|*^V+&MS(F$FivRx6_c`JmzS&GR_h%b@*B|HH#Y6nYD>@w z+aeA_cI!4wc#V^LRQJ#|b^M|G9y=^SvJCxFYW zF0h_fNWj&or|nSkUUG;5!?DPcM<7d=E`O#IhF~>bxV+d3F^W+w4cHI)aFl9k zV;nq+Qc!Wq<|t)}L4N=pt+y0%0YZO}0B~t#@bm9T71VU_P%AGq2$S`t^LM?z*}v!- zU!bUVI+{ga<4@?iOaiLjwO+T09YXYw>RKOxdWZuhge|ZP(?H(`cpddjHiZm@(^2X; zuzznk>NA(341=&L%S*m#SED&hatwMp{)mqys48y#mR1TbdP$)PRVf}8U;}}7ry+8P zhjAhga$F#?kB3d0hrzW-ePHo1&pGSTi$5p|Pe$}cnK4gIJk(SByBMm%MSRhnq44Cu zg&+%@KFDLde&6r)Tg(gTQ^?<8DoOy`UVp#H`Q|vczm?Pi=WdaYlY}o-_WM0LQWa`g zkk%6*^)S`ZNFCN#6T&H*VTCJWB#l#pvuOR*pIh9wAIEjOF=6x}`ydOqH)DACqAM{LLziQq!fUKhb8i#g5hQ)bB z(&hG-GI}Wl9X4ZRLClVvD5fM4yc8^^Ktp2}W9{YLl5GpXCec?pAnrR-zFkukkt9Qh z#(U~8ORt#%ZLZXqq0+;K%IV>hf`5Ig83oytCsWQ9FtxlP#Yth!)3wD(mobF&Ej-B3 zgNWAzJV0aH1jKZDVxc(};{-4asB=E(1Uj9eRhrQ*@g5!FdsIZ3I*9d)Tu0CWjquo_ z`YSa@Cht9AX@ABvef0BB zIQvW^MUNoGw2{)qN$gm+lLDpswVN1L`Lb&_FuYtO#wfZ5Wye4 zk%4pR0F1RWg)M6n`hTPbIxa+0oqK{YOWkqNb0X#7+-`{3gkaX?(~+47;8W46jK}7Jm(}RS`&GD-m{&!uoaG*UQ~_TEv{L=`P}{=XwS+S!R3MsUU0;&WtHc2jTLhTtJ^b=2lFh zj1$ZUGn6w_;2;Z6PzH|?3#CTS0V^J2PG%AB_o)Q%KkCYQbZz?U{)R(H<6CHPL#ayg z#J^OQYJUgpCN_%321_p2?SpTfOi}P8NBRmaIJTuxYpGH|E>?M@0pP`8LO^KXb~})X z6WYWU18E?<_x+vnc;yysBwn-BoT+hSG2C%oc6U^bTbt^JxXkwHlAO)KiP3WSCizI; zr9TAmB_65M6!}}=Y8!I;Qz?dp;AoTZQi3c=`hNgnHun|g&aH6yr6CnEDDpX2R7|Kl zcxLb{X*$nWaIx-l+@{oAsl$XFLhv)^)gpV zE#+QZ92dKLTNqQM6D9D5xqRdywpkcs$mAyIJ0VB1Eg+hxRblt0p7>il(`2aAI(Xht zVt>WR#7bNo{e`L+`G{?)1hXop@$d@>E3kz{M7nNQvk%lgKuVct5bEE?HU95^iA6Mc zUSRa@wFp`rEz3pzhmuL_m=+?|EwnTLSjK!z{?#;N9*cI!BI7bN!I*>4gJ+n_P-V*{ zD{l1owxI<;yc3&f6{;f+k5E35T5rHiD9MNpx zAKGmA`sTZq*=S``!2)N4j>~t2YUKRG^5$I3Z&sTt zmR`JCrcui~Oth`Gc+8SYYhgY`Jo&DW*5RYf<6TyH#<sTYvN& zF+CfiFv*j}o6wk+Gm9k^Ek-wiRQwtA7k2`MY$0JaORadbJ@BQGy%&wOieqVuJ52Cg@J0 zPk_C#@y1SB*Ma}SzK}IOn7~RsQJMy7Q|KefJlWGKVc=2mzS6IkW7UcdDHM1n;$BdH zwG359?`D0l>8LO4dwH`R>vu1*@K-?<84F<+$b zQ_C^V8^H%-@+F(KzPtf`e--a*h_D>YNa+aRB`Mfg%VR8FP+FDA0h3=CO9lSG2i!@# zmCrag!VcNhf~OW^veuIM#(zZg0c_JcP$Q`{xiF$UPebP@u_50*Mv9UsXBR}-kg={+ z{~~)LUH5PTe?#u+tLx+Ziz&g#G-1tG`uIXZhso6gqM2bl?MI@ z-P)^M)tGXHg{*l3%C(cm&bBA1)K<_i5iJYr{@}9p6HZDvZzqwRZFm*gjvR$q{8XUvg|d>*+NI$7tpJc zWy%Ua8?`2%sfn_p>9`DUdMvjZvfM>O%SSQw(y`2{#sh1WSH{yk4;M8(8)MEPoDGyap>9 z!Kz%49|u+k5q`1e8Cd%etgE*UTqK-CkVq0mqKR8&eE=^PB!R*bP*edn)Pjv|R$vGj zN@9P=FybY#B#y+B;VPTD@!94ku!TKu{R_5*f$eN^M+n$i0E*{=T`o{E5A3c2rFC{% z0vSn0k_JPq2?w_ItsBJaDiOlyP*2vOu|;@a4m0Am9f_eBdalC<2w$pz1xS zX1!w^_VMrFL>D+&3QpyKng($C9jJW;&Ln?;x^__i37qAe&qacU7;ruXT*v^8v7m_; z+RWd&7y(*%c*_@X=__b`2`;|@SAKx2FTgck_jNXYqZ70}0PQ{CW)$ei1h?jc+pKqo zmG4%9d;bUKBZ*|JO6L<>^Q*i~vouaMxWiJCs;|M)c-8DgOA}PfQ!OQ{HmjwHs*fTc zSxQm;Tw`gHYG0S7$?AmmSeioSsN=@`9DhLm0S@YcpqDQz0v`%AGdBt)B}Gq03ifZg A+yDRo diff --git a/ws2019/ana/uebungen/ana11.tex b/ws2019/ana/uebungen/ana11.tex index e7d4f3e..6d82877 100644 --- a/ws2019/ana/uebungen/ana11.tex +++ b/ws2019/ana/uebungen/ana11.tex @@ -16,7 +16,7 @@ Für $k \in \N$ ist \[ f_k(x) := \begin{cases} - x^{k} \sin\left(\frac{1}{x)}\right) & x \neq 0 \\ + x^{k} \sin\left(\frac{1}{x}\right) & x \neq 0 \\ 0 & x = 0 \end{cases} \] definiert. @@ -73,7 +73,7 @@ \xrightarrow{h \to 0} 0 .\] Für $x \neq 0$ gilt mit Ableitungsregeln direkt: \[ - f_3'(x) = 3x^2 \cdot \sin(1 / x) - x \cdot \cos(1 / h) + f_3'(x) = 3x^2 \cdot \sin(1 / x) - x \cdot \cos(1 / x) .\] Damit folgt \[ \lim_{x \to 0} f_3'(x) = 0 = f'(0) @@ -140,9 +140,9 @@ (-1)^{k-1} \frac{\pi}{\Gamma\left( \frac{3}{2} -k \right) \sqrt{\pi} } \\ \implies \left( \frac{1}{2} \right)^{k-1} \; \prod_{n=0}^{k-1} (2n-1) &= (-1)^{k} \frac{\sqrt{\pi} } - {2 \Gamma\left( \frac{3}{2} -k \right) } \\ + {\Gamma\left( \frac{3}{2} -k \right) } \\ \implies \left( - \frac{1}{2} \right)^{k} \; - \prod_{n=0}^{k-1} (2n-1) &= (-1)^{k} \frac{\sqrt{\pi} }{\Gamma\left( \frac{3}{2} -k \right) } + \prod_{n=0}^{k-1} (2n-1) &= \frac{\sqrt{\pi} }{2 \Gamma\left( \frac{3}{2} -k \right) } .\end{align*} \end{enumerate} \end{aufgabe} @@ -229,7 +229,7 @@ \begin{proof} Der $\sin(\tau)$ hat auf $\tau \in [-\pi, \pi]$ genau zwei Extrema bei $\tau_1 = \frac{\pi}{2}$ und $\tau_2 = -\frac{\pi}{2}$. - Für $n > 2$ gilt: $\left| \frac{1}{n} \cdot \pi \right| < \frac{\pi}{2}$. Damit + Für $n > 2$ gilt: $\frac{\pi}{n} < \frac{\pi}{2}$. Damit folgt $\forall x \in [-\pi, \pi]$: $\sin\left( \frac{1}{n} x \right) \le \sin\left( \frac{1}{n} \pi \right) $. @@ -253,7 +253,7 @@ \begin{align*} \frac{f_{n+1}(x)}{f_n(x)} &= \frac{(n+1)(1-x)}{n} = \frac{n+1-x(n+1)}{n} .\end{align*} - Wähle nun $n_{0} = \left\lceil \frac{1 - x}{x}\right\rceil $. Dann gilt $\forall n > n_0$: + Wähle nun $n_{0} := \left\lceil \frac{1 - x}{x}\right\rceil $. Dann gilt $\forall n > n_0$: \[ x (n+1) > x \left( \frac{1-x}{x} + 1 \right) = 1 \implies \frac{\overbrace{n+1 - x(n+1)}^{<\; n}}{n} < 1