From 49f4053379aaa3cd91eb6cdb3948059ca594ddad Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Petr=20Veli=C4=8Dka?= Date: Fri, 21 Feb 2025 09:51:57 +0100 Subject: [PATCH] make varepsilon not epsilon --- lokalni-existence.tex | 2 +- skripta.pdf | Bin 60617 -> 60620 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/lokalni-existence.tex b/lokalni-existence.tex index 7101995..d1cc912 100644 --- a/lokalni-existence.tex +++ b/lokalni-existence.tex @@ -69,7 +69,7 @@ Následující věta nám říká, že na nějakém okolí libovolného bodu exi Položme $\lambda = \frac{1}{n}$ pro $n = 1,2,\dots$. Pišme dále jen $x_n$ namísto $x_{1/n}$, tedy máme posloupnost funkcí. Ukážeme, že jsou stejně spojité a stejně omezené. Stejná omezenost plyne z toho, že $\| x_n (t) \| = \| x_0 + \int_{t_0}^t f(x(s - \frac{1}{n}), s) ds \| \leq \| x_0 \| + \int_{t_0}^t f(x(s - \frac{1}{n}) \| ds$. Ale funkce $f$ je omezená, tedy máme $\| x_n(t) \| \leq \| x_0 \| + (T - t_0) \cdot K$, kde $K$ je příslušná konstanta omezenosti $f$. - Stejnou spojitost máme z odhadu $\| x_n(t) - x_n(r) \| = \| \int_r^t f(x(s - \frac{1}{n}), s) ds \| \leq |t - r| \cdot K$. V poslední nerovnosti jsme odhadli integrál součinem délky intervalu a konstantou omezenosti funkce $f$. Stačí položit $\delta = \frac{\varepsilon}{K}$, potom $\|x_n(t) - x_r(t)\| < \delta K = \epsilon$. + Stejnou spojitost máme z odhadu $\| x_n(t) - x_n(r) \| = \| \int_r^t f(x(s - \frac{1}{n}), s) ds \| \leq |t - r| \cdot K$. V poslední nerovnosti jsme odhadli integrál součinem délky intervalu a konstantou omezenosti funkce $f$. Stačí položit $\delta = \frac{\varepsilon}{K}$, potom $\|x_n(t) - x_r(t)\| < \delta K = \varepsilon$. Tedy dle Věty \ref{thm-arzela} můžeme z posloupnosti $x_n$ vybrat stejnoměrně konvergentní podposloupnost. Zbývá dokázat, že její limita řeší naši rovnici. diff --git a/skripta.pdf b/skripta.pdf index 73aea09e47b9430791ff3bab46749d20cfbbd0a1..c1d01b8858a9cae98aac7cf66a66664887e2792b 100644 GIT binary patch delta 7442 zcmV+t9qr=D*#pek1F%{We>gZkJ_>Vma%Ev{3V7P>Tv?AJw-J8NuTcCngFtPvcssN6 zkiZENBS4(Yo7WH4OccP51NZ=vpP#RIHM>Q&q?uV?SsMdsTWof*SjSffd-u&w?gajq zJ4ZVw!xZ?x-Q(A|_D}r%are82-G_e&JN#nF>fOU}=hPH5Vf^0Gf8?yTYWMK;*_{yf zAcXPvpFjK=w-B}WNQ_iO22t zxN9fYZqh;7%=nwHZhNcIQ}9w{2HqTRdlMN@7{WZyUU_@-R*>$_NMd3U(Um`)HA^|U z;9aPTHsH4vaj|)499AhQ?GPOjE7UQ`f_&`m%$^_)CIlyUe|tP37=v|;m*poX9r;Wr zbr%yRErKlj_m}Qk#Nd)sDXD44F)qD?hSL2`C2u}XW$P&$ zXZ;vO&vv^9^U-BtAYm7NU&?|$hSA$%~lgwh^Z+k5A6V@u#%psk) z%-pq4%o)992c_AzgZcg_A#v|L8f1x@a$zwpDjK(l-Lc2{it8o&hWID^U3r*Qf>@D`*%9EXE>i`YSdYIW`*uoNY zH+oB(mdi|u53xP(E8#t~3KkE*RWjrolYl!btH%KJ);q>@r>4tJtM(0iUwY>?6aX@ ze~;zwW(F`*1|Cm?<3{G0q+& z-h33tvXPjySJqF?GPHF;{S9XD*mr`61fAqc? zT0-q;LRo+d5-NDcivlUK&ZV*NQWbWrJXOYP@g8Fqo?-<66(n8zh-*9nf?6euO`xs& zG$J*HMR)GAWi70t>05%w2?2yJ&e{(nCh4TN?@C3lQkhHa^`4x&dDEo;A%ir|`4{-3 zl?Oc~VGfi*h@M0U`UFPOKTgqTfA#YVB&rZhW6;_~ryT>LshF|g%qWo+?=)yKW5^PG z6>I|pzMm1^p@ndV7D1_;!D0Nf2)wh5)8j=Lh=|2WtAc^Fn~6%W4W7h{76a*F2F}8) z1mC(&H7}XPnvPxUq9k0R*R5klz!_$t@vB^ylA?IKXPKR)d;xWlt_t-Vf7z8RX(lY) z$hCDEq=>>~z)81Ccde%61XfM@G!{(juo0NE#naKYZG_C$PpmWgTCkHsoX~`V74YP% zv*}IoMlsA77e$)Iyyj3>5jssqQ4P@_ZuiFcaYj>-lwlM1Ko30N5ksGK=y?X2o%IqI zc?b!~CE)@q65xuYU@r0ye^S{ARD0jKuX>1$-32K0!Fm7A?fYv%B5bu+XCeaxoYuMM zZQmIn>ZgMoHqo5RwYf<*$)JJLZW9jOE#c}1M!t96*E?^6>y{kr4E%)%!o1jf%j03z z{o;r55dKB2d@(y&>~d=+N`gZkb-s;Mn3HDem>__C>MAiNnv~_lfBY32{nx&?LrfVs zRlZ;9CYgbN5y9d11M4t+_+3R6r>R%vywX2f)ELx2MjOm890tjdyCMg)EEc9LH!Wt#c2oVgXfsp|j8+jW$ zL0j-Z4TtQFUlyn+F5aQtj z`zo+1BU4!0=m7}!lNY|q&JS)Dc)bJH1Mo@CBIL2S6p&^KU zgMK|&qwLbGZ@?O9q51D&&8x#2C*Y^`_2{=XS! zpUHWW*kbq!D;o&^@x$($-K2u&8r6R=l3RT&e;mqQVIL5l0mgid%Rs(XiqKyE*Y5AT zPjdw>te)4*5)c%aRD-KUs&(%#v=8V&OmTulOCL73fvouelq4moB?w6!#VYmPFH*Vj7q)|? ze~~O*ZTt51)2RC%8(yRH=ky1!&{GHs;6r-yT$<)@q%0c;nd-&x^qIHb>9fF0F z29h6+>ZAZ(+OI?;^gEmDbfr?58W_UdcKEOfbNiS|HLRz; z(8|*(xok{^Hl}RZOGe$*3Ri4UPFSJL!JWKmOlg8vxxIlI8h_b3G9tA_lsy-ZE;%#u zb7|3Eu5>xkR|B8w5COZDgIaM+Oh#Jq9ELqVEnp_$IPyN97VwiYKD^HgymnT=e@w&` z=F_ae8or20C`sWwd{Kl|3WtDeuM|TTZdChvrxfD=(z3lghA-_bKr&#O4I7$gC%9!S zwSg}`ZmDI-PXp=#6h{}1H(>08sbI>{;2aVi;m2wZN~DUwwtf8Wr2-glVu z0aS93hj-n<)#r^2Rx~XXa!_#M9E;{0%km7yCw=9XPKD(Wc}geZY{?K0YpkSok*ZGy z2f2YSwLaC|bec}jo!2FLEZN5z{A5~|>CX~tgFRu|%K)_$ex3II z1zR|PCAfceL9I>);ePz3emgK2&Pg3!9=`&I+v?W_;j2V+!RKI>^DH52!l3G%1d-nk z&WuX|W6is+85#V%3dUN>nWAxMNE3ur=<_HPa1g8$T!>Z9B4)6Te?vnO+%CnZWtz)l z7~0O_iA%5kmlPUmDKePhZ(okU40u!ngzpDNMDg+Jt8B=(19#pk@|O&=%-E|#6@{GZ zAs6}bm`jzE4oT-1Zgf+&1saz_6zQf?XY3u^?zquCpPeE7MNz2wlQzW)rRys$| z!(^?!&c#e7DZURuf0P12nQ+eh=He(d1=K?sm5#dPr`Qw^jGq)wX^DGd%b^;dJb9Wc zo(GKIE=4E>2^1Dwgih)H z0mB{9y6m@(nPGNTMZXWa0lEPaR;0pi4+)$Q(Fnr`?`dAL#t@`OTpo<3%;@28YFb%#oYS$NEZQO3lIN0!1(!nj~xuRiU#X1c0RFW*iyznLg+!4+>NN_Q-MTi&$%N}J*% zwa8*Z1Z5V1PS188YU#du*!c}HtNy758C=c$*@Zb5e|x~Bt^70>jn8Wx850qNIey6=BH4>zR2VL@_98JS>+!645+biDNwsEYy zn5LeUrOdL}7Lal>^>rgg-c@KAEAf%kf|~MK+x&vZa@71Vf4Y!9`m`-PH8a8X4k|&uv1Bn4R z=#-uQ=$2sz;5ZLuV*ar~SFY+G!#l+`AH@fKge9g?`kOwUNjjgI2^4snk3O|Iqvkr9 zwGu2c6vMfTvx3ulyM@r`R`VWrM5ard+ zh@!NRzI8YeRp#G^E)YwjxBsMB@R#(TI(3u#>Qa|=Ks@~Z6mDzQ{p?3^5+ahT@Yg+J ztkv=ip?IJa zrsV}aos_8i)Fx#M^TDt*nVhQ620}wN3{zSWT67;giPB9CweYCct)Lt`2=|e_JUkTu zSXnkITDHbX_6%LY7$E9B-xo|O0{6iiMv~%CLTZ$ed&wI#19^jgOO?b(rCLs*KX_J2 z;hNfsQD|~9HCb=kiBaBis3IrPQ%!3fD^5=*=~*zo^mLe-$q@1vyotP13&;0a=p&NP za6r>qC%*;q8^*Au^jKPv=L3;2!j1I6oh4&l_-+<29l437!PbFqa$;acVpHIrXp-~j zq>WLtJ7<%z6D@ClMyT^Rl&+PoL()ZJt06Kd+v0o3ePZfpwBxAH?aXMpwy8Z4Ahb=4 zglU$%3kWsor4k{a9~Lm*@1#$qmMq_y!1K_8)-2HysTnbif}o1@L?6@9CENwZ5Ii_- z;@ghte&&J>$u7c!^f#XMqZ^5!#tGJEDvw5Bt&&&@9S(edV`f_AY30##ZG4j=SZfjs zmY!rpqGq&#bfDh$}N#wT(XLkhZN?WCorjxW1mam0#v3KElx3}AOU?H7*3@T zk-lsjZl`5`2kH>wBC>T_D-IPTidUibDJD9>o-oc#W1~x}5fk#*scpYO{5V0x#N6$V)TqfIb z145+_p5neix{5EEj1-tn4nCQHaqW4wPEWh$&({urjt}XYIz+zTnH1c6{Ce;B^M_wf z&(24K&QSW@^Wyft-x<^Ji*J*%7#t68MtqjI?M=$@x1GK2=&ESsZ|Ol@Z+N?N+{MYE zSAJ_i(%oIZ7|^*w;2cKtW2eb7i)xg5y^l&;koI zKSmPYyWSh#_KQs9NBz$2By*Ve8;-9pko6Bolj4P#{j;L>@8#4sIytPzk47Bz`h}*~ zH0h89L<<}?m0Dr;+)W$$XLkns?(6*5;q|D03s>ZowggzW10_auuuF%+{Brbu*u&@o zh&`soYO?HqKK^j_d9I9lMzScqOcuK#SzMJYW=a-kYh-a1vgjIF;2e-e(3?L)Wbsc) zmbLg*$ztkcaa)jO-Td2<#Xlg6+@37QpN{rEzMm-*o|7wnnJe*UxZ>Sft^`%(3X+?D zU4>m&8hRJ~)PZFUd{wH18dbE|Ca$cWep{x*2TW1hGv)GMm#@!HXS%p&WXkkS&w?rG zx!jL3CFpfban4kjVx=Xf@N4y_XUaPGDooL;&J?o+Q`StsEmNd?Koz|`Ret~V{rk5E z^9U!NlPh+aEB>c;ptv@wm{^S}n$i`2`UQ@W{>gWswfI%Z;_74xTaabl{9CidJz$I3 zo-KPPuMa+-RQALDQNQ~fR9Vh!jrW8KW}D=8<>cE^MLi5Gc6+LPIQ;PH^1MRaXQGN(&TGw2<+W0OnzdBX zwPYo5vO+(F6igNU(7VNh zFCB6&zgtXoB^SycoxJ<);>|AyXJ^M!JfMXA4g(hcbx1g-d~l#ZjSp34?n=`X^K@2C zUF$`K51P(7^vhTJg{!xsU2#-@Na2j^fTMQuZqYA0d2u)C^+!XQ_dA2D?)*Bx!wML6 z^Xt)I(8>G7WRl;~i@E}e=g0X;ewM$>FY``*m3Q--e3*~(NnYk(bF7s)b!q@@>WwI> z-iUI^8)D%M6BfRB_u=yL^aP?Ve)kVu;eNmsV(tjB-V3zHKlB3Sg^_-LS4ls^^g&g< z5LMj^gWm9dl6S`A(O+XyA-@~<21NzLs%0BvsB0LSWky#(jCuj$Z+ov!PJV}|^HqHI zPt9ks$f0J*PxblEjL9M{GdbEihHnSOZ708@ci;WurcAXdnth=AcQGF2qhXPke~rLs zqBs3`sh?ksJJ){{CEqlEt@CXa_^X&~48NjBmXX$UhjBYb3j-{;uJDSS}`uV95fQlcjFQICihp9qO4NANIPnNg-jLS?p416 zd<~6_k&jj6r4UQd|+0H74&g)=9N8a+TPL1%=L>RW z%$E&{`z-CwnZfUDK;q9xs=&R;!4yLVx>6XAWdI6@bLH4MzK$Y?)=9rJ`BDq8)%|~A*>-9B!QRou=hI4Bm@IgX#C`c8qxq?a zjCTD)MpHZDA)2Sh0h(T6Ja}}(gx5|+jes&|`;R;tSO;Y_fv-PswA(yzTr>T)DSUd8 z)vhh4uuTq+!*y--NwI#@t*C$a`PIqZ5pOc~;o?)zwc~<+CJ#dxPQ1GjhbbJWkh@1& zTYOTk*l{aMt!l`|n8|whm&@TFlwFx@zxwU$^~pZ1>lL8s=ZaWiRm76BG*KhSFGno7 z6!$4iRWPfVu{mV*ATQm93C5e_%Y%!5LDsLIRwNkCYzms#IKA+z;=7@!LrI;nfXk~o zTuTAIheYmOcHHTg#ngrq3UhRFWnpa!c-k$|-%rhP9LMp`>ktPe zO7g2xPN$NR{HEWcqN9W&rBg@Z6meXet&4Dxtt(p>HkZt1Yp!hl3AQfWG*>R%_z#RN z@8{m<=CRlF`(xjIzuyqT`g$bfg?gxiD(Hj`=z@!I8M+|>S0HFD69P-n4_6iPfBjel zYT!CtgF(0fLvS1Jz$lEtIHX_-CSe0SfHZ7`8AYD^gL0r2p1@<+1e+nMf=0h?3+#q` z*a17C0JcIQ?1CcL2gR@r_P}1KfcjL|Ic!;$Md9y<}HWc8zW!**eQ+vv>eMm?X;4dok)A0QEJI@a^CZTrlP*i zDwF+3mz4SC>zw9FeD!GVwXgG9`0A@yZ?b2bwD{6jpB8`kYSxFpz6SL1I;pP~EtmVc zspae)ty+osy04X2zS^`}uEi;Y_9snWxE!ZZQHiZWk0!v^FQZxF3#P#`(Au6 zKKJj5RoKy0*cyE-K#)sJ4B5ri84R*R_FTI#RElM3cH27AaqY+mm<2;f#FA%O%mgIN z;KKIX>l+-WJyCVzp_O0~a&oS&wJNOZ+K;9T6|`wUGrr||ED(|ITT{|^_ZftckNe*F z-m@0z=d0^G-l{i3;KTdr2tCx~O-W=2Uv7#-tb;^^eHH^4A}3|tA{-7yp9VbD#WlOfmwFhnFgPDJ^|B;ogE@%oSc4*L^>+ZI^<&vgicKL4tRez z^7JzK_(jFZ1b>B!)NdH%Sa76L%1}<~$rTo--6;{DWS?ywgr{{E2bkC{OgyK1ba#mp zon4V#jq3u-v9!h`a)sl%ZsIGFPzt`Jkhs*`oEo@CT||gHW{$CF0?YHkf?d6RvB<_F zHgR{_NEOEWW%<8Z`w6U|L&=`@)PJ+mc+62wSw|zio4$WWP%UX1!@|;Jb9%EIjeStX znv@M@)=L{!`v_x=tj9?5H0a>F&GlwfrKU)b_Dq5AcQ1|u*m;dB!#)O}n>7uFvw~fV zJu}UPWZ;8yBT~P^j>hMQ_jM+^UI32+9G{c>iISLX$v~_Od8v=;S}L!M9{++Rd96vYjxV38 z0vu3@^(IWqhRLa2t><|dDLa}ouBwi%aZv}^Y%4SRFlfa>w-3(_Leir$`a5FEExqBG zx4UT)%JhRW8m@gQ_I1yG#MbzWWDTA8OZ%=XDs<2;^Y^ZnZt{2rmX-3uWJx*=y^^!+ z5U#@n(uIdQQkF-0B!%zv+@o_b1)xtX>n*@yxN|CRI`N6sBHGX^4qsn)WtZI#(0QM} z%j{mecnPJ}+nzG^WINC%Zhdqqh8i9TFJz&?I+-f97qNdM`*`}R5Lto+A8S}q>{$l! z?|K@|FHQKycQ&f4^6&Ed9SqT7%#ko)s_?xQUAd?*yd+ml+3@}oY9pI+M}tFj{2Qz3&iKVxA{8rI=W@fwrkOrXOay;t&@M2 z>G*>w#Cv4#J$rE&zZ#;`3E%>2hrraURW^TRSB1KGMbWnj1LvW=-@Zq`Gv9osup@`{ z`B%47y3O+z;J5u{V@N0Yzv7mSz)f12A?XFCk#kZ1^DVSe4Z#|$;Ly-ZVEKaTkutOR z6t*Z69x1Ba5u`Ohr{$Evr|z+sS(n;>Vv&!WPNGa^>ml zVFx)xe8fNzDtjX{yNDKxZ^jMv{loM5q6>kNwKaN^e+#$WgojAye7M*9i`-DxUK=Wc zIe9;nO`Ljj3Pf|P3K0T+) zi4}>2>SxFKP@dh4-SyH_du>NE#>#<($i~-)V8AVL1<`tBSrlUq%&LUnW$MM)EI`U1 ze1$9}M`$A>mO(_uM_Q3qx863^APW+FU{4J7GY`_E$?2)Oq*|8CQ zwEgcpe>+lB6-VRxG zjCn1(YI9d}M(vEW0gDM@S@7I+#)uPxd+FxuY{3(3EP33Aji&UW0<|dc#5;^SI`f(P z(bj34Y~c-IUgLi*sa*A0G*D*6H^?hL-$v0TF5OUF29Uw^PzQ+po4G~iYCom5hhPvT zD2>T|qv^p_p4|feQu4IJm!p9kO>QypTTs*p^3Zn^*B>7PrA^&-nB$>gRE~ThO*yC- z(en~yUmKR4?ff{6xPI(4{0r`nqg8+F8Oh0dJj-exbdxi7{q!y=1KvLGydSq(d^-)j z!FUew=+gbo(z1~mp0P8-UbG=gvRpveZH~B;9jX`=j^_>}qyg~xA1>z?7#gw)$>hy# z3hq><(vB_rJwhtv?=xzHUZ=^vrnf;d(ed2lCI{V|gX${rzK{W))4xWVe+WE4~#)sn41b5fHNMOKuIZ&ftm6 z^iU!YHw^;J{MDnKOqF*Q?|`tYRKgL0t@Vh-oB;!n|X$>Sj91V%O-xN*`(L>Llt%w%fePh*PY*m4q3l+a?16)^2w zT!Wp_+5S5rlV@iiY(th_L^}rb%=%NLW5Rgs1O0Rd~D-kurt+-e@JT|*q3tE=L*4Q z#cuKiUN@hV=fhI>_Oj+zXLNH9(aiSm^*K(Gc&@65^DQ)aDj&L@t0uO?s!;jzclLiEv|yk$S?_A+3BuK9PbjDDH}}2hVy|}iv8EaE z={Lw5|7SUL8c4y9;-|2*%h1_cwNBp0!nb&09kSRqqm8C7lrwPpVSlIV;u)i>)T`w$ z97~Lt3U%__N*RR8VOpZ7X`65EL}+Z&MUG4lOp0(x)`D`zNqy%PS|9EV=c8pR=cxe0 z9|v_4GcD=UuBu(EMpU8!j9OI9=jyb{vPP;4CG!Pqgl@8XC?|6TT7TSh{KTQ2mEr$B zHely)>3B2GOeveel|!rgZP5)?!nX2>OE9SO?gs91ASc#zTTht=XQ6bheM}dxT0#PN zbWu%qT@iMH+uiHt{yt9m=u?10T_5lzAsLJBMF_U^=QP>pimRSydPx2Fcm1#bN)evz zGJAOv3lnF>Fvv!x%=e1D0sYt?zu%n|&oG)JAk7*$E^yHD!I2*2g^(y;+g#VFKI^L` z7?7D`#RVk1n6=+F5qFHdSVDGhLz(R;ehNX+S}AwHQ?}-(xop4DC1?6!3In>tqmace zhi#GRwyWN2ye!}=aE&H9ID%1a7m-V3eX&9X zc?aC5BE8pA->R&`9MT#+gHu@D^4=_~rR=-0{#~>1 zrpn#oK(V)UkKa+QJB~)=qKYqSub3FCDgFU1q$?W_5oH|%m_qLG0JMA9XmAg2D|*Pf zN@Qw_JW0DaK&fNYJ3?G2=JB$P)#wv}up`Is5R$IDdZ#Lm}W%eGE-2)jbU>qgUs>#hIFS!H&oq`X;E8GlJ_idfdA?v-W!;PPw=K5OHV zc&kWuSXQ(ZICXco1*($2DJF-BR1M=G7u?TM)S088=?*g%ebI!8Z~^WmZ?U$fAq_3NeH7G2ow{_M5*SEM@OMgozwrJ5^bP74vFy6)7${&?B@4uty6n zZ~A0RRb(ymdRMFt&x}2U?L%x!~QQJ+ntc-MTl^dw-|d9U;i|xh`N@m-dNWykR#vLMb_Z z_()YwNgn`Dm44iu1Y0)P+hpA^q3wsubi3uUu9QcnO}Dm`Hbc3u3V|aIbe2Pnfwg#5 z#1fwX6Zr4>_icGzKjcw@wbIWRa2zoTG}ObuR~Xy^0Q(d9KX^LDUONB$~qI zYYpVj(c176BhB3tJ*YnxV1lpcWP=Yp@5^mR_^~~p}TV|fBPzH!{PD1oS=kd1>aG~LPm;RXpy{;v)(gFC+d9VLmM79 zSmDGMKrF*UJPo!vY{ro*WzNk$E%8|>6d645G<`2{Ra6qy*Fj$lJ|Dd`JP|Q{ac7n+ zmCsGsp-WOLlFP3NjzO7boSNk=m4V|-?Rl4g z7TYRE%FB0A_vW(_ekgWmd3j^oh8}_7rLs+9U$d^3-dl z>X+o?DKFd7*Sw0@%QvP49h85WEA5Y`#hiFrFPt+FN2uvIrTx0Rj60h25s?XeVvP zYt#|F^oKsoa&jVpuZ>R)3lW)PR_l<6VeggK2EUgy@%gipj0{V9yTTe=*oqp!ReJ2^ zc<@)8^YI%PHuL1VU@x!vk*WYc$lR2+=ba)x_S>fvT~u9<=ef=~5i%+qx1SE}9o;&g zo<}m8Y%bYN3|%(U@LeW2?$+O6n4U2LJ6kMzH7Jia6V4WuJjVVdnrc2KS*NmJG#NMn zUd?+h%wJimDa-?L-S?zdtjoGS@5gE}_&uV3AQ%qC_a=U$!>(VXBJ2d9+@Fjn-^o9C z6QY_u1|4G=Iis=UI_oCKa-!R0-u=`&I@#lQ`|)B!SPxjLbZqw7h1T;S%Yh-d;S0(B(_fko5 z{%+S*DXbM_iFPz&gD(XF|C~oW>UXWrbK?Z41td8ut&{l`nEO?K>@a-(X8ab#ponr} z5Q_Z`{O)@UQ_JK1Pc;J`Q1~d-;BT#FbJIf@=)Y3kZL-*$_QRp4q-sS5HmQg7NplSS z8dSN9@$IX?`rw2#vF4#R-M6`4jDd7Pu!IgGgw;OdGMh_HA)P>@9S@(;GGBIe#SXI# z=U8Kw4dw4fqaEhzr?E`!+`d_s2GQmqsEP)H@4_o7O|HK=E{;pqQY4#7Zq2 z%&4&13;6Rpk~g4qab3Hoc5MIp>-QjY1&QXc{0*$9%O4=BDLVmS^>UmD1Vxa(@yxtJ zc$=!7dJ_jZs#4-;>5DA|k9j{wA?_cU$KFJS%>^}-SnEyXvDzR<^40Nv4QT&laDE=x z=C}Fu3~m&3-qx*eZvmAfc?XDw5skJ$masLoLxMYwzXi}#h|(-OzOvb4y%_0b) zqt{5hFUle~U=3w*#u8m@SgPH-b4|#G-&ax%Elo$I1bQ<^gc}`wazezYo>38aD%~Ik z6x|?H1I5@vTrV*UBKp04X6?b4F% zIPRQEL>}1J6K{)0wEk`CTLk$5x5N})IJ4Ws4_FjwWowg(E)VfD1~$^wnw}`g(pbw9 z>*9X~FkDz#LtbhuC$7x1;_%t3sJA8F5fQCbJclyZ5T|JbIIfCcBseQb)PEz&hZM@G zI#y0vP{vd!R&HRm4$_v(d&z^gykHVv1Rh#(*uH?hXZLWvBV)}h2kdGRVV;aP?Yir` z1G#9FY*YrtjeBYmd7ik|Ic9m-=Sce#JbY~pM4XhdS!+YKeYzIdiVt`+gbKA3A-Gth z#z9{_lfYb&6p^~pF5HnbIVldgj`&{uHA1aPmz*R#h?tpsz}>mj0Jx5npF;Cxhw+lQ zp!_LeR6bz!?@l5T!E(0c*(H*>uaE=l*EKb~&xGiSon*8SFX zlZiQ%_^F3IeR_Y#wxf@4Km{O>fx|v4#kv~h|)+-@ZB%oMVF<7mLr$HaN z9>{WbxShiC4z2h1!Cdg&wc+vR?Q8n=Z^N5$!*KC_+^Z#L+7c$6xQ%mw>AY=jrv11W zjm2iCT%C5IByW(<34DLI&P6L`8nKqmnEKJ?w)`o$3I#?D(P-@~`@8yIV9(PT=OZVePo# zjd&yMVB@#J0c*|Kq^cQ!-7)G;wF|cLzteT-l~(>IJ%jZU+F;_rc2(HR_xib@j{GwA zW%DR^%-O=`@3b1cWpbkvyq-e{bGCc#7xwy-dkliM4CXOLt3my~U^&EkqCRAUucEEI z>om>G^xG*+ry$eXb&Z4m-B6v>BiubPY^@1rV;euF>f||~qA;S0mN_61_l2>*wCtF^ zc+48HT(h*e$s%*X{ot>GhKzXbn1@^3;w7pAsP#os6Qi;(F^-eRd|F8P};lCS7yv zhMl+<5&4n;vZXl!Nn@f%q~GdjWV=GOIMGQ4sNp8 zccMh(=gk&=1||w^)^!3Yqy8Bd?lu^+1r!Wm{imBVL|8hjdaoX^7K@ZY<5lu2oA$8y ziszI*I=jT_Bg|kg{7l;U?-HLMT&6sG4!hYMdBZ^dIkMx`22K{c^JG}*?bUjt^*x{A zxyTC0awzXH88rX?Wc1muJG)Q)3I5e#EBhj=8?0L0CoV!9KFEZ%-_S|?>@j-hJ-nU) z+sA}e949ds;WT@Y8*kc096Q6`+MR&|PCk`c`*zE0UiRc({ert=^TRgEr-F(-L^qI( z{!Q?qWcrV|*W=0E#VWfc{y)p|+}KPhhi(L*Q&+Zg%-fczT{iry)eCJ~S*66st;C1)lkafIs?e%R4Km;!-*vP1kFYZTU0@1fqH{cla%EM zUMGBCIkrel$P8am>ipgssCpeOG-d{dPC`s@w?E5}DxOBc8yYX}>LXu^(l|F8?W=UN zbZPSsXE3f0-uL(Kn~3`v3SEjD*fx(>!=I~m;)U72ai_o7Ko7XTfZ?XH?!=P_KQ>pY zeyh*D&g`cTQ&c`JEt%l1*cs!BP?@a@IFJo_2}a(2z;9pYEhd6YuoD$ei6ptWc`bO& z%=r2Ex%qg(=7JzjumFe;#LWTb5VVxp(dD+*w6z(+6Y zRhCBv^R#b@k05<;+@!1|s3e3)09oe`{MrtY>lZ9_N7z7ZdP!3x`(EV`s&2dh0bnD* z{y3n01_gbG619gy1L%;FLr+>LPd1{{!a-q=rC<=k6Gml z*$qIVJ?G?oIIZ_&FW5iQu*=4E`2ZE~rP7m7@GUT%XieB?YjaAWD zi>i(@mCzq}$#!z5_?kq&%XGVI!vd@bF2z~%7#^YA*+;xw4uX99t7WbM)hI;3)mhun zvX9Y%W^1*(^5l#4%z0h4itpHFXF)4^R~ygcuugs#*mra3vAHXwLG9qoAREuw!Ei}^ zVi}v&_inKwIkLL=a+m4P$(g^H%O=rlpdE47NA_U-kUY faxk<0kAZ@mT|J$_7RbDUf&!e#w6rp+vdI4hPD-t^