From 631cd8d9934a3814cfb12000ebe27b02db6af277 Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Sun, 17 May 2020 12:19:11 +0200 Subject: [PATCH 1/2] hw4: almost done --- hw4/assignment4.m | 66 ++++++++++++++++++++++++++++++++++++++++---- hw4/assignment4.pdf | Bin 0 -> 44485 bytes hw4/hw4.pdf | Bin 117732 -> 117779 bytes hw4/hw4.tex | 2 +- 4 files changed, 61 insertions(+), 7 deletions(-) create mode 100644 hw4/assignment4.pdf diff --git a/hw4/assignment4.m b/hw4/assignment4.m index c1e8c9c..0072451 100644 --- a/hw4/assignment4.m +++ b/hw4/assignment4.m @@ -1,7 +1,7 @@ %% Assignment 4 % Name: Claudio Maggioni % -% Date: 19/3/2019 +% Date: 2020-05-17 % % This is a template file for the first assignment to get started with running % and publishing code in Matlab. Each problem has its own section (delineated @@ -11,11 +11,13 @@ % To generate a pdf for submission in your current directory, use the following % three lines of code at the command window: % -% >> options.format = 'pdf'; options.outputDir = pwd; publish('assignment3.m', options) +% >> options.format = 'pdf'; options.outputDir = pwd; publish('assignment4.m', options) % %% Problem 3 -clear; +clear all; +clc; +clf reset; n=10; f = @(x) (exp(-(x^2)/2)/sqrt(2*pi)); @@ -36,12 +38,21 @@ p_10 = computePolypoints(f, xe, x_10, 10); p_5c = computePolypoints(f, xe, x_5c, 5); p_10c = computePolypoints(f, xe, x_10c, 10); -figure; -subplot(1,2,1); plot(xe, p_5, xe, p_10, xe, y); -subplot(1,2,2); +figure; plot(xe, p_5c, xe, p_10c, xe, y); +%% Question 6 + +y = [-0.0044; -0.0771; -0.2001; -0.3521; -0.3520; 0; 0.5741; 0.8673; ... + 0.5741; 0; 0.3520; -0.3521; 0.2001; -0.0771; -0.0213; -0.0044]; +figure; +alpha = b3interpolate(y); +x = (0:0.01:16)'; +y_c = spline_curve(alpha, x); +plot(x, y_c); + +%% Question 3 (continued) function x = computeEquidistantXs(n) x = zeros(2*n+1,1); for i = 1:2*n+1 @@ -88,3 +99,46 @@ function p = HornerNewton(N, x, xe) p = p .* (xe - x(i)) + N(i); end end + +%% Problem 6 (continued) + +% assuming x_i = i - 1 +function [alpha] = b3interpolate(y) + n = size(y, 1); + A = zeros(n+2); + B = [y; 0; 0]; + for x_i = 0:n-1 + for j = -1:n + %fprintf("A(%d, %d) = B3(%d - %d)\n", x_i + 1, j+2, x_i, j); + A(x_i + 1, j + 2) = B3(x_i - j); + end + end + A(n+1, 1:3) = [1 -2 1]; + A(n+2, n-1:n+1) = [1 -2 1]; + alpha = A \ B; +end + +function [v] = spline_curve(alpha, x) + n = size(alpha, 1) - 2; + v = zeros(size(x)); + for i = 1:size(x,1) + for j = -1:n + v(i) = v(i) + alpha(j + 2) * B3(x(i) - j); + end + end +end + +function [y] = B3(x) +y = zeros (size (x)); +i1 = find (-2 < x & x < -1); +i2 = find (-1 <= x & x < 1); +i3 = find (1 <= x & x < 2); + +y(i1) = 0.5 * (x(i1) + 2) .^ 3; +y(i2) = 0.5 * (3 * abs (x(i2)) .^ 3 - 6 * x(i2) .^ 2 + 4); +y(i3) = 0.5 * (2 - x(i3)) .^ 3; +y = y / 3; +end + + + diff --git a/hw4/assignment4.pdf b/hw4/assignment4.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b1929fb2457b55b8c8321bdef48e7bddf8321739 GIT binary patch literal 44485 zcmd43Wn5Ov_cwf@lpu|OARPiCjdaOHcS|TO-Hnu>0wN$?lA?5pfCv(zASzu-htgfr zJhL}?e&_h}KL6))KmQkxFV121%&eI;d)2J<-4~-8QsycbJ1-VvL19rb78eaCjiadz zmWT)zhnBUQodpfEq^qm7rGveNgBuMm3l@izi-n1sqYDi)($Va?qoXa2vWc6OuA_^s zE6oi}B^pgPZ#xTDD+>!ZR~j8Ic78r~?yL4-geNotOjmPpG#V**ULjsz|D7%BgDoS`73cP29jvkV4#?+?MLydS2(#Z zb8=p0!4eb0vT!hm_RamD{nOBJsL=3Wf&VF4I9R$_(eQvi4X7<+ZRZAdz#(G?$RaJw z9L+5t60UB5HG3?tC(Fr}YFc_Efvb1Yh;T-gW3$@iv+toZ%rZ#6$dmAlwa#jjR5y1* zmBA;Fe@K%Zcs3GOQbwKNeIPS)X0{4_a3D8sBpoZO;&jy4;=S@pZG!{LgF~;?LRZx{ zt}BHG+w}z)f#;~$@2Jo~s&d2&x^m@6nqiYy5Ofh`w^$&8@vP*8cfy|aTTq|HMM z2>)S*ARJj;#2j$#jLr@hnch;O0@x1HnoZ~+tvT^2-4=Z*Zj zmyUwphL#X`9|F;H0!7r&$t5tz&S)B=*agx%oxRnFhZ%E^S@jIw#YJ zOhW(EWK%}<`JTOr=NG)OGqO$cP9#t8Rzt2#qVMvKoHM%X*!+}}+8>RB4iOZXNl#25 zgvs=lYVtm&B$Cc6G%fTx5-l60`@x*#d72ga-B9m$vJB+%(%g@Q)(EOlcF1{h=LRb@c_>eBkTz!byHSP*4=7swTvKKO0)s*va`}q2pZOFH&B4m2I z7pyd2VhfQ`hIh!2K8?y8?oRA(o@3dfLl2IAYh}j9bt}n93F|pR^S)q2&Ap3FN@ill z=Wd`C(GHxW2>NciS$5$Ii5-Urs&Melmar);=NSQxOB9J=SDN^ye4IR;dtX`9k=>$n z4ExdCI^{WiW!-+)cK7mbHR>73Fk_iD+IRRK1gz&QG5IlL??oZ8)WfY*&oe$J{6VBj z$U#I+Fdievz!!tVaxd=skg5&SO-50AKm4qRa}QrVER%MTwm}L>%OG#XI-6_1xaxgH zFx*g5OtxMzo%DTJLD*3EgYXsNVggYPA@9n`#{nekS z%69e7m27$WoI7Ikdvq*+UFEG#5Pi97wz4VGR9-`-M<#||kNGRRAnTs|L)!1m7cRR? z6eup_^J?fT4#^7L+|S$MoAMW4;|Zs;fu=w36}d9{?el>MacNrhpy zVf=`8jkovfx(MGQ-x}Z2!zX97n-pkevJ59<-YPy2XigyL%roEOAtAQ7aFbZ`!s3Mx z;>7sA_`~>(xS{xT*6H}y>rZqjej@7Cwj_pCIrCt4<3 zrXOlvR9pN!;5^7zL@`La@MR%q{^f$wLXW$s+ooHso6%za-2FMqE{d*$0>>ii((ycT z6A3OSo)0{8T)Vw$pB`J>Fxxd2wSLpPKe9c#I8J6w;ru9)T$J}Y`5p3Az7?KhQ&XF_ zJk~tPJlb41JjHyJmOV*hpG{|Qc;b@N^h&;>RYc7&&9KZ^wx5)Bm4{YXl_!m6*59qj za&C@$7Gv?s;tk{YLz4@)-9JBcEViO_7&Sf`k@@<-zSo?`tk9am_Ri2wWm-LDXGya| z+t`SI$dLucz2FL|tmen`@4Ls#ht`v0V>IJ53wwhq%3Z5AElA_SYd@hRI_nDRCfHUi z2Bqc?<5v57W*;#9u=z1drB5XiMWys^#~UUD+$*|jZ7(HJsNR24lCKN zW#jpYq=^-LB{>N>%@0$zTiZjfsfLh;klc&3S09fl|30HLU%YO>;Xv<*J?_DN{sIZv z0O5|{$dx);U#_=&+1E_(T67A1_IY1rl2|b-knn`NLn+6?T<@D*oJC(#UX9h_+{3v& zvV)|BB>r;ta`$pY=UdK~r*)>zP7f26oijN%TSM${sXCzcaOcr-bJ$QATG%1OW{#9R zwut8~P42il#v0-KRr}FYO8s(2?C#v{ti?>rQa%#>lQzmK;RVm@!t@?H-&tp8W>r>Q#@ucA{XIukguVB@ExcEEP*>*L5h88}6>(GfPO;r`{m?V|C3<>U`^c*Pw*CwKenKR!OD1nz{Vap^rQSPl zt-qY+AU{ivp~9)k%P-G=fzMB{O|(t?`<;S@uPzH___lw?-3cuN;*&%Jh3@0U?nY@f{}-wH)^o{^YA9b#$oV_Bp0(VS0tXIZ9wR7Ye zG1lv+bL`$+`Y|*}UK()RJQLM)KlA=Pe;xmo0Fs7ypF10A-L>`7tM4sWiN-iMF8Dv* zcUqv|oNml*ROR_JpTU)JrNPR7esAG;yG`Nk$A?{L_tL zyW$>qD^COp@8TOqMxC8V#0hMrL=+UDsD+~jUP^h}CPz@@Qq6N?n25)lOTC+Aku1OM zk_43J7_BJ~QBumy)_};%Z?|1I36M zH-8P_`#AtaydX5x@^-SI;ZU}4Gch-DGl8CHx|_Pe4{skE1>#v- z3pW~53rlMUQM%RI209vRb5S~7K4ng2Cn*anYdIem3r!ytEi)fmGeL7YaWN4uAuoF; zd$1afm%W{XtB{u{oe79zErh_|&|?ldF|a0$h>Q7cAq}L=&rzTyN@wNf<|M?y;pyqg z?#aXM=wiviB`7G!!O6|R&3zS&xa#Wd;AY}=)xnkil*lhSNDEgp7i%XsDBwO_)!gha za!&3pb}%t>GY$(o3n=P$<=|rH;-I5}_F!V_=xTh(HwoGyZQ(!q1U1Kd(w+;L7%M?0=s9 zUorfF%5P7Qby_4Ue*dZL1*r0of`pPCnjWrilL^ zrbxxa9+)C|doHktpLXens;P~InH%)r=bcsnHbbEOr#tQ6t(Uw#P$FoeuC+P13-ev$ zgc@=d)|OV_Ue67;u3Njgs#&;5IodlpIzab*0cebrqn)FRCNORmH2-Zx|8Z>|T~^dy z+@D(yut?i9m4AnxEH6)@Ir=4-$d@J?M*;aI{1(HK;VS~s4-YK1g9XrX34ji;DZMb21|{?AK@S-BBP_DhlYlp zJ$p7eIr;J9$D+~#jSJ!LZ`J#@O3Bg@v7SyyM_dQeN)t?98Pr`p_*dRZCmD4L7rKL2??!?Hw1k95FJj+ z;lIDdvgsIxVRX1g6mWKd6McRBtx!ldddOW|GyIdQq0}KBbE~V4*4FA2nwpyBH`tPu z*f}@~wR@h*Mmadld(;OS1D_{r*+za#x-;X*)SOMrYm^^LSnhwgKiLb6X+%tZ?4E%f&F5l6Lt z`uMSq$a8vnnvs!l-cb$vS$TPRQ3E5D?|QT9Ax>U1+1uT#d1rDS%DqP9i`xiy{+Jac z)|OXj2-QtLL%j^xpn`eGMJ5vXqJA>K@cu4!#2jFCcDORxCRUJ!?(S()qEX6;BnaI<~8=-Q%l_LHQWk(6`xQjvQG6MCou-(Kcna zEVN2=c-ZB+4aJQ5SSrjdY3NCL@@V4CzJ`t$Lg5IQ5A$+Iy_Xk9JwM!!V-B65kA43s zyZr{U2@k{%Uevs@+i?!5!ppHHDR_3hbELy$bw-XX?E@x^9GHNBdn7w)$LQ&Rn=>L(GLjO>G zf?3{!9(n;)*<|-05_wk9K@AT^FXN=L<*Uxd4H(v>eu$1@D%4A4MyJSHyM2GtGtD8x`g5>bx^-fB%5dpKTh89GaM zh!$)A=qP1WD^b|HxjE{LjbtUJ;h~|O<>A4uuClbkR>f zltGR{52KmtvgZTnee>2WF&>_2 z`$D1bSHppQili1Vxt1;-;J-E39Up%|yQVLjWzk)$eSdhB?vhxLBv%i%y(vuDRwy_FaUiHL}jjqL61N5PIJCNhU^85)YX&2^2vx#PL?rF(ENaa%|U z{quXjQXh>5;*QLeaIgzqUESBOv~+b#?jcoXBCcGB+pp zOoLj?m-)(-gWYwrnMHhGsqU_4JPr!nq~Ju4kB<`)NGiGb_<-?@Y0F_1wTXAzS<>T3 z-`v~;#?{7oYq7M!!^5LMG($XKtin=t|8siCL~7`Y!x5JW(7kR9ZNHS1l&SSs`Ws%C zS6W(I6%`b;ybHdnsjEA0&L~#~cz9Hml&BPE^`&3YEir$C+r*EH<1UkVc0?@)DTz8R zu*FGs&qH_`*39586}wB0T#B`6lJNVNe8pfwCzkM*4IWAZhCak7A-@NSCxwZ-f1If~ zI~AijJ|s$|;;UH_hQXXEp36~2*~jqo<0SpzyE11AK1KC7gN?wA;zB4CFEjLeqt=hB49*?n!E=a=LLA6|X zjA6Nq3iC*y6j;uf&`Af;^WOs0u5(%wz-EEIC0U3#w8G=G>s`{8slg2e*+0H+uQ1$g$yGleing})CeZ16sSmW6SgJdiszNvgam=D5n=h-Y z@y`1;T%nDYQ&Lhg_XnoPeYPVlI5^m*wzydNYNpnakn7Cna%=)_(X^e z@G&Pxlz5e0__v*XR-CZwv`7l;hJ_=S=R)?{= z{GHOGq94e8rLsNxK~b94-2gkkQUa$P+%Rrdn1kdyA&f$Xv>34cp(WVTSl+;|ap)$?lrxjjN|Hj$8HjSC3l%+ILUKUu#u3D z$8M4~Ns46DxC?jD?kZhOiX1jO?W=w_h3TOd!i6Jf0H36RsWYl=8FC2L-%iIpqeL%e z#-b+E07dAy#KqHbQB$2RSf94=I;xw+*rGS+e^=G+BQ>rIefjEbiM5=TJRCS{|io|wZ{!5%CQ0}a8%FT$Pf2sGoH1Yrtj?R+`YS6@dHW6 z#l;o9r>*Vhu`(hkAdpuzJ2OL0K|xAN8gL^j>HHx+e?_wcTeJ(XVO?N%gM+j7Kz;w{ ziwmx~ySa@t1e|<`rYGgNb+|RJ34HM(^^~=H8C7fvH60)^K0e;llYQ;^^XI1JH@K6v zp2-%+bfk+ezIXNauYdVc>uNZasE^wnDL=m>5F&{AA13Vyp$71b`o+WWGB~O5rAa}jwUg`}x3s_XBNMxNZ z#+LI!lDRrrz<#Nfuni7`2lXzYh)C>uLJ|{@rpkcv>fx3!G^%q7bFB=if0^+i-R|de zcX`Kf!H`IdjQ0eO!x&?xS=1Azn`S?PMxk6}E>e#j>M2W}R%A%j%oa!t2Ge(G6BFgw zbCJ#+=+Lb2Pl^ndn%THdaKystX-E@G=>$^=-6w9ZfgS7#)*@Ya7;WCRPgDq;sroiU z*w!AGcMbz0!+bAQkpV+9TkO(#FfkL~;g+-uC*wkMB{URkc7m|2GA{4JAhcoTC&7vg zgBsa7Pq4sH<61)lI>C5C_h~I?jp6S@gl+k8c@&;7Z?l`54MiPM3Q=@)D>+D<8Y17^ zF-?iUC;D_Xv4nizRG=yK`SVY<@ad*h6#5?1H(5HP5;vzY#?Z5b$3y=jZ0s#`TPh#IIdT*cKwe7_Xd_I1^pIi;fkpY3c=h@4stnp=h<^xDLg5 zy&uJ;OGWwlZ9bF+!K)q1cO;qox8Liwa&)97+f8=_7lU{2kTbdl1_t2noKJ$B7sbZL28u5daO^v< z_(kaAjoszT^R$Qo1;6Cu!$S@of1mJT7A`H|ZvFiH#ys5Jx4(VMsfJ>Ky1JrkAoi}d z9b|Vj&@Ktaz#FUeblhCLanYAVR^rQ&0iAb|Qkco&Z&&z&o~JtUmYwO_Q-b2+M*8}% zU%lD|4s&MacIA5WU^(|LTVx>S4ezjlk8xzze#XvTdV0y8Qx3kqbvZf6_}mnn3l}c% zJC11`s6G)Wzw={7+|kO;?z0c&K~Mv9@sHYKMjYtXAYX7WlE15mq%_;ag0#?&rJE)J zZT#aoip;AACH?vdgzct!YyEcZAJeHN3=*zJvnht%E@y!@FsE0Ae8Iv<-ufJ#ZTVH3 zAyp&Wx#%2_YKe!+sy8Z`7*jr^Obl;DB0ih;61JtpS<%@Bb;>Ftf{K#y^L>YJy@^0ZZWhJI%$GXUJ4CQs0%kXiCxh+{FoU3k^MIzoJiTWGjD$ z%t0et%hl-53SkLcSHKw~rjz=2?R6AhK9X$WcgRLF9(~zMj{XdJKt}A#_lbKE$s6e2 zyAr|2%sA8<2c5)c3$^KL7M5;>=cwbE(-$R3f)nJP=OC{n69Rb)>{d$FNQogVHb<^0 z3&x;)GyNE={YWmXsz-bmy?6`do5{O(8Ygj9Tt~rW>r*FY;JCjkB^@}0=gvV zzR(BqT$Q$iDXFOy#|bk{M#qDU+p<2yh@h+!Lix=&tB2|7>0@KJ9Bb?9d~e@w8P;+C zXh4F4gF`9qFS6)pO$D8tUG~X$<95w#mZ9Nc4D@;4Po8X5<>uxlCMNc`)OxOZEDwFR&z%>!XzYyo zMRt=hrUPihe@DZ8tn*7>D{kGJot*{z0Y?s!YaNiWydghZf}WX_4Z#vNmnT5mA=OIePkg4?|ZO|w)t8qs{%qj{CWR=PNMg?F{(E~>_Ti6Y|K3$UuVbZmxQ0~Ce&GH%zn(t}l6Y9_g=q0x_xES2 zD!4snf;VEZZ;nvb$WG-wCuTnuIL)g^XB|U0nj^JVDRk$>e6$L?}r9X+(W`tE}W z{H!)@21j++kJ45$;800v2N6um!^L%9_gu8%` z+@)=*GFXVF$)A8|k~XzCf<`o0FA)WiCaXU;*J4!j=977hC7aK(N!o?$lJx`_k!M1YZT%@J`wyq+}Chk^Mcw3u-% zUHUtXC{*c-zC(2=yZGPb{Q(VTm~ratC7VpDa}U{e>55YkZ1@&CnN)lKyvR;}v*Au& z0?zYbadEM=rKK+&XjMl?$I-!$!GQs>J3DNP?sVAT%n^rSFV<6izBPaN5L&?TH@yRn z)4{>v7$mqRCXAJpBZf9XAPch2;O-^nwKmas_pDwik@A}LCc4_A4-mh<2f#ZK6#VI8 zeg)TNW@Z!=75iH@XFH>)L|?ynkl86f`{me{p>Ki&ap&g6t!W%v8~CBR}b@M2Yug}k7}3S{p#Fd@vd zsPvfDU<%|WhU7rV=k;HAF60~PLaKw=&Qg~JE}?5e`M zU>A*Wmrv6*f6XXVvIQ~{)O$BRFnSMHEbTmEOG{~*ADD^NiQYDZgh2t-UTq2>{EcJxZh#CY0`-Gu^##z~ z0C!u~*Fpxt7ZrLZ`M14V`baZrd-Ag1|?os5iIwc5%Wo1K# zjNr(3X1VUp4+oH)H8Y#;%~h?q)oLWldixs!ABQ^D6382rR(|*Qv~s;)ZB5PIaJ3`I z`+#JPO(*zM+dCyKP5b+NSC*7sX~igXRE@-9?CfCix%v6|<>l=vkifOPeY~r^TyA3;y{S;XUfUl z#ANbYTkLFShVC%fvkl6#g~4PV%ScLLkXTFDO9Q#9fP>Kl@i-tfw^`O^;@rsX92_Ez zwG!rmlV4oitOB0)^5x6DMjGx56HUQDZ8nc}wM(!`*}aD0)RV12ED8{iB#T-_h5=xA ze*CC@`7$peKR^ElTY?f3aLd}o-n(mA1*_rV;q$9??u!Zf7xQr{c{V?%TN*(2zfoX# z0hrLe%mP(V{t}_$xN>Fl&#L<0X{Y&ZKD^`#H02Op@QeAUCo-ahapI86YUJMQB?qggPi*o!B6ixw zA^D2C49l0_h9@Mn-?1ROVHa=t0+^JzE6~Quw9~^0fV|hhsu%tpI*@bwCTsKN2;*wQ z+(58re==Vn_>B%VRF<-$gf9i5(DKzp2(0y~uqz@xEDKNaVX$(ArcH!CTQ_>~d>jzf zgSAcgno&9+`VkA#n33`tSf6=z-MA4>$^-JWfSb>del+sw!6fyz<(wE0G<2@4Dqxea znvKMH=LZ4PK?xn$E0zj~!Ut@KoMFN-bRIMu5HC)8K4D0pQ&4<@X!VAvu|gezgOCk% z+Y>)dSai)tG_n*>9j&e~&8xpETn#u<1w@Vgo>nM6LxLyaK&IDn>XjOfI!|atu{^um z5D06K7nmYl1>VLSqibY@O&CGhLUU|zV^*zig0suy{8R(l`g9jEU#pPdybBVvAgvZJfyFp!#gkuCrD4ZlMdPiTUQ~jM(tS9e2n^QH_qN85sqvjMPL?i+_wDM;KH5;pY17T^JFF#E&;0 zfJFW0hHMgrWJ=G!pa6|{;3{x*v>Fh#)cW1$#k-*)eM`$(upL`l+YHe=Ox)<-S4L}X z5P+jGd&6!#W6>YYyxA9}N@RVJBIl#2$-g(;JS4*pApLexD()+d;=4V3>yzUj4JZ49 z4af7U0UE1KAN;LKE-Q+Fggq!Yy}_Xt@p77IM2Y5tlK*e`Zd+PvvkSS+scoHI`{t7* z!)x|oJ2WaZgmy#f>0xy^_1US_AQ7`rZi<|+Z&Eq=%@g4500SDueO!~ywtl#`3Ci-I zx{yb&*?wNS#p!pG9OaK#^rn7J&`##R8yYqDXR@tVIXF%RADDB`FYmRA9}{>S@&rJ8 zW@BeBX!~+n>5H!K=bz$wSy`DcpW+jhr{%Qtr5pr~p%{-(Ovt+*HO20QMyY+Le((V= zVrHD({k)xEpxQVZS|sg~-D6o)a4O)|qfBwU*BRh*hb}Z=2CseFJnT~?`Hwr$nyIZu{<7yyGcDKA@pqbeu`V|6KbQ5r$J)t>^tUoU@yItb{)BRGpPkw1 zyoqJ7NS^Eqy2yZmSjN5wo{cB_i=}C4X*Sa!s=TwabZ)y&xrA%vwYy^bxk3KtmRae- zANc~Ash>yNo=>G-Cn#uFB`4U+s-bcuJR+kLMt8xXdEqIU&V@|Id7&0%4#Y$H;z`kE zMX#uMwF65_%f*F-zDxPp*;60muh5y@(9;w2+uxdO4(U%UF>0VXLiEsiCCQXP?8S8$_MMXiIOT>v)xJ)d|%D|`&xEUPI3MsEM;s<^8SLEcyvT~DWhb!&4j!HoGLBx%y=Sg2lNML zTMwk|25t@Q&bzd``-$1!v~AxP`d@fQh)9LM$9|Ew{eZJ51p)~!S|k@K5RH_-r9||M zw4Z?j+Z*3K4RbpL!tu8cDr5@ZY5yBOELh>$3O4tqLCbze} zhk+2s#>S3S>0kWsio!S{Uj6Rm7oy!z zGy?iO(pNiOnG0g#{0T}$(8Pv-M-+uTKE7%gJN+(ZLN)Hj$B=!;WAbmotTF`*&x{O@ zUKtxvN5U01Oyup0Sas;%v;Ea*uW)2KL3Iae=?&IsV|6(^iQ-fwIwBRW)9pU*-#}_V z`*wihc$cIHjp-GPdT9>X3X*I7#BU=vhQ>r$!0_l4&2v>qH(P1SlE8!+DqTdKDej0O zPaNyjW(gWjG=>XyQCIVeRz~w#Me#H0)(Z=A_EK<#U`)%0P6H#kC_2Qg>R`P^VPq6k z7o{g9QtkSlDCT!r6n`!pdSb>?%Tb#bkY?dda$3dwU^@Q7is$v}W4Q1l%M_j{om#H; z_BPB9Rd4Ftp*%;N)tg|r&T8g2eQ6XbXGdK>uZ$}PAB5AwrJL+eYEK8$er|Vlt8rSv zUCcqMqmJtpVfu% z9kLffH?fFe0ipSVxGu?z!;P3@$t#liP+%plu#*IEt?J z9QN#T^ql3QsAdujf><7o5GzNNE4ZLQm8I%bR+$l5A{rg}fm3+VhK>yJ(?<1}B^jt{ zX)n!%b>Z@g9bD@yMsi_M491)`mZi8JqBFKBz`N!yAX^u?`yo%QSI+SiU_`XS^N@MxC*{hI2%w^E+Co;kBUy%u?;mQiIH+WPc0!|YW0|27)H2>0v*2u+X z1IG15PI7?L`7M(CTWZzN9@I7t=enPQQfYxpzO?4RVqVI2^|-WvFKV!94sAMu7_46R@?t|3r+t8}W0%5{Td#cUJ(vscHn`!z|`J%cd zhM?3GM`+D$D zNidWc&yWuj5>NiO3x~|_KNyNAe4v3u$tYk<>h3Fa6W-M!>IUQy44^sUs;oL2u`dH_ zeah32CMJ<;$pCn?rQx*~=QBofPq9Z=S6@VLG2@G9#{i_zRY+G#ud-8oeTXeD1k1k9 zR8i#UUtP596g7cSr9DK@J_d0-DFlOrW47R0zI(F;eXbL&JU_h172mFO+k}>Y$?I7< zFb85*wJ!KkRlPXyOpFZsZ%nJ#W44&rDB%aFIAiQjj|`~c$r!v|!jTmo`|#$4pemV5 z^t6C&?m1|C9FYffljh=y_QM!^lpK&^FF%W^HV+ zRa2y0rXgiw4T86W+0jRf*D-VswjXp!2s1^t3~A`#59a{2`^G(Vuvw6O#N*D4Z@G+w zyF$6<-&d8zMUp{Vjw0;LZ}#B28}_r6!F+dls>Xu=KDFpcBE|{43>g9#cu0NmcFkG@ zz#hS%!zlDU)b|q=OY~9-TmLMH$Yv`t#ex*c!OpU+m6g}#Ohol2xQBp%8(c(GGsN0~ z2vMVSRA+iE;7|EFq5_b?qeqYO2rFK`9&r1e zfpj|Lu@pXW`Vv998@U#Xx)0loHC+1i*S$^`os^XH;SdyK7dz5~F)%RtJq!#|^(FH6 z-2v*^tThr`H7)!v$KF+omcVCokj0`n)#(4+7kh?*nYjgm0*ckKH^Y^-i9q~+=u2^UvR zF+`8MS_0QNhtJ_b;(>S8{yZi{R0h_$(^iJly|VRrIa(MzbJuL@p;exRKm^WRSX`Q> z-igsYC6P6JejY~2;Q_!OKG%$R6D*!6*%2t+(_uJ&YDrF7pMU(8h~f;agJ;Wy#6N~8 zNZ?1{NS-b z5q<#Z(WyHxU- zshuxxVW8F-^+(3MDtx4&&}K=4!ryXID(0MlyRNS#b8}Kwx01kpj$!W!mH?!k+@%UW zNeF#3U>x=m*58crjRy90HH90&?~fDU`5y^EB1H9>!Q>eTX5TV)8=h)&-@pL-{}~Xhva0+3 z3LNWmL(|#WB8{&k$ay~|^!qJmYp2tqu(r1s1wg+UAFX)Wf^v$$ z-lei6p*n3*a8ye3yt|v7;);JEfGtIckLQGJNt1N z71zaGT8G;qqk!e>3R_wKS_pksrER}3H}@*wq3f03m|k;V##v zJLd82I@QTD?vHs^haSbv1s8}ZD=V*i)fE+e7Ix+r6T79Mk^ND#{lUYB55d9TKE6UgvEO@T>SaCDq&N<@I{Go9nX=J0u&G&gq__;+T|MKLL-bX+Hmy<`*vHQ?_MgD z=(cNXXN*Z@Q76KMkOyj*{E2e7&yjutWY9#~k-CT9j1~GqY@wmV9Zifewk+ybN}w#N zngB+kT*)(JvA_i|w9%uc3hBbixX_Xh!3C;5RWinCvZ(W~phCPsAA{(nQO8qBBS+-J z$Xzb)s=Ub!L_ow@#O49Wha5`MA|o-m!En|RyTu^*Smvly((b>@uu5yeNSRSTi0 zFcY>F8EQ4NXO?j5w^u1?Ob0sGmsG-1Dg<^18k`XX9;%ycDq)lIf4^uSJKHh*j`i*B z?#`;OsIT|8vjg=}Hr#sK&bz2|X6rb_5LgNL-hbCtojVTW^~Ki!eCe8izzt2!*lnRI z^zpReAAxiEKIV&rZ3A&+1^*UJkGyi82I^vC20jJ?K7P^NzXN}C4*U=UCT5i1kEG$0 zb5FbBl&t7D6ZF|o*ymDEQvs<_&8wk3cSwE}O!tZg%iX!l)`nGkq!2~%``kx+dqzcv zTO5CygFj1!X?byNP#!h z50X-3^G9}_1C4A+HthO%qH7WcJ2wj}*%(vaznCceni4N69$3X@K&J=ck2IkTbXVAg znQJQWMk>RzJ6{yaU3-R{zCi>5TZmKB%WRYtc%%86s^=+DlHNaOjNxPHMh>86L?s|l z^>}DsLRh5834_gBi1QZhtr1_N32`hW0s4p@AyZ+dZzZX~n~2s_T{KF}8Ar(@g{_fY z!-iSUN2-DH^jW=#>EbmmXheUE%dKO_XRw| zc!2|fkdh#C5UBUafG#(bjB@Qh3msbaaI?DgB(YY*p08PbNS9Igza!K@7g3- z41`%B7N3t&*s3YKjADUKBI#3Ri=(DScD&NLddXIvKsNNN(Ms}RI^2v7_cLBb zM*X(XiBuPcvmf5aT`nYxCbWMRu!f^QfA&LjXxKA=6f)h%7%#f5j;JB)j?W2D;JwZo zzp%0r**Xn;ncG&Byxeb|_{DL;F$_7u9h(l$ku66%(#$6K9|44xM_1c}{{p_DN}Vvq z%&>G5|DW;tUIXJjBGAcwL7seSO36iP{)FUrfL})W=e@V7s6jiPHv2A#(n&sX;BwFA zLnk4e;!OT;K)I8 zEe$f3WicH$afEV_!GHQ3uRx^VE-x?dmG|atz{s^-nUMlSBSzY4ME^C$*Akr*i@vYU zFsD}$sfcQ3({G)xPuCSeL6shMU_5O@pcaqKm5BZ_6Zy`#IZT}3PSLVJ>@7n&>mWXXYP-KUTu zd_6iaQ2G*t%u;(b2g48m_6XF1u}J+mAPh+yv{<8uv`E-904d^zil%F{A7c*z4RyH8 zFNXXILqqh|r1kkxZqR&Ok`&$e=S8-)t>2vcD{!7WgM))n5fS;Kps)t2GY3CbON)#B zz^x2`eMq1Z?kiH;`FHcI#p%-(UlespBy5Ww`CnwnBl$SY?-AnWe#4(?ul zYZFa(&td?OObP^A$Rvw~5fi@!zz}Fb#BX2A{&RbKd_qD&1Sp?T1H`(8g|v1F_%^O+ zFpLS}jah?tw5E@q`ZOE$qk!?JA7y7_dm`e!0ZMH)cYq%QCI8Z9@I1rUsZMesE) z-wAt8Ru5R;D@(<;*IMn_yT|_b-UDyi^ZwZ%DxQDLNQD`5O8^Yml(g$m#&-1YDXF&a z-z_Q?d;oBPb=^i86hDmujw&lEH0^x>1`2#oIL~t@9TP_DOab3g$f)tfULA1q>ndy0 zHvJ2AFX%k=4kQvz>axnq)t58U(uxZUHEr`iWDI`>r}70BFg3alm%&>$fh7e6s}_du z^g(A#bHjvx5rPzvL+=dr%98(0P^({q*XFz@)D0vB3$D9kO68HB0b>3ZhLT+;yai(- z6`}=fhBdNAO~HwC-!rVp>sW0+bOSu8Y1YMIK<^IasW&NbOE_|OlS2#YU-jPX51+SjZ6!tLI?aYMj+>txjmfG6=<6$jF1Q{3c=M!>alr42zcW+Mb~cVusaM=(Gg@Y{n&T@00@}!a5IG z^v~M67pZ{ky&aS_i9$uZZQhp)xDV^w^D9meJm#eZOu~a1hRa!fg0VNa_dk~aoLkj- zNUfHj@B6>cCIM0WjwhC(h}+0yFvAOI0@M-J4Maj)#sFAu14hV|sUAbJmMPLSd)FPY z6x#r+v@WEp#)TrW_MpoUW}tY(@DYRpx&RrI8*I9e_H^L|t6n~fsQoU41~#_UJh9m* zgz)&ZEg%f0wL?K-Bbqpjaviz^O@mNJ?||k4)Df5_3qCiG;X49)4WLDU^O5qs4OF1= zIxq?;vK_|%BsW3GDcct+SU`0xV2ol(8w_&#SRfljkAfgb)^rA|xxtg3e^$Hz=+8r% z+NhiiQ@929yx4-z!yD%1QxEK}IiNsQpkon&Ou^n7Q3pYTGe4bdR~u$1q9Ih&x91va zQ*&e|OX^P!@c;*Z!Wjje01W#|b}R~_2V?}^af|_M+`s+mL)br+*R2(Ntr{7LTiI}W zWqot!h}Rkbu#BA)#H-87 zwgQlI)FnX|U_^jEojXs8FwXz&Tl4<_dy8N7+b|!cLBQlUS@reDApJ!x?jJ|NhiJU^ zhdXOq7|~^q)v=>l9FU?<4v@YY$1S|_7YZPd8hp33e=vSR^)>1fK)(EjxmGei{}Fs2 z-;V$sy%~O$P2K3T5@Lx1%}|e{j52RiMDOd2&7@P*z$IzZ{7#eEDmY-s{>HP{F-;9?b~^yb8X& zNkEcAp7Ei$g0Q2x+L}GD@_#U25k?x@1$Nk+{`%H+Ck!}?0SH^2i)UxL6uc{r=i(irVy1l zDs^5L^^CJcsDm>ZB)cX)MX!~b!ArLx5ouD<|E`R^uI3pM!^b=_G*H5r<{pUx8|N7k`F8D{-{`j*+G=Kk1 zp>_>r#j{-aZ#qkpo>HZBDAiZBDrj%%ocCfDwH8P9RLq|k=kHz0u6?FEjLJQ^&gAXw zeaARQL{@(axmNtZFPuw8l42kKAaLo??)gOa1qJk%b3$|$zEocWe)RR9QLp|wb%j@A z7c<;KW}Ihbo#LbO&1vk<7}w_tUJ^J=us7{C(Gxx2F0D*|gcB!gT=XbD47W*Zdg`{1 zfUjMOPrq+g?M+jk2M2w3@+@jzgmm25cabwqrMRs1^2W|w0h=gUKjw69igT6l4vu^i zIuTv8>s3vx&2N!b=<5u8O}Jb(etQ@cD*oSGOgln2e4Zxd`;^C4=+&0x3I!m>O6U9k zaKOKQ;Fz25+Akm%*RS8bJN*ee4GULCcNa4YR~oMWr@gO$it_9B1`9+5K?Ffyh7y#X z89GL~J0+x3x*MC64rwH%1w|SRN)Qkb5NVJuM?_k_GlSUgz5n}t@4D;WwLV-+c+PWT zpS}0l&#C>}Xz)7>;_?24E~Re}Q1YfGmWCpB?ob`{eJ(B_3vOPR9tP{l(a!#_FGQ?O zO-=s#M$OUE=C3c53?0!JQXFW2Jq!_|cDBxFJUJ*vAu!;1-+=o#4$!a!1b%1;8VyV3 zA83fCj@^U=vG0s#QryGvQyoLzHy#q$lnvTI(j~;LvUq~D&Kc01J2Oc4sJ_nJnzh6y zBsOjtpPDOUHJNvmcXa3~AH_M}tz|;|BXSzaaXt`#rRyj2^o9toGbwJo(@TFheuHv( z{OfxBV=a=tmC=FmE3PFk+YWu5dxw8(ls`=I9e<-QWP~rP`tZk<4pElVUql<53`RCx z-d65@IBd?IkgT!*S3Gu7=Na{<%yShN44&LD0kvy9o84L-T z6{Qy|DkFvKq7p&D(B%8kT*sAOv-8jh|f7NY4IB9dw4_rg(!@#_QMKQC2S; zp;obzQ=S}6(Y_H)(|mdiZ4O0~%dE?U79I1w?`ROqrPNJL{sIABNM-h`o&C!mH=kM8 zA~cYSTysqLSHv6->v)=s6Xn{IpSs!I=o$aP!OiRN>;f(?)3fVdG7nB%P?Ntw#4AUX zc)q%;r$UoSkVTG4JSSf7f{!sn2jzn!C#x(8SUsXl%0@nHO|~`3$;XD-e6=HSTcQqF zr%h*{r+UQ8pzo3ZmC+p}*tr{ZoLKkt7+1t(f$bqPX&GhXXuDg=C_}>2guzN;%tzxZ zu1#EhaIvS{C)4rTMHs9iT<&n%QuN3CdCv59nvhY*NgDmFQI)Oi-Rw3tTr_C72T3Qi{Hx}W# zMXZW$ODQ%~mPNhR`NzWm=Y)5q9#{7{&P9xE_Fi% z%r*1UDVbz7@!r^*BBESZEwunj7paZO5c@>wRJrA-hNAt^VdbjN^f3+#_#9J5uARv! zW)!YQxqWbmj$Dp^!G&Ge6Y)_`K~lDX@M*`9p^UpeLI~ThBf2c7cQ zpIV89NG(QmL^v>xlzMJ{XmV%OJB7tdaI>k-wMzh(+^g-Z#*oRKre;gN5n<+`4>YT5;ck-AYeG-!xxNyVH#FT@+dkUk%T3_Zx#pofrNOFG zJTc0fv%Ct)y)?n}7yEjE!NXv0b&PmZD?={VBE{|ny~PQ+L?j1pyfj=E{j`>>XZ+Drek9>s zaAOoN@O_x(S+!7xZp`I2uD~nww8Om4A5FzxZ@n<@dHK#uU4%JfB%6g&Fqk1tmAn~g z9HrBTpLyjs^wLgN`$(PkQ*=n@-k&1S+x^ho3? z=Sy!UyR(R9;-T%U!;~5y3G3){L%Q>4Eu&iA)Xl%ZUtzR1r)XT_$)7+)N|l7_r@xV) zI5imOl63l5(tEy$F1{!0vr;eDNRytcTgcMJFOan)INrXKyV)7_Hk!)FAe2L5AcNGs z$CK`=K>!`{Jzf%ADd#5Dq6k0!pnE|mpN6%7hEjg}oN@0CEpPtgFZpNr-5<12u{@WW zew~|tqVXZ^A%%eDnbj3p{M5+w(#m=2)au)BRN!Zx#kw6z%_wj(+>?t+6n}DrpvOM3 zMV`65i`g#!LgD>8G~RmTbff}ttY5wKJ!rn3K=;8R!>CN9dJBRJtCgXVQ|~IKID!v7Y3OcA z5&ju#fcB9IG$@TUtjFlaVfP?=zxuCW#*4BDxnDcKF3E0mowQu#=+S)mT>JD(y6?`j6<~gmbgPISvTnM8Ggl{{}jKOb!1d{&HY&nt$RiH-;~NaQDYF z?R#ziQLnynmz{f`y9f|&i5K*d>mPF$L)M?%g`*{c{xvlPaGVCCT92Hgal{%86NbPm1y@rVAx ze^{1(Bxz1K7b`C}0{V>-P6P+$@!vtoKeP~Ng28socMAdIMX>T9xc{&aaI`7;FX+xU z4)ehGISl&60|VWJ{bLUAZ*u*~;s2tg!f==u%DGPnhQZ+J-$2LDmg*o9**A0W7tr=! zcRz=9?9Mk+`j3101l1mwE8~Lu?iJL0lPfxyBZEdMH3^|qIwpo!sqHmKwNV+rA zOHg?CAyREHRL8t*^JUqxzv{rF=Ni2)dq-`Hg!Y;=B(^vbqE%d+Ms-Qv$Jd?9iB>Oy zRoNLS$$01w3#cQ%e67+Ag=8zPIjg@~Q|xUY<>s6VH|EV!w2IU^+TY)lL)`Opqy3|+ ziss|(`F887))?k>mkODD*Ky+QOXbEr;qr4Uq>puXa;rM9m$vHW2W&P)bECaPb!DC> zu1-i!y>Em?!Fb=h^!f5rT-IW`(@l~kD8mrhcPxuiMlzDMA1ZZ5#wI+AC6l6D{ zUzU19_a!6Cvn5Xww6x`!3opcKoRPg2kj2^1O>SZtktH5Wf~oCdi^|{Y!``Rwdwq;X z7lX8dXkya?vMh`)CTIm+h&2kxI%jn8*p+k#i14J)%_g5C_qK7#*9j5Jv=&2>Zz{sl zFfSrkzKje+J3|jQM2>#YiBZApNR8`RaxQT84RayiGSp+!oE&iWBAUy|n;E2S%ZBKx zv~Ijrey;r7m3Nf%tmEX9B}O+c-()Fmrk4|;jkH16+9zI2U^m_~a%h{tRZmPOkBJ;Z z(%YU2<|hebWKC$i(|9hES$JaWH`+JS{)n94xh$u`B<@q?dd4*1!|tWGSMXPR;;@LHy}SzbDi z3X7AvGDU9eKpx1CW@j%9!^}ImeA)B&^O*7?jd16pCXP4O8agyj5H{X2b1+0*pHMET zFm=c#kGC?)3(UQcshwSduIkJcXsXbkt&a+P@TKs3TFQFB{ zI4=~?t9{{!SX?Bb?A=kide%$>Rhd@j=mm24>lFe8QAt_u=yCtNnZ))md@{2gVi%bS z8_6Ii3y2a1{o>-4JhOkar_j0^MU1 zbdR$W!t<1+dKojVh*dc8a>?(h-IK_Zpj9^x&*PBlJ(q12pJ$pEo2$J~9y{nQMmztm z?cb$SD3Yf^t)3sA*DKX)m1)JPQg$p?`<7ava9-bU+K%on5RY+QO0H-2{62NZ_Nj{i zWG37Pf;pZWB$AgNp2wJZi99z*A}>86&m!~EvD_fxJfr*H$&0zia{Rmw5!X{XYisFu z?;;;ocsr!eM&jJ0+4YhIM0ZerdjVfzjmoCpI@fC0xaw||C!DNkrRni} z_4vz|?p&0qq_4|Om*iPzU76nd3hGYXG(G!9v$eK+yg-U9r3G0&L(^v+JR}erw^lX{xeuRa9ptUczuKeBd-Iren#~<=Gd-GR$ zjkuWlghv&5QLu7Hsj8EiQx%80nyuutca&05_Q@y>T4uSenFtj64>niwdu|U|+bMqC z){5Y+jaV;h?>RnfQr_f-WfX*%VRlDbGaTL1T=A&QcZOst^R+~1%*5<((l72O{ zNmFf89|$4Hr(GrjTPqrKO72c9i(j6ScG;Szu-!#j2$Hc1b*C~cC9}_OVjUFuN*KAF1YqWQ+gsLin0!eg~tapNX>Dfcqg zjFeLptC!E|mfIOvRa+Q_>)r^eHs7{hPbAb!aq00a@wvJZdNQ$nbJ*wcW1r9!*Vu)@ z9EatJmr0hjGo!Z1&OXUGA0`5v<)+}Fv4M$AQzz?M0jq`%Kfmc$CSQEr$`rz#U`{dP-anPf)ubj|!E zzs~1daj^-t`7Sr+3|lU6$Edpqz5=@d^d^!87Rf9>1Z9VYEvR^XbgwXjX>_VYwJ z>YMB(X0^YV$%A}S)GM57?KOL$*kU&n<2LuXU*n6>_G+@><2>z4dqJ8jS0714RS9f! znW~K@j^*u?8tZOGiX~d1-n>1Xm2r36Wc|aXg)(k#I#a$<{iMA^wvP?8^QAM2bg!fp z#8mISy=y1LlBOp#Dl^}km7SVZouJnDk?**C{o6d+iYtf(I``D#yY(kMA)oH?BIl5= zacVU^R*+`)yB?c%3Qb=~U+@wtQP-&Au`lbyx2am%eQEw-hx; zq^}m+e9^-smlo8fMEh~cC3sye?;cjDoZl81b)L6KfDJ^on&8yM?Y(kE_5mN=S1bMZ zUou1`)jn_Ie&AS8zxgOK+I5B5*&G>1V6>UM{bXkoGQmST_2-ri{9y47`=fD z*3de5n9T|mJQ#Wg4t}9usNiANw%`|LTNMwp4g*j7ht%->bCd6};V|1G`)e8W>G=QH zlXYSACV>+a)Ae7E+Tr_s|FMCC__4G{cj+L=6yiUEeL?ImVnO!5ELhM5zwiD0w?vuh zyhdT3(u}HI+$&)SrZ|*M7%GL!Vrjql@OddaHO`zt2!7(NQv=tJ8Q!>HFu;3X-oWUH zVf%G40Tf>I*g59$cf54Bj2dT&+r^CzQIq2s^D|m#+k}a)+zNjCS)*KgSY5GqeOsV& zyAf6&igkNmx4R{=v2go4k$XGK)hl&7pV|Pp%t~%jA1Uqeqw3RRq|GsM zRY(dds`gmDu*=AB`?Bj+=gX=SrXSQ^cLb_-X%a)QjKtERK?GPFG|HzNaju_>U<{Zz zd^tplSPX>?m7v2*-@rOeOLSD(8!Lb?@Ft@WRzrZ8@PBu);jNbwqhFWmmv@>g>yvg? z_1pKh*a{PxyK#=b6GqcG8)PY%4Jq64MByCPlEkFE(L7&&-%vSm@7d>zaNSFb?$-rQ zqdUO&wvJ3(PS%Ro^N{X-+1r*9X5MdK`3?A4#>5_R!*uhDskU6vo?sddnjgmc7d7s@ z8^Myqym~L&f93V*IYlPFa$ok#BN%qr+<)3VUaAbgD`iiJE|GDMnMs&oZ$}d(;A5&_ zOSb-K8%9l(B#Nfa;#DhC;5nM-ykhBRtE;O)sGTXtN|jt@HnwJcfWNT`{Ct3`z}kZc z4*)(XABs6BEVh&&l;&i_7zs^RP*xUwkHgdQasW8Iy}dnI?6)};*BnYYI54mYe2qrk zC+{^pW)w=e;+=T(YVeWr(}5bV!B0nrMA-J$78f^nM)RomYP*hI?HT|=@5uo{hEm-7 z-o1Ob;-@-0F2bR6D+8D1_*Nqa4_X_Ug#ZwMmX=lkmFA@waQ$%>1>%f!Wk~A#to6oS z^|Z9LWkoUD9t$_v6`Y-ww8xFGXao-YL$=`7DR3RhXomQF?)vcI1Bkw+h}?ZUIJgbK zMVic&z3D!bOuXFoeR&qJ07M4@vBRdHH#JEBn3P^C@Gm41zz(%+8)*z`?X@fDzG1#eiiT~#oUJ)CqIEunz>5y~Shw=VL} zV$>;sm^~HL@fkA|#PH75m~0~G)6QXP@w{Qz&>l@0jd}O>M3S^|D}F0RRpyV{`Wky5 znZoFZxUbb!?COo{%$TRjy2XcNs;+M^Bki!}&h*Q;9M1e83eiH|6u~qFr9T{1%;l@>NVjh}4rdycvc4c9 zdP0q^2Fvn(LGYWfv{RUBB)-^?5xE;bdoL|B@&|KzgUA(p^o{+W=RR;?@PCtguEL5TFZ06X=?h z$tr#Nltj>XrFQ+b{VV5ufazGj`(+SZmA3OH;XZsrsiKEJ+tu&{HVsiBKve{Q>A_S4 z`1nP|#gbw2sas>W2-2JV2_T`{YPTnFL^jD!sk$oN1bz4Q_4W215&+|xF-E3LyHL9@ zVR>~fCp94=VsopnPh(QxD!RZD=#WRX7!T->+&UR0B}-fz;6Ycz@cg`$qGBhYX4GRQ znS0c`$9o}4QAipv4b#dbEhEN!QopHjCw>4CK8}&nbHAkRHi9s>%&>IIr-;k@i8nq0 z(TUPoUq%Myc|bIL-=BiO zD8T5#dZNxPe9dfP@#%m!Et&tIqQj{!ICapgv6M(>MMK0 zhP0P%#-T^VsBuvZ(NjWEx)!vmdxuq42tI2xzN@(Z&ep~(pu+w2$Kz*N> zP*9IGjwz<~((bCca6ur&2q5V4dPU6@n%Pqnp=%1sgy~V=O@zeXV zYruV{!W}gMuF#XGilb`Bd(7_Q?&3Dmf+DG@5D0{ufQyR@fcq6{>%MM4&8q@I-%UqKY9XV>+2vpIs!of zdsqFmb`s|TpETNIQ~=_g+)uPun|?T;DsnMm|KqfRt`_LC7t*^7KDE6miJL1_LCpKX~?&b+>}GM~(PDE{K9(-=~i9!fEf#VuJq zB5RSe1Qs7y4LxCTBkbLW`H<({$;CfX>a8_*lUYvHVQh)poyYMHu1bA%)9FlG#(9Am zK%sb6a8bJc9Yy8Nw7!*tH1?BDOOhAo$;MKl`!aZ#2F8_R`yTO{NnM95Ml280z^JVJ zl0m$d3Gu*uy(!NcAY0k?dTe41f|K`+nK-ts*~6Lmz1&eq40WU*hA?r%qRg3f;gVd^ z$rn4K8~4!t)FU1~{Z0z?nwobT=aRuFdN(|roRY#C*A^pV>fyor)w56|F)0b5522#E zdgTiGJHgNJ?zysH>P{ytK*`^OjI1J*&n$qoPp@2mkY){uMm z1Oe_Va3BO;2fR(!e6etMZ05CqZg@Pxl#TB=tOA_-3suNR^ zM%Sf2Q{C%OHV)e~PGevs_qB|p%oB>Pfc#J~Q;YV9^gUHu=$ zQyVdnCD%1>^nZ0`P{%Lgug|L;3?*g{wteq z4LFo~E@aDA?=e$fe|z)uC8ex(4w=xAu&RWXvq){s%y;yxb*NRepc)I88wx3}DN2>S zxL-gsocdm=BHxvyr$mYyw$*S|R~wAoVQLAN7|1tt<4~|CVixB%X)QX1>A}%NGIG;B zC|xxz`Sa>K#PtCiM=}7vz_$}`dV$?aCdrz9&CpGE6j$cP>q7 zx(PR!)23E+^+Hb>y&3Ge2bpE%3kb|VZ9v!Xv(EFtP0L9!tsDxLY%l#xyAYMCf=Zd1 zfr(jRMqTn{XN0UE-K$);fX^As_hN+I_Q&TDE;vGVSi?~}Y`OSfvpD@#oNHDmb8 ztKIZg3Q?&vFYhtepOBSe!B}74*ZGU|dZv!#qyaneF#IImvV{F~YqNK8!UfaZ} zO9OtSTgn*0^@lZP)MnmyXf3|q$kuF<+H!U%yNCoWB&OyGWiQVWnI{|d(J`ZU^iE+` zHN_MTU02KFlk^3HN2v@!24LGI|25q!KMHKgi+z&Pu(Ts=NmsjwV>!eaT(60+p$EyU}SZX;*r zBQtSkd?H5bpD-r-y~I!4~-M~ULbpLhM7pr5>7lq9UGTPO{W$&AB|&|`=A`Kxi_ zV)k@kaVPQ{CzLhci#bQIO*)oCj)yj@qv6v5`xi3J+oG7>XGUcVJ{oK>u3#w3dalMi ztY)Z?bpcIm@;QpEN1B(t=M4c3x*$v^02sEhvtwgr{j$E!P$!EzJ7J@yDl4Y@rC~usb7dt#!FR9D zDK#!aa-D(5i4z1gGc)lv+p{F-~yBWc9di(*$wDRH7FCa9SyY$sellYck&rb10*oG+cl z=tQDwK#|(Hgb2u|YLlRWu;d=pj3!C2O@c4o!A;)DjcV>6B3LtW077`;ZbBc4zIJ{>cEt+1ly#c!Af@*IU=y7xHny8KF zY0!@7sU!(Qt}#4wZPaMXfE;ht^3GmiHbS+W=k=Kvd)}WA12Lybzkj2?&wU9HWtK{aiqj?X^R3O^0%ynK;%%4I@y=5)f06TiPLT=*XPl+r4Ds$j?6kv^7!5Z z!Uf5qwv3IF(JE;Wk0wuK95+o-ey*{6S# z&OOq(OkTnDQR;3^4}f@3@{WyqZ=6$J%$)LNfHIi)$6G?+FAy4>_3$oZJFP5W#ydCn@R8 z3obEJZMIGr2)~kVyUOxvC!1VCf^aLM8jrQkQrfda=Tt-g(QrtxpRS;$JOV$cIx@vj zIb@U1?X14Hg!H}Wp>-O9r%%0^cG8+=^{UG#)#;1}=0J?8AVt}r*_w32;24Kfw`Ofk zFY@n#uvREb=?@|61xX)I(*}*EV!19~Ix^ji)LmmB{ zxBYREQYWKQCa$?lM$>Q{KXY?Yr%qfbd+2bkxy+iF(RH=6$cM2F@^(-d{%u~xZfT$T zoJ&Na8HF1+tm%z%+<0Iq-8Xaz`?BgPdr1O?m9@N`u)ltEm)v6f)M%o|1LGoAh;dfw zbh;GMU_Z-Tx4wQ`?!|4|$ycxI;@)=0D45e76R|-FDp+4wxcOo$BHG#QkQHqg%kSHx z#l2QwBYk=SM3V#iGJ)o~EOnJ9eTFqY?9LM0D@|$|sxlEq5_5Oy z1)wX{AZ`W#H9OPUt?R9?&(KC`r&MbFE*oMBru1oG;R(Vz&pPPqkAfJBpEo{DwSk=? zP67Z40pKK_H$09e9j&JvLu6oOrIZ>oCW}|u76}k>g5_X+jZ%IufP(^YkQE@v|mG5qqf zryN1-d9u^{E7beNsQ@(xpykwi8Jd{Tp_ny;R+?S;Q;&Jh>Q-%^PEdCJOk2*Q=1uU0 zE1ca>ro1t4quUmqX(_A8#hVfBCkL`B&e$^%2~O&whHa!9WLs2{%9m5- z^PIgH#&qen8Y(CFw;b3DsH}Knw`1zV~gR2B_DEMk)jFGoQlUVDR z*27Is%s%g=cG0&Ive;nnLHB8cxt8kMehgEgE@$z(WQVyGAz@oG?nZvmB#p zDk_?0+zAkq+*I~1JO3^^Y*^_LzFpGW5q;jgeD9C6M3htVD3>~^SQH_-i=*EQ9ghzW zR!LUm>8e_CCazWIo7msJt)Or`L79rL{+v&c*o?b|w>ryp17Q({6J&WEhIVhllE(B= zF(Hxfrtigm(Q7e)i&6nr-}GzL{3Kl}^Pc9(zL zzqa5&H$UTs64xf$#a&u*x%;s-LzhE8Cw0A%t%9Co^rp{}OQbzR6w?>@EU5_UMD+}U zL14Z7UH?@d|5sJIE<8IYsfl{=9&FqxJO1eirIoF;T;JKWC_)R|h=&@`Mz*n2nl5!W zsGP09*vT%(N+#wFmwJ0KX<}wKo$kV3SLO71x?_?@4ZJgTl^Q#XP1#lbZ8=Bs;XIcl z7}#V_ULqYL=1E1T(Sw(BxS)lk9S- zZ4Lfs05|LPgpRfJ?c-zNK5RSew90`7!FxF|_2$;nW?EwKLqjh%q#i#Dw$oOrl6OPU zJ`>+A`f`jau$0j^sFJ@g1WR&hTSW{#wGDHA7qhs02TQ5u3;aBKraIaZ;we{cNws~S z%cR4Qk8te97g88l+Jb|Q&!3$ub&J$4f#_R-E6aXd_E&mgp4dhx;b z^4~Vb_Kw;f!@8Z_9(#HB2Ogi1^_6OEJWVd(C@0bsWP0*941Nh8kb1g_#B zdZNOIV9zW~jWrJR7DNKvKd>R+I}BIS-rp`bj4Pk0*+>MoafPm9EtM@KqSqeSM=Xlz z>Ch6uISp-+!~+MB*zd2&9!5{V596w~9TZ?M(-pxiXR!>^A(o;j?Ay6hfJ98W>{z=3 z8T4)}*3Di4Q_R-WFMsV^`qn^c!)eUg4f{58$!*O3EtW{rh7>^%0n{Y0^IQElt|gUX z*1K3g|K**r-`I8cR?LnaHq_UU8AIgd-E#H)`mgitN@0VuR>Gc0^bQRKi{>-!74%u^ z-q+MUCLi4KbO`q4#rnM&)j3gI(p}vDtFzgyU=UgP!taO3KhB4JfBm3x`z}206gm%J zEXc?z0b=OHm9607y(1TXJ3-z83ylBdbVSw z$Y-mAf2C$~I&oSj)%#Q6XTw7WHwP|faM?|#iS2KNN%9B?)RdOe`0#jv7?ZrdI~xGU zCX2vC(PkuB^cH|0Nd+5T0F8tjXA}G{3sL9>LDfh7_I77W-sfkdpe|Q=?n$2Oavi4& zRoYaajISMeEgbgyPFbu_l3@JxOEEG4+%8HeyzhNIzyC3P-_5BK<7PQNpRb!Ub~UKG zpT~FFe!7ZqVFb8u*H|SGs0`le@xDzcNUE?(IA3oi@VGybch+I^)6{Io2QWzjz_HOn zNqqp179dZ@gR`J(6$RQ>d&?B$-~gE$93c0BH2Idq^Ja4|H%WWkUXQK=gm(wIi$Cgi zG5xu23v|p;0B;|!D!z4Do4vo9ti#U6=3?cyTJIl9(0QW?j+V3Ht_JD9DQqUy_alaUutW3%DJ4o?uGz zo;i`-G?>d}hB3}G>HhR%nme(#(5YP`1QKKa^Ca%x|-T8d1GjI60)Zx zgk!7F2;!GYL&PO3tPVZc%lkK?!Ca5~O7Ohx6aAmpvoP0&(6^^}IXDmp*Rv2mJK%hC zhd_MuE&0psqF>!1GzJ~tv!h*H7gce(4V+6%qFx-9V~bMY?|Jh2GjD;C8B08^#Ql6}momGFIDDRle$}mTA(+X0RT1zmj@*t^HQ)5>B zB9d78{+z^~-V7Z=j_9UTFRR*|%KUqVIjuPF`G$drg4froWF-epVyrY9KRvIdwsD(X z+J1Ahy`4syp4x_674reH5wul|oD84G!o+mG_Vm7QJm~;sxKCW$vdWvo3Zt2jnyHMoKVERo zO(yyn{s!-~S_Wxuvdl~Efhc2$CXsH0hQlgJsY?)(Q;n$w4>(vYFju_B?iQ&Scx`D< zrHIn`jYIC$JBoVm-OXH>Sa%N z&Vwt~A{?FxnZ_Y`92PGGYo6<$IXr|L@k;cC!f8f?4h}B&A=+%3GrnSs*zDX`kI!9h z?~+TLKkRu-`wrqvu1faaiHZ{eiPnrzjYAXelqb1)=~QdX>>2F?!yGeU97)Xf3>5d1 zay{HFsqS81f4?R7BGP1ZRm$E$cN^y+q{b#N|4}%ZA9c zdfDQzYxI*{wdX8gL>{ki)Gyr)Hi0LGjI$1Oj9shV&KeFRds?~@@}}v{fZbzXlC36< zfQNV6RpF7uSz~ zY=|G8SpTDM8-B;}!Hngfy-g8+@%{QI-lklS1234BF=@+Gr$wl;+_su>!A zUv_5D12Qh+LD{}x4FPvF^c7EVtq6aZ^knmYdDc77eu z|GXV3Q)^dKXG>#477;sZlYiXYkI{!N^3R*w=b?(7jiK#7Z3~0y^ z_(=xnCJtn);9>)$$c`xn5CgKT7SM|-H-_y`~i2i}veouq31NwhW zgK=|#e*T`u^Jf|l#~)?kyj;Kx|5_H#%kz602OP!&I`m6E4mcdRp#Pc%gK_;W4eg`w zOIa9<7x?o3k;eUd8Un`6`#UWN7$=&Yzt#um;0CJsYZ?NMc6|6f4Lt^bOXEb_?qAaY zki#EkIT1X6q;bKy|L7YRocGT(PTt>XhI6p9|5+aw_wV^&Ts(is9LB|qURL}{GdDXL zAmg_*4zy+XEe$aLk2E;4Q;S~Ez80AN8bpf;2eMO z1;N3Mwokv+$MeVdMR4$$2V+NnF@8(q__ID33_VZ!Egu~1>hoI~;t$RkoC6%waEXI~CnheA z5a*TP=3wU#=jH{eqGH0L9753lJPYO{pla!9iZ+D6I665SIyz%aBLdFB#eqjpFRmzo G_kRG3#yx)k literal 0 HcmV?d00001 diff --git a/hw4/hw4.pdf b/hw4/hw4.pdf index 376cbc4037782b5be0efd060f55f074e020a39de..581436ec0ed2c0769e4a26940b179daab88547a3 100644 GIT binary patch delta 5300 zcmV;l6ie&mmj{!W2e3E+0ob!S0Zt198ta!yvx^K`3js8fK~gAx?HtQ;<2Lf{ui(+H ztgt|LisY|6^85cqc-h2)UGhW1!7H)?g~*0EWWz!R1PhXc zN|VIIRWt!fUbvL#9WHi;WCa<+7D@|1+OA{`X=r~X^Zh^oov-?klA(uYpe-o5jk8jK zRhgWnzOd(iZms5ce`~H%MmTNbN=P!;3(ry5jDs=Tp~=Nh*B5_W)LWL7VmQDeW_(|VZEARQaPS+y*wfgVHe=a`M;opG&5tOt-*0|S?bSV`0 zcXAc;U24~&fE#3rbR8;o&`t7z-`Zyp-NXm#@nM`nh*+U4)iJ7T_J!-z1=bv^pagYW zP+jWM%Ex;SRyFSg;@6ugw5O!P5hYdckW?@vjMGJE89`-9s-Hko)vl=GO_NL-y8-8u z2pf8TJXeQD3dAUbeV$e3X zB9WS|j&g=dAkWEkP(l4%+uEk1>P{`+9bi5-)s3-m5CEwI1}BJpSBCDe@lY1FdRjuE zImcaBo;7C%4*3J8yueGP-lsf_$L$1!H`NIUue1{oUQH(;ymtbgfLL#<6A<1z0ZWsA zHQJT{21I5vhD#-3N3 z;-o2dvi2@ld2b2wwy?V7r5BiDuM4y-C(tPE3d<4{Z((H#^f9^_-)hBgV?`lRLP)j) zN}*I4E7!tG2*tRL!2OFG-Ht%Fr6Ul3h2cP|2=)=ynh`35J&eLW#e2twnkA$N_7PSV z?^8X2<)W_s&Sr$H2zB3Jtq`%<`})i6)eOO0e!Y9RoiFP5_VVfPr{(Q^`2Ox8yxz~3 zi@U$V`_GSy`>^uv>H9!uf|V_rF$n?n=7O?fv=vrltk8rbyYd@Y+6dItC-shh)2A(d z1{JfU+knY?n1Nz$HUDnnm%+Uczcaeu20N5fZ!0h?XYHz)Lkure8?O43jNlXric+M) zZzil%TUN>*OxnFtg_S0LN)ZPIvQ)F=bc=;U+SmkbY=Snr2}-=d3YpW!lq2|>lLUT9 zm*gXo;aOY<{8${FNFEL@UHh$n?Uze}D;LD|#UiA_nuLxl94{A%fCxOBX<-cXO3;Ij z&QO9bt-nL7J<#7xm35aPOmSU@U3uVCS#VGm9G*&0X5cNmAG2zWr?~0X#ogR4Zx{D> z5A%mBc3sQx<>Tkz$(CQ{%XLq8_Y{7W;p5X|cwGjNcD43OU54ehSjop)y$fZ_ z0T}0!T9vUXEvz+_$vyrnCxiQ5nX=`8lt36Nn-S5lw^a+z9&ku8XO6exO z$!LdC!a+(&9A9(r$oUwIRLKSsZ@uUZV^j4LVSTw4V>ll57E%ZGvu-9r&Hu;X3ZNb# z<#0W#lc>;;#0>a@Gqp2Ss}7v=9Wle4vj=<;cXMSFn(&p5nBl7#F~fiN5wjBz>ursg z;robLX_CCc5i>mAh#5!st0?~*&n4bR%+8K#N4VCczWYd3it4m^WD|M=3hT5P4?G|^ zyGu>72G=}4(3T-$1FB2J$qu55=w^Fl*fyfTnM4WdwXoKxcNSHI zDIQ}Y~Zig_&4n~0K~iKx-@ScxgC4KQU7rrAD^1*vfgHF6$Hmm~Pt zc`VAE$Ab5HtnQeNE+rQQ=CN=paoTqti*o0&;C&uz{5%%>)p`xmct-5#;>tGK`9E&k zlTIQj12-};lR;7_f9+gLZ`?Q%zWZ0`*v3wC$g1M&kV}A_1?Dzb1ACav!AN>A1`->a zc6N9E{TB6VNv0%9qTC5Gr*5|?@~h%kUlkwSZ|`n?{0Xp7R)Eq}-2Gl?BaFwwS`Yf8@WcuAWf0N3ibjrfLeN z+Z2@0-t!Y>%RlQTR9JposuF{+(iKM~EF&pf$=H}qNX^(Ut)jyz5X7Ii=bx5af4aZBrHM~H4oUjt9+5+#m226C7gE+jQgqy${q|wx(DN( zlroA}JKOZ6f8r(&puUZC;}{V3#=}&zrK)U4IQ%Wj6cP^Yz}SHeLu2J@7=lAXWPU@g zGb43(e#m^hgoljM+6%bwa&BIw0H>F^1HNH*Kz?`NH|!4VDjqj3zzQTA^xU})`EUMy zck}NX#gEBC6~viI8edo<0ooMrKi&NLn=C%?KYuQye<16M|CUWZv7WmU(#PVTH^1Dx z4M}cbT+N*IQI4Dh~Ul_0ai1luU9-5=bXq(0Qwyz|5xaHPIr~XLR3dO_n8{uqSd+A)26ip0$ zsif#Df2*XXP+ug9Mk*YuqjmM0}t4+pFlsFlTZ@b!S=3%efEmro7^jcb$dnd-Ru=Lt48TYvImDVQl`x!+|v@+!4h?qZ7;QhCD+;LktvpZ zM|(7C-d0*IF@H8wf2pCvtK z`A)m8)Qr_UD7mDeE2r(=cs!xr#LVptcH}vetXh`EOHHVj7RKuW!rw153BaiSqBT1n z1#L7Kk_?WpAg96GX4s|~Olbz2p~0Khf6OoP5=@-5!JZYzYx5-Q3DmUPFjIMhU$NVk z?a<(iR0DfOtNozdrE*5ua2@Y;|!f6ey1 zNhu5Mjr&mpFz z00pT$#ns#$@^rX8HsI;b+et@GKNybVfPvxW&6;D=ZgF=iZ+9@;9jv&uf6!;G6B2Y` zTuGlq(&B0&_emQ`h2T|qaZMfZCWg5_FfB#dygu0npgH6W_UImG6|iyG*S4{vWt%|K zrd@|sY2B&3-Nnz`QBavOIwX~G`YEytHFGce9kHAno+GqEn!3w>-8wj&!aZtxc>d!Q zzK#Os@a24}-u&bK%lZ6pfB#q;H>jkzpux79Jbk=9YI*qh@af_4zWioO?5ak%Bizc} z^HVj-`RU_-ZW%y5DC4LAo&P*PpFW8yPb=bJ4iw{rtu@V+2zuGIp>B!LBvz~@wO`JG z_uR9>TU#tPdH!dFbcljilA>SA}`ljr?Z4R67oE_v2c$U`R0Wt z&4bMm>MhuKsBaGQe->d~J-c%zERt=7{mu){z*8#2DWz zr4z3YxM`jkCu;;RNfw3pf~aY+<_K}JB&5QAi1(rf<3wDbNiEHD~=$`1q|07$~QD>f|MI-i7x`R(UJg|YQSn9Ft7M|t?2Rq26m=x7MvIb zYB5ol>=STGpfpSF!dPnDimU;8 z#H|$%RLz44e@SW0G9WaVtvq0zmYO>=90h$z3D2?O(s)nVKf{3LeWs#Ji^2X|2C8YtJMN%5TDzS&qD3E@+pmHmPTgRbdiLJqdEcrd4`0G|4ccfgUB1 zxt=Ief3`f(8d+Vvf^IR_{bt=#o|ud5({Mk@V4fXiKv`bf>Jb}27=K0CTJA2Ia96m? zq%s{ifHFdAHSVrC)dG+ze*dXL%Q-}V&2(JAV{sUxLWXadoy4_8={$(KKx>0Xq3u-PutX(UAgStQLvuf9uzqG>N zjNO-%W+h+LwA8ll_3>QWIwptEp_Sz?LG*-9HKjo)|L5wn(dDG!!on=Pshc4*r?`|w ze{I7)JIaCD0_3C#IH74yR=VdnG#Z7J$#Xe08?_hOU0t)ocC$4>+stT|$| zA$4+$4oOYnw=u#N1W2J|y|TUggqf@*mIV z>Js7o=TkIcX)gaRTesQoaIgNUB%R^ADWk*r@n2?2o1Gt`xUoEr-~QFtHNPmeG~pXV zu3A@hhpT?#vgX(Q;X#vGTnxawO}v*woL+vdtElV6%p9RX+9!G6I10Rb>a<%16` zw+}4=jF18`IhUQi0VxMJF)%ebGdY*dy#X8tH!(0ZIWswz@Vx<7DKR-9C{$%wAWUg? zWgs#zF*qq8Yj9z7Xme#@Aa-SPb7^mGAT};DE-<%nz5yc)Q8F|!LozuxMnpn5F*GtX zGBYzqF)=YjH!?*-MngkIJ|H|ZGBhzmGC4O!L_#<*G%_?YGc!gpF)>6pGDSm1LqkPA zT?#KuWo~D5Xdp2+I3SZjQYe2N%qa>201$=Y_uTh=-}em!k65HK02ZV75UmDp zV9+cEi#PBFRuPmp<>Q}$836b>fCvQ$M4@VmhNy^|Xp6e2imvF07I{6{SzkDXg9sgL z=N2v^`>@d~JVaw*V^8>mUj#%@ghW_GL{!8?TqKC$)E*>>*~ltItP?R-X<|3A$`I#` zRhIZXt#V}RW|b$$bE^Wmd|MUC?boVA?q61AF(5BTkoOh-CAJuvm%GaWAPF@v3MC~) GPeuwNkRbg4 delta 5255 zcmV;26nN{CmRJ`|s033`R3UkOVmxlH%$jzMvlT`+;tJ zxV`$@KfzQ=39mI&x1TGey)cNC)lOJ%s@uEj=j%Uje!2aLS5jq#w^G#$wXo7wGsUYg ztWsf>vR!yiNKI8|f7Ra`SpBDhAkdWzOI}idtkC?tX2nOTrQ$#8W(L-!VWQ+6Qv8p3 zSc0;Hf)R?Shr&lGWdc_tv}>QOQV%0x5xQu8NPuy|@z34oK?+kNEYqU5E5?>F1-Sl8 zc!{_J@;9mNFcZlRa37$gNcZ+4PlH#)1k#v*7ijfrVHtUw{MVGh}_kO9Gh zB%#tIF>w`5K$15uC3=U8ogrC4#;}Fb0+6;VSwkAyU&(ww5J2avKBQ#mp&4ikN^aw< z6kt^*XQ?mjf4N($Io{uztCSH=+qe>v4EDlv6gJ~v%ywvU^^e=D->&K{%StgEV3Bf_ zW%ryD${9LDN5&l~J=jz{AAL{UL!2bwJS|MxP>qoj2 z3j8~{iuo?JYf-=rGDW%$6+7rA`M_`OtB7vm1NHbYLQX`iP?qW#)iwLVb?O3Zj#W^C zx-FGQsIb_s&_~#7!tmpErH!Sxo_G+d`+ za$TMLtYAv66H3l|u?W{&SkrJ_kK}rtpg~50#)&jXHT-^tAfzdjr(K#0>XFI2?!V32?!U{2?+0PC(60b`>q;cLKiWzGs&Ej&D+o1*24#qW1#8RvJCWvm=iUA}o%LA%?N% z)uuRUik+;z%T?Z6g1jxPE_vwO5SuxrQD>7DS!jWD14J>U0YU-1E$A9V5 z7C(cES<-F5)_?ZPCBc;o;`(9{QejO(M;4Bki$p*Kp3Sr{26`pv zK}TmOL6_Fwq17Jf@21MS%MhlxuEVZ8aH=dgC<_ixB`7oSmfeq8wZ>E2bnD{o?w9wA z$A_o+(+#_>W%%;>bMR!#FZ1QPr+at_zsm6OjRl7-`DOWCBnV}1RdalD> zMl)>YQ~3D2xMP%;_s!Op&*9f^yzMU!pK3~c3EzB_ivH-~XRd$yeSWu?FL_wk5a z5SGsu&6>Jkm6UH(f!mG!_$V<{#8Wb)B>$62Y0MzH8rC9TNi?j`)ofK9` z^d20Oytb1fA0syXH3<8+T45T&|1^JU;^pOhvFwD#{1t2el8wMXo$gtmmJiQQjaB*l z%u2Y3_Gc{+H%%b^Lz8^pyp&uw0FGb}y(G2$zFFfQN{45Pe8Fixx=%L<|-5+Fs z<=%y|JI8_Gw z2`iG zJf(D#-ej~xDd8ZcB#y5+c;tKxMyg~3iML+#hOw#oiLkz0i!mIJdJCz8`dK#QCQk8DD3Kw+IW zgd!A*u%?m^nI6bey1Ai_GjTZSJeK7h zNLyqa%Q!*G=1gNn8fMI6VQQX3O)-yUdJ|EyGZ8g<9xE|rwE?E=!8F_Fu^=^0p+?SQ z>2d@gJC8-V^H}gckJTNs(WT^~z&sXCB~JU!V^Qur7QD}6ji1M2zgkZX8qbIwU0m5l zJO2l6vXe?8DFZe)F_$5o0Vsd%TupD}HWI!2SLoQrrWuk|WRZtl2G}f+Td)T9kjcTA z=^zGXZ0y*xyZQH9)K^PtN}?pH?Ll(tcAHJsD^|UF{7}EUyZQb{4TVAhXmxS-TVbrQ z9tt858u^ot#V?1Szn-379-ba=k3i;NZhyV|>H8lQ6-o(jO!+lg95sIl?S1tPb<2Oc z!w*$cZM#ji5{BE!s^uSb6D&x0lB%>pkaWdS3G!IAlCd$JkcMqPw~7v@v_Sr(9_~5T zkr75=)8f4}GS&}@@TP9`zlhcBIBqa&H3|aVwVDTP-BrHG@IrAksD$&bG2^~!0&2d2J@bmQ0{=#2-du%)W7gFgH%8Yw6o&;aZJ!w{^z1`rqm zk@<*pZ9qQM;zo8 zS0J0mK}}y>2f-2trP1^Sc;!dXyP5DXoJH197W8djN#Ws^qmNGg5s8H2Vfl@4R1a%% zDJhs3`mChjjVXVrDXY&E1tUsIZpfK52s4W&I$c2f;1foz{ZxkkVUr=en$}T3W10sfOB#ITvfUfc3-mSyw>Q|4=S(td5tEmmU@ar8Hw90uv6q!Z>rws%vO8{G zTpO_50r0jNrZm8q2G|Y(Z(1;4)FnM}%%=BL9M^x2ldQ*2({6)I@&?|p+m>z5zzkJu z_KH#aQMK=vI&i*-_SA<^y=|vHW_VM9x9;Rz*c6Q}eoIzm;IbnIDH2xByH;p@-3o1I zEA(5)D9BQ!1(-ybFM+5Y9xD z))4N2sz5B|7a3=`9uvX~1)^<_Ps%1(zs7&RD*IoOy~fR#q5XUEOL8#MdE}%oMI6OZ zNx0Hckx1ciu+|rrXV&^oS`0&lA|bHivo=)9Ytd|`=M*ziXr-?or^D^BL7eVG9~^8EDaKeuc^Jt*5z8T9hMjM;>FH6sjjayT+OYVraxzs|4`tHe)5{18`rw+Hy^Ws6Bt{!|fbg zp~FV(MJqfn{PH;0aFqZWaqboM-w0OVS3{Yw0-NYnH|n{2#fxV{f*f3L70?F@z(K>Q zoh8Kqf#<=crFdhMz2KpFV{@cde{ETIRyYpO3K11p=4MxyK+ina28H!4PMSE$-&>R8x3$~sG zIRR=NAq_!UHc9sgDY1-{kWf*1@HDc(x&m>9jfl;}rF`HR_9<{sg3Ga~Ox&3fiUyIb zwXpsbd26}5aKe9G%Cc`Dle@HJ@ovZ6HAh?Afr?*ys;!kgC{iYCB`1zksgpZM@3YmB zBVrexjB6+Rhxo<4&PBCqrsJ5EBW6;yjU+TvwKUwB?_3y23&Po3cJuI{b}s9%uvwc% zo@Fe^3BM39B(v2oT)HG%PSu6KbiY&q6ZnBSb4v5IVz+<0FW0{V1S}lQwrB~vgh;fO z{sziFTtYOiPk(93-;Q0Aly+sjsA;Ng-RtAEwslMnq63oU4_x(#PBqdX=>Iu^HoBm+ zEG+E83%VInb2654ux+_*F9&R^bt$!^jhelUQL2i-H1K% zf`rJXG3^|n97G!yHJ-U3Mzk(GzHXCvcGlJ$4Am=TEm+2smofKxyma+1B$*74q>K*d$A6G1Nh6x?qIPq{NXqcU z>zW^u+VG%|tJYQB;i{jttmEtc@niH+F~`ytRy{m53%4Afnwzd1{L<>Gk|Wu{j==u` ze0BmxVuL*`w>>QZjF18{Fqg}{0VxMIF*P$cGB=m|y#X8tHZe6bH!?Su9=-urDKao1 zC{$%wAWUg?Wgs#zF*zw9Yj9z7Xme#@Aa-SPb7^mGAT};DE-|;Bz5yc)P&F_xH#9~> zHbgT;FhnsoI5tBdOHj N2{1GYB_%~qMhew+^`HO% diff --git a/hw4/hw4.tex b/hw4/hw4.tex index a2af726..d82a35c 100644 --- a/hw4/hw4.tex +++ b/hw4/hw4.tex @@ -184,7 +184,7 @@ a_0\\a_1\\a_2\\a_{-1}\\ y_0\\y_1\\y_2\\y_{3}\\ \end{bmatrix}\] -\subsection*{Question 5} +\section*{Question 5} $$1 = y_0 = s(x_0) = 0 = a_{-1}B_3(1) +a_0B_3(0) +a_1B_3(-1) From 2d2ad2a07df224ca7fc0fadbc0c6caff56944e82 Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Sun, 17 May 2020 12:27:11 +0200 Subject: [PATCH 2/2] hw4: preparing for submission --- hw4/hw4.pdf | Bin 117779 -> 118918 bytes hw4/hw4.tex | 60 ++++++++++++++++++++++++++++++---------------------- 2 files changed, 35 insertions(+), 25 deletions(-) diff --git a/hw4/hw4.pdf b/hw4/hw4.pdf index 581436ec0ed2c0769e4a26940b179daab88547a3..5023d69448bd60350ec9afb370db09e27fa772cf 100644 GIT binary patch delta 12822 zcmajDV{j!v*DV}-f)m@eZJmj2+qOq?6`=(}1@0WF7H&0X^@UkIGUS1PxKMxQ75E*yw(v zx?t8iEi)HX6Uyg-G`5x>6kh?+m*JU=6PFpqp-I09eafs2r zZW(})+(Go*yHqu$^PB0GQV|Z2{aYb44-a?Dmwar*Y8{P_75;___4_-m3$rFT%bo-j zRof@eXhNspPXX)XKdV8?WT7@$P|OK}MogdH?su!9aM75Uz;Ph7C=p7F56BmE_vc$} z%~_=(-Aqyaaw*GgkC;x^U1dk*XWQ$?zWSQrd=#EC0giN5bJfU|gDS}>38l*n8i^Yf zDvmesaxhKX65elh;E9cwkWJpf8n^Y?4S$4vgzFIgg6H1BhiXvB=qD;wi=sdlc1d?# zPpE{=odkPE^y>V6xYoR15(sX@wU9mcTQkNdzmq zTTa|$byZJAhnHTQGKfEBQo5czPhHgu=+pO;&C9J}W+=D-jr5$ot~dlt>h)S&hF2#> zxqD2T4=kKmu$}4H_J_%WCn4mjy#msJDlc^yz@J&RsGx(4nKc)sitP#*_*l~bbNeIHkM|U&=AJyFoozN zXQ1F*JF8eclMqhA+NC<;~ zAH%T^1}XI zq1btK?9n`aF{?YyK`jGm=*^*8on!%Et2y)T2-u^unJvA*-XT73i1R$N8Z=*R4Vd0} zyoXO#ctU%py;e+@J!_LiT;KmSD!6#6adK6^% z3BJ%pWrcULQ46FWRrUJT!)-9RM(XO1hu}Q?Q_&PH@3<&^C3T-~%|BNWalD8t)h&uP z<%m{zZ}AK!Q)Zd+vq|CDDpNUqsUaz91YcH*2n1$O0H?dZ;T{HfN~lezbt~FnS1qoS zdD0upvO;iyfN-;zEJiE#W7d8|k?bUzJX4y!5wmO^beZ)+=|m^W2>Q} z)(~}+)~;S${CRmYB-Zp~ctxS6PI+H>bJP_cEzLfEGu#ND{0)wk=9(EEwLpXr1<+a#jYxSLoRTO_JwiE|Lqg9f%i~utuu1CT>#P)eiu}ovwhDi! zy&@BOb2y!Fhh$q6N?Eq=UTT=BdM4}IW34?v%%|-SnmhBiX^Yw_1f#OlBM(`U=Tjr4 zpAMubJy3KH7yL0E=tA4aNx5iOXdhryRE6-@}Ez?Kvn;Cl=bD+lLAl;<*Z3kL*e!BTy z#%U~FOhjVFVxpXn1lOeG%XOlTA>+z0K)KLYp)^=c4xL`*7N5|AGSXZ;L3!yE1~-ZC zClry(TNi>%dSnQSf8PPaCbG>~A8X6;)n!HN+$hL~$Upp7aHBXy8rL|ElwKEA2-Q~Y zP*z*BmK{E{Q0~=}zTVD`pQ~$|9nOV6bS3jHPCTjwD%NU<%XWQrzusQ=u00n0Fl7XU z*U%N%vYtHQ36aLG@)y>9L0E^`5(zVm@KT+IuZY}EOjmNn-HYE0;vp2$BNgL+c-deq zxk`ZN;z2(UDEHZ6oO4Ytc>eUOMsv6QD;W+Vc(&PI;l_)g^r#F!J{t1e2=^5}H0`xr zdUEj-A6qnAHaz}$m^OD`Q6 zi*-L{JTRs%yV&WN2`a;x`piEln)>jqCp(H7`l<=U<^RsByKrnjo{U5}bJ2Y|kGv%( zK5i10gn`jceegT%oPrR~MqyDB#f#yq>?Yre4H+Qy0*&~MJmxRrSApcy@S$L2vd{a@ zi3GdMC)+`D!@tw#kcQQ#)po$sTu$qRF_uj<7~K&Up;VmxdzRCQ zg~q^qgvwJdC>E#o?byLVd%%MiEJ(Pc@IxOgUt@_OaF-*{sKqs_G3WhnxAnx8{DZ)Y zh4Fdg1Lj4tDB3tu|0x$@)$NgBquNDe}~~KeR$MB8H-cNKnN~i}NargIjzye#TzIn5Wfm*BQ^6V;T`>)G8I@<=WDxD&kWo@a=ZKIG(lqP1UREAn{BZw6uy$FW z!z|GoPT<3& zm@?fs7c2AzsX3YUxffm`58@N@Q_-Mv4LJB}C-CjNOhoVzmu0$X}IFe??P6YiQiz z__Gzi#iipo1g@sq=m^O-%suOz0m63+`Y7?Bj0$AsbQ(au+XjsP1{H?{F>I8S z${36G<@SC329dy&`&YJ(o9s<8+BLk2YtT?0Og)Vxe+K2OE`A`_EtfHLx}jltV`+S&ld-=PX7Q;8m=O^fJ*NLeJAZ zrE*x^Mfv8?PUB@V_{Ma$xXHugnD)weay)ja;w43HOEk+P8~3WB>V4==tK?`l4GR9F z%{FvTW-3_Ufl>DFkwrg3{??gM^>rMC59_@keMGB4e;a4Nu34eA;JP1~d@cXoUILR` z@+>#Gj%}rFgltyKxFY01$EZ8W#U&xv;_Guz;lRbrO+@)zy_WWTUJo#ClAoZOdR7- zqGc-sJjjBnlhZ*i^GEyT{tY6uNaVz(nMav;eN*U&-GPg@3->ld0PO%9EFNM# zsWOmWhCWqxbIqp1-uctz!hH8Zv`&f*7K0zDYi2*`sj~~rZ+~Rrayqjg*fHkcMrrsy z9lsTCmDwj#59&kAa;d7$~wy+UaU63)+fzY!)r$@z|_-f`0|Gf`u4^}sc>G) zOZ?qKxVoReo&z@uXy_$xHEWdTe(!8F#}3URtVIhrD^ju7X$s+v(U2rmx8|Xfk!Z%- zeBc?s;en)tOsWeEBG(Nk@oEO^Z>s<~EQGTTb>?MAOjdR|dg}&(jOoy5Pu~<|ny`Ax zUdamcpq#~6Q=JaaIzJK%I2R}uZ{iQbrK47WxtwukZH5xrjF+j#`WtI8_I4v&9R@$U zM4`|wR(hNZ`ZtW%*j^}=X9uKOq&QAq*<;+XD@5f4=h*8pa#xkHb{R{E z83N3K-zt*#udiEQvPQ#~UEG7qZSjTmKyev%@8<1SmL1`xuZ-~^q3VMRuBfKYdhUHb zi+y$jn}jE zr0zGUF~2aY z_x1dK@6fop-&VHzHNZuRZ0K;#$|IlqmG)tY z?jD(bThi_X@_2x6(>;vE$xRJLSDp2QMz#hE&%5L-g%PO+&(?wbULC}+5;!c_Fkqrf zx3leO98&!W$obHP`?-E6%&7Sk;N7`t_tG-*iYE1ig&m~8vt=q^PrOp;qyE-4Lb6%( zRN9Kh{Ur5wv`ankuJ0k#YzT#dJZr-u;VH;Xbh!W;M7P5`=HyPzJEJtcS|sJw+fr@q z-|GR=Xp31|M^56^3f6u&mxu6|chf`{3Z~ju=NMlOsb02@2pch3;YN~-wbSxH&zjsD zhajRq(?H6-E37)^N_BMDCUPNczOeNtClxkh_fGL02V|AYYVlYN$WI%gIQO#_qZG}D z4t>Ofg4@W!7@2ENbVEwnj6_3@__NmTb0pwVlojV6b1c+VM|oP#pT;7)q~0P){jlm2YSd&$v%A&#o>UsK z9Xr>N6jx&!gDFSCT);)%!2m|@xYJs*cW4v zwpw!VEk+o^>z760S%$JXeMA>ra)OJ{y}Z}=^oRa8pd91C&__^WPVDX%cxPD-C|Qn- z8j_cX<-a53(LXZOQKEEYV_@rbi5x(5CiTD!~c9q^%YQF zp`xu`eeP4xwB*NaYA>?K(9`F107ghl`5yK-Ycm8@wc)n;Di}eRYY=q0dilYzW!}} zs2C4g?yU*v_=6f#y<;pb>uVfi^T&;sv?}3{ZcfZ*6+VcU^p(6mKU{715_l^;iPbX( zG)hj?`o*B309>6296g7gh8fsOS8|l5S7#0t@-C}hnv!|-5t9@*?FK_4z;DvGn)6u7 zRbJN_10dX3O*dHf1`Fm9K6(tAs$<1hcYcCuTKvF1n-#iQOfiNC=+$BV834G2rF1y5r0SJ~t0{{*JODmv!(F{vFRx-{~U}eqtRaT`|lGOBv2~mXv&h7A$R~Y1> z<56}Wy##oRm(lYAqEos8Rh6TB$QUA3x@UlEfrqszLS|z;d#@Kwy;Z4r#@g6|HCpQt zg-_zm4Uor)scSW3GZGq?!JipNQAshwaLjY(=1DPes56OB8u3b0Y_u5noLPRHwS`7A z#Qc!9Y6UU&caip+a5Ltw;$xe;5}>9RF~Z+}Lt1vlp7nsbEaoB<1anhnnA`&=@5=Z! z${TATZU~~c`eh-T3e;zOg)PPHG?tQhg%cZ8csGo;bj;?6x6R&+JerwmBec)TR8fBo2nmy+dkMyfmtBpG~fH4o)4w_$b{HhY>xf;_o&fhy6*I`0$&U zu-su3`?(E-7?R5}LIN_|XkrYJ%C}m|QXpzZ+1wxg#U{J2*Sm(31{vss1?*ola%sPaz4ttC{ z$tUnz%T7US(c~uN9+keiN zR1fH&dHmbmV_YjCCn_O#>iNQCJ268f&&J;hk9q6wvZcb!!nm##t*k@!@sg5|EG`f} zR&YIjnpjY+so<=tqMBhGs&}6iMEoGDRugTL5Vb8?W9YgkU+9E?fXl|Sj|T3{XPOf^ zv_H}r8r5YtfI{^Ri!uyjtaX0N$=FX7$z{kPq?xPO=u;Jca%HIDn_$G+;TlI27qdzk zuu(J6jhl~1T$7A%$OJnnY`R_4hW5d}o= zTp_lONgScj%vVv1R8(o>9>in}rJjNGsf5wvHC*EXAhLd1QeSD?Jm>JpN2<%YaDN2J zTc$;Pt@Q1_9jfj~RjD(nmqeK&v9oWvq{{SkksUx(HGUvOy4;e2?0pR)MJ1afqd~NjHLB5E|Fw2W6Gn;|mUC-u4@l7gcvKod)$&aN;$q);k zO%|~qpr~_MjbvCdYJuJZ0m&igmJ_)(I~TD1a+%J3CTSaOwroz|dNHu^N^NpRxpOrS z`%qqC)4&pzh%M6CpY>7(7nYGRW)*WOD;+%vgND*@2%HecCuY7^RKEGl1R(w9z8tUc|K$)pH1Q%UfFN5;F?c)tr#L zgQR1U>>1#S94bd4mn|Mh)vFp3mV}mxK)^HjDB+45|3jO|9E<3HWAd6i(NWuo2KSe> zMo3RppABr#9wVY2Rm((DeX7OqN^5zG`g?HN@QR9Nc+9ALqoBjWx-FfIo>~gw2&5Mg z$hxv3*kUq2SyO;_yu8y{tZ>V3SXEw@JG5{cBj95mLi1OpvDMSc;a~mFsh0c%gY0srDJ-4dC&>kMlQhNvsb2Bv>8t2g$hzfcX)rrt!z#}wPUqKXoJ5o1 zv#-z9n}i~pJ%?Hf*T8~|q&Y+AbU%(E2$x9vV8t+d$WA&LjgF4XbQX zl__sOsYcM^xg1>T!cQ29&tL0FoFOlz8t-RtDRqg3M5qv#5>q&yR!rBoPI zN5fMfUeEh_AL)PcceT4eI`H?cy1#Js{k+ffG|bB*z7}d8_~t^ZAYL~~eM9VstYaW7 zBp~S5Q90Xu*kThxI6%1B?dRa*{rUHG2j3$9>hWI%pM6yhXB{+tSC)Dw+pv;y3{mC{ zj+7EO*1DEb99T<9%1&x$kTECd%&Hge&#GUXF87SnepQ*7@@uN4oYEOm?r^FdiCyaC zLlRm5;|k)aIZlglc(QCg#q4cWwpEDnG;ZTYh4HvUKfAynI#a}AgT`R))%b#%Me6ra zsaHx9$8a$O3)9kwPgjgCOFAAhX0P0db+LOg_zaj31kpNOHG$>&xEXE`G0wJ5|DwCp z8DJ(ZL;%5I>i5H-O&pkY<|-g=n>EYxaav)URd-;#en!DallcaRY40e<RsO1r%oyMW&Gl>)nwVBFhQULwB{%3l?@DDf#N?84u^_%*g> zJF8eKZ{#r*k@7SE4!B$(b2a+tu-Ed}snPC6GXn9%z&yXppa< z^fh9TEG__>BbTwa6KUphf{-szl6sgjv~LDy1cu?)t$n%^%MCbcLzlKs$B+bDxxg5a z=SmniW27Cpp&2*Vv>KKChVV48OJ>~FVS(p%{8LBR2=$W$*SKmMQ;uwqvXA1vlR(a5 z!{5v^haKhjHwiV+Y=-6U51dEF4J=QX)97`bZ~!FyZ(#~uw1%2?OTvhE*dEz<7a@ zF_auXgo|F0JCFlooJHE7BGY{2iY1h-JelCO4?aScdyC~S@q+Mfl;A0Qxrz$|C(VCiN>!p*~)jy4HS3mVpxSKJUm?LN`GWCfc2{GIbV9um9AWmkif zdpD3ILIzqbYN>a}`SRzQ-Cs0~^Ks{A)9o(*2M@n`gKT{j7KsFjDsACR9FjjqVyJyI z79U16>23{CGmP~xZq%bQSs6P_gK?Zs|1>ST0^ufCv13OnsCpLWa9*zMS#L+j=(X%O9 zVyI`%`<3|?k}E$H|5UxKAsfqMrxr&Sq3R2WqRY zrqG~L_mI$z?=J$06=yjQ`AS1J!t`-XR=IWhzzxR4?(ZtQ;3(7>mFC#$*uBc351=9n z62VA2y4Nl z8R|o>3pJn>U!BM0m^HfEI(;@iq222TT3-^rHf%nLnVkNX)ce)$4dR_oe-U2R(%SB} zZ@^BodOy*%ZL2)t;Q&_=SvR;4(OXUpXml2aoXm*?h|l)TbqgHsyKedn zSGko|nBWWd)$lK*p{e6frmjYh3@>#Wet|E@uaPA8|3L?5X;fz1p@86EOIBvYY?M}+ zA_nL92QeCHT!?7Ex&8xnGYN^odHzB4#+Y_#Q3y_!|0JLP*`|fy{2wAl>ox^=0RDfN zSgo!i;2^U9Qd?n7!Ji5K!^LO?1cJ{IKydvJ8Kd$cPzo= zT-wSEH#yzm6eB7vD)PpDmf{TTINISKb#mTV?q(CLv~Dk{8+AY7B6Ba^k~kn`cOJeDrPS3J}=Y!0dvNMgmhr2F$7_+-e5Y% zEQHg$r#k`UkmO1exdO(1P`Vbk@LYn8?eMD|J~TXwu%%1n!50kOaH|l*FmsX=VC^*TvLdSm>Vh&&X{0*W zlZv-H@%zb`eaT$@nJrjX-T&4P#lRNO^sI(!F`iDo2~PRtNk299k9=05Np4liVb01+Y3WqtQ|TVr6oy{OQCQ=qgae+A}IrfiO45WhM3_-k?P3~>9-m|?n`~QL5x9)_Q}6CQO3~< z`-Y_nUJt`h4Xnk&2tiIq(;I^Ai|n(738o9w=5Y!g^a>?f+uo&FWBJSS2Q6#3?Ga1! zD6|-Cbxk#2KfU-XW|TrV{b&Ry!X=e*m`I{vrEM&`CvrV0hR8Dc5ziGwH=dHM4U^eP zC759ImiPleX6PqzL!~|Ag2!&A4>PgpPK(ZdBAc%_1;9ZR79QW@BGIp2hUC1)qtE9-k0ZR2 za?wX;5g~m@6pUdYWmOA;dX|Bv=iw8y}*S@NohK8Y8D(F#6{{i|H;N z>QdtqQzhrB9DiV@C{m~@uA4hz9y{#!xLqz>x0ksNtqZ8YN>Ue5?l+Ic4~gD}amUC{ zQB2d-Q#3=W=pSzmHI6fVr!Ba2qiCSZnMY0h!L9dqcO3b~-Kn8dVME4D2?ac`3xP3c z)IZ^@4`vz+`T`M@8U(RrU4_Qr#7c=qSvP>KrDwx(ijX``xRDz{xGNr?zbW4jUpuvO zpp2SIx^+lJgGq8F7?y|pV{=u;3zw*ZAr=E84OIo;y+c#s3}q2i-)12m$Bq1bgb#`H z8%-<;apWQMx<)i-(l?A(1EsM-CLjWxI9#K6)4EN?hyWo7$?LccN130H1*UAkXw8Np z#*WWAXo6$8-)C7yF+kEH_B$Q|-cF@zg47wb-k&uwV2*-E)mzc zjP=|_u+1{MFW^*x)gg;imFo-1N2u`2HYoTAVH+;(deY8T4FB%xF_<|%>rCZ7p!afd zyZ^KNc`6Jm{a&ilA3XkQ(l3Wsueh^zsXhG+N*Q)RXYY@ih5U4L4l`Wq%bE=&j;S?N zdm5(PgyG(`g2}u3?)U1#o4aJGSc;t-6$uV%t*Y5Rm>1eD^#)-4%0F-s7_GJb9NxJX ze*FpB!~D_#L9%vcJY*`q#mtsq_Kq+|9rMI9za}yF&c06fE}BSl{$wfm3v*_bB{}S2n?hcA*}EIUK?6Yi0-nWyH^Z)}$q$U))bN zbX7V`b{@IAN;_DEFWB>^IT*8e$E2)aV~2h%T={tfxV8w?Vy`COq4c!1ci=H!{BAmX z)!1UHJbNpU!XOl`)c@<}_4>Sg=X(19nxB+sg<&4|joN;>)$R5*mD|BK^XTpY%FPj1Vx&44rtCcu=5v~6ax(0Cd-#2tvK!`GfG{sZO9Wqy zf?$~6;sV5nowYO_6>1unEW#gly{3F zfh(Ej))1H%ghmbgm`wiEX;iXU+XYSbwD2xTx)2n0Qz>LcPs0(5n>p0P{KK< zNcD@WTt;ky)J?Em`xf*{U|Rl1L;kIxN3x zSW4YQ955JgieWLoyYYR~NBA-f^ikKbi88+xkNuYt6t100ARss@F!A`|>9LRG#|>+_ zd3!_Y(1~u_`~DHneIrcf&`R1Vk*u#QEJFa(&5CU7$`AC|AC8|j{E)P~xr51_t1*^j zCW4Oi+{hN*x}eRdXArxkcyXJAk^%PIuC6kK9MmUR}Tac{wf1Wln2@ImNwQ zJ@6w>ToYb8nVf|F+&8s6`1sResrM3lfS@@dZfAe7W&dK^lWV`xYA+72H@?!gvE@|@ zaq;`vw#9U`#mYYG-}yvQ?v+>VEPvY|WI?qqz|DT6*{D%}qoWx_=-Ij@nC4ruDmI-t zz_HDdMx0=}W!hxv%{Z5^s+ndQ^!wVO9mMuzR|~CUt%I<-p4Pc>O^6}5=xn=ax|_ezNB+-9>+7dlfrN-P04M9KsT@X^!CGFruQx3?1hN#!%MO_PlNXS@`LTj z2x&^V6Nkcmm{}I@6Q!LjS?B-O;9MN(RukZ)AWnS*W>sq+a}ut9bG#;rE-MK;3G2U@ zild_&2^ZTxp{|0!{N2IAk%Wu=e>V~&x&q>C?4oSqqT;+fl00HuY!Z^9ygXvuVyrCU zY#icTtUN*_|L>H4x5&Z&SN*f{{;&F{oN2d*07nhodjWH0n6yrt{Q1rjo`5Pjg!W%>k@Y`d{-+dg4q<3x=cG)`~-|N9>982916&9%px zd$0Aj=G-fy2re}XuCWyvk}I`Iix#+Jumd3A!2muPHzl&Wf>H#qmGY3BZkQm}TP>eY| z4^u}(lXG}={!}pwK5**jV-y2n5BPgMg3v@b3|XaX)3$}AV}U6Q!|}EG z&+}ymH7E(EzK=APW`Dh%ft4#u;PO}J&N|?y$A{^_ZeZm=;5wm`S0+^;@ZOvYO?$@& zk4sWsJ*_axsyzE@&9{`6vniXM=c@?a_p~^h4#_uk_-^K_A{6er-ifL)E6n{3`b@apr7xtzN^2r3LMvIE> zlNF&6nAM8iIzq2qoVkl)b(fJXXT5METO&KlQmi(e?K~@n1awlz{aO${a8S;29qz_4 zYZ^1pk~8Q<#MzrbJOyInnHV^<`ZR@*Y2m3vpRl884WgJT>qyrQkM($$q>r8QSr!ez z+~L!N)J!3$in(tkXg9-)M_|TH)4|J`QN}?nPo6`KnkyF~V3*gNoXQ^6w$qk>Q4Owi zR#So?s=7q>(-fO>iMqBU`HdDF2ybAq*q@$W+DBw_^Ar1oD<0TQOU^{1a?BoO_^v^J zYD|+Zv8#dFrN);tb4<^ad4EHbNstB|uEtB)rk^!yZ;Da3m<1lY>zYZMcbxNg!Z~#t z^e*y<3C2*vf*EEF{sJ3H#4j{6pAtMYWHW4|#-OkY zsbBKxN?y^_v;@wB$m~XX{V@a{bos&~2jN=hLS+~(6a`rc$cht!(?k4N-vK1%(+6g) zmlku|2#4Teb1-{v@lL-O7n4{v>1bu1QEiSvQ+RNs+&|Z|!`tDb z8j4!*sj56o^F}fsa;iSkq9`i_8Jj#)S(kau#iGf!Hx^kOaQGbe5G|v4bI+BA% zqG{;~A+Hkx!kOoMdgl(Y?jdsyQmb!LKk`iKT+^1uu2PeKFpyWyKzT6KAhl&fKiwNAiU|+V$i=a?M5B7Xsbfi+=A6+DsUIcb%59FLz;}6EThqDQlXFE3;0#C6z8z zB!hV*3@mOKlBPhmhNqD*CDdl|*qvjp`w8z#e-iqrVXQp|zF-6K@KL)|47*fV-BO&r zeg_`&`VQ`J9_g&B40ch;Xf-Y=Jw`bR#FZI;bSfS;7$QYMiU#~^>Tt%MqJf#mc^gDQEIy^de^aE>S zb{_i!UaeZXs)PZbzIuC@HzHVhs)YKd16SLG1!~o&z^gx7vw^DoYbBm5Xlp%@i7Jsw z$mj|3o8}n`NVN~6P8hA0-$cg0Kmys@xE<( z7_)@t!iRX;)sHq)*2D!DiuC>zDO#BVL^KzCjhR201Bi&@s7=Pf8Hrh&8{nh?XPy`=u}6xuHg|s(9=>U)<9ExzD$aFEPnz@m67|)} zhLDq=Ah|$wgGx1;l(Mt^Fa!c{$5&=pmBH*gOyA=AH1XztET}}dq+^Vmvd85Jni7_y zk(*^AaYX+1fg{yro*Wk>aKs$7uX^0 zDZzf|R&$kd9248&J(Fj%*!&^dE-ETYokU=2n53&t4GVZmV>apnwE`!1h*+X~cPSF) z5wL1D*ZgV+B##Q7pQsvFQ0aC7K4y13 z1J(ssSrD(PjYST=ba<`RZ+8pz>pbvz=g8JI+VpIg&w3etygdK@bn5w~>BD>?iBM;M zD_77?pN)E8TkD-A5%RU$iGyIxLdwj zZ{6DDGB=2-CgA4nU^rM)p>d)c>qP#pd=~B4N@O{#V_I$ zo9ZjaHzo5nDia9(SVNDD_uXU}Gl^hYQ{GX)$xj5;W(f+Cqcfd;rKg_uz5vz-IgLp8 z2nPjX^`~g1H41M4x!HCZPSn-Irx4>4296s|A%5JH;a-|`qa{TpVA`|{+ejQaU4pwP zi+6E|q-(EI0GKIA!D&dMf@`&Sd@~j!yTEF*g>2JSACtFst6EZJQB)+! zErBo-u?zzqi?E|YL1`;pD)Mz<{P1^Ft+D}8NPB5Qp-RMU>;I`0dt5m>4%0<>ys-qc zhohMY&Cs4OC4ri4nLy+R81!QlC+_{v;y@j&IyfI);QPltd$S~RBOhT&@}pn^r128l zr<3jWPMMW%-0q|=aY<|@r!TYot>ZUC2rA@Lkm-Z;>OGzzGiNu$thczWR!HF&gL#NB z5miOH#Ze3TGxZ$xNDH%W8HWU+!{qy({qSmpJ#5oKet_-d7cRx#&57DOBGR%CsZ6P` z5*(@;;NKe4-V^=qE-BQlGKkR~AN=bZWC*Te9LUjA$WdH@4&kk?`Mn(ez1`)#lie9Y zZpm>&HXXgwxN0xcnE{T~4Uv|l*uLT|{B$1ahZII0Nm2P_H>s+V5;?7)zbuv!J>3>i zbYZ5Sf(qGL>E$uFs~vx74MGr6X$%;A20UIo&iAI!GxVlc8&!s$$~n zfja`IZJAQ15bX<#$|YZ4N#(;fL?M*2L9#5If3WglM?wJw_b?r>eh#**OaDF}K?3MQ zB2j)HhfUF71WU_8>}FxEpIM~n8C7y6?};&bZ;XZ*-boA(F*HMkPe$-rjEKX2K}rde zPaPfAP$N$V)l%%fWaTZkw+5UthYe1wEUxsZ#)G$)tz_)?d#Q~c^$#>vh3?f^WWFwLfN*8rp8p(aDfKP`WfkFyo2P&^q_08 zm-D5MF_wYp^9bNuS4zmU+@2Q0%}bZ^8u@$M>&+ODS$O$iFX%6NTC4bK=USpW9oDm# zEB3vi-CA)Ffkliv5xwVv(p~W0rZxJ`DBSsP(_CPbP7&h@Tebb4@QY=bRl=IWFCO}; zhUY)0C;U*gxNg22uI13RR+J8-Q)%8LX|PhpElJtni&@H57l^8@)K?xV5%fORbyc-% zW!WumX%4n8T-5u>MsY$Ss!>t$i_62i7f261csh(35fbVRLjDk+9zy7{Mp<{lmm@gP zZIcJ?6N}a9j$_>g0*_YuCvyz_pW=!c4&C)tb#p}^d}QGO(hcN#35UGv?r&+nChApi z_wqtks^|FM9GGVF7n6SzjA`&uac)fCGQ1@J{vKvoCb~i~E?r^n#Y^-RQDg6y1=f@& zZ{AXQbATSFTnwf(t_e+J6$I_@EWF+1%&85KF*{t}e-SY>sn5;Uj3dfHzF+nL9)Rve~@0f0?j6%Ht(V!an zq%DJGt179|Y1#}r!e5W%jB>+Cawnz09u@fg#%DKi3|*R5cS~6;(|Gda?D_=P2zk=^ zDR)(|4}-1E@{)69radAPVeXaqCg-mZRnV|BQ32W!*NwkbaoTsKC zK)7pf>^>^P%`$61YGtBXpuIu_VPFAFEfF5ry7Bky#sKD0WZ%VQ+o9!)H1HYLhUMYM zIuiyzVhMHS`RqOWG7(}xMg4GWL?O+;iXv!}u z*vyIRuhU`XP#@X=NB17AK$<#$ftAtvs``fnGK&U5{cCF6J{CfZTIn!=x>iG9YnoVb zvrjENojOn+9S){hqvgmwHe6zJ{fnbR2zNJveD4tAkEQW6is$UqFM+a|M-(Xsx(0CX zI)r($#Hh@MZ*^Ch3Ly-*z|qGs^3vh#umnaz7To9{c)=W7Q{)Hvh0ugK)ez>o0f`!+ zg*~Nok4!Q}7sE(HpM4EoXH^`+e0f!?eWo6g7Q6V0&R?rzJKM|v+$S%z`$z^)L(uIp z_;i0N!}tW`q2l#~6_|biUNZMKgiZ2n`y+1{aonyucX3S|_;!B*^e6k~OC}&2Y%$Ed z0k9?V#i(<NJbj=ytj<6g;&y^G5TLeNw*-wD-kVT^W}X~zBB5NHPcrfCDPA;Yi!~J88u?8;6jmne9ZCnLX zId%Hpg(9wT<+KW=k@qCuv+lP9_J=}*vm&ELonw<|htE@N&h?cvbTOzD zW=L}%q-2+ekDF4f3?&{v1yHWTuPc^iE3FoPH zWoG6Q>|{DuOZ(qsHDk_OeJn72p%E`y@zDfHfxJBmO3koqf+7nUWMDv0I2akZ}G zQeYLShH<6)Pdx?6)aa(_5_{nNcjHG@~XcyzdxReu> zqLvq3xp=P=O;>}$YKd09A)E+P<$}oNO01Ac@x1QCIGr~U^5j!{ofXeag$r}iR!1sulE8WBbF%h9YiaSPx57bvuAIQ0V>c*gY`N+qYM!cm*!tZEI&< znVVW3Osv8cRm&P8WEwJ@a3OuAnE9{SV2jhMHGd`lx zQdaFC!~4!OjfH`FHMky(B4K`Pj~5~#VG$@%&q3kg6*%eN8x|S&a}%la*44?V45A{l z3%cLbOEl1HpXU0=6Oc*<~Rekkc7X#+3x4?$GB*K>aOOkP^0a$~%P_ozpw2;%ILNt>Lp z*Y*ZV-_GMznTk)@Rm8E8JAwlTL%&}J=~zn00ac}E48WSwa#dr>7-@RC^qQ1nhjc6| zIu0tj`!NUI3Op&^6!Y4`BSHzevSsCrN+QOtx!g%2#^4&Als0<8Cph@T#u<|5Z=Zu2 zK+zbH_d>?FRWZ+E)I%+4?D6}kKy!QzuVYo!)S>1hzU;|6>960{N<1>)$kNZhk<_^4 z;;OwejstlNJ>n-g;K3;oF&s`_{;cl+8q<4u=Goa1?S&yuL@;%vp64z z1ilc@W=nbV!n-6h!GB%-(S?FH0V70n!nn#4-~p_g?}CKRsV%5hX4l#Klg|0Xz->~T zFNnc+@X}JEWof%!2j%P2LJt*{$-u?U*IYm)Wc{MtQ@?r!_b=Rv9j^rL3Xs_cv$=TJ&Ve8@%8b^o&ed{ zFk*LOU&yZUhO}dP!1W?~yL$C#b@X;Ki3d3Qr&o>+>FTH#p(cxC-5ZXi9*cG^M;Z^$ zaD?F|JXMhd)6;LcoXdYJ6BQHTgRA#uX;_|U_4evv`5sTQ_`+`B&27oaD2w1F zfP;b$PG+;smz-R@=mB1=TST>;ZNLTVI!p|AQvfITe-~O)pgPor;-Q|_8 z8Hvk{<&~%@mmr^_RXn>J1l-~0%^eo9^Yi`$WA!IMo!lP#SMR=qKS3|=&cMGq1U!|c zorOc~*@6MRLUumS&$qu0e)qpyDO^@dcT#Lm@7$|%O~3pC1D)_ig=oQ*5P5$EMGAzY z=~WmGO;EAGVRc-y>F~ZAZADJe3zS;5)NRId!AG~0^9NI`al{#JXDr8rJPpOBt@+TX zEOks?Cm?$mT*=T znX+^^FzTro7A$E{D95xM|kZZq_S%f8rTe zLtmA_jrYo~JH084J#cD0^i;b-NmO8*+_JAxY>x^F(yIf3nIVA`1cSUrJk7&t7m`L` z7tI3IwwMBuWjAO?%-ZnmQ+IX;0qa@rh?9#4L&GW?)-1L+s#U83B6vw~{_X>i@Zs_odIBr3*K@Xkt% z5SBAQ&NeSa45+4u%9Mr`iFJyuOT56`^T|C&(NRf3(!+Q&-7>?NQeMWG*zuN@(k%g>6v zTyhqfGRY^h6xB7_I!7lE#RE!h=v^~sJ8O4>O4*5cgV@FGxUTE@R-lsT1OHZugZLza zX59@2&0VP1kQP1d*AP99YS+UXCfia!r~^YA+|28;f1&3P!Nd9H$U`u+Ef=-iN&yDK z!dlj@fG4vfn_UJG;dz^!zZttoY2M9-k6{N^Sh8_n;j!tn0T9i=x zCmZOc$}~=e2ATF{Q?9S9tW&Va)~?vf3s|e77NOL9V^Pwu>FQKF0PQ323q*3o4!7y_ z01>U#n>_mj<2~}qN1Og|N->SnPpEt$sO2+jCG&8NLzPoH=wKVh`663n(bwGf27Y*X zcw0JMLhFkSIl29ewpaGz?Mt2R1@a2*-jU3oo?{$*oT_u0+A{xT*6-N3ac=mmqRcz! zp(`1A`E6_^pOZ&Y+}R|Uu=lyL#x+3OCVN02QLm@%5m^&>*nG0~0ggQL8Yj$b?>yJn zXgSFMZ=Fmtq#9?d5@UD zQFM}dp&a?S?31!2Zr!ceXr-JS23wCHJo&GnwipZcc&hU)HRB@i>v!8^3jk(w?jE7 zyzyvT8~UYMcZT_tW;R4{mS|%3m-mCSCzUYm`RxW83S^N7iuxxnDU(|^DhJ-jl8VEA zR9ej2Q@CL9ih!UP2jU*LZKC)ZC7QdE_$n7gFtgHKyOyCoq_F@cwAnF=X9=acmc?9(Vxf9_^6R7shpd6*yJ5x~LZA{i|=0LsU%Ij|RV_(rgsi@5HhFv|k z#36DV=O=g{dT~j0iEEr!QPMZ5EFM1xShpoxx4e0DRk0&Z7kl9+;CGR4cl(;Z*nYQJ zME&Y%0Y9=no%WNpBL#Z=&D6#ce@HbRKU-NVmT}cg`RHkXBq#azreJFv+|BvcB|X&88bk|^LN#2BtkzA@jrLsDcNX+(UqDDLPt!VvmoUH)nf z(Oxw)M*t$B1H6c&NTbiF<%5o=ydebK8!LEhURFPqPEy(hsuMDV22Gf zn|l~wwMQ|W0<|Gc!bW|UFR99`3C<-$pQH#`h8goKTPT3W^c~!S@=BxCjsky0Z=Lex zjgkG~OZbGD31{^%>L?5Zt|Kau&f6t;_m*m)>sb_hCG2y29_;@u%TpVdd)dgrOz`u# z&FuO(Gzu?&R}HF>5Bg|VqGBc4Uy?;gCjs0`qfg2P*V^<I%1xHO$bn+_|8&&=e@4Ajh1Ndx%UWYDmII zS8wZs4VAvg(yl6V-lj1`b@-{Ms%o5tL%WzK6))NlP8H1?{0n=5Y~S3~Ioy7V=xTCQ z`L;^Ne+zK@ZUNE!LlPe+Ge1 zLV$!dp>mkafeDeqSXr-_9puqy*fyOCB-f9Z;z!Ny{9XYJl74|m;NytDw}Zmq!T134 zj-sc*b4gQR5CX>w_brX#Han05qh{Xmf?SiKC<}WTuPj^`)pukF<{>c%6GKhwLTz&# z>5>lZRvOj)N~FJRPSs*r5*v5Lwa&Gdg`MuAiac3edjFcb1m<^q)6W-}KLU5|#1|8l z++kBOBI*M_!MRg}mbnx?l+EM11K+g$^{ln!8`E2KEe$ghuiS{LSA8Xk9zEvnTEs(s z`_=TsUuvndNe)Wm==ug>w62aRy1&;C>5@1gz=KK{NuZ5%^KGA>%h&B=^OTK$Y@C<3 zBp8$M@8DT}U#e{E9`aFzzV2A{QdMs^->L3E@4Eoj!ThG`kG$|1x6|-hbrjc0UVPQn zb+T=dJoF=+;86hIMa1#D@n*PX*}Gf^gQD5{z{T9I-Y|h6@eNe^u^%H;EQzhK=7gGr zQqXl+#^0XbNVF5B80MX}5HxOn$i{}RQiEpFMR%YvR<5e;IkzU|RTeL?3Q9C{cK!MA z*UW*NU1cIgt~LRf!OeO47e{QB5UOwx_O@#(NX0*$sPwIcNNNCl&)avEy16qGxS)+* z|3Du${tB82K)#5d_d`J_IP#|T-wNU4hIl9YAsKMF>$I-ZT5wv)B1D@6?r!Mt%bxq$ ztTic`G_Fea?+x5;B~wyU*!gQ)LH}+-rEwq)TDf(qbY-xApTit*WKlTjW>ZchEG2=Z zgim}jm3hLI-rAMh@d&1$YL-of`ACg2>KB)uFh58CH+}2n^5M?Hhu_FRQ5rZk zcKq^U!8hCI8?_$p58>ZTwP+ z%~PdkZf)r1;ZfoxH{{Yn+WFWTkp<|g&OtJ-Ojn7Mx_B7dn*v;M&fA5%PQ@J%=J%a& zUPAc})2de^AWpEP+zNB5^O^bctQrn<4SlgVbTbbOlxj4(+LqLgBCmK*QLWOy+z<9= zELdPB2I1bt`-9THTgtH@9U*j?u;;WV@KQOM<~6_EyK*Mau1s0DE4$pj+yiy)1`@pH zj&NPKB=b|j<>ti&8DXn(nwV+?;<%}x@OSk&bcNY9D|%cogX&7A&l1*fit#azWY&An zk9ps>gQtx|@WcR-u=mIty4PhSlW7LguJT>o)5y&_8sOqX_2a{vykL6mpS8LFi|($n z>_$Y7W^U5U&2_x&eRZS{7id55`I3n{&?yI} zW6t`fhDgv$N%;9 zW6qpE*3VzwzSm50z!S^oY62d6;*D85{>1esBhDDjDVT1Ec>}F&7TC*o6}%H`SwQmn zdVPx!x`T~{wYvd6<~SG-x-&FE;W2Zhp5^;xldPP>!Hh92=fg%)@FWjCfE(Q>B+Pr* z7tqqf32*pPb!nIm+k@#VdBkP3xY8Wy0H>6^3Q|$%m!hfi_LxV1E`t7x zcE0G{uVaRyVT*ip5B%;|B6bOwr~t7by=$2Pz&VeOd!6bz<4wcqzth&*S3cW$?E6u+ ztd`nUxSUJY#;-oddFUHawwyqo*Be^)TfV?80?VHfSI#B#ew`m^lKKOb>y3Tb4*h_w zddJ8Q$EU>Oe%k?TyIsw4mnA)JktFjEbN7)ut-|0n?F_eY;Kn7u=iXxTAH}B4wr=>F z$L8Ljaj_;-{_Rco-Mt5DTfq8mObel3sTTeK_1tYsDomi4YqZy8us^r3VHDbNCUN&W z^=R6jFP>FIym1|2T-2Slb!c(L>&nz$h|6!rIRCF;;ASIbCH)`eASj5KxjqMp3uHG$ zWKp;EwItOIJf=k#n*FkaDyBH;^RN7yLKRvq`Y? z@kmRG^RjWWbFy=ANVBoAN%F8uNk~gbNC}hv|5N_GqWG`G|B(c2JiMg;>&p|(&>=!W z{QCrb+I|YydqVk=Xu^i1HDq!Nfm8DIwGI|wvcL!%%X15>g`UQ93@!-9hj``-f`@^Z zEV7GPKpAzgOiUTQF-b2~i^-;CYsS+w{q;0nGvisX&eh_=DxGG+tp=*gUmttwYOPxG m+E~kyjg&S32ixrWIN=ii<@&5qCZ<#>= diff --git a/hw4/hw4.tex b/hw4/hw4.tex index d82a35c..71619c9 100644 --- a/hw4/hw4.tex +++ b/hw4/hw4.tex @@ -89,7 +89,7 @@ $$y = y (x - x_0) + a_0 = -\frac{23}{6} \cdot 0.5 + 0 = -\frac{23}{12}$$ \section*{Question 4} \subsection*{Point a)} -The node coordinates to which fix the quadratic spline are $(1/2, y_0), (1, y_1),(3/2, y_2),(2, y_3)$. +The node coordinates to which fix the quadratic spline are $(1/2, y_0), (3/2, y_1),(5/2, y_2),(7/2, y_3)$. Then, we can start formulating the equations for the linear system: @@ -100,40 +100,38 @@ $$y_0 = s\left(\frac{1}{2}\right) = a_{-1}B_2\left(\frac{1}{2} + 1\right) + a_{0 + a_{1}B_2\left(\frac{1}{2} - 5\right)=$$$$ a_{-1} \cdot 0 + a_{0} \cdot \frac{1}{2} + a_{1} \cdot \frac{1}{2} + a_{2} \cdot 0 + a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot 0 = \frac{a_0 + a_1}{2}$$ -$$y_1 = s\left(1\right) = a_{-1}B_2\left(1 + 1\right) + a_{0}B_2\left(1\right) + a_{1}B_2\left(1 - 1\right) + $$$$ -+ a_{2}B_2\left(1 - 2\right) -+ a_{-1}B_2\left(1 - 3\right) -+ a_{0}B_2\left(1 - 4\right) -+ a_{1}B_2\left(1 - 5\right)=$$$$ -a_{-1} \cdot 0 + a_{0} \cdot \left(\frac{1}{8}\right) + a_{1} \cdot \frac{3}{4} + a_{2} \cdot \left(\frac{1}{8}\right) + a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot 0 = -\frac{1}{8} a_{0} + \frac{3}{4} a_{1} + \frac{1}{8} a_{2}$$ - -$$y_2 = s\left(\frac{3}{2}\right) = a_{-1}B_2\left(\frac{3}{2} + 1\right) + a_{0}B_2\left(\frac{3}{2}\right) + a_{1}B_2\left(\frac{3}{2} - 1\right) + $$$$ +$$y_1 = s\left(\frac{3}{2}\right) = a_{-1}B_2\left(\frac{3}{2} + 1\right) + a_{0}B_2\left(\frac{3}{2}\right) + a_{1}B_2\left(\frac{3}{2} - 1\right) + $$$$ + a_{2}B_2\left(\frac{3}{2} - 2\right) + a_{-1}B_2\left(\frac{3}{2} - 3\right) + a_{0}B_2\left(\frac{3}{2} - 4\right) + a_{1}B_2\left(\frac{3}{2} - 5\right)=$$$$ a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot \frac{1}{2} + a_{2} \cdot \frac{1}{2} + a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot 0 = \frac{a_1 + a_2}{2}$$ -$$y_3 = s\left(2\right) = a_{-1}B_2\left(2 + 1\right) + a_{0}B_2\left(2\right) + a_{1}B_2\left(2 - 1\right) + $$$$ -+ a_{2}B_2\left(2 - 2\right) -+ a_{-1}B_2\left(2 - 3\right) -+ a_{0}B_2\left(2 - 4\right) -+ a_{1}B_2\left(2 - 5\right)=$$$$ -a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot \left(\frac{1}{8}\right) + a_{2} \cdot \frac{3}{4} + a_{-1} \cdot \left(\frac{1}{8}\right) + a_{0} \cdot 0 + a_{1} \cdot 0 = -\frac{1}{8} a_{1} + \frac{3}{4} a_{2} + \frac{1}{8} a_{-1}$$ +$$y_2 = s\left(\frac{5}{2}\right) = a_{-1}B_2\left(\frac{5}{2} + 1\right) + a_{0}B_2\left(\frac{5}{2}\right) + a_{1}B_2\left(\frac{5}{2} - 1\right) + $$$$ ++ a_{2}B_2\left(\frac{5}{2} - 2\right) ++ a_{-1}B_2\left(\frac{5}{2} - 3\right) ++ a_{0}B_2\left(\frac{5}{2} - 4\right) ++ a_{1}B_2\left(\frac{5}{2} - 5\right)=$$$$ +a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot 0 + a_{2} \cdot \frac{1}{2} + a_{-1} \cdot \frac{1}{2} + a_{0} \cdot 0 + a_{1} \cdot 0 = \frac{a_2 + a_{-1}}{2}$$ + +$$y_1 = s\left(\frac{7}{2}\right) = a_{-1}B_2\left(\frac{7}{2} + 1\right) + a_{0}B_2\left(\frac{7}{2}\right) + a_{1}B_2\left(\frac{7}{2} - 1\right) + $$$$ ++ a_{2}B_2\left(\frac{7}{2} - 2\right) ++ a_{-1}B_2\left(\frac{7}{2} - 3\right) ++ a_{0}B_2\left(\frac{7}{2} - 4\right) ++ a_{1}B_2\left(\frac{7}{2} - 5\right)=$$$$ +a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot 0 + a_{2} \cdot 0 + a_{-1} \cdot \frac{1}{2} + a_{0} \cdot \frac{1}{2} + a_{1} \cdot 0 = \frac{a_{-1} + a_0}{2}$$ The linear system in matrix form is: -\[\frac{1}{8} \cdot +\[\frac{1}{2} \cdot \begin{bmatrix} -4&4&0&0\\ -1&6&1&0\\ -0&4&4&0\\ -0&1&6&1\\ +0&1&1&0\\ +0&0&1&1\\ +1&0&0&1\\ +1&1&0&0\\ \end{bmatrix} \begin{bmatrix} -a_0\\a_1\\a_2\\a_{-1}\\ +a_{-1}\\a_0\\a_1\\a_2\\ \end{bmatrix} = \begin{bmatrix} @@ -155,9 +153,21 @@ $$y_0 = s(0) = a_{-1}B_2\left(1\right) + a_{0}B_2\left(0\right) + a_{1}B_2\left( a_{-1} \cdot \frac{1}{8} + a_{0} \cdot \frac{3}{4} + a_{1} \cdot \frac{1}{8} + a_{2} \cdot 0 + a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot 0 = \frac{1}{8} a_{-1} + \frac{3}{4} a_{0} + \frac{1}{8} a_{1}$$ -$$y_1 = s\left(1\right) = \frac{1}{8} a_{0} + \frac{3}{4} a_{1} + \frac{1}{8} a_{2}$$ +$$y_1 = s\left(1\right) = a_{-1}B_2\left(1 + 1\right) + a_{0}B_2\left(1\right) + a_{1}B_2\left(1 - 1\right) + $$$$ ++ a_{2}B_2\left(1 - 2\right) ++ a_{-1}B_2\left(1 - 3\right) ++ a_{0}B_2\left(1 - 4\right) ++ a_{1}B_2\left(1 - 5\right)=$$$$ +a_{-1} \cdot 0 + a_{0} \cdot \left(\frac{1}{8}\right) + a_{1} \cdot \frac{3}{4} + a_{2} \cdot \left(\frac{1}{8}\right) + a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot 0 = +\frac{1}{8} a_{0} + \frac{3}{4} a_{1} + \frac{1}{8} a_{2}$$ -$$y_2 = s\left(2\right) = \frac{1}{8} a_{1} + \frac{3}{4} a_{2} + \frac{1}{8} a_{-1}$$ +$$y_2 = s\left(2\right) = a_{-1}B_2\left(2 + 1\right) + a_{0}B_2\left(2\right) + a_{1}B_2\left(2 - 1\right) + $$$$ ++ a_{2}B_2\left(2 - 2\right) ++ a_{-1}B_2\left(2 - 3\right) ++ a_{0}B_2\left(2 - 4\right) ++ a_{1}B_2\left(2 - 5\right)=$$$$ +a_{-1} \cdot 0 + a_{0} \cdot 0 + a_{1} \cdot \left(\frac{1}{8}\right) + a_{2} \cdot \frac{3}{4} + a_{-1} \cdot \left(\frac{1}{8}\right) + a_{0} \cdot 0 + a_{1} \cdot 0 = +\frac{1}{8} a_{1} + \frac{3}{4} a_{2} + \frac{1}{8} a_{-1}$$ $$y_3 = s\left(3\right) = a_{-1}B_2\left(3 + 1\right) + a_{0}B_2\left(3\right) + a_{1}B_2\left(3 - 1\right) + $$$$ + a_{2}B_2\left(3 - 2\right)