From 32ae555c290f27d083cc4a59617eb2e97dc37a93 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Wed, 21 May 2025 14:08:18 +0800 Subject: [PATCH] fix wrong math --- main.pdf | Bin 118445 -> 118472 bytes main.tex | 8 ++++---- 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/main.pdf b/main.pdf index f2674ec0e09479d7219728fbd1b06e7ef135965c..05359dea5a81f65fcab95bbda427237c85c3ba7d 100644 GIT binary patch delta 8992 zcmai&Lv)^jqDA9Bw$a$O8ry1YHn#P}Y;6588rzL++iIL9Y1HW6wceY&(VL#Nhi9zC6`}aqGSuuL>3}N(UAMy#jKE#vU|tOzH;yMY`i#j@0hi069K)JChMKAn zYVv4COjS|&jfXmwSx9tfGl=s)We3kv5W!jCGbi7F*%8j3N&A%YS^-LyUC6q!ntc?fpv3V5#@Gs^DVuw1*}zU?6YKFq5a>trPOdedLBje;DGDOe+*ngbx?V|H{{Dv^h$${ zWKE>XC&?bWqpzv6Cv{i>^nbGzY<+@zcliXs8wyWPP6*k^vFlYw{zkcMJg);ue=UqU zdS|z>06;+exr%CkwK(8sX25x2XO}?@5-T(O;;%mVwZ)A&tJpqFwlnmdwc*C6X|r{| zE*(XkJfK6DpTvqZ0wv%iuv>Y?9a$?o0!K0sTkG7!VeHb>QltKyUdM(|i(jG@GUvUt z6I!lL1)c}jrpyKcLlmm?dtByVjoM7eqZK6U?A8i}qm;8}8n@RjiDQ#izibSj^?~(6 z4`8DRWQ-hoq=Q3iN+O1_1C>oJhsi$od^)C_0DUni%B~D5YV^ZkOr3^C9i7QzWa6Cd z8c(eUhMi2`l|6H(_q5rbYRB_ff6YKl-zyg0{7tpt%koLIkNHN1jf~;S$M;9X@adWR zWdW{=H~lJs0X|D~&DX>J4T+K#kWb>{DV;NUzk7ctA2w-;3Jol&f4IzP%53B%2%o7G zcsv<2S$S@3z6ro^ma?kr37lsyaB#x!Q|+7VU7!f07Kq3QA-ge_59abtg(JYcTxlVC zd^u|ZMC|&gw}NzGqkm&!u+79yH#OzEw1#D@j3;eac-5ZK6eG*A_3`O{uLHjqV^ud( zwhrMVlN1H7*_=(~j{Wk&<+S9t{rVRkkVzPN8!|My(Z`o;ey|rD7@-NTk;%eXmR}(s z7fX{DOm5MRg?8_iUm#hmAfc5Nqbn3^z)+>Byk+aZ{L!HwK~8ZXY)C*z3>reW87Wd0 zx)mUfGKp^vOQMt6x@(n-B1At{Z}*HLktIZTphj6MwGdqp2<06bP|U#kaAC3o2B$b- z3tKNaRg$~Dyw%&OuHI*^$8AqvkG!}#8-^$uXs^y3wk2q+%=Y*PJ(%;@ zuLxv+;IXQN<-SL~w8LHo^}$HsTeK-qn?lsId{_6A`@uK8x>>pJ(^aAE1Tcg7b+n-y z9K2dZeo;_q)sZ~HL>P*hJyx>L zzkGzN)%%DM)7T}6YI^XC`Y08RygX*E1_EO{Nl&3LZ zc_@e1v6vL%`r&@7ht`yl$~E^Xy|}ca?U}ApEUUD>`@-Axg?LEa6)s35q<;CJTA++i z{SwgqSGB}P+ZXi&hA|2Q;PlK{BaU(ER1$2W>(7d_#rl=%=r`P*JB?ED*wBJ8Xw7c7 zd;Iw^a}qLSx2PR5Q5LwV1@IyK9QT8k&dVY?AR9r-<*ork)ahIpp}3BU3C_zD;>;Qb zKM@~3z~PFJdt-<6gXmz_D8F4K(0Q+AenO?fQx>OWt$ zJAYmVE`Sl!R9(K$gu9hd)W!jBI$~j#U*mM4tB0KM2GG)P7DXzWy2glJjlS z?3V+l;z@-C-381w;Lb5RmBLRhkc(B6FoQ53ZBcR6WtbM5c*FWgYxm;W)k()I)j=fG zV=F-%C4?`Df-!YNix)ZQhM1(b2D5r<20-1W z&K{<$Z$^w5l66*3s0_i+f&UT!C^xd zbuOh({*s{IJpK*JUqFn{cCdAx+lp{%KjseD=k2P+(27>8!Z^aCw`oGqf|oTwq>41s z(3E9M7}xv3p?=&)7VwgB1g;#Y6lilst-@i!gz5*tDvHoQacf4aA9&)zG+~&I!zu9l zSYaK%mf7ayefQIPCv_QO1b!e1D~#iNLl5pO<(foX3Y`M&ce}(J|KOX~B`8+h!>0r0 zL-Dm3ZG>J&2(t9oolPWGD!WncUZHdO6S{>|MluzZsB#8g28A99UG()`}GQ7l!{=NV>agk9n1;YyC< zD|ni}HVhs8W4i{7CkM23=uJxk?Qg zB>8i?hTCg&nPRDD^!=O#fml}7q}m@pGdSAE-`$Aj;V$7+nCYV+K(kzg`0a@rQ5ui{ zd(OgpJeEg>X8>P!jPI{(Vug*BOOxuObEw1GTpM==IO6pkMcfolc0{#g3txQm2!{yq zT~hN1?xOnL>qz!$;RD4jEOU6i{<`u={MR@!6vy{9u#!Ez^V+|5-gU3RoX$LDW2DeH zg)EVr^tb)tGb+*;K?zTlDQp<)kiYCeQ;TX-oysNuXzb5m{t(efblThPy%FnI&DgI>C~K8dN_=Pi`VlJ;YKe&SOID^HuMUVX_7` z&#a~Sn`615ZA=Jl`PfhSTwC`tcuMk}U6uU!4Tdhu=m*O~G?D8h9TyB4+V*uCGcuM3 zEGjJ~{q6aP1`AnNPxlIW!*1WZu`G~74iUq4)-`!cDMPgVwx<7`V z!!!+kh)!}tp4wIg?+ZaC*HB3P*Cm>`GfcqwXtr%)r_J-Kzx3065`6RR!ovd()pX{< zmNWMPcVM-M{WS|KPKEGa3j$QR;TmVM=eHVILV8=-_S|Xjqp;?Jzgp;_w0$PWIPKH^ z%_CEAtjUk*^=-#QwmcC)B8T$P_9$~ehVbeK`+S_**VaS1-;-IXl({adwh@+|B%22u z6On_y@YXQ59@$ovnaYu7ougkok{Se*-k$}g2UA!c=aoB$QjUg>Q}RWd^LZD^-$Az1 z73Q7^tsbNH&Eid^G$><;g013<#AJS{Tprs=Mv*F!!wGYdk7jP zC;>Lqn_&M=_m%_R&7r=|Tm#nM6VTv^0?@6X){c+hr}!x2yUP2kOFfya-XN9Vty}Nx zx9q@*0^sR1ViY?6=%w6Xdp9C}@jXe!HQ3k6vCz*;7o*=fMK9QGub)B#ua9mfoOid_ z-QA<38m;ILV`zgk!ZsU#nN;+9S+u2F&UZWc9 zT_`^G|Cy%4`Zi8PGMMi(#_XhC7*PSnqV+-4`aK7f>HfoPAxT{6?S-=8C1!rWd(SFo zs+D(C$-(ddxT~`(UVB+{JN2$IG5oHxzB`jbz`(u`nKYmfs;+P@W>v>})Q-VD`6Y+F zOqB(f<{2DQ=?r=00#oDsAcc&vs!I3MCp7dM0~PS2U@kz??P8wBOp(?nbWnFrau%$q zLVY)imqoopwNShm773%5z>4sGP@JxZi!P(k?H>I@AwHf+o=z+{-d8?b+|DzOy~VTp zwp|W^0(%4BfV>8>crK*1AWW;7+H?qnB#vxotpZDx2t(q_Q1H4(&mt`K7k=Koxd);n zJ`orT=CC=(A*N=)za)^G2x6oww4fA2(8n(=)Qm=pVLS8Za@Q#gQ0NN_6C z#LMVY?40sYfOLrIBzk#I8%y-t?{YG1YXXmlU?FVaY%C`#XGld=l43Q+c4etJf&tmR z06B9y-{b~@%^D|$PUBpyVV==e3rFstWd$gcwTCps8;6q8i-bblEB{s^3ripct*h{^ z5q`NTDnMYWu?Dr5wDXtiM{z3qI!n8oiO&2N2CEMft28ooaSeCWDK)$wHd(}4v}3XhOr6!LP+wZ`>eZ}UD@Ll^)v zWDH|#iGoUv2WI>AU~qI7F^HypF5%SNU5GgF`u)O~@r+DySM62p z;q0pyCQXoeMPkjQTR3-6e7Nm&kiW^i|H3} zr3m52jhvoo#P3>!cUIIDc8r|pRhb4CZ}rf|*Q!{(ykzvBCV0y-!&tX7Bf zHyNvg>wE70KK#N&Q$3&YvOwa89QOzM$q%i7=z@Lh`cc_Sdt(j?>qRe)+V6F_KWL9{ zmfNaa!^i_=OjJVP4Lq z1mM?^7eAu72Aaw%D6dG=FOdZHlyOS|BT-83!X-VcIiv2l?qUvjFq;iOv0%FP8%SWf z`mcDxm?ejlv9BUVwa~krwiI0XuZIl}jPwD05D zg*S?8ijicdA?vc=e*ssZ-Ymt@9+jM7(HMkB9=ln(HeW5{g2kPuBr>Nm&c<0}7c7!8 z8Si;3{zR-t49ZpCd0*PoXYJL}m5#_o^4}G}B;{7J43(Qof~==t!s#6Q2Dye3(fHRg z*z$KP0Yy8jSko+Vw%uLX?@(pbU0KSNJ=!_P_3&d%bi50uA3&0o8vqwwYY00nC6GC2 zua&PnhN`}%_}yW5c-pLhPm%<4)psaexsN*tyEW(^En_g24s>wQ*i*h=e)}A{Hy`5Z zU2qXDzBEeA27g~a)Hxzr*AZiHG2gv>AveSN!KZg3bi$udJJEJTpYz|C5XcbGW~AM} zpDMn+Ry0B{2mszlP)_gQyYCKT+@DZi89iq1z@J)YJA@$R?55Sd`|BoXc<<&-f@d;J zEq`qYSqcZ?Ri|@KkE84+XlE^wmXqwJyCq9uzl+yaeqMo@?|AIGPJC?-xclav{B5JQ z_XO-!quuaDYu7Vwl8q*90;|@UPeLLFe~viwQ;KPDz5^U|YoJQxJs0m%HGJ$pLu+VW zB68HGptM!oY~gdq)jalng4Q+`jO=Oetne;YpRh9B8eZgD2#;aJxthSp0U8HBUg6}~ z`;{uzo`A|}eZ~%9K`ChF{7vV2`o=U*Vh%OZFE|trWKH9kB>47_G}+Fm6?p8pJD=07 z^M5<2WPm-X&MCCOT1d?Ls8hA-?G^AwAHZFGZAR=YSjY1`kGena%Ef^)=3TTzTpgvw_>S zt99|Yo7Kc#S5_8R%mj_=N;lKClbe>nPuPAzjc;B33rj5rTi2bGk~x*Hr9!P7$!=$r zW!qkPYmJpj>V5clyB;DLh;6}L2)i{wV7F_+&4<4$13uq!O}aZDl6?FT`RDEfT#>ha zfB*q#JS+GgJ@b~%=0D)GGjB||R642dOAhhra8^Z&X89rxW#k(1#{tQRVDc;fp5?7} z)ABzwjR_Oz%p{=42LsQGbh}yR?W_{cbKKdhtll`qqFMw9_^5lMJ0h|sC3+(K=E11$ z%_`SN>l|JKKP$oYT&Ek{&$C2cU1v7=KEQd5r`TM}e5A-V0s)#YTlarTiUvrfFPZtwKZ9R z{6hUu9?4man#OGMn1^L~vz-7sU18@))D@u{ev6cw!EbJBcB6+1$Yc%oe^Uty z4lOqwNO>Qi)k03vuZGQFRAUzh_6A>nESc~HPH8*kG^lUK3&zUv{yXq=0t_w`wrpWR z=|bKa4%W>Rfye$kLMO?1%`^P?x0&=<^?f|+F%gsJTdlM+4v$8T{;+1!m{_NaqfE@B zN8I`nt3*#(Am2`l6(%Q?5eb)!bmNZ6+=`r2yUGnicMZa9f3~0_t$W9IlpNkdW^>Uo z|N6qouXEQ5t+ps&I*gJwAh!ltEiao@dNrMEi}8tDKnHrmnv;VAWk=uUVnvusrY6$< z76-(vfNcTOj7m`ufKKfO!}R5dkurqNnuGv$JyhM{9!JKcxwYrg>$InVU%EyfiE0n7a2;B=GU{Oa+8@Y@wy}nDF!nj1F?5BTwGjnA%Oe zcMG?Vk^+}DFe;_&v{dL$xnOSr#>mh`Q1QK~ z<`|eSTQlPZz7qUA$2Fku+7Fu#M1c-OzhWm06?!v(qm74Un>xPM3sN#q9)nc$* z7-l4?d!kq+Tvz~>q2Z^hSX(N%12lH*3^S?`v#*syY3&EU0&!I#MT{bir59Cf%eoHH;rDmO2$ApPiWv?8~VHH=SJe1#S&YtagQ}vrHNV^ z4Vjd{IGSSJsP2m|q^Vy+$9K-2g3fDB_Z2ANE?+(_IPuhb55Fx7b!z@ITv92?tm1HGcl}#QWi&

ar<^T{m5Lvd^8V=fpg=Pf{`#qi;B0;y*XoW(I!Q`nHlA{elQXubA!^v^SC$9OUl5{m z`5ampcfl?C;+f_He7<*o|46S17Sr7wS_zv)Y$q*y64TLbCTk8iI|?TU)>KC({%vPm zQwHt_FET*A!hsKqv96wwA^k7bH+$ndIp=S_I?9jFttZw_D2mjQElkh4*HzqfX*uj0 zU^*sJM{0i$-Moy0=c4R1Lw>KAVAA}{Uf|w7F1IdznFNZXf8SQ>&m%xV2555_V5LW_ zF4oQohG!+P8ubLcT@V(M2IOcK)0b6x07W_W!)9>ZjyGA z*=1E385I0egv?=G%8XR)iFho*8?Hy%sjuI3*gCt zO`CGpJP|bE#Z7HwU?%D!{8N6)jr%X$+^U6M-buHmj#Bx<#bp1fFT-8JN^QjM@bg7Q zc9h(MrA|POQNCgoK{C!%cV=#+q2Ud-?yj&vpx+T7W?7cNM}tRJhtC_sb zp3deHhSE<}+RsFu@(0G}YDVOAdk-Pi)h`{DnXB$f9@^e+8c^c9Ip+qc(y^s1ZylO2 zxb*J0I68aHj5+Lx*&;c|A%IrEK(DIJgwb{WDh|!3Y^kQ`{pzAp!=#A0wY(Mn@@{%} zen!LAza&Ouv$74=a?KC5X?5Mj;J^>8*tf&l!F~$?%4$Td#NO1J$fwYz^yB)pdCVyCjCPBN?OtZI|YQ77McNfa)#x7Q7|Edi#+b6i~%5}^&; zDpw+{tLy4o{Q`XxP38td{al9ag~|n_>bo2BbRCbbrcRCMri%0Wv0lgfUAS;vtCgFX zOHYppiVPcsE(cNXWiz+%mqDSmt-?kflNBbZV4k`djTMw8$E@{s!6yCk``FxULH;Wp z5f-;>&)@D<7T!sd)BxTgVn0gGcyoKRf`NcfLPe*elF$6hpy3SKlQnC?@~ zMS$y-6fA3WzekQfY}-&`VAh$RqJG!WVC)o2cu0v>aWD|mF?1)J6_8MBOiH2EOMBBK@C2V3BTvCxRIn@uS#7;PqTXN3Ua zQu~ux|Gq8szBe!H^Iz}F^u2i5HxCy#BO4zr8yhV>BCC>=Xtr5q;6j{aff0i*Yljb^14n&?7LbKhST?h4 zNH<$s{(fzKnR1&EywCO8xfiT+yZ)E$H6!T4>=9o{wbTff?~|Z+yQo0!+8E02f31?-4#M07Q*w0vfGC#siJP02ucoj80cGb+MC|e zOrRmHAr^P63!6k`(#{wrq;n6E-$;6H>ANp@zw~uCMR*|;Z#F*V?=t0 zPzIf|gdW^iot9T_HhZ#&Hd*?Zb5UnzmhOxt|fw<1mUq$AR*0}4EvYWXOGpAD79M;qOm&=gIN z=?h{>;l}&Jt~XA5rc{3_*#5NB)IST w8OBRlP$CMWPL8the`+9lw5{{}rgQZHsh&{vmJv1sNru41g-A^;r6P^^AG*Rw=>Px# delta 8965 zcmV+gBl_IPod>O*2e36v0yZ|60X6|Bf9+gbbKJHSe)q57Td1iQCGNmVzRIwHtNe+lBL!NNbVI zI_>6FT^F01s$6W`=-bL&b9Y%SOFyl3v8|W$KZVT9Fd;BBPiAoCm^-2*WErNRSjdWj z;9t8bc39Df5SSuCR7ekCgcTuKf6{5@=I7?RERThdVs zKSEvHfWQ>xwpe&U@hvyix@ZKtZ00bp)^CkyFRQXA++&l{S@;esk-o{Ye=IU*^BVi{ zZnpuv7P%xtS^;!GHBky_DiXmkVj5O{4fvkfl>;J|`Hj1ATr2Llh6l^_O=wcSn%|Ekd@9n3DA;~hcvfWb#;4r_d3U)CF#INsAPERdHBb`M89V{Z4nI-wG<$m{EVH#(q{ zzBALxVBJ^C;;!iKYKHR4)oxP6qCD3aoh~WkPerxGCYOn0yzeQ-$jHjp3`Mf-# zH2u<&bYxEL-xv0YyFVKt{B@M@>(Rc1r_UW!_gBkR;iLs+WG65j_O#NZ*&?t6R9J-9 zX0gnDvsmWdEEeTPe`l_jR)?8~enX$zAFrb`9fOs`VFi&Ng_T%X8!|gs=@hJL3RV(_ zRiwx?py^+#dyw?{1ZUbHO%?GV0!X0rj1jZAHOiTj-Ynqdx!qdWu1J<#C(YgW``fHl zFoGZknmAa!A6{wQ-=*Prouy}BD~YBdgBor$1kPaE}&M(Y`E} zwd=Q2C^Wsumn+|(a&L|5@Z*$TRLzeEYY!gR9{54pqocsN0MiBQVo`jMKA9AJ@D(1R zPrewkA7HSgf83c5ozcdZC4KJ;gDaAre)Fu2Y`WXS?v8Yf36o!r}|>x57wzyN2- zDmi`jk4~`Cj@S~TFtm>bLZ_uOb1Zh=S;&;(sPWJPe;geWvlDxHy=$QnR>rgK2pgRg zwjO1QWh+_xBnMS0=}6X2Ee_EAI!bp;qjsDi3891n#fCwd$i~A2voq#yX^Jop`*dY! za2yarq$mwK^~7s`Ae=Fz1dK!&VurVMMAU#N6Wq3wbWhTNXe(*JF2Mtm))~AZ=4ZN? z@Q*`Te;a0=N(i+@pLrbJ_27ryfVzIaTsP4O$LZZNcQj`&zpI@ zeEYF@W?)@4#J1hp{(OpYs9rnl@eFna4%sule;`-s##!btcEAWz(+eX70%@BY!5ijc z1Sm@s)I!EtGNa7yOqXJ4An9!`Mv)fGd72DKp-s{R5!R#+PG(s0^xZD> z-L;31BumK(g!6&X(wPe1Ea4tmT`{ z^0HhP<#x1u)rL3_dr*iP0+`=VE)Ij|1R|rL+f-$?Ep@f^1(TNaa#X zkWhSNFbpv#V}ymW!ee7Nag!`TGu;`>W;rs#6^@ z_0i>exytKyEv-;6#uAN4egtMVa-3w>0Bia=v zb{C08cqX4Z8lfUG8dx^NCHD5$lPc{E-anS*gp9K4R= zg+Dc3zval4gKI*PKgR?pPWya?;t*cuT7Mojtnu!?dw|<7)9j!4}CuMMxyNd~NJ$8d-Ou_^> z!{NBVgqQ-He{gV6Ams0LoH&OGy2zf{4tkwVjwN7GA++p|6N0=(S4Nt&fvOrWP3)jA;2VJ zqQ~XC;a%NMr>};=$i?g*W?A~<%d<&7RA!Czk(en6e+n_}V`ffArQe*G%E!y8ftgul zp`&6NNwUGSO!Ri(q`qg^9B5`oGRa_xN!IHX$(kSq2EjzFB6@&}Q3YfS(i$X3J<=@f zsQJk~fmvW8L!jXf)g1N^X@=A9E}ZhTdfq6}+-q5*-iP%zktj1?p;*0rZ&Qde-4xsL zgHjk%f0S_@c0YZ9|EF}D4({(LDoss~M132Dr+pjDxf7i3Jl|IK1S#<8602bFz>s=r ziYQ%E(8oPG04kX@0azmwWQF~vtPNp^(y-H{$Pk(#bGoKKRP-#CGS)rcIwX9;SAyuQ zRx3a#!N$P@qo?#^W0r=P7na53H&JCA1sp@+d0Y7^!zxX4#D71j zZ4PFc<)OwOAYn1cf{}j_&eO&9e>0kL7(BF$@L&{%APk(}ojU>=IV2>(urkReO{%olU4YERy z7OEqq33ol3UYF;QF63cSJM4K=BOanFwI5Z$_>%~jcwzxFE~6gHsKeG))HGJKf1~-V zEAeE?j8LRuC4TG5jD9j@_Un|jab-p%v}cDYGv+^}>^dAFZ#Knd9(6qq_W=jvNSptf zu`}jB#!iXykrciA>d*ujq}dq|1nu)hLG==DGT`72bMi<(x20x|=r{iW;~`0yz|q}{ zMJXBea43BcW*pdM_yEip6Ud6fY*Rqi5IYpHhlB?9kY+6Rc|PHsPv$VsXEsGNk`=QR zLQq?X+mDDK7ScWz+?~4I`_Ca4b-pf|Pkp=pKg7RpKR4DRA7{bKBRIHIv~b6NB1Xx7u0(CRqYEIF|u70VsdPTWN3G zI2!)$U!jk?7(^6r5*RGdq)o5sF->|+Q4ADEO}yjS8QYn*`|J04C^?aQJJZD?Ath5U z$@e}eTRNE)nRF^Myyz@gTudf4Ke|kMUVWw{uOU;G*O;kTq-R>|OxloX?|L%G%zDg6 z^Gf3Cvfu-*K9hyw4H}`OkJx`f0yJQ{;eG+l7QDfzB{IQr<)SAe@#8>3i8qn(cpR}* z1S|(59)lQF03PJ1!p02$s)&)hg96PPKw$TtEOe%n1U`kujpdEQO$;C*Q(EH+YBewh z7nA`cuvCVC0S^oSv2~`LR=5F(SKxrh@}6u1B`RQPht-3z8IVu`q*;GJMnxV0qll-n z022#BdRzesChG!_kcF1Y0#iIlBJmheiA5Mu1~-u-ZDdZ z3mC1iprMu;NXh~o?J%EA*MZEzmYOOjxxSI$wBe1`CZmZMOCWzsrycRlJ^z`zO^b`;>{-505i(T78>kw`81tzpn70yr_EO2qaab*%Uq@ z;n@>!s_FP<(yyk;U#6iK)%l=29DmC`alk=?d{m#YO*wzTTxw|odoUV}vC^mHzljQY zQD1#?4b8P?!vaQWeovg-oXyhbUk0N;dg5?AIj<&Z=i;+?BA$vPd;q&Y6W2b7gqO;} zG)u`TnxW6>d2IA~p#k+E$?PN(kH@Fu3C6gZ}vHYJ5JJ zeg7SvApU=oZ2Td}X(jUlPQyy*9zC%&c>zakmCj+}>(MJG<4O5@_3gSGVT<44n9YyF z%6OAU_(&^ZjPRofoH;z=cC6kHhQmR5H3i#FF0aPlr$9`~zk`@eA8rwgCZ`2xAP8fz zh0Gb6g3G-$E77X{yeVgc@hDe6jy(vMDBkXL!!$ioxjn!w(j0CqpJyWH^1G1 zgTWzx*$PvroGxM;obxevYWtqn0M5(Zd|=x(b$npgJGOmb*SFnyumy+MaiU%C*z$q* zb$hW`c6(tTx7X(1jxjkbM9vuDu(pFiDVZs4g@dPYh1?2fz0a-o+u`6NwAWkVkcH>o zZ3ll-A$M^*8OSh!-gfSx3RoS~7vPk0Ir8J}aJtBuAZ~+0bc7XcrvoWr4i~xBiq;!J z$G*BIwaz>&d^uCpzhx12$s(e@r$t1wn?-cs=k)`|3y-z%6@NAX@+SDAXEQ{(rpmg>yFpsjr zK`!aa=cpK+l$eb|Hv2y|V&LF@aP@FkK81dVzzzX`qL2gkH11@`QDl2$b+8sV-gk;rY z=NlWTn$3O*h0T|=Lo==B8#b3vb;$zIt{y{O6Swq@jii1)d-cV&yIQ<8lk46}3TU}+ ztAmMwvy|nQk^FEd5Xpxki7e89ffm$6QJp=?MeK#~zd1MPoF{Tryg-xr!JmI?g{n=> z8}7SeBsqb`q#CCZk(IeE{qj6(bZU2Qrh#ePow;>6?U7oJ^qt9{&_+@YoRly$`lAii z+Npw*ZR%05xS}921zG8fq_{y59b;)UI{6CdDN-C0(;O;m(8&X6uEbB$(p6KtV;$eJYLb-DF`m7vJe3M} zj=OQ+Er*_R?pn9&@Yd4vAQ@nsyt9$HfAY!Eb39r)Zu$e26c=Lf^uB+=CgsK+0Wcxm zL1>O}u}DG3(ossDfdrTI7=|ATf`vf*12PP3Y3A`(@x2Wfg*6dxo2FJF7PNX1Mz7$p8bN!K>jxSlF$l zJxsA6DW63N34k9CdpKx}C+(5ag#;wD&=ovKMt02DRiscDC?(}nJ%c(nU6X`esB4&_ zu3sV~?lda{%!_}irc5}S6i$<*EVJbl8^O_LXGlt$T1qzh#7n*yqXc-U5+ZfWra639 z$m%NTfZ>TZF(`Nfp7{vrYMCr(Zf9*wN7*-sM=gqQyS{j=s?n}`7%M(cyV^1Bwa`oZNeeFT3?h^Yx78o zKs{gAa#nvm!Y5kQlGJ=v?rQB|4QPT%nvq~IZVFS+cD`vfACkmZG)k6QkTjsxr~&Op zEtzY;Vk2HjBd7(fmb4&QP^@LXqIr62<7zu@wWI^BX6du$I?!#OTL*$Xig{Q~`;)=- zY&@xtW?q(8d~*2s*`udF9bJ}_Sv5I4JvbZ>&jEihEH9>+sUZ*3`QQP+`*EP)Jf%WR zh%UW|oA!B32-nmTN9FaC^a744N>6YS{)O#C8Vh?nA(crD(D6LBiuic*}3zBm_^_#(ax#f7*O zm*0P{FRPIlh(E+oT#1nwi}9!u*W#K-nZxSKtiGFYH*sA}2IF%v5mPY}v#(=uBfg4n z|Evi<9so-vP5bO~8zUdQKRkN=rp3rN z1^cTSJnby1(Rn$&q(}z%|Er>CzKm}sUHSaGkvq}ga{E*Kc~edK zk`OmzFg!h^G^f?opvHDu{e}IegKw1Z<)o@aO;0WD8@|t*_Q#V-{6*1=@8Un=Kh=L^ z+|>cMj}E+e`s~@OcP$;LWwlxfAk4WEcovXFO9^D`C_yMXN>Fr_K+0P!P&;YC+@$X0 z|BiCM0&l0~=)4hlE$X>wTQXfM&^t-)B)_tun-Q@u+ZXX*~v|r(MGKiVuIUUmpGX z_|;D@%Go6V1{b4)qbCP%s*9UpIXU2HjnTmxH7!7@VZqOGU#`*8V!MvE1BKy-I|^T2 zRTrf=y{u+sm(IO>;KwJ2@83LIL+7hgxq(0?I6STJ&iqdZ=Tb!wvc( zH8Nf7|@ zY+B8btRV6tI1jtL-pjMTeRTTv=-nD#|6lErZ2O~{-Leq(H4_u7yCm=Bb5GvAefaKh z4atqa-R^o17-o!Ye6N4rn+5WY2e!?F(B{E;HJp{Ax}FY@y2`oQh+%nkc3z5MP2DPn z?B$$~-o1bR;MdIz*{TXC>gBz3VJAkcQ-&_p`+4VwFJBzL-Vl20(c0fUSgQu!X9qlE zk#y3sP%STQV(7RW8~&C-yg55N>(Fafc-2RvJ}Rcs1$G*I`2v3zGvXI}KABkOV#uTG ze2nQCQd4n^^j18D;KVOkD8%b57;*Y}r!@l#IgpL~KhvRAD713;RZ%rFrl}*EQlU=% zL2XLwSKwJ!7%bW9dZ4a7QdKdp>6+>;bNaRf1i!b{xhRbmOI_G=s|d`Z?;I+yz-d=D zV-z%hHpd06`hsajl>Y5`ULJZ1@1yFaRXl{ky6_KWzdHvVm;(96iW{gK_RPFchzHnHtdh=&guLjrg|f$9qETp8#UG(7| zaqB)w_W~VeIgM%45n5yvei`W%5sY7qpYogWH#|C8>51oqbELO*MgsZOKV@rvIynFB z%>3K`&3B#`GPPGXvmw&cX@h0f7(S8A=7F}}PI2OY0iy3yHkSdC0TY*4#Q}pVFflka zGBP+TFgPbLFefPrFHLV`L}7GgASgsSGB7eVFf}qTF*r3cGB~#d#sPu_LohQmGB7YX zGBiXpFg7zqI7Bi;Gc-9kIW|N^IXN~sH$EUdFf%kVFfchXG(w%0F~lgk!`dZ*0A{rILvS@=`N!1BuS zS`O^1EU&iwE~tjBun~5^PPSy|dFpGR7B<0V*aE!FZv%Go^{^dS#oH?E;hSJLG{7Ec zguT!TRpMgb?SJF%gZ;dXF;W5!z(MBNP1!>$z`mDC;0B3CY28WNcvD}B=GWG;MG@KYD(nO3?sjQk_Fk20};rDJjjOvh(aM0K{3Q2 fF7h!E{(g?S|NMxnz?YxS0U!x6Hwq;sMNdWwInyF{ diff --git a/main.tex b/main.tex index 47f9dda..194ecf2 100644 --- a/main.tex +++ b/main.tex @@ -160,7 +160,7 @@ s.t.& & (x_i-x_j)^2 + (x_j-x_k)^2&\geq (x_i-x_k)^2 & &\forall i,j,k\in V\ \end{aligned} \end{equation*} -This SDP models \uscut{} since every assignment of $x$ corresponds to a cut and the objective is the sparsity of the cut (up to a constant factor, but we don't care since we cannot achieve a constant factor approximation anyway). Now we consider a relaxation which is similar to \lp{}. +This SDP models \uscut{} since every assignment of $x$ corresponds to a cut and the objective is the sparsity of the cut (up to a constant factor, but we don't care since we cannot achieve a constant factor approximation anyway). Consider a relaxation which is similar to \lp{}. \begin{equation*} \begin{aligned} @@ -171,12 +171,12 @@ s.t.& & \sum_{ij\in V\times V}\|v_i-v_j\|^2&=1 & &\\ \end{aligned} \end{equation*} -To get a $O(\sqrt{\log n})$ (randomized) approximation algorithm we need to first solve the SDP and then round the solution to get a cut $\delta(S)$ with $c(\delta(S))=|S| \opt(SDP) O(n\sqrt{\log n})$. If we can find two sets $S,T\subset V$ both of size $\Omega(n)$ that are well-separated, in the sense that for any $s\in S$ and $t\in T$, $\|v_s-v_t\|^2=\Omega(1/\sqrt{\log n})$, then we have +To get a $O(\sqrt{\log n})$ (randomized) approximation algorithm we need to first solve the SDP and then round the solution to get a cut $\delta(S)$ with $c(\delta(S))=|S| \opt(SDP) O(n\sqrt{\log n})$. If there are two sets $S,T\subset V$ both of size $\Omega(n)$ that are well-separated, in the sense that for any $s\in S$ and $t\in T$, $\|v_s-v_t\|^2=\Omega(1/\sqrt{\log n})$, then the SDP gap follows from \[ \frac{c(\delta(S))}{|S||V-S|} -\leq n|S| \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{\sum_{i\in S,j\in T} \|v_i-v_j\|^2} -\leq |S| \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{n} O(\sqrt{\log n}) +\leq \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{\sum_{i\in S,j\in T} \|v_i-v_j\|^2} +\leq \frac{\sum_{ij\in E} c_{ij}\|v_i-v_j\|^2}{n^2} O(\sqrt{\log n}) \leq O(\sqrt{\log n}) \opt(SDP). \]