From 0f8083a787da90c048c8848f8423f7116d44a1ea Mon Sep 17 00:00:00 2001 From: cast1e Date: Thu, 3 Oct 2024 23:40:41 +0800 Subject: [PATCH] paixu --- 排序问题/report.md | 58 +++++++++++++++++++++++++++++++++++++++++++- 排序问题/res/t1.png | Bin 0 -> 2643 bytes 排序问题/res/t2.png | Bin 0 -> 3720 bytes 3 files changed, 57 insertions(+), 1 deletion(-) create mode 100644 排序问题/res/t1.png create mode 100644 排序问题/res/t2.png diff --git a/排序问题/report.md b/排序问题/report.md index 8afec96..a5c76fb 100644 --- a/排序问题/report.md +++ b/排序问题/report.md @@ -11,4 +11,60 @@ $$ \sigma_{i}^{(k+1)/2}, k为奇数 \end{aligned}\right.\end{align} $$ -我们取 $\boldsymbol{\sigma}_{i}^{\prime}$ 为 $\mu_{i}$ 在 $\set{\mu_{0},\mu_{1},\cdots,\mu_{k}}$ 中的排序 \ No newline at end of file +我们取 $\boldsymbol{\sigma}_{i}^{\prime}$ 为 $\mu_{i}$ 在 $\set{\mu_{0},\mu_{1},\cdots,\mu_{k}}$ 中的排序,对于相同的值则随机排序,即为一种综合排序。然而,$\boldsymbol{\sigma^{\star}}$ 不能从中得出,因为可能有相同的值。 + +### T1.2 + +![](./res/t1.png) + +对于 ABC 三点,C 在线段 AB 上,$|C-A|+|C-B|=const$ 所以我们可以认为这两个点对于 $\sum\limits_{i=0}^{n}|\boldsymbol{\beta}_j-\sigma_j^i|$ 与 $\sum\limits_{i=0}^{n}|\boldsymbol{\mu}_j-\sigma_j^i|$ 的贡献是相同的,所以我们可以令 + +$$ +\sum\limits_{i=0}^{n}|\boldsymbol{\beta}_j-\sigma_j^i|=C+x_1 +$$ +$$ +\sum\limits_{i=0}^{n}|\boldsymbol{\mu}_j-\sigma_j^i|=C+x_2 +$$ + +其中常数 C 是共同贡献的总值,于是 + +$$ +\frac{\sum\limits_{i=0}^{n}|\boldsymbol{\beta}_j-\sigma_j^i|}{\sum\limits_{i=0}^{n}|\boldsymbol{\mu}_j-\sigma_j^i|}=\frac{C+x_1}{C+x_2} +$$ + +可知常数 C 越小,比值越大 + +假设 $n-1$ 个点重合,点 $A$ 距离重合点距离为 $x$ + +![](./res/t2.png) + +$\mu_j$ 为中位数,所以 $\mu_j$ 与 N 点重合 + +$$ +\sum\limits_{i=0}^{n}|\boldsymbol{\mu}_j-\sigma_j^i|= x +$$ + +$\mu_j$ 为平均数, $|\mu_j - N|=\frac{x}{n}$ + +$$ +\sum\limits_{i=0}^{n}|\boldsymbol{\beta}_j-\sigma_j^i|=x+(n-2)\frac{x}{n} +$$ + +所以 + +$$ +\frac{\sum\limits_{i=0}^{n}|\boldsymbol{\beta}_j-\sigma_j^i|}{\sum\limits_{i=0}^{n}|\boldsymbol{\mu}_j-\sigma_j^i|}=1+\frac{n-2}{n}\leq2 +$$ + +得证 + +## T2 + +## T2.1 + +$\boldsymbol{S}=(7,8,-10,-5)^T$ + +## T2.2 + +$\boldsymbol{S^{(2)}}=(37,-7,-17,-5)^T$ + diff --git a/排序问题/res/t1.png b/排序问题/res/t1.png new file mode 100644 index 0000000000000000000000000000000000000000..3865660969e5b4ef68f44b75806eaf50fcc3886d GIT binary patch literal 2643 zcmd5;`#03x8vi1fVoFghx0LI7$<%8?NM%yFi$#>nl)EsBaT$zDE+wJS>lVg!Oy1ED zNiz(3E6PmcGL;cCcypPdG49h~IOF_q&OdP0I%lu7pS_>Y^X%ud_kPyi>&d$2conoy zc^?1(ARB8d7XT1risr#GQX=`vK~GgQh()m z);FR6;1FWBi?IUL<3t$S#>&z)&WFo%3ig^-?p+FU8P%abs{KvU!BlGfqBvc5Hgnzg`Xb*YI1fGwdfL9{>ZD{OJ# z*4`>vAVoVJ3;>1lBmnS${;xo-?U*>8vyLN15xI>-3y|npj6z80iX(tEJ`Y}=NMohW zz8<#`2Bh^L&a|HIm^W^@$=){R_m;GZ}S?%Dw_zm;AX zx);&tzUL=W7>(w2DiC*)phxlDVUO9rO>$lKwgI?X|O?uauc?MB9~VjiIg=+e)Y1%t~4_Xi9V zdUJ`CljboeeF6vhH|Zd^g68%E-QTCi$qg-D%(iO&;xUUkRB(D=Rb8ia;hts z{_Jmz_xrg*&#{E4{<4Dtu1TW|qedTjs$)Wmm%wxD$#jOc$+4SXg4Lz#YZFFpz0ENp znWW7x1Ta6yHHdqErZXwKjre3@ zsNV}sgnJ1q8PA`b+aDlA`I!bDh46oB#Bg4zbleHE*Ft-YQ8hJE4=;}!)|wYZOFC89 zVtWfxcF7P8X|%4;5vP6*Xi7wuKQQq%2t5DzTO;o3RIfTxPRkY=duF~-rQkHhZEHA5 z^q~H-KMDYLGTs1i@&rN*Xi6VPq#A*XC^oh`?Ru4a5Q!yqKN|3Oig_CLG~M7O!ptBo zEp3Rw(BdaYd)b2<3)SW9$^6{Yv?Ksd;;OMQqyo zuO&<3WM-MF*%>VfJBz#6`v*zdx1?$_!$aqvyOZz0!J6ZVkmtkaO(XpuO2C{(K4_@0-(g7MIClzUy0@GJhLQ&5M4 zW9hSJfBqVQPihZz@L@-UYn3Zk81Y|{L_$@ltgO5Y@hay?4Z4x(xE<{Z)Xery?T0P3 z)}eStl3rLvsV?*iEi(Ldf01P=CXm z<;8aiwJ%;6B&)9T(qCNV2K}1<&l=4z3;V973XoaEzfA05;@F2*kQ4e%{DPSmO=Dz0xew9 zW_4}N;MgY*^wQG3pcDu15f6E-Qtn-;2%2DqG3S;Nyv?6{u*{+2WU%qmF_iwf+?@xeCVqgk6;ANqe0)o;4cKiMc< zZt~Z+nsxoD3rAl^FhnP6?A%Zfxnrg}$KP*s9`gi)ImY2~KdV0wn%|yx-c;HaQ@^7r?ZJKldW$8zxkZa%`9t)P)#3Z_`FEC3l}IZ-${#DU{j+)50&~6Rg!>zc37uaU)`uZXs-bDg z)0vsU?(uU}o?2KfRCP6`3N0b+-$(40%+VnSwT*^6wot?uf3AilGr9J946@h5>FHt) z_d@bEhgEZAYNfw@<-qR~)ep|HFUh#XvVLREe?t~vIe%43f&C6RvoE1;-gFTt{ao{J zVJ;8DOvQ&El)(2E>xy&8BoIhAOpS}q+~l^UHN(p@qZZpPA8**I5g>H8#>yUVFJ0Zv za?KcWq}v4)t(4S<8N(zVVShA`$(~>z#o6r9Yum~=%IElS`e=G2@!%j=JfhRIPr@Tn zhIwcZ=VGz^vC`EY=P%p<*Vm6=lI3dm+X~BTT3RRqdj@PgWpVZXS!3fFn*UQ98`DXB zH@AFZqG__v{JAHR`fg`w`@Rn2re5~OvHK%M@eI?Jf8;?T=Ug4(WFu|!nA&9uBysan zJ?9twR5G1T7X(bvXpTKZ#y#$*p}T)0!&>tHqXhY{6+jsucf@d>sl(q+E98j^ PLBQs+qZI*q>;69hRV4<_ literal 0 HcmV?d00001 diff --git a/排序问题/res/t2.png b/排序问题/res/t2.png new file mode 100644 index 0000000000000000000000000000000000000000..3d0335dd0a7c5d6aa7a7274e7b72411b9a4c6948 GIT binary patch literal 3720 zcmeHKXIoR*77hWa14IN1EfE#RK~Z2Rp+*G*G68g?2q7RiD1<5{NRfmlih^E85TwW` zK?evR2101ktVlpgfCOU@iJ=6LkkB(H_}s7exj$gOoU@*_&ffdotGsKS8!k>rMR|33 z5D28`U~hXC1d^Zw@?BYJ@$>t4Dc~g$c@}vBMC#Gxf1}Le zWVAzDN)`e`;1T}d@>HyUjC0{&kf78uBo&^LB#WG)LB8Cn)Pn5(7P$1T&uF>WpC%58H?7tId<_jr6pm$7wPsP)cnwJ{0?*DAWr4w+G9VhhdE1nrSZn!v6C)7+hQY4dwK`L(n+1+p+zzI~dGv|KvZ$p}L}2)5E0EdUXM zr93;x{rAn-qh9Y*v0<2qaobL>FuR$N9Sg($k;TnUD7&VGQ=RkS*(oCh0NnS7UDB|g zHiqzqd2c%4()-*FI}eLTE`q|7#yCWraTU8XiEsnUjOO zad^ad?LDXl2QEiX#VS^rXXgC9ubJPa5?W0n1nmGzJ%@Pwj4l^8FLF3T_s(S&GU+~5 zA?@V!{u9vodTVliM()u@4q5scN*40Q*0&@a^_7g#z9tyS5Pn)Lg-0`V(T(;;`0Q(q zWv;nd-s&WajbJ4HmZ~yJni(bh=>0!9?U4+Dph%C(S>^XX zUKqpu9>EBS_aCDc)`X6c9uyVb)N?0}4?L@(u+yK#@*0|Jj*c139tSpWm-_g&BiF~N zb7I2w$-Q)>_K4a;Q{}79Udh%MPwY?MowQhCdntg$V>41rcm(Rq++di9|FsIIu zS|?5Qj<7wK2Ono!o7e7^Y~h%tX&zGxOP-ZtE{5)?f8f}5XmhbkkRBz*a# zI3;5!rF1Bz$VukRn9^?_BRO-9^^&!rGMMSFS6H}e%8D#82y6J7 zGfa)f79OGxo7jlfBhgK<5p5}x*W&0rIEps+?u9{y#Ga?rU)(t%R)uB-mT(oCO_=vv zzu&S63Un@)yBiA49&2=doo~G428P>H}Gfp&CsWcEq=D`mX25;Av$wX973hd28g znxx4s8+d(!&5C+}vjEL-KG(4%=geg+7nM!QZIV{GWf7a}R^Bwv@|8Q*0_xI`=#|3J zrrxEgKTIEgwVo%%gU}q?$9E++`}+DujOclJc_YO`NmZK(OH+fb>%tXAR9xs%V^GD{ z&Eh6`x_NQE8#yaG>=v87R1@*#)mMGi^e&)HDuiP8mS$%|Gudomkk5TboZb5JmsWov zuP|-q0{jpCm1N?|1q%^vEZMRzW|X+Pu`Zc5B$*7L*U|T0y;Mm<_b+B?o;P00n@(5M zAiT}-WTY@v8dB11AGtn zstdl6i-`+#U}-5qtpTTuZ5|2SiwSm+%7TiVD||#%qjMpL&}C&dH$oV7haD1A_&Sc- zB=~0{?EvQ5)x*4I;bppG2lx3~lqsq!&&ZljE5QYMBoJ6hqK$k#*Jvf}Re0>P$lmip zUxV9&JpA+G(dhx#OAo2!GIfiUL3p4&O*B(W^SknWe5Mc*tsQu#UkjH2rCBci_WE8< zYouE3vlKSf-F^+e_gQUHbX+g>10o$ul?kR5^^L5dp|QSfhgf#I?UQiwSxS$HShpA9 z+pz6ph?%pcfx2iJDco^=$|-}7ii#Hwc`+N9YwjPXl`m3=(xv#sZzXpr32?vZA<3i$ zR&=P^*yShT#tWzOdx8!iV2Q_jge)X8DTz}{uWC`h(a5{`pr8Vm<5*`Z=9FK2m9@~-{D0!M+NfUG1nV)07@}s-Fme| zckWvFf=oK1{9=`6v{ zzvOey_$63jTkz@}cdfENAFtX3@acpe481se`<=AMBl+Q|^!(6y=j^Fl0)C*+Wy6=h zcaV1J04W{#r=g6qZ*cJ8kI8?kR&%d}%QItgZ^}5o(%$jL2SJ%+?Nd{#M#v9utO;rI z%fxDlBH5T`Gl22Y?^P}B!GmDpi|V)Mr(w~bT5|MZXveuntL_yDGX79ugPTxKeprnW zYEjoNl{zhteL8l25BQJ=5{VQwX`+vWpVmtg&N4W9({868&UVQI)2SY%A*K>dcwuO^ zEol1P=OMMuL%vo(J9GztFf{HfK1G=OqjCw1Bffv{G%I!N0>>Z@W{l~dKu~t}>DnpX zM0mc8ni;OiVu|vTh242ZwZhgOk#yPdi08#pS9U>24LJKmahq}}z=t0p{UNU_ZcI#I zO5$mqg^EX9)q^l}CN?iw+}L0^z_an!2ezi@E+7=+o