暑假隨便做做3

2023-08-04 20:40 作者:臥煙鈴 0人讀過(guò) | 我要投稿

赫赫，今天也是閑的一批

1.(西南大學(xué))

$%E8%AE%BEV%E6%98%AFn%E7%BB%B4%E6%AC%A7%E5%BC%8F%E7%A9%BA%E9%97%B4%2C%5Csigma%E6%98%AFV%E7%9A%84%E4%B8%80%E4%B8%AA%E6%AD%A3%E4%BA%A4%E5%8F%98%E6%8D%A2$

$V_1%3D%5C%7B%5Calpha%5Cin%20V%7C%5Csigma(%5Calpha)%3D%5Calpha%5C%7D%2CV_2%3D%5C%7B%5Calpha-%5Csigma(%5Calpha)%7C%5Calpha%20%5Cin%20V%5C%7D$

$1)%E8%AF%81%E6%98%8E%EF%BC%9AV_1%2CV_2%E9%83%BD%E6%98%AFV%E7%9A%84%E5%AD%90%E7%A9%BA%E9%97%B4$

$2)%E8%AF%81%E6%98%8E%EF%BC%9AV_1%5Coplus%20V_2%3DV$

證：

$%5Csigma(0)%3D0%2C0%3D0-%5Csigma(0)$

$%E6%89%80%E4%BB%A50%5Cin%20V_1%2C0%5Cin%20V_2$

$%5Cforall%20%5Calpha%2C%5Cbeta%5Cin%20V_1%2C%5Csigma(%5Calpha)%2B%5Csigma(%5Cbeta)%3D%5Csigma(%5Calpha%2B%5Cbeta)%3D%5Calpha%2B%5Cbeta%5Cin%20V_1$

$%5Cforall%20%5Calpha%2C%5Cbeta%5Cin%20V_2%2C%5Cexists%5Calpha_1%2C%5Cbeta_1%2Cs.t.%5Cbeta%3D%5Cbeta_1-%5Csigma(%5Cbeta_1)$

$%5Calpha%3D%5Calpha_1-%5Csigma(%5Calpha_1)$

$%5Calpha%2B%5Cbeta%3D%5Calpha_1-%5Csigma(%5Calpha_1)%2B%5Cbeta_1-%5Csigma(%5Cbeta_1)$

$%5Calpha%2B%5Cbeta%3D%5Calpha_1%2B%5Cbeta_1-%5Csigma(%5Calpha_1%2B%5Cbeta_1)$

$%5Calpha%2B%5Cbeta%5Cin%20V_2$

2) $%5Cforall%20%5Calpha%5Cin%20V_1%2C%5Csigma(%5Calpha)-%5Calpha%3D0$

$%E8%80%8C%5Csigma%E6%98%AF%E6%AD%A3%E4%BA%A4%E5%8F%98%E6%8D%A2%EF%BC%8C%E8%AE%BE%5Csigma%E7%9A%84%E7%89%B9%E5%BE%81%E5%80%BC%E4%B8%BA%5Clambda_1%2C%5Ccdots%2C%5Clambda_n$

$%E4%B8%94%E5%AF%B9%E5%BA%94%E7%9A%84%E7%89%B9%E5%BE%81%E5%AD%90%E7%A9%BA%E9%97%B4%E5%88%86%E5%88%AB%E4%B8%BAV_%7B%5Clambda_1%7D%2C%5Ccdots%2CV_%7B%5Clambda_n%7D$

$%5Cforall%20%5Calpha%5Cin%20V_%7B%5Clambda_i%7D%2C%5Csigma(%5Calpha)%3D%5Clambda_i%5Calpha$

$%E8%80%8C%3C%5Csigma%5Calpha%2C%5Csigma%5Calpha%3E%3D%3C%5Calpha%2C%5Calpha%3E%E5%8D%B3%5Clambda_i%5E2%3C%5Calpha%2C%5Calpha%3E%3D%3C%5Calpha%2C%5Calpha%3E$

$%E4%BA%8E%E6%98%AF%E6%88%91%E4%BB%AC%E5%BE%97%E5%88%B0%5Cforall%20i%3D1%2C%5Ccdots%2Cn%E6%9C%89%5Clambda_i%5E2%3D1$

$%5Csigma%E7%9A%84%E7%89%B9%E5%BE%81%E5%80%BC%E8%A6%81%E4%B9%88%E4%B8%BA1%EF%BC%8C%E8%A6%81%E4%B9%88%E4%B8%BA-1$

$%E7%94%B1%E5%AE%9A%E4%B9%89%E5%BE%97%E7%9F%A5V_1%E6%98%AF%5Csigma%E7%89%B9%E5%BE%81%E5%80%BC%E4%B8%BA1%E7%9A%84%E7%89%B9%E5%BE%81%E5%AD%90%E7%A9%BA%E9%97%B4$

$%E8%AE%BEV_%7B-1%7D%3D%5C%7B%5Calpha%5Cin%20V%7C%5Csigma(%5Calpha)%3D-%5Calpha%5C%7D$

$%E5%AE%B9%E6%98%93%E7%9F%A5%E9%81%93V_1%5Coplus%20V_%7B-1%7D%3DV$

$%E8%AE%BEv%5Cin%20V%2C%E5%88%99%E5%BF%85%E7%84%B6%E5%AD%98%E5%9C%A8v_1%5Cin%20V_1%2Cv_2%5Cin%20V_%7B-1%7D%E4%BD%BF%E5%BE%97$

$%0Av%3Dv_1%2Bv_2$

$%0A%E8%80%8Cv-%5Csigma(v)%3Dv_1%2Bv_2-v_1%2Bv_2%3D2v_2%5Cin%20V_%7B-1%7D$

$%0AV_2%5Csubset%20V_%7B-1%7D$

$%0A%E8%80%8C%5Cforall%20%5Calpha%5Cin%20V_%7B-1%7D%2C%E6%80%BB%E5%AD%98%E5%9C%A8%E4%B8%80%E4%B8%AA%5Cbeta%5Cin%20V_1$

$%0A%E4%BD%BF%E5%BE%97%5Calpha%3D0.5%5Calpha%2B%5Cbeta-%5Csigma(0.5%5Calpha%2B%5Cbeta)%2C%E4%B8%940.5%5Calpha%2B%5Cbeta%5Cin%20V$

$%0A%5Calpha%3D0.5%5Calpha%2B%5Cbeta-%5Csigma(0.5%5Calpha%2B%5Cbeta)%5Cin%20V_2$

$%0AV_%7B-1%7D%5Csubset%20V_%7B2%7D$

$%E4%BA%8E%E6%98%AFV_1%5Coplus%20V_2%3DV$

2.(西南大學(xué))

$%E8%AF%81%E6%98%8E%3A%5Cforall%20x%2Cy%2Cz%5Cin%20%5Cmathbb%7BR%7D%2CA%2CB%2CC%E6%98%AF%E5%90%8C%E4%B8%AA%E4%B8%89%E8%A7%92%E5%BD%A2%E5%86%85%E7%9A%84%E4%B8%89%E4%B8%AA%E8%A7%92%EF%BC%8C%E5%88%99$

$x%5E2%2By%5E2%2Bz%5E2%5Cgeq%202xy%5Ccos%20A%2B2xz%5Ccos%20B%2B2yz%5Ccos%20C$

證:

$%E8%AE%BEX%3D(x%2Cy%2Cz)%2C$

$%E5%8D%B3%E8%AF%81%3AX%5Cbegin%20%7Bpmatrix%7D%201%26-%5Ccos%20A%20%26-%5Ccos%20B%5C%5C%0A-%5Ccos%20A%26%201%26%20-%5Ccos%20C%5C%5C%0A-%5Ccos%20B%20%26-%5Ccos%20C%20%261%5Cend%20%7Bpmatrix%7DX%5ET%5Cgeq0$

對(duì)其使用合同變換

$%5Cbegin%20%7Bpmatrix%7D%201%26-%5Ccos%20A%20%26-%5Ccos%20B%5C%5C%0A-%5Ccos%20A%26%201%26%20-%5Ccos%20C%5C%5C%0A-%5Ccos%20B%20%26-%5Ccos%20C%20%261%5C%5C%0A1%26%26%0A%5C%5C%261%26%5C%5C%0A%26%261%5Cend%20%7Bpmatrix%7D$

$%5Cbegin%20%7Bpmatrix%7D%201%26%20%26-%5Ccos%20B%5C%5C%0A%26%20%5Csin%5E2%20A%26%20-%5Ccos%20C-%5Ccos%20A%5Ccos%20B%5C%5C%0A-%5Ccos%20B%20%26-%5Ccos%20C-%5Ccos%20A%5Ccos%20B%20%261%5C%5C%0A1%26%5Ccos%20A%26%0A%5C%5C%261%26%5C%5C%0A%26%261%5Cend%20%7Bpmatrix%7D$

$%5Cbegin%20%7Bpmatrix%7D%201%26%20%26%5C%5C%0A%26%20%5Csin%5E2%20A%26%20-%5Ccos%20C-%5Ccos%20A%5Ccos%20B%5C%5C%0A%20%26-%5Ccos%20C-%5Ccos%20A%5Ccos%20B%20%26%5Csin%5E%202B%5C%5C%0A1%26%5Ccos%20A%26%5Ccos%20B%0A%5C%5C%261%26%5C%5C%0A%26%261%5Cend%20%7Bpmatrix%7D$

$%5Cbegin%20%7Bpmatrix%7D%201%26%20%26%5C%5C%0A%26%20%5Csin%5E2%20A%2B%5Cfrac%7B(%5Ccos%20C%2B%5Ccos%20A%5Ccos%20B)%5E2%7D%7B%5Csin%5E2%20B%7D%26%20%5C%5C%0A%20%26%20%26%5Csin%5E%202B%5C%5C%0A1%26%5Ccos%20A%26%5Ccos%20B%0A%5C%5C%261%26%5C%5C%0A%26%5Cfrac%7B(%5Ccos%20C%2B%5Ccos%20A%5Ccos%20B)%7D%7B%5Csin%5E2%20B%7D%261%5Cend%20%7Bpmatrix%7D$

$%5Cbegin%20%7Bpmatrix%7D%201%26%20%26%5C%5C%0A%262%20%5Csin%5E2%20A%26%20%5C%5C%0A%20%26%20%26%5Csin%5E%202B%5C%5C%0A1%26%5Ccos%20A%26%5Ccos%20B%0A%5C%5C%261%26%5C%5C%0A%26%5Cfrac%7B(%5Ccos%20C%2B%5Ccos%20A%5Ccos%20B)%7D%7B%5Csin%5E2%20B%7D%261%5Cend%20%7Bpmatrix%7D$

最后不等式左邊被化簡(jiǎn)為

$u%5E2%2B2v%5E2%5Csin%5E2A%2Bw%5E2%5Csin%5E2B%5Cgeq0$

其中 $%5Cbegin%20%7Bpmatrix%7D%0A1%26%5Ccos%20A%26%5Ccos%20B%0A%5C%5C%261%26%5C%5C%0A%26%5Cfrac%7B(%5Ccos%20C%2B%5Ccos%20A%5Ccos%20B)%7D%7B%5Csin%5E2%20B%7D%261%5Cend%20%7Bpmatrix%7D%5Cbegin%7Bpmatrix%7Dx%5C%5Cy%5C%5Cz%5Cend%7Bpmatrix%7D%3D$