From 19b684c2b8a4cfd91fe649694c493a649528380a Mon Sep 17 00:00:00 2001 From: cast1e Date: Sun, 6 Oct 2024 08:41:53 +0800 Subject: [PATCH] compl t1 --- 排序问题/report.md | 70 +++++++++++++++++++++++++++++++++++++------- 排序问题/report.pdf | Bin 0 -> 115827 bytes 2 files changed, 60 insertions(+), 10 deletions(-) create mode 100644 排序问题/report.pdf diff --git a/排序问题/report.md b/排序问题/report.md index 6cd1b66..fead24c 100644 --- a/排序问题/report.md +++ b/排序问题/report.md @@ -34,7 +34,7 @@ $$ 可知常数 C 越小,比值越大 -假设 $n-1$ 个点重合,点 $A$ 距离重合点距离为 $x$ +假设 $n-1$ 个点重合,排序 $\sigma_j^i = N$ 点 $A$ 距离 $\sigma_j^i$ 距离为 $x$ ![](./res/t2.png) @@ -53,22 +53,72 @@ $$ 所以 $$ -\frac{\sum\limits_{i=0}^{n}|\boldsymbol{\beta}_j-\sigma_j^i|}{\sum\limits_{i=0}^{n}|\boldsymbol{\mu}_j-\sigma_j^i|}=1+\frac{n-2}{n}\leq2 +\frac{\sum\limits_{i=0}^{n}|\boldsymbol{\beta}_j-\sigma_j^i|}{\sum\limits_{i=0}^{n}|\boldsymbol{\mu}_j-\sigma_j^i|}=1+\frac{n-2}{n}\leq 2 +$$ + +得证 + +### T1.3 + +$$ +d(\boldsymbol{\sigma^{\prime}},\boldsymbol{\Sigma}) = \sum\limits_{i=1}^{k}L_1(\boldsymbol{\sigma^{\prime}},\boldsymbol{\sigma})\leq \sum\limits_{i=1}^{k}L_1(\boldsymbol{\sigma^{\prime}},\boldsymbol{\mu}) + \sum\limits_{i=1}^{k}L_1(\boldsymbol{\mu},\boldsymbol{\sigma}) +$$ + +$$ +\leq \sum\limits_{i=1}^{k}L_1(\boldsymbol{\sigma}^*,\boldsymbol{\mu}) + \sum\limits_{i=1}^{k}L_1(\boldsymbol{\mu},\boldsymbol{\sigma}) +$$ + +$$ +\leq \sum\limits_{i=1}^{k}L_1(\boldsymbol{\sigma}^*,\boldsymbol{\sigma}) + \sum\limits_{i=1}^{k}L_1(\boldsymbol{\sigma},\boldsymbol{\mu}) + \sum\limits_{i=1}^{k}L_1(\boldsymbol{\mu},\boldsymbol{\sigma}) +$$ + +$$ += \sum\limits_{i=1}^{k}L_1(\boldsymbol{\sigma}^*,\boldsymbol{\sigma}) + 2\sum\limits_{i=1}^{k}L_1(\boldsymbol{\mu},\boldsymbol{\sigma}) +$$ + +$$ += d(\boldsymbol{\sigma}^*,\boldsymbol{\Sigma}) + 2d(\boldsymbol{\mu},\boldsymbol{\Sigma}) +$$ + +$$ +\leq 3d(\boldsymbol{\sigma}^*,\boldsymbol{\Sigma}) +$$ + +同理,由T1.2知 + +$$ +\sum\limits_{i=0}^{n}|\boldsymbol{\beta}_j-\sigma_j^i|\leq 2 \sum\limits_{i=0}^{n}|\boldsymbol{\mu}_j-\sigma_j^i| +$$ + +$$ +d(\boldsymbol{\sigma^{\prime}},\boldsymbol{\Sigma}) = \sum\limits_{i=1}^{k}L_1(\boldsymbol{\sigma^{\prime}},\boldsymbol{\sigma})\leq \sum\limits_{i=1}^{k}L_1(\boldsymbol{\sigma^{\prime}},\boldsymbol{\beta}) + \sum\limits_{i=1}^{k}L_1(\boldsymbol{\beta},\boldsymbol{\sigma}) +$$ + +$$ +\leq \sum\limits_{i=1}^{k}L_1(\boldsymbol{\sigma}^*,\boldsymbol{\sigma}) + 2\sum\limits_{i=1}^{k}L_1(\boldsymbol{\beta},\boldsymbol{\sigma}) +$$ + +$$ +\leq \sum\limits_{i=1}^{k}L_1(\boldsymbol{\sigma}^*,\boldsymbol{\sigma}) + 4\sum\limits_{i=1}^{k}L_1(\boldsymbol{\mu},\boldsymbol{\sigma}) +$$ + +$$ +\leq 5d(\boldsymbol{\sigma}^*,\boldsymbol{\Sigma}) $$ 得证 ## T2 -## T2.1 +### T2.1 $\boldsymbol{S}=(7,8,-10,-5)^T$ -## T2.2 +### T2.2 $\boldsymbol{S^{(2)}}=(37,-7,-17,-5)^T$ -## T2.3 +### T2.3 $$ s^i = \sum\limits_{j\in T_i}q_{ij} @@ -91,27 +141,27 @@ $$ 如此求出每个 $s^{(2)}_i$,即可得到 $\boldsymbol{S}^{(2)}$ $$ -\boldsymbol{S}^{(2)}=(\boldsymbol{M}+l)\boldsymbol{S} +\boldsymbol{S}^{(2)}=(\boldsymbol{M}+l*\boldsymbol{E})\boldsymbol{S} $$ 对于 $\boldsymbol{M}^2$ 若 $m^2_{ik}\neq 0$ ,则必然 $\exist j$ 使得 $m_{ij}m_{jk} = 1$所以 $m^2_{ik}$ 的值即为 $j$ 的个数,即为 $i$ 与 $k$ 之间发生的间接比赛的场数 所以 $\boldsymbol{M}^2$ 保存了所有间接比赛的场数 -## 2.4 +### T2.4 我们易知 $$ -\boldsymbol{S}^{(3)}=(\boldsymbol{M}+l)\boldsymbol{S}^{(2)} +\boldsymbol{S}^{(3)}=\boldsymbol{M}\boldsymbol{S}^{(2)}+l^2\boldsymbol{S} $$ $$ -\boldsymbol{S}^{(n)}=(\boldsymbol{M}+l)\boldsymbol{S}^{(n-1)} +\boldsymbol{S}^{(n)}=\boldsymbol{M}\boldsymbol{S}^{(n-1)}+l^{n-1}\boldsymbol{S} $$ 所以递推得到 $$ -\boldsymbol{S}^{(n)}=(\boldsymbol{M}+l)^{n-1}\boldsymbol{S} +\boldsymbol{S}^{(n)}=\boldsymbol{M}^{n-1}\boldsymbol{S} + (\boldsymbol{M}^{n-1}-l^{n-1}\boldsymbol{E})(\boldsymbol{M}-l\boldsymbol{E})^{-1}l\boldsymbol{S} $$ \ No newline at end of file diff --git a/排序问题/report.pdf b/排序问题/report.pdf new file mode 100644 index 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