|
<!DOCTYPE html> |
|
<html> |
|
<head> |
|
<title>MathJax Example</title> |
|
<script> |
|
MathJax = { |
|
tex: { |
|
inlineMath: [['$', '$'], ['\(', '\)']] |
|
}, |
|
svg: { |
|
fontCache: 'global' |
|
} |
|
}; |
|
</script> |
|
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> |
|
</head> |
|
<body> |
|
<p> $c^{\prime}_{i}=c_{j}$ </p> |
|
<p> $\mathcal{W}^{*}(E^{(v_{1},u_{5})})=w^{*}_{1}+w^{*}_{2}$ </p> |
|
<p> $|w_{1}+w^{*}+w_{2}+w^{*}-w_{1}-w_{2}|=2w^{*}$ </p> |
|
<p> $e_{i}\in E_{m},1\leq i\leq k$ </p> |
|
<p> $|w^{*}|$ </p> |
|
<p> $P^{{}^{\prime}}\subseteq P$ </p> |
|
<p> $P^{\prime}\subseteq P$ </p> |
|
<p> $k-i$ </p> |
|
<p> $c_{1}+c_{2}>1$ </p> |
|
<p> $c_{j}>0$ </p> |
|
<p> $e_{1},e_{2}$ </p> |
|
<p> $\displaystyle\pi^{\prime\prime}_{v_{i},u_{j}}$ </p> |
|
<p> $n_{R}(|y-C|+|y-B-C|)$ </p> |
|
<p> $\operatorname{min}(\mathcal{E}(M_{R}^{*}),\mathcal{E}(M_{L}^{*}))=\mathcal{E}(% |
|
M_{L}^{*})$ </p> |
|
<p> $i\xleftarrow{}i$ </p> |
|
<p> $\displaystyle\leq$ </p> |
|
<p> $\mathcal{E}_{L}=n_{L}\times\big{(}\underbrace{|x-w_{0}|}_{\text{between the % |
|
vertices of }V_{L}\text{ and }v_{2}}+\underbrace{|x-w_{0}-w_{1}|}_{\text{% |
|
between the vertices of }V_{L}\text{ and }v_{3}}+\dots+\underbrace{|x-w_{0}-w_% |
|
{1}-\dots-w_{k}|}_{\text{between the vertices of }V_{L}\text{ and }v_{k+2}}% |
|
\big{)}$ </p> |
|
<p> ${n^{2}_{L}}\times{\mathcal{L}}({\mathcal{L}}-1)\leq{n^{2}_{R}}\times{\mathcal{% |
|
R}}({\mathcal{R}}-1)$ </p> |
|
<p> $M_{L}^{*}$ </p> |
|
<p> ${n_{L}}\times{n_{L}}\times 2w^{*}$ </p> |
|
<p> $M^{\prime\prime}$ </p> |
|
<p> $f\in\mathcal{F}=\{{\text{MARK\_LEFT}},{\text{UNMARK\_LEFT}},{\text{MARK\_RIGHT% |
|
}},{\text{UNMARK\_RIGHT}}\}$ </p> |
|
<p> $7-21\leq 2=L_{2}$ </p> |
|
<p> $\displaystyle n_{L}\times\big{(}\sum_{j=0}^{i}j\;w_{j}+\sum_{j=i+1}^{k}(k+1-j)% |
|
\;w_{j}+(i+1)\;w_{i+1}-(k-i)\;w_{i+1}\big{)}$ </p> |
|
<p> $x=w_{0}+w_{1}+\dots+w_{i}$ </p> |
|
<p> $x>w_{0}+\dots+w_{k}$ </p> |
|
<p> $L_{1}\times L_{2}\times w^{*}$ </p> |
|
<p> $e_{2}=(v_{1},v_{4})$ </p> |
|
<p> $e_{1}$ </p> |
|
<p> $H$ </p> |
|
<p> $S_{L}-L_{i}-S_{R}\leq 0\xrightarrow[]{}S_{L}-S_{R}\leq L_{i}$ </p> |
|
<p> $j>\frac{{\mathcal{R}}}{2}+\frac{{n_{L}}(1-{\mathcal{L}})}{2{n_{R}}}$ </p> |
|
<p> $\mathcal{E}(M)=\mathcal{E}(M^{\prime})+\Delta_{1}({\text{MARK\_LEFT}})$ </p> |
|
<p> $w_{i}$ </p> |
|
<p> $u,v$ </p> |
|
<p> $v_{5}$ </p> |
|
<p> $n_{2}=n_{R}$ </p> |
|
<p> $u\in G_{2}$ </p> |
|
<p> $e_{i}\in E$ </p> |
|
<p> $\frac{\mathcal{E}^{(\frac{k}{2})}_{L}}{n_{L}}$ </p> |
|
<p> $\alpha_{2}$ </p> |
|
<p> $w(e^{\prime})\xleftarrow[]{}w(e^{\prime})+w(e)$ </p> |
|
<p> $V_{L}=\{v_{i}|1\leq i\leq n_{1}+1\}$ </p> |
|
<p> $\mathcal{E}_{L}^{(i)}=n_{L}\times\big{(}\sum_{j=0}^{i}j\;w_{j}+\sum_{j=i+1}^{k% |
|
}(k+1-j)\;w_{j}\big{)}$ </p> |
|
<p> $n_{L}\times\big{(}(k-i)\;w_{i+1}\big{)}$ </p> |
|
<p> ${T_{i}^{L}},i\in\{1,2\}$ </p> |
|
<p> $\mathcal{W}^{\prime}(E^{\prime})$ </p> |
|
<p> $e^{\prime}\in E_{m}$ </p> |
|
<p> $\displaystyle n_{L}\times n_{R}\times|x+y-A-B-C|$ </p> |
|
<p> $M^{*}\xleftarrow[]{}\operatorname{argmin}(\mathcal{E}(M_{L}^{*}),\mathcal{E}(M% |
|
_{R}^{*}))$ </p> |
|
<p> $u,v\in G_{2}$ </p> |
|
<p> $\Delta({\text{UNMARK\_LEFT}})=L_{i}\times\bigg{(}-(S_{LM}-L_{i})-S_{LU}-S_{RM}% |
|
+S_{RU}\bigg{)}$ </p> |
|
<p> $\mathcal{E}(M^{\prime\prime})<\mathcal{E}(M)$ </p> |
|
<p> $L_{i}=n_{L},\;1\leq i\leq{\mathcal{L}}$ </p> |
|
<p> $u\in G_{1}$ </p> |
|
<p> $|w^{*}|-\left|\sum_{k=i}^{n_{1}}\epsilon_{k}\right|\leq\left|w^{*}-\sum_{k=i}^% |
|
{n_{1}}\epsilon_{k}\right|$ </p> |
|
<p> $\beta\geq 0$ </p> |
|
<p> ${T_{j}^{R}},j\in\{1,\dots,{\mathcal{R}}\}$ </p> |
|
<p> $\Delta({\text{UNMARK\_LEFT}})$ </p> |
|
<p> $\displaystyle B\times k^{\prime}\times\underbrace{(n_{L}+n_{R})}_{=n-(k+k^{% |
|
\prime})}=B\times k^{\prime}\times(n-(k+k^{\prime}))$ </p> |
|
<p> $P^{\prime}\subset P$ </p> |
|
<p> $0\leq i\leq k$ </p> |
|
<p> $e_{i},e_{j}$ </p> |
|
<p> $C\leq y\leq B+C$ </p> |
|
<p> $c_{i}\neq c_{j}$ </p> |
|
<p> $\alpha_{1}=B$ </p> |
|
<p> $x-A-B$ </p> |
|
<p> $\displaystyle\xrightarrow{\text{setting }x=A+B,y=C}=$ </p> |
|
<p> $v\in V_{s}$ </p> |
|
<p> $\mathcal{E}_{1}$ </p> |
|
<p> $L_{1}$ </p> |
|
<p> $X<0$ </p> |
|
<p> $V_{s}$ </p> |
|
<p> $|w^{*}-(c_{2}\times w^{*}+c_{j}\times w^{*}-\epsilon\times w^{*})|-|w^{*}-(c_{% |
|
2}\times w^{*}+c_{j}\times w^{*})|\leq\epsilon\times w^{*}$ </p> |
|
<p> $e^{*}\in S$ </p> |
|
<p> $c^{\prime}_{2}=c_{2}-\epsilon$ </p> |
|
<p> $f\in\mathcal{F}=\{\text{MARK\_LEFT},\text{UNMARK\_RIGHT}\}$ </p> |
|
<p> $S_{RM}$ </p> |
|
<p> $|w_{1}+w^{*}+w_{3}+w^{*}-w_{1}-w^{*}-w_{3}|=w^{*}$ </p> |
|
<p> $M_{L}^{*}\xleftarrow[]{}M_{L}^{*}\cup\{e_{i}\}$ </p> |
|
<p> $\begin{array}[]{cc}\Delta_{1}({\text{UNMARK\_RIGHT}})\leq R_{1}\times\epsilon% |
|
\times w^{*}\times\bigg{(}-S_{R}^{\prime}\underbrace{-L_{1}}_{<0}+S_{L}^{% |
|
\prime}\bigg{)}&<R_{1}\times\epsilon\times w^{*}\times(\underbrace{-S_{R}^{% |
|
\prime}+S_{L}^{\prime}}_{\leq 0})\leq 0\end{array}$ </p> |
|
<p> $\mathcal{E}^{v_{i},u_{j}}_{1}=\left|\pi_{v_{i},u_{j}}-\pi^{\prime}_{v_{i},u_{j% |
|
}}\right|=\left|\pi^{\prime}_{v_{i},u_{j}}-\pi_{v_{i},u_{j}}\right|=\left|% |
|
\mathcal{W}^{\prime}(E^{(v_{i},u_{j})})-\mathcal{W}^{*}(E^{(v_{i},u_{j})})\right|$ </p> |
|
<p> $\alpha_{1}=x-A+A+B-x<B\xrightarrow{}0<0$ </p> |
|
<p> $(u,v),\;\;u\in V_{m},\;v\in\overline{V_{m}}$ </p> |
|
<p> $w_{1}$ </p> |
|
<p> $|\Delta E|\geq B(n_{L}+n_{R})+n_{L}n_{R}|x+y-A-B-C|$ </p> |
|
<p> $v^{\prime}_{1}$ </p> |
|
<p> $\operatorname{d}_{G^{\prime}}(u,v)\geq\varphi\left(\operatorname{d}_{G}(u,v)\right)$ </p> |
|
<p> $S\xleftarrow[]{}E_{m}$ </p> |
|
<p> $\mathcal{E}^{v_{i},u_{j+1}}_{1}=\left|\pi_{v_{i},u_{j+1}}-\pi^{\prime}_{v_{i},% |
|
u_{j+1}}\right|=\left|\pi_{v_{i},u_{j}}+w^{*}_{k}-\pi^{\prime}_{v_{i},u_{j}}% |
|
\right|=\left|w^{*}_{k}+\mathcal{W}^{*}(E^{(v_{i},u_{j})})-\mathcal{W}^{\prime% |
|
}(E^{(v_{i},u_{j})})\right|$ </p> |
|
<p> $7-21\leq 3=L_{3}$ </p> |
|
<p> ${n^{2}_{R}}\times 2\big{(}{j-1\choose 2}-{j\choose 2}\big{)}={n^{2}_{R}}\times% |
|
(-2(j-1))$ </p> |
|
<p> $\mathcal{E}_{LR}=0$ </p> |
|
<p> $w^{\prime}(e_{i})=w(e_{i})+\epsilon_{i},\epsilon_{i}\in[0,w^{*}]$ </p> |
|
<p> $\displaystyle+$ </p> |
|
<p> $i+1-k+i<0\xrightarrow[]{}2i<k-1\xrightarrow[]{}i<\frac{k}{2}-\frac{1}{2}% |
|
\xrightarrow[\text{since }k\text{ is even}]{}i\leq\frac{k}{2}-1$ </p> |
|
<p> $|x-A|$ </p> |
|
<p> $\alpha_{1}$ </p> |
|
<p> $V_{m}=\{v_{2},v_{3}\}$ </p> |
|
<p> $\Delta({\text{MARK\_LEFT}})={n^{2}_{L}}\times(2i+{\mathcal{L}}-2i-1)+{n_{L}}{n% |
|
_{R}}(j+j-{\mathcal{R}})={n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(2j-% |
|
{\mathcal{R}})$ </p> |
|
|
|
</body> |
|
</html> |
|
|
