|
<!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> $w^{*}_{i}$ </p> |
|
<p> ${T_{j}^{R}},j\in\{1,2\}$ </p> |
|
<p> $G=(V,E)$ </p> |
|
<p> $\displaystyle(k-k^{\prime})\times n_{L}\times|x-A|+n_{L}\times k^{\prime}% |
|
\times\big{(}|x-A|+|x-A-B|\big{)}\xrightarrow{\text{no edge weights are % |
|
changed, set }x=A}$ </p> |
|
<p> $E_{L}$ </p> |
|
<p> $\displaystyle(\underbrace{\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}% |
|
{2{n_{L}}}-i}_{\leq\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L% |
|
}}}})(\underbrace{{n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(2j-{% |
|
\mathcal{R}})}_{\leq{n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}({% |
|
\mathcal{R}})})+j\underbrace{({n^{2}_{R}}(1-{\mathcal{R}})+{n_{L}}{n_{R}}({% |
|
\mathcal{L}}-2\times\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{% |
|
L}}})}_{=0})$ </p> |
|
<p> $x=\sum_{j=0}^{i}w_{j}$ </p> |
|
<p> $y=C+\epsilon_{2}$ </p> |
|
<p> $E$ </p> |
|
<p> $z$ </p> |
|
<p> $\overline{E_{m}}=E-E_{m}$ </p> |
|
<p> $\displaystyle\xrightarrow{0\leq x<w_{0}}$ </p> |
|
<p> $\epsilon_{1}+\epsilon_{2}\leq 1$ </p> |
|
<p> $C_{L}$ </p> |
|
<p> $(v,u)$ </p> |
|
<p> $w(e)$ </p> |
|
<p> $-L_{i}\times S_{RU}$ </p> |
|
<p> $V_{R}=\{w|(v_{2},w)\in E^{\prime}\},{\mathcal{R}}=|V_{R}|$ </p> |
|
<p> $v^{\prime}_{4}$ </p> |
|
<p> $S_{L}\xleftarrow{}\sum_{\forall e_{i}\in E_{L}}L_{i}$ </p> |
|
<p> $\alpha_{2}\geq B$ </p> |
|
<p> $w^{\prime}(e_{i})=w(e_{i})+\epsilon_{i}$ </p> |
|
<p> $e^{*}=(v_{n_{1}+1},v_{n_{1}+2})$ </p> |
|
<p> $i$ </p> |
|
<p> $j=0$ </p> |
|
<p> $w_{2}$ </p> |
|
<p> $\displaystyle=\mathcal{E}(M_{0})+\epsilon_{1}\times\sum_{e_{i}\in E_{L}}L_{i}% |
|
\times w^{*}\times(S_{L}-L_{i}-S_{R})+\epsilon_{2}\times\sum_{e_{i}\in E_{R}}R% |
|
_{i}\times w^{*}\times(S_{R}-R_{i}-S_{L})$ </p> |
|
<p> $S_{RU}\geq S_{LU}$ </p> |
|
<p> $c^{\prime}_{i}=c_{i}-\epsilon$ </p> |
|
<p> $n_{L}=n-(k+k^{\prime})$ </p> |
|
<p> $T_{i}^{R}$ </p> |
|
<p> $|\Delta E|=n_{L}(|x-A|+|x-A-B|)+n_{R}(|y-C|+|y-B-C|)+n_{L}n_{R}|x+y-A-B-C|=n_{% |
|
L}\alpha_{1}+n_{R}\alpha_{2}+n_{L}n_{R}|x+y-A-B-C|$ </p> |
|
<p> $\epsilon_{n_{1}}=w^{*}$ </p> |
|
<p> $\displaystyle\geq$ </p> |
|
<p> $|\Delta E|=\sum_{u\in V_{m},v\in\overline{V_{m}},\text{ or }u,v\in\overline{V_% |
|
{m}},u\neq v}\left|d_{G}(u,v)-d_{G^{\prime}}(u,v)\right|$ </p> |
|
<p> $n_{2}+1$ </p> |
|
<p> $7-21\leq 2=L_{1}$ </p> |
|
<p> $z=w^{*}$ </p> |
|
<p> $x=\sum_{j=0}^{i+1}w_{j}$ </p> |
|
<p> $M^{\prime}$ </p> |
|
<p> $\displaystyle\Delta=$ </p> |
|
<p> $u\in G_{1},v\in G_{2}$ </p> |
|
<p> $\displaystyle|\Delta E|^{\prime}=$ </p> |
|
<p> $x+y$ </p> |
|
<p> $u_{1}\in T_{1}^{L},u_{2}\in T_{1}^{R}$ </p> |
|
<p> $S_{L}=S_{LU}+S_{LM}$ </p> |
|
<p> $S_{R}=S_{RU}$ </p> |
|
<p> $\mathcal{E}_{R}$ </p> |
|
<p> $w:E\xrightarrow[]{}\mathbb{R}_{\geq 0}$ </p> |
|
<p> $c_{j}=\epsilon_{2}=0.5$ </p> |
|
<p> $0.1$ </p> |
|
<p> $e^{*}=(v_{2},v_{3})$ </p> |
|
<p> ${\mathcal{L}}\choose 2$ </p> |
|
<p> $\mathcal{E}(M^{\prime\prime})=\mathcal{E}(M^{\prime})+\Delta_{2}({\text{MARK\_% |
|
LEFT}})=\underbrace{\mathcal{E}(M^{\prime})+(c_{1})\times X}_{=\mathcal{E}(M^{% |
|
\prime})+\Delta_{1}({\text{MARK\_LEFT}})=\mathcal{E}(M)}+(1-c_{1})\times X<% |
|
\mathcal{E}(M)$ </p> |
|
<p> $|w^{*}|-\left|w^{*}-\sum_{k=n_{1}+1}^{j-2}\epsilon_{k}\right|$ </p> |
|
<p> $V_{R}=\{v_{5},v_{6}\}$ </p> |
|
<p> $u\in V_{L}$ </p> |
|
<p> $L_{i}$ </p> |
|
<p> $0<c_{i}\leq 1$ </p> |
|
<p> $\overline{V_{m}}=V-\{v_{1},v_{2}\}$ </p> |
|
<p> $(n-2k)w^{*}_{k}$ </p> |
|
<p> $u_{1}\in T_{1}^{R},u_{2}\in T_{2}^{R}$ </p> |
|
<p> $\alpha_{1}\geq B$ </p> |
|
<p> $e$ </p> |
|
<p> $L_{1}\times L_{2}\times 2w^{*}$ </p> |
|
<p> $\mathcal{E}(M_{L}^{*})=\underbrace{\mathcal{E}(M_{0})}_{\text{The error % |
|
associated with the empty marking}}+\underbrace{\sum_{e_{i}\in E_{L}}L_{i}% |
|
\times w^{*}\times(S_{L}-L_{i}-S_{R})}_{\text{The sum of all }\Delta({\text{% |
|
MARK\_LEFT}})\text{'s that |
|
transform }M_{0}\text{ into }M_{L}^{*}}$ </p> |
|
<p> $\mathcal{E}=\mathcal{E}_{L}+\mathcal{E}_{R}+\mathcal{E}_{LR}$ </p> |
|
<p> $\epsilon_{i}\neq 0$ </p> |
|
<p> $\epsilon\times w^{*}$ </p> |
|
<p> $\pi:\mathcal{G}\xrightarrow{}\mathbb{R}$ </p> |
|
<p> $j$ </p> |
|
<p> $y=B+C$ </p> |
|
<p> $S_{L}-S_{R}>L_{i}$ </p> |
|
<p> $21-7\leq 20=R_{1}$ </p> |
|
<p> $\mathcal{E}(.)$ </p> |
|
<p> $e^{*}_{k}$ </p> |
|
<p> ${\mathcal{L}}={\mathcal{R}}=2$ </p> |
|
<p> $\displaystyle(k-k^{\prime})\times n_{R}\times|y-B-C|+n_{R}\times k^{\prime}% |
|
\times\big{(}|y-C|+|y-B-C|\big{)}$ </p> |
|
<p> $w(e_{i})+\epsilon_{i}$ </p> |
|
<p> $w^{\prime}(e_{3})=w_{3}+w^{*}$ </p> |
|
<p> $c_{i}=\epsilon_{1}$ </p> |
|
<p> $0\leq\epsilon_{2}\leq w^{*}$ </p> |
|
<p> $x=A+B$ </p> |
|
<p> $T_{i}={t^{1}_{1},\dots,t^{1}_{k}}$ </p> |
|
<p> $X=\{x_{1},x_{2},...,x_{n}\}$ </p> |
|
<p> $T_{i}\sim p_{\theta}(t^{1}_{1\dots k}\mid x_{1}\dots,x_{n})$ </p> |
|
<p> $t^{i}_{j}*\sim p^{vote}_{\theta}(t^{i}_{j}*|B)$ </p> |
|
<p> $X$ </p> |
|
<p> $n-1$ </p> |
|
<p> $V(p_{\theta},T_{i})(t^{i}_{j})=1[t^{i}_{j}=t^{i}_{j}*]$ </p> |
|
<p> $n_{i}\in N$ </p> |
|
<p> $t^{n}_{1}\sim t^{n}_{k}$ </p> |
|
<p> $G=F(X)$ </p> |
|
<p> $X=\{x_{1},x_{2},...,x_{m}\}$ </p> |
|
<p> $t^{i-1}_{j}$ </p> |
|
<p> $x_{i}$ </p> |
|
<p> $t^{i}_{j}$ </p> |
|
<p> $F$ </p> |
|
<p> $G=(N,E)$ </p> |
|
<p> $\{t^{i}_{1},\dots,t^{i}_{k}\}\sim p_{\theta}(t^{i}_{1\dots k}\mid x_{1\dots n}% |
|
,t^{i-1}_{j})$ </p> |
|
|
|
</body> |
|
</html> |
|
|
