output_mathjax_example_4.html

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    <title>MathJax Example</title>
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        <p> $E^{(v_{1},u_{5})}=\{e_{1},e^{*}_{1},e_{2},e_{3},e^{*}_{2},e_{4}\}$ </p>
    <p> $x=\sum_{k=i}^{n_{1}}\epsilon_{k}$ </p>
    <p> $v_{i},v_{j}\in V_{R}$ </p>
    <p> $G^{\prime}$ </p>
    <p> $c_{i+1},\dots,c_{k}$ </p>
    <p> $S_{R}-R_{i}-S_{L}\leq 0$ </p>
    <p> $n\geq 3$ </p>
    <p> $\underbrace{|w^{*}-(c_{i}\times w^{*}+c_{j}\times w^{*}-\epsilon\times w^{*})|%
}_{=w^{*}-(c_{i}\times w^{*}+c_{j}\times w^{*}-\epsilon\times w^{*})\text{ %
because }c_{i}+c_{j}\leq 1}-\underbrace{|w^{*}-(c_{i}\times w^{*}+c_{j}\times w%
^{*})|}_{=w^{*}-(c_{i}\times w^{*}+c_{j}\times w^{*})\text{ because }c_{i}+c_{%
j}\leq 1}=\epsilon\times w^{*}$ </p>
    <p> $w_{i}=w(e_{i})\;\forall i\in\{0,\dots,k+1\}$ </p>
    <p> $u_{1}\in T_{1}^{L},u_{2}\in T_{2}^{L}$ </p>
    <p> $\pi$ </p>
    <p> $\displaystyle\mathcal{E}_{L}^{(i+1)}=$ </p>
    <p> $i\geq\frac{{\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$ </p>
    <p> $\displaystyle\mathcal{E}(M)=$ </p>
    <p> $n_{L}n_{R}|x+y-A-B-C|$ </p>
    <p> $V^{\prime}\subset V$ </p>
    <p> $R_{i}={n_{R}},\;1\leq i\leq{\mathcal{R}}$ </p>
    <p> $\pi_{v_{i},u_{j+1}}=\pi_{v_{i},u_{j}}+w^{*}_{k}\text{, }\pi^{\prime}_{v_{i},u_%
{j}}=\pi^{\prime}_{v_{i},u_{j+1}}\text{, and }\pi^{\prime\prime}_{v_{i},u_{j}}%
=\pi^{\prime\prime}_{v_{i},u_{j+1}}$ </p>
    <p> $u_{1}\in T_{i}^{L},u_{2}\in T_{j}^{L}$ </p>
    <p> $\epsilon=0.4$ </p>
    <p> $e\in E_{m}$ </p>
    <p> $c_{i}$ </p>
    <p> $e_{i},w_{i}=w(e_{i})$ </p>
    <p> $\displaystyle\mathcal{E}^{(x<w_{0})}_{L}$ </p>
    <p> $V_{m}=\{u_{1},\dots,u_{2k}\}$ </p>
    <p> $\displaystyle\underbrace{n_{L}\times n_{R}\times|x+y-A-B-C|}_{\text{between %
the subpath of }w_{1}\text{ and }w_{2}}$ </p>
    <p> $\mathcal{E}^{v_{i},u_{j}}_{2}=\left|\pi^{\prime\prime}_{v_{i},u_{j}}-\pi_{v_{i%
},u_{j}}\right|=\left|w^{*}_{k}\right|$ </p>
    <p> $c_{i}>0$ </p>
    <p> $d$ </p>
    <p> $x<A$ </p>
    <p> $\frac{\mathcal{E}^{(\frac{k}{2})}_{L}}{n_{L}}-\frac{\mathcal{E}_{L}}{n_{L}}\leq
0$ </p>
    <p> $\mathcal{E}^{v_{i},u_{j+1}}_{2}=0$ </p>
    <p> $\displaystyle\xrightarrow[]{}\mathcal{E}(M)\geq\mathcal{E}(M_{L}^{*})$ </p>
    <p> $|x+y-A-B-C|$ </p>
    <p> $E^{(u,v)}$ </p>
    <p> $x=w_{0}+w_{1}+\dots+w_{\frac{k}{2}}$ </p>
    <p> $w_{1}+w_{2}+w_{3}$ </p>
    <p> $j\xleftarrow[]{}0$ </p>
    <p> $S_{R}=S_{RU}+S_{RM}$ </p>
    <p> $v^{*}$ </p>
    <p> $c_{1}+c_{2}=0.6+0.8=1.4>1$ </p>
    <p> $S_{L}=\sum_{i=1}^{{\mathcal{L}}}L_{i}$ </p>
    <p> $c_{1}+c_{2}=1+\epsilon$ </p>
    <p> $w^{\prime}(e_{i})=w(e_{i})+w^{*}$ </p>
    <p> $\{e_{1},e_{3}\}$ </p>
    <p> $G^{{}^{\prime}}$ </p>
    <p> $i\xleftarrow{}i+1$ </p>
    <p> $\Delta(f)$ </p>
    <p> $\displaystyle n_{L}\times k^{\prime}\times\underbrace{\big{(}|x-A|+|x-A-B|\big%
{)}}_{\geq B\text{ (Lemma \ref{l12})}}+n_{R}\times k^{\prime}\times\underbrace%
{\big{(}|y-C|+|y-B-C|\big{)}}_{\geq B\text{ (Lemma \ref{l12})}}$ </p>
    <p> $i<\frac{{\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$ </p>
    <p> $\epsilon_{2}\in\{0,w^{*}\}$ </p>
    <p> $\displaystyle\mathcal{E}(M)$ </p>
    <p> $|V|=n$ </p>
    <p> $V_{L}=\{u|(u,v_{1})\in E^{\prime}\},{\mathcal{L}}=|V_{L}|$ </p>
    <p> $e_{k+1}$ </p>
    <p> $|x-A-B|$ </p>
    <p> $n_{L}+n_{R}=n-2=\left|\overline{V_{m}}\right|$ </p>
    <p> $T_{i}^{L}$ </p>
    <p> $x,z$ </p>
    <p> $i={\mathcal{L}}$ </p>
    <p> $\epsilon>0$ </p>
    <p> $G_{1}$ </p>
    <p> $v_{1}$ </p>
    <p> $E^{\prime}\subseteq E$ </p>
    <p> $S_{R}^{\prime}\geq S_{L}^{\prime}$ </p>
    <p> $M$ </p>
    <p> $e^{*}$ </p>
    <p> $u$ </p>
    <p> $u_{1}\in T_{i}^{R},u_{2}\in T_{j}^{R}$ </p>
    <p> $y$ </p>
    <p> $\displaystyle|\Delta E|=$ </p>
    <p> $v_{i}$ </p>
    <p> $V_{L},V_{R}\subset V$ </p>
    <p> $\displaystyle=\mathcal{E}(M_{0})+(\epsilon_{1}+\epsilon_{2})\times\sum_{e_{i}%
\in E_{L}}L_{i}\times w^{*}\times(S_{L}-L_{i}-S_{R})\xrightarrow[S_{L}-L_{i}-S%
_{R}\leq 0]{\epsilon_{1}+\epsilon_{2}\leq 1}$ </p>
    <p> $\{u_{1},v_{1}\}$ </p>
    <p> $\Delta({\text{UNMARK\_RIGHT}})=R_{i}\times\bigg{(}-(S_{RM}-R_{i})-S_{RU}-S_{LM%
}+S_{LU}\bigg{)}$ </p>
    <p> $(u,v)$ </p>
    <p> $\alpha_{1}=|x-A|+|x-A-B|$ </p>
    <p> $|z|\leq|x|+|z-x|\xrightarrow[]{}|z|-|x|\leq|z-x|$ </p>
    <p> $B<0$ </p>
    <p> $n_{R}\geq 0$ </p>
    <p> $21-7>1=R_{2}$ </p>
    <p> $v^{*}=\{v_{2},v_{3},\dots,v_{k+2}\}$ </p>
    <p> $w:E\rightarrow\mathbb{R}_{\geq 0}$ </p>
    <p> $e\in E$ </p>
    <p> $i=\{1,\dots,{\mathcal{L}}\}$ </p>
    <p> $\Delta_{1}({\text{MARK\_LEFT}})<0$ </p>
    <p> $0-\left|\sum_{k=i}^{j-1}\epsilon_{k}\right|$ </p>
    <p> $u,v\in G$ </p>
    <p> $\mathcal{E}_{R}=n_{R}\times\big{(}\underbrace{|y-w_{k+1}|}_{\text{between the %
vertices of }V_{R}\text{ and }v_{k+2}}+\underbrace{|y-w_{k+1}-w_{k}|}_{\text{%
between the vertices of }V_{R}\text{ and }v_{k+1}}+\dots+\underbrace{|y-w_{k+1%
}-w_{k}-\dots-w_{1}|}_{\text{between the vertices of }V_{R}\text{ and }v_{2}}%
\big{)}$ </p>
    <p> $V_{m}=\{v_{1},v_{2}\}$ </p>
    <p> $R_{i}\times R_{j}\times w^{*}$ </p>
    <p> $L_{i}\times L_{j}\times w^{*}$ </p>
    <p> $\displaystyle\underbrace{\mathcal{E}(M_{0})}_{\text{The error associated with %
the empty marking}}+\underbrace{\epsilon_{1}\times\sum_{e_{i}\in E_{L}}L_{i}%
\times w^{*}\times(S_{L}-L_{i}-S_{R})}_{\text{The sum of all }\Delta({\text{%
MARK\_LEFT}})\text{'s by
}\epsilon_{1}}$ </p>
    <p> $(u_{2i-1},u_{2i})$ </p>
    <p> $V_{R}=\{v_{i}|n_{1}+2\leq i\leq n_{2}+2\}$ </p>
    <p> $E_{R}$ </p>
    <p> $w:E\Rightarrow\mathbb{R}_{\geq 0}$ </p>
    <p> $x=A$ </p>
    <p> $\epsilon_{i}=0\;\;\forall i\neq n_{1}$ </p>

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