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<title>MathJax Example</title> |
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tex: { |
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inlineMath: [['$', '$'], ['\(', '\)']] |
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svg: { |
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fontCache: 'global' |
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<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> |
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<body> |
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<p> $v_{1},v_{2}\in\overline{V_{m}}$ </p> |
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<p> $|\Delta E|^{\prime}=w^{*}$ </p> |
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<p> $1\leq j\leq k$ </p> |
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<p> ${\mathcal{R}}$ </p> |
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<p> $\pi\left(G^{\prime}\right)\leq\pi(G)-d$ </p> |
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<p> $\displaystyle\underbrace{n_{R}\times|y-C|\times k^{\prime}}_{\text{between the% |
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subpath of }w_{2}\text{ and the vertices in }u}+\underbrace{n_{R}\times|y-B-C% |
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|\times k}_{{\text{between the subpath of }w_{2}\text{ and the vertices in }v}}+$ </p> |
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<p> $\operatorname{min}(\mathcal{E}(M_{L}^{*}),\mathcal{E}(M_{R}^{*}))$ </p> |
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<p> $E_{m}=\{e^{*}\}$ </p> |
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<p> $\displaystyle n_{L}\times n_{R}\times|x+y-A-B-C|\xrightarrow{n_{R}=0}$ </p> |
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<p> $P$ </p> |
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<p> $v_{i}\in V_{L}$ </p> |
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<p> $M_{R}^{*}\xleftarrow[]{}\emptyset$ </p> |
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<p> $\alpha_{1}=A-x+x-A-B<B\xrightarrow{}0<2B$ </p> |
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<p> $M_{R}$ </p> |
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<p> $u,v\in V$ </p> |
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<p> $n_{L}=3$ </p> |
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<p> $\left|\sum_{k=i}^{j-2}(w_{k}+\epsilon_{k})-w^{*}-\sum_{k=i}^{j-2}w_{k}\right|=% |
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\left|w^{*}-\sum_{k=i}^{j-2}\epsilon_{k}\right|$ </p> |
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<p> $n_{L}(|x-A|+|x-A-B|)$ </p> |
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<p> $\mathcal{E}_{L}^{(i+1)}-\mathcal{E}_{L}^{(i)}=n_{L}\times\big{(}(i+1)\;w_{i+1}% |
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-(k-i)\;w_{i+1}\big{)}$ </p> |
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<p> $w^{\prime}(e_{i})=w(e_{i})+w(e^{*})$ </p> |
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<p> $S_{R}\xleftarrow{}\sum_{\forall e_{i}\in E_{R}}R_{i}$ </p> |
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<p> $u,v\in G_{1}$ </p> |
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<p> $x\geq A$ </p> |
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<p> $\Delta({\text{MARK\_RIGHT}})$ </p> |
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<p> $\operatorname{min}(\mathcal{E}(M_{R}^{*}),\mathcal{E}(M_{L}^{*}))\leq\mathcal{% |
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E}(M)$ </p> |
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<p> ${\mathcal{L}}-(\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}})% |
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=\frac{{\mathcal{L}}{n_{L}}-{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$ </p> |
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<p> $u_{j+1}$ </p> |
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<p> $T_{j}^{R}$ </p> |
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<p> $\varphi(x)=x/\alpha-\beta$ </p> |
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<p> $\mathcal{E}=\mathcal{E}_{R}$ </p> |
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<p> $S_{R}-S_{L}\leq R_{i}$ </p> |
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<p> $e_{3}$ </p> |
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<p> $a\geq 0$ </p> |
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<p> $V_{L}$ </p> |
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<p> ${n_{L}}$ </p> |
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<p> $n_{R}$ </p> |
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<p> $n_{L}\geq 0$ </p> |
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<p> $c_{2}\geq\epsilon$ </p> |
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<p> ${n^{2}_{L}}\times((i+1)({\mathcal{L}}-i-1)-i({\mathcal{L}}-i))={n^{2}_{L}}% |
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\times(i{\mathcal{L}}-i^{2}-i+{\mathcal{L}}-i-1-i{\mathcal{L}}+i^{2})={n^{2}_{% |
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L}}\times({\mathcal{L}}-2i-1)$ </p> |
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<p> $e^{*}_{k}=(u_{j},u_{j+1})$ </p> |
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<p> $|\Delta E|=|\overline{V_{m}}|(w^{*}_{1}+\dots+w^{*}_{k})=(n-2k)(w^{*}_{1}+% |
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\dots+w^{*}_{k})$ </p> |
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<p> $e_{1},e_{2},\dots,e_{k}$ </p> |
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<p> $-\left|\sum_{k=n_{1}+1}^{j-2}\epsilon_{k}\right|$ </p> |
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<p> $\overline{E_{m}}$ </p> |
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<p> $e^{*}\in E$ </p> |
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<p> $\epsilon_{1}+\epsilon_{2}=B$ </p> |
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<p> $n_{L}+n_{R}=n-2$ </p> |
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<p> $c^{\prime}_{1}+c^{\prime}_{2}=1$ </p> |
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<p> $\mathcal{E}_{L}^{(j)},j\neq\frac{k}{2}$ </p> |
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<p> $L_{i}\geq S_{L}-S_{R}$ </p> |
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<p> $j>0$ </p> |
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<p> $\Delta_{1}({\text{MARK\_LEFT}})$ </p> |
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<p> $L_{i}\times R_{j}\times w^{*}$ </p> |
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<p> $e_{j}\neq e_{1}$ </p> |
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<p> $x-A$ </p> |
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<p> $j\leq\frac{{\mathcal{R}}}{2}+\frac{{n_{L}}(1-{\mathcal{L}})}{2{n_{R}}}$ </p> |
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<p> $|a|=a$ </p> |
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<p> $2\times L_{i}\times(S_{LM})$ </p> |
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<p> $n_{L}+n_{R}=|\overline{V_{m}}|=n-(k+k^{\prime})$ </p> |
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<p> $M_{L}^{*}\xleftarrow[]{}\emptyset$ </p> |
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<p> $\frac{\mathcal{E}_{L}}{n_{L}}$ </p> |
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<p> $\displaystyle(\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}})(% |
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{n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}({\mathcal{R}}))+\frac{{% |
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\mathcal{L}}{n_{L}}-{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}({n^{2}_{L}}\times({% |
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\mathcal{L}}-1)+{n_{L}}{n_{R}}(-{\mathcal{R}}))$ </p> |
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<p> $\displaystyle\mathcal{E}_{L}^{(0)}=n_{L}\times\big{(}w_{1}+w_{1}+w_{2}+w_{1}+w% |
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_{2}+w_{3}+\dots+w_{1}+w_{2}+\dots+w_{k}\big{)}$ </p> |
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<p> $\mathcal{W}^{\prime}(E^{(v_{1},u_{5})})=\epsilon_{1}+\epsilon_{2}+\epsilon_{2}% |
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+\epsilon_{3}+\epsilon_{4}$ </p> |
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<p> $v_{i}\in\overline{V_{m}}$ </p> |
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<p> $e_{4}=(v_{2},v_{6})$ </p> |
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<p> $u\in\overline{V_{m}}$ </p> |
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<p> ${T_{i}^{L}},i\in\{1,\dots,{\mathcal{L}}\}$ </p> |
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<p> $\operatorname{CONTRACTION}$ </p> |
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<p> ${{\mathcal{L}}\choose{2}}\times n_{L}\times n_{L}\times 2w^{*}={n^{2}_{L}}% |
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\times{\mathcal{L}}({\mathcal{L}}-1)\times w^{*}$ </p> |
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<p> $\displaystyle n_{L}(|x-A|+|x-A-B|)+n_{R}(|y-C|+|y-B-C|)+n_{L}n_{R}|x+y-A-B-C|$ </p> |
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<p> $e_{3}=(v_{2},v_{5})$ </p> |
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<p> $\mathcal{E}(M_{R}^{*})=\underbrace{\mathcal{E}(M_{0})}_{\text{The error % |
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associated with the empty marking}}+\underbrace{\sum_{e_{i}\in E_{R}}R_{i}% |
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\times w^{*}\times(S_{R}-R_{i}-S_{L})}_{\text{The sum of all }\Delta({\text{% |
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MARK\_RIGHT}})\text{'s that |
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transform }M_{0}\text{ into }M_{R}^{*}}$ </p> |
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<p> ${\mathcal{L}}=2$ </p> |
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<p> $\displaystyle\geq\mathcal{E}(M_{0})+\epsilon_{1}\times\sum_{e_{i}\in E_{L}}L_{% |
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i}\times w^{*}\times(S_{L}-L_{i}-S_{R})+\epsilon_{2}\times\sum_{e_{i}\in E_{L}% |
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}L_{i}\times w^{*}\times(S_{L}-L_{i}-S_{R})$ </p> |
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<p> $\mathcal{E}_{LR}=n_{L}\times n_{R}\times|x+y-w_{0}-w_{1}-\dots-w_{k+1}|$ </p> |
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<p> $w_{j}$ </p> |
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<p> $\displaystyle 0$ </p> |
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<p> $\Delta({\text{MARK\_LEFT}})$ </p> |
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<p> $j\xleftarrow{}j$ </p> |
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<p> $M_{0}$ </p> |
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<p> $x=A+\epsilon_{1}$ </p> |
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<p> $\Delta_{v_{i},u_{j}}+\Delta_{v_{i},u_{j+1}}\leq\left|w^{*}_{k}-\mathcal{W}^{% |
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\prime}(E^{(v_{i},u_{j})})+\mathcal{W}^{*}(E^{(v_{i},u_{j})})\right|-\left|w^{% |
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*}_{k}+\mathcal{W}^{*}(E^{(v_{i},u_{j})})-\mathcal{W}^{\prime}(E^{(v_{i},u_{j}% |
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)})\right|\leq 0$ </p> |
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<p> $L_{i}=|\{v|v\in{T_{i}^{L}}\}|$ </p> |
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<p> $R_{j}=|\{v|v\in{T_{j}^{R}}\}|$ </p> |
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<p> $E^{\prime}=E-e^{*}$ </p> |
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<p> $e^{\prime}=e_{1}$ </p> |
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<p> $n_{L}$ </p> |
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<p> $E_{R}=\{(v,w)|(v,w)\in E,w\neq u\}$ </p> |
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<p> $e^{*}_{k}=e^{*}_{3}=(u_{5},u_{6})$ </p> |
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<p> $(S_{LM}+(S_{LU}-L_{i})+S_{RM}-S_{RU})\leq 0\xrightarrow[]{S_{LM}+S_{LU}=S_{L}}% |
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S_{L}-L_{i}+S_{RM}-S_{RU}\leq 0$ </p> |
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<p> $S_{R}^{\prime}=S_{R}-R_{1}$ </p> |
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<p> $\mathcal{E}_{1}=|x-w_{0}|+\dots+|w-w_{0}-\dots-w_{k}|$ </p> |
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<p> $S_{LM}>0$ </p> |
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<p> $\Delta({\text{MARK\_LEFT}})\leq 0\text{ if }{n_{L}}\times({\mathcal{L}}-1)\leq% |
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{n_{R}}({\mathcal{R}}-2j)\xrightarrow[]{\text{Rearranging the terms}}j\leq% |
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\frac{{\mathcal{R}}}{2}+\frac{{n_{L}}(1-{\mathcal{L}})}{2{n_{R}}}$ </p> |
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<p> $\displaystyle\underbrace{\epsilon_{2}\times\sum_{e_{i}\in E_{R}}R_{i}\times w^% |
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{*}\times(S_{R}-R_{i}-S_{L})}_{\text{The sum of all }\Delta({\text{MARK\_RIGHT% |
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}})\text{'s by }\epsilon_{2}}$ </p> |
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<p> $k\geq k^{\prime}$ </p> |
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<p> $c^{\prime}_{i}=0$ </p> |
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<p> $w_{i+1}$ </p> |
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<p> $(v_{1},v_{4})$ </p> |
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