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<title>MathJax Example</title> |
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<p> $\mathcal{E}_{L}^{(0)}$ </p> |
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<p> $\Delta({\text{UNMARK\_LEFT}})=L_{i}\times\bigg{(}-(S_{LM}-L_{i})-S_{LU}-S_{RM}% |
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+S_{RU}\bigg{)}=L_{i}\times\bigg{(}L_{i}-S_{L}+S_{R}\bigg{)}<0$ </p> |
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<p> $|\overline{V_{m}}|=n-2|E_{m}|=n-2k$ </p> |
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<p> $S_{LU}$ </p> |
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<p> $\mathcal{E}_{L}^{(0)}=n_{L}\times\big{(}\sum_{j=1}^{k}(k+1-j)\;w_{j}\big{)}$ </p> |
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<p> $k\leq\frac{n}{2}$ </p> |
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<p> $e_{i}$ </p> |
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<p> $\displaystyle=\mathcal{W}(E^{(v_{i},u_{j})})+\mathcal{W}^{*}(E^{(v_{i},u_{j})}% |
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)+w^{*}_{k}$ </p> |
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<p> $S_{RU}$ </p> |
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<p> ${n_{L}}{n_{R}}(i(j-1)-ij)={n_{L}}{n_{R}}\times(-i)$ </p> |
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<p> $w(e^{*})$ </p> |
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<p> $S_{R}^{\prime}=R_{2}\geq S_{L}^{\prime}=L_{2}+L_{3}$ </p> |
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<p> $\mathcal{W}(E^{\prime})=\sum_{e\in E^{\prime}\cap\overline{E_{m}}}w(e),\;\;% |
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\mathcal{W}^{*}(E^{\prime})=\sum_{e\in E^{\prime}\cap{E_{m}}}w(e),\;\;\mathcal% |
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{W}^{\prime}(E^{\prime})=\sum_{e_{i}\in\overline{E_{m}}\cap E^{\prime}}% |
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\epsilon_{i}$ </p> |
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<p> $|\Delta E|=n_{L}\times k^{\prime}\times B=(n-(k+k^{\prime}))\times k^{\prime}\times |
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B$ </p> |
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<p> $S_{R}=\sum_{i=1}^{{\mathcal{R}}}R_{i}$ </p> |
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<p> $\mathcal{E}$ </p> |
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<p> $|w_{1}+w^{*}+w_{3}-w_{1}-w^{*}-w_{3}|=0$ </p> |
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<p> $P_{n},n\geq 3$ </p> |
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<p> $\displaystyle>n_{L}\times\big{(}\sum_{j=1}^{k}(k+1-j)w_{j}\big{)}\xrightarrow{% |
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\text{See the proof of Lemma \ref{induction1}}}=\mathcal{E}^{(0)}_{L}>\mathcal% |
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{E}^{(\frac{k}{2})}_{L}$ </p> |
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<p> $e^{*}_{k}=(u_{j},u_{j+1})=e^{*}_{3}=(u_{5},u_{6})$ </p> |
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<p> $n_{R}=0$ </p> |
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<p> $\overline{V_{m}}=\{v^{\prime}_{1},v^{\prime}_{2},v^{\prime}_{3},v^{\prime}_{4}% |
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,v^{\prime}_{5},v^{\prime}_{6}\}$ </p> |
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<p> ${n_{L}}{n_{R}}\times(({\mathcal{L}}-i)({\mathcal{R}}-j+1)-({\mathcal{L}}-i)({% |
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\mathcal{R}}-j))={n_{L}}{n_{R}}\times({\mathcal{L}}{\mathcal{R}}-{\mathcal{L}}% |
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j+{\mathcal{L}}-i{\mathcal{R}}+ij-i-{\mathcal{L}}{\mathcal{R}}+{\mathcal{L}}j+% |
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i{\mathcal{R}}-ij)={n_{L}}{n_{R}}\times({\mathcal{L}}-i)$ </p> |
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<p> $w^{\prime}(e_{i})=w(e_{i})+\epsilon_{i},\;\forall e_{i}\in E,\;\epsilon_{i}\in% |
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\mathbb{R}$ </p> |
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<p> $0<c_{i}<1$ </p> |
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<p> $e_{i}\in E_{R}$ </p> |
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<p> $\displaystyle|\Delta E|\geq$ </p> |
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<p> $\epsilon\leq c_{i}$ </p> |
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<p> $\Delta_{1}({\text{MARK\_LEFT}})=c_{1}\times(X)$ </p> |
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<p> $A,B,C,D,x,y$ </p> |
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<p> $(\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}-i)({n^{2}_{L}}% |
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\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(2j-{\mathcal{R}}))$ </p> |
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<p> $|w_{1}+w_{2}+w_{3}+w^{*}-w_{1}-w_{2}-w_{3}|=w^{*}$ </p> |
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<p> $v_{3}$ </p> |
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<p> $w^{\prime}(e_{i})=w(e_{i})$ </p> |
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<p> $v_{n_{1}+2}$ </p> |
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<p> $e^{*}=(u,v)$ </p> |
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<p> ${e^{*}}$ </p> |
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<p> $e_{i}\in E_{L}$ </p> |
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<p> $S_{RU}=S_{R}$ </p> |
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<p> $n_{L}\times\big{(}(i+1)\;w_{i+1}\big{)}$ </p> |
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<p> $c_{i}=\epsilon_{1}=0.4$ </p> |
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<p> $+L_{i}\times S_{RM}$ </p> |
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<p> $V_{m}=\{u_{1},v_{1}\}$ </p> |
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<p> $R_{i}\times R_{j}\times 2w^{*}$ </p> |
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<p> $\mathcal{E}^{(x<w_{0})}_{L}$ </p> |
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<p> $1\leq j\leq{\mathcal{R}}$ </p> |
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<p> $\mathcal{E}(M^{\prime\prime})=\underbrace{\mathcal{E}(M^{\prime})+c_{1}\times X% |
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}_{=\mathcal{E}(M)}+(c_{2}-c_{1})\times X$ </p> |
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<p> $X=L_{1}\times\bigg{(}S_{LM}+(S_{LU}-L_{1})+S_{RM}-S_{RU}\bigg{)}=L_{1}\times% |
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\bigg{(}S_{LM}+(S_{LU}-L_{1})-S_{R}\bigg{)}$ </p> |
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<p> $|x-A|+|x-A-B|$ </p> |
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<p> $\epsilon_{i}$ </p> |
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<p> ${e^{*}}=(u,v)$ </p> |
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<p> $c_{1}+c_{2}\leq 1$ </p> |
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<p> $|a|=-a$ </p> |
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<p> $v^{\prime}_{3}$ </p> |
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<p> $u_{1},u_{2}\in\overline{V_{m}}$ </p> |
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<p> $v_{j}\in V_{R},v_{j}\neq v_{n_{1}+2}$ </p> |
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<p> $u_{j}$ </p> |
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<p> $A+B+C$ </p> |
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<p> $\pi_{v,u}$ </p> |
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<p> $\pi^{\prime\prime}_{v,u}$ </p> |
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<p> $\Delta({\text{UNMARK\_RIGHT}})\leq 0\text{ if }{n_{L}}({\mathcal{L}}-2i)\leq{n% |
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_{R}}({\mathcal{R}}-1)\xrightarrow[]{\text{Rearranging the terms}}i\geq\frac{{% |
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\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}$ </p> |
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<p> $\epsilon$ </p> |
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<p> $-\left|\sum_{k=i}^{j-1}\epsilon_{k}\right|$ </p> |
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<p> $\displaystyle=\mathcal{W}(E^{(v_{i},u_{j})})+\mathcal{W}^{\prime}(E^{(v_{i},u_% |
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{j})})$ </p> |
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<p> $S_{L}^{\prime}=S_{L}-L_{1}$ </p> |
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<p> $w(e^{*}_{i})$ </p> |
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<p> $e_{0}$ </p> |
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<p> $y=w_{\frac{k}{2}+1}+w_{\frac{k}{2}+2}+\dots+w_{k+1}$ </p> |
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<p> $\pi^{\prime}_{v,u}$ </p> |
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<p> $\mathcal{E}_{LR}=E_{L}=0$ </p> |
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<p> $x-A+x-A-B<B\xrightarrow[]{}x<A+B$ </p> |
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<p> $\epsilon_{2}$ </p> |
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<p> $j\xleftarrow{}j-1$ </p> |
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<p> $\Delta({\text{MARK\_RIGHT}})=R_{i}\times\bigg{(}S_{RM}+(S_{RU}-R_{i})+S_{LM}-S% |
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_{LU}\bigg{)}$ </p> |
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<p> $H=G-\{e_{3}=(v^{\prime}_{3},u_{1}),e^{*}=(u_{1},v_{1}),e_{4}=(v_{1},v^{\prime}% |
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_{4})\}$ </p> |
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<p> $R_{j}$ </p> |
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<p> $S_{RU}<S_{LU}$ </p> |
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<p> $M_{R}^{*}$ </p> |
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<p> $\Delta_{1}({\text{UNMARK\_LEFT}})=L_{i}\times\epsilon\times w^{*}\times(-(S_{L% |
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}-L_{i})+S_{R})=L_{i}\times\epsilon\times w^{*}\times(-S_{L}+L_{i}+S_{R})$ </p> |
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<p> $P_{n}$ </p> |
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<p> $\displaystyle\mathcal{E}_{L}^{(i)}+n_{L}\times\big{(}(i+1)\;w_{i+1}-(k-i)\;w_{% |
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i+1}\big{)}$ </p> |
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<p> $T$ </p> |
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<p> $e^{*}_{k}\in E_{m}$ </p> |
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<p> $G_{2}$ </p> |
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<p> $S_{RU}>S_{LU}$ </p> |
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<p> $\Delta({\text{UNMARK\_RIGHT}})={n^{2}_{R}}(-2(j-1)+2j-{\mathcal{R}}-1)+{n_{L}}% |
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{n_{R}}({\mathcal{L}}-i-i)={n^{2}_{R}}(1-{\mathcal{R}})+{n_{L}}{n_{R}}({% |
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\mathcal{L}}-2i)$ </p> |
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<p> $w_{1}+w_{2}+w_{3}+w^{*}$ </p> |
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<p> $\left|\sum_{k=i}^{j-1}(w_{k}+\epsilon_{k})-\sum_{k=i}^{j-1}w_{k}\right|=\left|% |
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\sum_{k=i}^{j-1}\epsilon_{k}\right|$ </p> |
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<p> $v_{i}=v_{1}$ </p> |
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<p> $n_{L}=\left|\{v|v\in G_{1}\}\right|,n_{R}=\left|\{v|v\in G_{2}\}\right|$ </p> |
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<p> $\displaystyle=\mathcal{W}(E^{(v_{i},u_{j})})+\mathcal{W}^{*}(E^{(v_{i},u_{j})})$ </p> |
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<p> $\mathcal{E}_{L}^{(i+1)}-\mathcal{E}_{L}^{(i)}<0$ </p> |
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<p> $\displaystyle(n_{L}+n_{R})B=(n-2)B=|\overline{V_{m}}|B$ </p> |
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<p> $M_{L}$ </p> |
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<p> $M_{R}^{*}\xleftarrow[]{}M_{R}^{*}\cup\{e_{i}\}$ </p> |
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<p> $S_{LM}$ </p> |
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<p> $e=e_{3}$ </p> |
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<p> $x<A+B$ </p> |
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<p> $d_{G}(u,v)$ </p> |
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<p> $c_{j}=|x-w_{0}-w_{1}-\dots-w_{j}|$ </p> |
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