Datasets:
Hyeonsieun
/
TeX_1st

Dataset card Data Studio Files
xet
 Community
1
TeX
stringlengths
8
256
e_{1},e_{2},\dots,e_{k}
-\left|\sum_{k=n_{1}+1}^{j-2}\epsilon_{k}\right|
\overline{E_{m}}
e^{*}\in E
\epsilon_{1}+\epsilon_{2}=B
n_{L}+n_{R}=n-2
c^{\prime}_{1}+c^{\prime}_{2}=1
\mathcal{E}_{L}^{(j)},j\neq\frac{k}{2}
L_{i}\geq S_{L}-S_{R}
\Delta_{1}({\text{MARK\_LEFT}})
L_{i}\times R_{j}\times w^{*}
e_{j}\neq e_{1}
j\leq\frac{{\mathcal{R}}}{2}+\frac{{n_{L}}(1-{\mathcal{L}})}{2{n_{R}}}
2\times L_{i}\times(S_{LM})
n_{L}+n_{R}=|\overline{V_{m}}|=n-(k+k^{\prime})
M_{L}^{*}\xleftarrow[]{}\emptyset
\frac{\mathcal{E}_{L}}{n_{L}}
\displaystyle\mathcal{E}_{L}^{(0)}=n_{L}\times\big{(}w_{1}+w_{1}+w_{2}+w_{1}+w% _{2}+w_{3}+\dots+w_{1}+w_{2}+\dots+w_{k}\big{)}
\mathcal{W}^{\prime}(E^{(v_{1},u_{5})})=\epsilon_{1}+\epsilon_{2}+\epsilon_{2}% +\epsilon_{3}+\epsilon_{4}
v_{i}\in\overline{V_{m}}
e_{4}=(v_{2},v_{6})
u\in\overline{V_{m}}
{T_{i}^{L}},i\in\{1,\dots,{\mathcal{L}}\}
\operatorname{CONTRACTION}
{{\mathcal{L}}\choose{2}}\times n_{L}\times n_{L}\times 2w^{*}={n^{2}_{L}}% \times{\mathcal{L}}({\mathcal{L}}-1)\times w^{*}
\displaystyle n_{L}(|x-A|+|x-A-B|)+n_{R}(|y-C|+|y-B-C|)+n_{L}n_{R}|x+y-A-B-C|
e_{3}=(v_{2},v_{5})
{\mathcal{L}}=2
\displaystyle\geq\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_{L}% }L_{i}\times w^{*}\times(S_{L}-L_{i}-S_{R})
\mathcal{E}_{LR}=n_{L}\times n_{R}\times|x+y-w_{0}-w_{1}-\dots-w_{k+1}|
\displaystyle 0
\Delta({\text{MARK\_LEFT}})
j\xleftarrow{}j
x=A+\epsilon_{1}
\Delta_{v_{i},u_{j}}+\Delta_{v_{i},u_{j+1}}\leq\left|w^{*}_{k}-\mathcal{W}^{% \prime}(E^{(v_{i},u_{j})})+\mathcal{W}^{*}(E^{(v_{i},u_{j})})\right|-\left|w^{% *}_{k}+\mathcal{W}^{*}(E^{(v_{i},u_{j})})-\mathcal{W}^{\prime}(E^{(v_{i},u_{j}% )})\right|\leq 0
L_{i}=|\{v|v\in{T_{i}^{L}}\}|
R_{j}=|\{v|v\in{T_{j}^{R}}\}|
E^{\prime}=E-e^{*}
e^{\prime}=e_{1}
E_{R}=\{(v,w)|(v,w)\in E,w\neq u\}
e^{*}_{k}=e^{*}_{3}=(u_{5},u_{6})
(S_{LM}+(S_{LU}-L_{i})+S_{RM}-S_{RU})\leq 0\xrightarrow[]{S_{LM}+S_{LU}=S_{L}}% S_{L}-L_{i}+S_{RM}-S_{RU}\leq 0
S_{R}^{\prime}=S_{R}-R_{1}
\mathcal{E}_{1}=|x-w_{0}|+\dots+|w-w_{0}-\dots-w_{k}|
S_{LM}>0
\Delta({\text{MARK\_LEFT}})\leq 0\text{ if }{n_{L}}\times({\mathcal{L}}-1)\leq% {n_{R}}({\mathcal{R}}-2j)\xrightarrow[]{\text{Rearranging the terms}}j\leq% \frac{{\mathcal{R}}}{2}+\frac{{n_{L}}(1-{\mathcal{L}})}{2{n_{R}}}
\displaystyle\underbrace{\epsilon_{2}\times\sum_{e_{i}\in E_{R}}R_{i}\times w^% {*}\times(S_{R}-R_{i}-S_{L})}_{\text{The sum of all }\Delta({\text{MARK\_RIGHT% }})\text{'s by }\epsilon_{2}}
k\geq k^{\prime}
c^{\prime}_{i}=0
(v_{1},v_{4})
c^{\prime}_{i}=c_{j}
\mathcal{W}^{*}(E^{(v_{1},u_{5})})=w^{*}_{1}+w^{*}_{2}
|w_{1}+w^{*}+w_{2}+w^{*}-w_{1}-w_{2}|=2w^{*}
e_{i}\in E_{m},1\leq i\leq k
P^{{}^{\prime}}\subseteq P
P^{\prime}\subseteq P
c_{1}+c_{2}>1
e_{1},e_{2}
\displaystyle\pi^{\prime\prime}_{v_{i},u_{j}}
n_{R}(|y-C|+|y-B-C|)
\operatorname{min}(\mathcal{E}(M_{R}^{*}),\mathcal{E}(M_{L}^{*}))=\mathcal{E}(% M_{L}^{*})
i\xleftarrow{}i
\displaystyle\leq
{n^{2}_{L}}\times{\mathcal{L}}({\mathcal{L}}-1)\leq{n^{2}_{R}}\times{\mathcal{% R}}({\mathcal{R}}-1)
M_{L}^{*}
{n_{L}}\times{n_{L}}\times 2w^{*}
M^{\prime\prime}
f\in\mathcal{F}=\{{\text{MARK\_LEFT}},{\text{UNMARK\_LEFT}},{\text{MARK\_RIGHT% }},{\text{UNMARK\_RIGHT}}\}
7-21\leq 2=L_{2}
\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{)}
x=w_{0}+w_{1}+\dots+w_{i}
x>w_{0}+\dots+w_{k}
L_{1}\times L_{2}\times w^{*}
e_{2}=(v_{1},v_{4})
S_{L}-L_{i}-S_{R}\leq 0\xrightarrow[]{}S_{L}-S_{R}\leq L_{i}
j>\frac{{\mathcal{R}}}{2}+\frac{{n_{L}}(1-{\mathcal{L}})}{2{n_{R}}}
\mathcal{E}(M)=\mathcal{E}(M^{\prime})+\Delta_{1}({\text{MARK\_LEFT}})
n_{2}=n_{R}
u\in G_{2}
e_{i}\in E
\frac{\mathcal{E}^{(\frac{k}{2})}_{L}}{n_{L}}
\alpha_{2}
w(e^{\prime})\xleftarrow[]{}w(e^{\prime})+w(e)
V_{L}=\{v_{i}|1\leq i\leq n_{1}+1\}
\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{)}
n_{L}\times\big{(}(k-i)\;w_{i+1}\big{)}
{T_{i}^{L}},i\in\{1,2\}
\mathcal{W}^{\prime}(E^{\prime})
e^{\prime}\in E_{m}
\displaystyle n_{L}\times n_{R}\times|x+y-A-B-C|
M^{*}\xleftarrow[]{}\operatorname{argmin}(\mathcal{E}(M_{L}^{*}),\mathcal{E}(M% _{R}^{*}))
u,v\in G_{2}
\Delta({\text{UNMARK\_LEFT}})=L_{i}\times\bigg{(}-(S_{LM}-L_{i})-S_{LU}-S_{RM}% +S_{RU}\bigg{)}
\mathcal{E}(M^{\prime\prime})<\mathcal{E}(M)
L_{i}=n_{L},\;1\leq i\leq{\mathcal{L}}
u\in G_{1}
|w^{*}|-\left|\sum_{k=i}^{n_{1}}\epsilon_{k}\right|\leq\left|w^{*}-\sum_{k=i}^% {n_{1}}\epsilon_{k}\right|
\beta\geq 0
{T_{j}^{R}},j\in\{1,\dots,{\mathcal{R}}\}
\Delta({\text{UNMARK\_LEFT}})