Kyudan/arXiv_latex · Datasets at Hugging Face

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

E^{(v_{1},u_{5})}=\{e_{1},e^{*}_{1},e_{2},e_{3},e^{*}_{2},e_{4}\}

x=\sum_{k=i}^{n_{1}}\epsilon_{k}

v_{i},v_{j}\in V_{R}

G^{\prime}

c_{i+1},\dots,c_{k}

S_{R}-R_{i}-S_{L}\leq 0

n\geq 3

\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^{*}

w_{i}=w(e_{i})\;\forall i\in\{0,\dots,k+1\}

u_{1}\in T_{1}^{L},u_{2}\in T_{2}^{L}

\pi

\displaystyle\mathcal{E}_{L}^{(i+1)}=

i\geq\frac{{\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}

\displaystyle\mathcal{E}(M)=

n_{L}n_{R}|x+y-A-B-C|

V^{\prime}\subset V

R_{i}={n_{R}},\;1\leq i\leq{\mathcal{R}}

\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}}

u_{1}\in T_{i}^{L},u_{2}\in T_{j}^{L}

\epsilon=0.4

e\in E_{m}

c_{i}

e_{i},w_{i}=w(e_{i})

\displaystyle\mathcal{E}^{(x<w_{0})}_{L}

V_{m}=\{u_{1},\dots,u_{2k}\}

\displaystyle\underbrace{n_{L}\times n_{R}\times|x+y-A-B-C|}_{\text{between % the subpath of }w_{1}\text{ and }w_{2}}

\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|

c_{i}>0

d

x<A

\frac{\mathcal{E}^{(\frac{k}{2})}_{L}}{n_{L}}-\frac{\mathcal{E}_{L}}{n_{L}}\leq 0

\mathcal{E}^{v_{i},u_{j+1}}_{2}=0

\displaystyle\xrightarrow[]{}\mathcal{E}(M)\geq\mathcal{E}(M_{L}^{*})

|x+y-A-B-C|

E^{(u,v)}

x=w_{0}+w_{1}+\dots+w_{\frac{k}{2}}

w_{1}+w_{2}+w_{3}

j\xleftarrow[]{}0

S_{R}=S_{RU}+S_{RM}

v^{*}

c_{1}+c_{2}=0.6+0.8=1.4>1

S_{L}=\sum_{i=1}^{{\mathcal{L}}}L_{i}

c_{1}+c_{2}=1+\epsilon

w^{\prime}(e_{i})=w(e_{i})+w^{*}

\{e_{1},e_{3}\}

G^{{}^{\prime}}

i\xleftarrow{}i+1

\Delta(f)

\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})}}

i<\frac{{\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}

\epsilon_{2}\in\{0,w^{*}\}

\displaystyle\mathcal{E}(M)

|V|=n

V_{L}=\{u|(u,v_{1})\in E^{\prime}\},{\mathcal{L}}=|V_{L}|

e_{k+1}

|x-A-B|

n_{L}+n_{R}=n-2=\left|\overline{V_{m}}\right|

T_{i}^{L}

x,z

i={\mathcal{L}}

\epsilon>0

G_{1}

v_{1}

E^{\prime}\subseteq E

S_{R}^{\prime}\geq S_{L}^{\prime}

M

e^{*}

u

u_{1}\in T_{i}^{R},u_{2}\in T_{j}^{R}

y

\displaystyle|\Delta E|=

v_{i}

V_{L},V_{R}\subset V

\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}

\{u_{1},v_{1}\}

\Delta({\text{UNMARK\_RIGHT}})=R_{i}\times\bigg{(}-(S_{RM}-R_{i})-S_{RU}-S_{LM% }+S_{LU}\bigg{)}

(u,v)

\alpha_{1}=|x-A|+|x-A-B|

|z|\leq|x|+|z-x|\xrightarrow[]{}|z|-|x|\leq|z-x|

B<0

n_{R}\geq 0

21-7>1=R_{2}

v^{*}=\{v_{2},v_{3},\dots,v_{k+2}\}

w:E\rightarrow\mathbb{R}_{\geq 0}

e\in E

i=\{1,\dots,{\mathcal{L}}\}

\Delta_{1}({\text{MARK\_LEFT}})<0

0-\left|\sum_{k=i}^{j-1}\epsilon_{k}\right|

u,v\in G

\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{)}

V_{m}=\{v_{1},v_{2}\}

R_{i}\times R_{j}\times w^{*}

L_{i}\times L_{j}\times w^{*}

\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}}

(u_{2i-1},u_{2i})

V_{R}=\{v_{i}|n_{1}+2\leq i\leq n_{2}+2\}

E_{R}

w:E\Rightarrow\mathbb{R}_{\geq 0}

x=A

\epsilon_{i}=0\;\;\forall i\neq n_{1}