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500
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stringclasses 10
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dict | extracted_raw_output
stringclasses 1
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stringclasses 9
values | extracted_error
stringclasses 1
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d s ^ { 2 } = ( 1 - { \frac { q c o s \theta } { r } } ) ^ { \frac { 2 } { 1 + \alpha ^ { 2 } } } \lbrace d r ^ { 2 } + r ^ { 2 } d \theta ^ { 2 } + r ^ { 2 } s i n ^ { 2 } \theta d \varphi ^ { 2 } \rbrace - { \frac { d t ^ { 2 } } { ( 1 - { \frac { q c o s \theta } { r } } ) ^ { \frac { 2 } { 1 + \alpha ^ { 2 } } } } } .
|
{
"error": null,
"forumala": "ds^2 = (1 - \\frac{q\\cos\\theta}{r})^{\\frac{2}{1+a^2}}\\{dr^2 + r^2d\\theta^2 + r^2\\sin^2\\theta d\\varphi^2\\} - \\frac{dt^2}{(1 - \\frac{q\\cos\\theta}{r})^{\\frac{2}{1+a^2}}}.\\\"",
"raw_output": null
}
| null |
ds^2 = (1 - \frac{q\cos\theta}{r})^{\frac{2}{1+a^2}}\{dr^2 + r^2d\theta^2 + r^2\sin^2\theta d\varphi^2\} - \frac{dt^2}{(1 - \frac{q\cos\theta}{r})^{\frac{2}{1+a^2}}}.\"
| null |
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\widetilde \gamma _ { \mathrm { h o p f } } \simeq \sum _ { n > 0 } \widetilde { G } _ { n } { \frac { ( - a ) ^ { n } } { 2 ^ { 2 n - 1 } } }
|
{
"error": null,
"forumala": "\\tilde{\\gamma}_{\\text{hopf}} \\simeq \\sum_{n>0} \\tilde{G}_n \\frac{(-a)^n}{2^{2n-1}}",
"raw_output": null
}
| null |
\tilde{\gamma}_{\text{hopf}} \simeq \sum_{n>0} \tilde{G}_n \frac{(-a)^n}{2^{2n-1}}
| null |
|
\rho _ { L } ( q ) = \sum _ { m = 1 } ^ { L } \ P _ { L } ( m ) \ { \frac { 1 } { q ^ { m - 1 } } } .
|
{
"error": null,
"forumala": "ρ_L(q) = \\sum_{m=1}^{L} P_L(m) \\frac{1}{q^{m-1}}.",
"raw_output": null
}
| null |
ρ_L(q) = \sum_{m=1}^{L} P_L(m) \frac{1}{q^{m-1}}.
| null |
|
{ \frac { d ^ { 2 } \varphi } { d \tau ^ { 2 } } } = \sin \varphi \ ,
|
{
"error": null,
"forumala": "d^2φ/dτ^2 = sinφ",
"raw_output": null
}
| null |
d^2φ/dτ^2 = sinφ
| null |
|
| t _ { i , j } | ^ { 2 } \leq t _ { i , i } t _ { j , j } , t _ { i , i } > 0 \forall i , j
|
{
"error": null,
"forumala": "|t_{i,j}|^2 \\leq t_{i,i}t_{j,j}, t_{i,i} > 0 \\forall i,j",
"raw_output": null
}
| null |
|t_{i,j}|^2 \leq t_{i,i}t_{j,j}, t_{i,i} > 0 \forall i,j
| null |
|
s + t + u = - \frac { 8 } { \alpha ^ { \prime } }
|
{
"error": null,
"forumala": "s + t + u = -\\frac{8}{\\alpha'}",
"raw_output": null
}
| null |
s + t + u = -\frac{8}{\alpha'}
| null |
|
m _ { \phi _ { 0 } } ^ { 2 } \sim { \frac { m ^ { 2 } v ^ { 2 } } { \Lambda ^ { 2 } } } .
|
{
"error": "Failed to parse extraction result",
"forumala": null,
"raw_output": "{\"forumala\": \"m_{\\phi_0}^2 \\sim \\frac{m^2 v^2}{\\Lambda^2}\"}"
}
|
{"forumala": "m_{\phi_0}^2 \sim \frac{m^2 v^2}{\Lambda^2}"}
| null |
Failed to parse extraction result
|
|
( \partial b ) ^ { * } = - q ^ { 2 } ( D ^ { - 1 } ) \left( - q ^ { 3 } c d ( \partial a ) + q a d ( \partial c ) + q ^ { 2 } c ^ { 2 } ( \partial b ) - a c ( \partial d ) \right) ,
|
{
"error": null,
"forumala": "∂b* = −q²(D⁻¹)(−q³cd(∂a) + qad(∂c) + q²c²(∂b) − ac(∂d))",
"raw_output": null
}
| null |
∂b* = −q²(D⁻¹)(−q³cd(∂a) + qad(∂c) + q²c²(∂b) − ac(∂d))
| null |
|
d _ { k } > \frac { 9 } { ( k + 2 ) ^ { 2 } } ( \bar { \varphi } \varphi ) _ { 0 } ^ { \frac { 4 } { 3 } } ( \bar { F } F ) _ { 0 } ^ { - 1 } d _ { k - 1 } \ .
|
{
"error": null,
"forumala": "d_k > \\frac{9}{(k+2)^2}(\\bar{\\varphi}\\varphi)_0^{\\frac{4}{3}}(\\bar{F}F)_0^{-1}d_{k-1}.",
"raw_output": null
}
| null |
d_k > \frac{9}{(k+2)^2}(\bar{\varphi}\varphi)_0^{\frac{4}{3}}(\bar{F}F)_0^{-1}d_{k-1}.
| null |
|
\varepsilon ^ { \mu \nu \alpha \beta } \partial _ { \nu } B _ { \alpha \beta } = 0 .
|
{
"error": null,
"forumala": "ε^μναβ∂νBαβ = 0",
"raw_output": null
}
| null |
ε^μναβ∂νBαβ = 0
| null |
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