alephpi/FormulaNet
Image-to-Text • 20M • Updated • 1k • 1
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111,... | E _ { n } = - { \frac { m _ { e } e ^ { 4 } } { 2 ( 4 \pi \varepsilon _ { 0 } \hbar ) ^ { 2 } } } ~ { \frac { 1 } { n ^ { 2 } } } = - { \frac { 1 3 . 6 ~ { \mathrm { e V } } } { n ^ { 2 } } } . |
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... | C _ { a } ( t ) = \sum _ { k } ^ { \infin } c _ { a k } \exp ( - \nu _ { a k } t ) |
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1... | C _ { 0 } |
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9,... | - 0 . 2 5 \gamma \le \Delta \omega \le 0 . 2 5 \gamma |
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... | \hat { G } _ { e } = J ^ { - 1 } \left \{ \begin{array} { c } { \overline { \rho } W } \\ { \overline { \rho } \tilde { u } W + \overline { p } \zeta _ { x } } \\ { \overline { \rho } \tilde { v } W + \overline { p } \zeta _ { y } } \\ { \overline { \rho } \tilde { w } W + \overline { p } \zeta _ { z } } \\ { \left ( \... |
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... | \left ( \begin{array} { l l l } { \rho _ { 1 1 } } & { \cdots } & { \rho _ { 1 d } } \\ { \vdots } & { \ddots } & { \vdots } \\ { \rho _ { d 1 } } & { \cdots } & { \rho _ { d d } } \end{array} \right ) \Longrightarrow \left ( \begin{array} { l } { \rho _ { 1 1 } } \\ { \rho _ { 1 2 } } \\ { \vdots } \\ { \rho _ { 1 d }... |
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1... | \begin{array} { r l } { 0 } & { { } = \frac { \partial \tilde { H } } { \partial \kappa } } \end{array} |
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1... | \vec { e } |
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... | { \frac { 1 } { 2 } } { \frac { \coth ( \pi w ) } { \pi w } } - { \frac { 1 } { 2 } } { \frac { 1 } { z ^ { 2 } } } = \sum _ { n = 1 } ^ { \infin } { \frac { 1 } { n ^ { 2 } + w ^ { 2 } } } |
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... | t _ { 0 } = 0 , ~ t _ { 1 } = \delta t , ~ \cdots , t _ { k } = k \delta t , ~ \cdots , ~ t _ { n } = n \delta t |
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... | G ( s , t ) = \sum _ { x = 0 } s ^ { x } p ( x , t ) , |
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... | \begin{array} { r } { K \equiv \frac { 1 } { | \hat { e } _ { \perp } { \cdot } \partial _ { \Delta \lambda } \Delta \hat { e } ( 0 ) | ^ { 2 } } \operatorname* { l i m } _ { \mathrm { R e } ( \Delta \lambda ) \rarr 0 } \frac { | \lambda _ { 0 } | ^ { 2 } \mathrm { R e } \lambda _ { 1 } + | \lambda _ { 1 } | ^ { 2 } \m... |
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... | T = \frac { 1 } { 4 \pi } ~ \frac { r _ { + } - r _ { - } } { r _ { + } ^ { 2 } - \S ^ { 2 } } ~ . |
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... | 1 / r |
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... | \frac { 2 } { \sin A } = \frac { 1 } { \sin C } |
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... | R e |
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... | \left \| x \right \| _ { p } = { \bigg ( } \sum _ { i \in \mathbb { N } } \left | x _ { i } \right | ^ { p } { \bigg ) } ^ { 1 / p } { \mathrm { ~ a n d } } ~ \left \| f \right \| _ { p , X } = { \bigg ( } \int _ { X } \left | f ( x ) \right | ^ { p } ~ \mathrm { d } x { \bigg ) } ^ { 1 / p } |
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2... | E _ { \pm } \approx S _ { \pm } / [ 1 - i ( Y _ { \pm } + 2 Y _ { \mp } - \Delta _ { \pm } ) ] |
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75... | \Omega \rarr \infin |
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2... | F _ { i j } = { \frac { \partial f _ { i } } { \partial v _ { j } } } . |
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52... | \tau _ { m a x _ { 1 } } = 6 . 7 |
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158... | \left \{ \begin{array} { r c l } { \tilde { m } _ { n } ^ { \pm } } & { = } & { \frac { 2 n - 1 } { R } \pm m _ { q } ( \phi _ { H } ) } \\ { m _ { n } ^ { \pm } } & { = } & { m _ { q } ( \phi _ { H } ) \pm \frac { 2 n } { R } } \\ { m _ { 0 } } & { = } & { m _ { q } ( \phi _ { H } ) = \frac { 2 } { \pi R } \mathrm { a... |
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107... | X \sim { \mathcal { B } } e ( \alpha , \beta ) |
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... | \begin{array} { r l } { \mathbb { P } ( \mathcal { L } _ { < } ( \Pi _ { x , t } ^ { ( \lambda ) } ) } & { > ( 1 + \varepsilon ) ( 2 \sqrt { x t \lambda } - x \lambda ) ) \le \exp ( - g ( \varepsilon ) ( \sqrt { x t \lambda } - x \lambda ) ) , } \\ { \mathbb { P } ( \mathcal { L } _ { < } ( \Pi _ { x , t } ^ { ( \lambd... |
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1... | \begin{array} { r l } { \sum _ { i \in V } D _ { i i } ~ x _ { i } ^ { ( 1 ) } } & { { } = \sum _ { i \in V } D _ { i i } \cdot \frac { 1 } { D _ { i i } } \sum _ { j \in V } W _ { i j } ~ x _ { j } ^ { ( 0 ) } \cdot \left ( 1 + \sigma _ { \lambda ; i } ( \mathbf { x } ^ { ( 0 ) } ) \right ) } \end{array} |
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223... | \begin{array} { r l } { G _ { n _ { \| } n _ { \| } } ^ { R } } & { = \frac { \chi \Gamma ( i \omega - \tilde { D } _ { \psi } k ^ { 2 } - \tilde { \Omega } ) - \chi \omega _ { 0 } ^ { 2 } } { \big ( i \omega - D _ { \ell } k ^ { 2 } - \Gamma \big ) \big ( i \omega - \tilde { D } _ { \psi } k ^ { 2 } - \tilde { \Omega ... |
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... | 5 { ~ } 2 0 5 |
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... | m ^ { 4 } \int D [ C ] ~ | \Psi [ C ] | ^ { 2 } = 1 . |
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167... | \begin{array} { r l } { \vec { v _ { i } } ^ { 0 } } & { { } = \frac { \Gamma } { 2 \pi } \sum _ { i \neq j } ^ { N _ { v } } \kappa _ { j } \hat { z } \times \frac { \vec { r } _ { i } - \vec { r } _ { j } } { \left | \vec { r } _ { i } - \vec { r } _ { j } \right | ^ { 2 } } + \frac { 1 } { 2 } \left ( \frac { \Gamma... |
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1... | \Im \mathrm { E } |
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... | h _ { j } |
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1... | \alpha _ { i } > 0 . 5 |
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... | \begin{array} { r } { \left \| \widetilde { \Lambda } ^ { - 1 } \widetilde { V } ^ { * } \omega ^ { ( - ) } \right \| _ { w ^ { k - 1 } } = \left \| \beta ^ { ( - ) } \right \| _ { w ^ { k - 1 } } . } \end{array} |
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38,... | \left ( S ( | \mathbf { k } _ { 1 } + \mathbf { k } _ { 2 } | ) + S ( | \mathbf { k } _ { 1 } + \mathbf { k } _ { 3 } | ) + S ( | \mathbf { k } _ { 2 } + \mathbf { k } _ { 3 } | ) \right ) < 2 |
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183... | \Delta n ^ { a } |
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