harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}{F(\lambda)=(-i)e^{-\tau}\tau^{s_{_\eta}}\sum_{n=0}^{n^{\prime}}a_{n}\tau^{n}\timesu_{+}\mathrm{~,}}\\{G(\lambda)=e^{-\tau}\tau^{s_{_\eta}}\sum_{n=0}^{n^{\prime}}b_{n}\tau^{n}\timesu_{-}\mathrm{~,}}\\\end{array}\tag{A13}
\langle\overrightarrow{X_{1}x}|\vec{r}\rangle\neq0\\mathrm{~and~}\\langle\overrightarrow{X_{2}x}|\vec{r}\rangle=\langle\overrightarrow{X_{3}x}|\vec{r}\rangle=\langle\overrightarrow{X_{4}x}|\vec{r}\rangle=0.
cos(a_{0})=\frac{k^{2}C_{1}C_{2}C_{3}-D_{1}D_{2}D_{3}}{k^{2}},~~cos(a_{r})=\frac{(-)^{r}(D_{p}D_{q}C_{r}-C_{p}C_{q}D_{r})}{k^{2}S_{p}S_{q}},
L_{2k}=\frac{\partial_{2}c}{(\partial_{1}c)^{2}+(\partial_{2}c)^{2}}\cdot[-R_{k}+P_{k}\frac{\partial_{1}a}{\partial_{2}c}+\frac{1}{2k}\frac{{\calD}_{1}P_{k}}{\partial_{2}c}-\frac{1}{2k}\frac{P_{k}\cdot{\calD}_{1}(\partial_{2}c)}{(\partial_{2}c)^{2}}].
Q=\frac{\partial_{v_{1}}...\partial_{v_{p}}\left[\prod_{k<j}(v_{k}-v_{j})\prod_{j=1}^{p}\left(\prod_{i=0}^{q}(w_{i}-v_{j})^{1-\beta}\prod_{i=1}^{q-1}(\xi_{i}-v_{j})^{-1}\right)\right]}{\prod_{k<j}(v_{k}-v_{j})\prod_{j=1}^{p}\left[\prod_{i=0}^{q}(w_{i}-v_{j})^{1-\beta}\prod_{i=1}^{q-1}(\xi_{i}-v_{j})^{-1}\right]}
\mathrm{N.L.}=\intd\taud\tau^{\prime}\int\frac{dk}{(2\pi)^{D}}\dot{x}_{\mu}(\tau)\dot{x}_{\nu}(\tau^{\prime})\,\frac{D_{\mu\nu}(k)}{k^{2}}\,e^{ik(x(\tau)-x(\tau^{\prime}))}.
\psi_{m}(r)=r^{|m|}\,_{1}F_{1}\left({\frac{2-g}{4}}+{\frac{|m|-m}{2}}-{\frac{\epsilonR^{2}}{2\delta}},1+|m|,{\frac{\deltar^{2}}{R^{2}}}\right)e^{-{\frac{\deltar^{2}}{2R^{2}}}}\,.
S^{(1)ab}(p)=\displaystyle{\delta}^{ab}\frac{g^{2}C_{V}}{8\pi^{2}m}\int_{R_{\epsilon}^{3}}e^{-ip.x}ip_{\mu}x_{\mu}\left[\frac{1-e^{-mr}}{r^{6}}-\frac{me^{-mr}}{r^{5}}\right]\,.
\begin{array}{lll}{L_{IJ}^{(-)}L_{JK}^{(-)}}&{-}&{(-1)^{P_{J}(P_{I}+P_{K})}L_{JK}^{(-)}L_{IJ}^{(-)}=}\\{}&{}&{(q-q^{-1})(-1)^{P_{K}P_{J}}L_{JJ}^{(-)}L_{IK}^{(-)},(I>J>K)}\\\end{array}
\times\prod_{c\neqd}^{N}e^{2\pi\frac{g}{\pi+gN}\sum_{k=1}^{n_{c}}\sum_{h=1}^{n_{d}}\Big[C^{0,S}(x_{k}^{(c)},x_{h}^{(d)})+C^{0,S}(y_{k}^{(c)},y_{h}^{(d)})-C^{0,S}(x_{k}^{(c)},y_{h}^{(d)})-C^{0,S}(y_{k}^{(c)},x_{h}^{(d)})\Big]}\;.
(\bar{C}_{\alpha},\bar{C}_{\beta}^{\dagger})_{\Omega}=\bar{U}_{\alpha\beta}^{\gamma}\bar{C}_{\gamma}(-1)^{\varepsilon_{\beta}}+\bar{C}_{\gamma}^{\dagger}\bar{U}_{\beta\alpha}^{\dagger\gamma}(-1)^{\varepsilon_{\alpha}},
\deltaT_{N}^{M}=\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{-\delta\rho}&{-(\rho+p)e^{\alpha_{0}}v_{,i}}\\{0}&{(\rho+p)e^{-\alpha_{0}}v_{,i}}&{\deltap\:\delta_{ij}}\\\end{array}\right)\:\delta(y)
\stackrel{\lbrackj]}{a}_{1}=\alpha_{j}\left(\left[\partial_{\left[\alpha\right.}V_{\left.\beta\right](\lambda)}\right],\left[\partial_{\left[\alpha\right.}A_{\left.\beta\right]}^{\;\;(\lambda)}\right],\left[\chi^{*}\right]\right)e^{j}\left(\eta_{(\lambda)},C^{(\lambda)}\right),
\left[\left(\gamma^{0}\otimes\mathrm{I}+\mathrm{I}\otimes\gamma^{0}\right)\pi_{0}-\left(\overrightarrow{\alpha}\otimes\gamma^{0}+\gamma^{0}\otimes\overrightarrow{\alpha}\right)\cdot\overrightarrow{\pi}\mathrm{}-m\gamma^{0}\otimes\gamma^{0}\right]\Psi\left(x\right)=0\mathrm{~.}
\mathscr{C}^{A}\rightarrow\pm\bar{\mathscr{C}}^{A},\quad\bar{\mathscr{C}}^{A}\rightarrow\mp\mathscr{C}^{A},\quad\mathscr{B}^{A}\rightarrow-\bar{\mathscr{B}}^{A},\quad\bar{\mathscr{B}}^{A}\rightarrow-\mathscr{B}^{A},\quad(\mathscr{A}_{\mu}^{A}\rightarrow\mathscr{A}_{\mu}^{A}).
{\calL}_{class}^{\prime}=-\varphi_{1}\Box\varphi_{2}+m^{2}\varphi_{1}\varphi_{2}+{\frac{\lambda}{4}}\varphi_{1}^{2}\varphi_{2}^{2}+J_{1}\varphi_{1}+J_{2}(\varphi_{2}+\alpha\varphi_{1}^{2}).
u(s^{\prime})-2\frac{s^{\prime}}{s}\int_{0}^{s}ds^{\prime\prime}eA_{-}\left(x^{+}-u(s^{\prime\prime})\right)+2\int_{0}^{s^{\prime}}ds^{\prime\prime}eA_{-}\left(x^{+}-u(s^{\prime\prime})\right)=0\;.
\chi_{1,i}^{(2,2k+1)}(q)=\chi_{1,2k+1-i}^{(2,2k+1)}(q)=\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k-1}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k-1}^{2}+n_{i}+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}},
\delta\Pi_{e}=\frac{\alphal}{x^{5/2}}\,H_{e}\!\left(\frac{1}{x}\right),\quad\delta\Pi_{\perp}=\frac{\alphal}{x^{5/2}}\,H_{\perp}\!\left(\frac{1}{x}\right),\quad\delta\Pi_{o}=\frac{\alpha}{x}\,H_{o}\!\left(\frac{1}{x}\right),
\begin{array}{l}{h_{3}^{\mu}=h_{2}^{\nu}\partial_{\nu}h_{1}^{\mu}-h_{1}^{\nu}\partial_{\nu}h_{2}^{\mu}+2i\bar{\lambda}_{1a}\gamma^{\mu}\lambda_{2}^{a}\,,}\\{{}}\\{\lambda_{3}^{a}={\calL}_{2}\lambda_{1}^{a}-{\calL}_{1}\lambda_{2}^{a}\,,}\\\end{array}
\begin{array}{cc}{q^{3/2}M=M_{+}M_{-}^{-1},\M_{+}=\left(\begin{array}{cc}{q^{-\frac{H}{2}}}&{(q^{-1}-q)Fq^{\frac{H}{2}}}\\{0}&{q^{H/2}}\\\end{array}\right),}\\{M_{-}^{-1}=\left(\begin{array}{cc}{q^{-\frac{H}{2}}}&{0}\\{(q^{-1}-q)Eq^{\frac{H}{2}}}&{q^{H/2}}\\\end{array}\right)}\\\end{array}
\intd^{3}x\intd^{3}y\left[\sum_{n}f_{i}^{a}(x_{n})\frac{\chi_{\epsilon}(x,x_{n})\chi_{\epsilon}(y,x_{n})}{\epsilon^{3}}\right]\left[\tau_{i}\partial_{a}\xi(x)+(\omega_{a}(x_{n})\xi(x))\right]^{A}\bar{\xi}_{A}\;.
L_{\omega}=8\pi\sum_{\omega^{\prime}\omega^{\prime\prime}\omega^{\prime\prime\prime}}\sqrt{|\omega^{\prime}\omega^{\prime\prime}|}C_{\omega^{\prime}\omega^{\prime\prime}}^{\omega^{\prime\prime\prime}}\:C_{\omega\omega^{\prime\prime\prime}}^{0}\:\tilde{\alpha}_{\omega^{\prime}}\:\tilde{\alpha}_{\omega^{\prime\prime}}
V_{long}=-\int\frac{ds}{s^{3/2}}\frac{e^{-b^{2}s}}{8\sqrt{\pi}c_{1}sc_{2}s}(c_{1}^{2}-c_{2}^{2})^{2}s^{4}(\frac{\sqrt{\pi}}{L_{1}s^{1/2}})=-\frac{(c_{1}^{2}-c_{2}^{2})^{2}}{8c_{1}L_{1}c_{2}b^{2}}.
f_{{\bf{q}}}({\bf{x}},t)=(e^{-i\mathrm{~}{\bf{q.x}}}e^{i\mathrm{~}\omega_{{\bf{q}}}t}-1)(A_{{\bf{q}}}+B_{{\bf{q}}})^{2}-(e^{-i\mathrm{~}{\bf{q.x}}}e^{i\mathrm{~}\epsilon_{{\bf{q}}}t}-1)
\begin{array}{c}{T_{\mu\nu}^{abcd}(p_{1},p_{2};k_{1},k_{2})={\sum_{i=1}^{4}}T_{\mu\nu}^{(i)abcd}(p_{1},p_{2};k_{1},k_{2})}\\{=-ig^{2}e^{\rho}(p_{1})e^{\sigma}(p_{2)}{\sum_{i=1}^{4}}T_{\rho\sigma\mu\nu}^{(i)abcd}(p_{1},p_{2};k_{1},k_{2})}\\\end{array}
(D_{3})_{\alpha}=\theta_{\alphaC}(\vec{X}_{B}\cdot\vec{P}_{A})X_{D}^{a}{d^{CD}}_{F}f^{BAF}+\frac{1}{2}(\gamma^{ab}\theta_{A})_{\alpha}X_{dB}(\vec{X}_{C}\cdot\vec{P}_{D})({c_{E}}^{AB}f^{ECD}+{c_{\lambda}}^{AB}f^{\lambdaCD}).
I_{E}(g_{E},\Phi_{0})=-{\frac{1}{8\piG}}\int_{\partial\Omega}[K]\sqrt{{}_{3}g_{E}}\;d^{3}x-{\frac{1}{16\piG}}\int_{\Omega}R\sqrt{g_{E}}\;d^{4}x+\int_{\Omega}{\calL}_{E}\sqrt{g_{E}}\;d^{4}x.
{\frac{\xi_{0}}{\lambda}}=2~i~\xi_{0}\left({\frac{1}{2L}}\sum_{n=\pm\frac{1}{2},\pm\frac{3}{2},\cdots~}\right)\int_{-\infty}^{\infty}{\frac{dp^{-}}{2\pi}}{\frac{1}{2p_{n}^{+}p^{-}-\xi_{0}^{2}+i\epsilon}}~,
q\partial_{\alpha}\partial^{\alpha}\biggl(\Psi+\Xi\biggr)+d~q\biggl[\Psi^{\prime\prime}+{\calH}\biggl(\Xi^{\prime}-\Psi^{\prime}\biggr)\biggr]+d~q^{\prime}(\Psi^{\prime}+{\calH}\Xi)+X^{\prime}\varphi^{\prime}=0.
f(\rho^{\vee}\!\!\!\cdot\hat{H})\hat{s}_{\rho}|\nu\rangle=\sum_{\mu\in\Sigma_{min}}\biggl\{f(-1)\delta_{\mu-\nu,-\rho}+f(0)\delta_{\mu,\nu}\delta_{\rho\cdot\mu,0}+f(1)\delta_{\mu-\nu,\rho}\biggr\}|\mu\rangle,
D_{\mu\nu}(x-x^{\prime})=(-\varepsilon_{\mu\nu\lambda}\partial_{x}^{\lambda}+{\frac{\alpha\partial_{\mu}\partial_{\nu}}{\Box}})\int{\frac{d^{3}k}{(2\pi)^{3}}}{\frac{e^{ik(x-y)}}{k^{2}+i\epsilon}}
\left(A_{\muAx},B_{Ax}\right)\left(\begin{array}{cc}{G_{\muAx,\nuBy}}&{\tilde{G}_{\muAx,By}}\\{G_{Ax,\nuBy}}&{G_{Ax,By}}\\\end{array}\right)\left(\begin{array}{c}{A_{\nuBy}}\\{B_{By}}\\\end{array}\right)
[\frac{i}{2}(2)-\frac{i}{8}(8)-i(0)]Tr[t^{a}t^{b}]\int\frac{dw_{1}}{2\pi}w_{1}^{2}X_{i}^{a}(w_{1})X^{ib}(-w_{1})\int\frac{dw}{2\pi}\frac{1}{(w^{2}-R^{2})}\frac{1}{[(w-w_{1})^{2}-R^{2}]}=0
{\frac{m^{2}(S)}{m^{2}(V)}}={\frac{1}{6(4\pi)^{2}}}\left[36e^{2}+{\frac{\lambda^{2}\xi^{2}}{e^{2}}}\left(15+{\frac{3}{4\omega^{2}}}-{\frac{9\xi}{\omega^{2}}}+{\frac{27\xi^{2}}{\omega^{2}}}\right)\right].
<\tilde{\phi}_{\omega},\phi_{\omega}>=(\tilde{\phi}_{\omega},\phi_{\omega})+i\int_{\Sigma}\sqrt{h}d^{D-1}x\tilde{\phi}_{\omega}^{*}\left[2a^{i}(\partial_{i}+i\omegaa_{i})+\nabla^{i}a_{i}\right]\phi_{\omega}.
|x^{\frac{1}{n}} - c^{\frac{1}{n}}| = \frac{|x^{\frac{1}{n}} - c^{\frac{1}{n}}||x^{\frac{n-1}{n}} + x^{\frac{n-2}{n}}c^{\frac{1}{n}} + \cdots + x^{\frac{1}{n}}c^{\frac{n-2}{n}}|}{|x^{\frac{n-1}{n}} + x^{\frac{n-2}{n}}c^{\frac{1}{n}} + \cdots + x^{\frac{1}{n}}c^{\frac{n-2}{n}} + c^{\frac{n-1}{n}}|}
\begin{array} { r l } { \langle u | u | ^ { 2 } , b _ { j , 1 } \rangle } & { = \sum _ { k , l , m } \frac { A _ { k } A _ { l } A _ { m } } { L _ { k } L _ { l } L _ { m } } \left\langle e ^ { i \Gamma _ { k } + i \Gamma _ { l } - i \Gamma _ { m } } e ^ { - \frac { | y _ { k } | ^ { 2 } + | y _ { l } | ^ { 2 } + | y _ { m } | ^ { 2 } } { 2 } } , e ^ { i \Gamma _ { j } } e ^ { - \frac { 1 } { 2 } | y _ { j } | ^ { 2 } } \right\rangle } \end{array}
\begin{array} { r l r } { V ( x , t ) } & { = } & { \frac { 1 } { | \sqrt { 2 \lambda } \alpha _ { 0 } | ^ { 2 q } } \sum _ { m = 0 } ^ { q } ( - 1 ) ^ { m + q } \frac { ( 2 ^ { m } q ! ) ^ { 2 } } { ( 2 m ) ! ( q - m ) ! } \lambda ^ { q - m } x ^ { 2 m } } \\ & { } & { - \frac { ( - i ) ^ { q } } { | \sqrt { 2 \lambda } \alpha _ { 0 } | ^ { 2 q } } \Big [ \Big ( e ^ { - i \tau } \alpha _ { 0 } \sqrt { 2 \lambda } \Big ) ^ { q } + \Big ( e ^ { i \tau } \alpha _ { 0 } ^ { * } \sqrt { 2 \lambda } \Big ) ^ { q } \Big ] } \\ & { } & { \times \Big ( [ 1 + ( - 1 ) ^ { q } ] \frac { \Gamma ( 1 + q ) \Gamma ( \frac { 1 } { 2 } ) } { \Gamma ( \frac { 1 } { 2 } + \frac { q } { 2 } ) } \frac { ( - 1 ) ^ { \frac { q } { 2 } } } { 2 } + i [ 1 - ( - 1 ) ^ { q } ] \frac { \Gamma ( q + 1 ) \Gamma ( \frac { 3 } { 2 } ) } { \Gamma ( \frac { q } { 2 } + 1 ) } ( - 1 ) ^ { \frac { q - 1 } { 2 } } \Big ) x ^ { q } } \\ & { } & { + 1 . } \end{array}
\begin{array} { r l r } & { } & { \left. \frac { \partial } { \partial b } \left\{ I ( u , v ) \right\} \right| _ { b = 0 } } \\ & { = } & { - j \left[ \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 1 } e ^ { j [ a Z _ { 5 } ( r , \theta ) - 2 \pi \rho r c o s ( \phi - \theta ) } r d r d \theta \right. } \\ & { } & { \times \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 1 } Z _ { 4 } ( r , \theta ) e ^ { - j [ a Z _ { 5 } ( r , \theta ) - 2 \pi \rho r c o s ( \phi - \theta ) } r d r d \theta } \\ & { } & { - \: \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 1 } e ^ { - j [ a Z _ { 5 } ( r , \theta ) - 2 \pi \rho r c o s ( \phi - \theta ) } r d r d \theta } \\ & { } & { \times \left. \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 1 } Z _ { 4 } ( r , \theta ) e ^ { j [ a Z _ { 5 } ( r , \theta ) - 2 \pi \rho r c o s ( \phi - \theta ) } r d r d \theta \right] } \end{array}
\begin{array} { r l } { ( \pi _ { C | D } - \pi _ { D | D } ) q _ { D | D } = } & { \left[ - c + w _ { I } b + \frac { 1 - w _ { I } } { k } ( k - 1 ) q _ { C | C } b - \frac { 1 - w _ { I } } { k } ( k - 1 ) q _ { C | D } b \right] \left( 1 - \frac { k - 2 } { k - 1 } p _ { C } \right) } \\ { = } & { \left( - c + w _ { I } b + \frac { 1 - w _ { I } } { k } b \right) \left( 1 - \frac { k - 2 } { k - 1 } p _ { C } \right) , } \end{array}
{ \cal Z } _ { B } = T r \; e ^ { - \beta \varepsilon _ { \kappa } ( \overline { { { \Phi } } } _ { 1 , \kappa } \Phi _ { 1 , \kappa } + \overline { { { \Phi } } } _ { 2 , \kappa } \Phi _ { 2 , \kappa } ) } e ^ { - \beta \mu ( N _ { 1 , \kappa } + N _ { 2 , \kappa } ) } ,
\begin{array} { r } { 4 \pi + \frac { \gamma \varepsilon ^ { 3 } } { 2 } ( 1 + \varepsilon + \varepsilon ^ { 2 } ) - \left( \frac { 2 \pi { h ^ { \ast } } ^ { 4 } } { \varepsilon ^ { 2 } } - \frac { \gamma { h ^ { \ast } } ^ { 2 } \varepsilon ^ { 2 } } { 2 } \right) \log { ( h ^ { \ast } ) } + \ensuremath { \operatorname { O } \left( \gamma \varepsilon ^ { 6 } , ( \gamma + 1 ) { h ^ { \ast } } ^ { 2 } \varepsilon ^ { 2 } , \frac { { h ^ { \ast } } ^ { 4 } } { \varepsilon ^ { 2 } } \right) } } \\ { \leq 4 \pi + \frac { \gamma \varepsilon ^ { 3 } } { 2 } ( 1 + \varepsilon + \varepsilon ^ { 2 } ) + \frac { \gamma ^ { 2 } \varepsilon ^ { 6 } } { 3 2 \pi } \log { \left( \sqrt [ ] { \frac { \gamma \varepsilon ^ { 4 } } { 8 \pi } } \right) } + \ensuremath { \operatorname { O } \left( \gamma ( \gamma + 1 ) \varepsilon ^ { 6 } \right) } . } \end{array}
\begin{array} { r l } { \phi _ { i } ( \lambda _ { i } ) } & { = \mathrm { v e c } \left( { \uppercase { \mathbf { P } } } _ { i } \right) ^ { H } \mathrm { v e c } \left( { \uppercase { \mathbf { P } } } _ { i } \right) } \\ & { = { \lowercase { \mathbf { a } } } _ { i } ^ { H } { \uppercase { \mathbf { D } } } _ { i } \left( \lambda _ { i } { \uppercase { \mathbf { I } } } _ { M M _ { e } } + \mathrm { { d i a g } } \left( { \lowercase { \mathbf { f } } } _ { i } \right) \right) ^ { - 1 } \left( \lambda _ { i } { \uppercase { \mathbf { I } } } _ { M M _ { e } } + { \mathrm { d i a g } } \left( { \lowercase { \mathbf { f } } } _ { i } \right) \right) ^ { - 1 } { \uppercase { \mathbf { D } } } _ { i } ^ { H } { \lowercase { \mathbf { a } } } _ { i } = \sum _ { m = 1 } ^ { M M _ { e } } \frac { b _ { i , m } ^ { 2 } } { \left( \lambda _ { i } + f _ { i , m } \right) ^ { 2 } } , } \end{array}
\begin{array} { r l r } { E _ { 1 2 } ( a , c , W \rightarrow 0 ) } & { { } = } & { \left\{ \begin{array} { l c l } { - \frac { 1 } { 2 } \cos 2 \theta } & { , } & { d = 0 } \\ { \frac { \pi } { 4 } \sin 2 \theta \cos 2 \theta - \cos 2 \theta + \ln [ | \tan \theta | ^ { \sin ^ { 2 } 2 \theta / 2 } ] } & { , } & { d = 2 } \\ { - \cos 2 \theta } & { , } & { d = 4 } \\ { - \frac { 1 } { 2 } \cos 2 \theta \left[ 1 + 2 4 ( 1 9 + 5 \cos 4 \theta ) ^ { - 1 } \right] } & { , } & { d = 6 } \\ { - ( 5 3 \cos 2 \theta + 7 \cos 6 \theta ) ( 3 9 + 2 1 \cos 4 \theta ) ^ { - 1 } } & { , } & { d = 8 } \end{array} \right. , } \end{array}
{ \frac { \partial a _ { k } ^ { s } } { \partial t } } = \frac { i \epsilon } { 8 k _ { \perp } } \int _ { \mathbb { R } ^ { 6 } } \sum _ { s _ { p } s _ { q } } \left[ s _ { p } s _ { q } k _ { \perp } ^ { 2 } + \frac { k _ { \parallel } } { p q } \left( p _ { \parallel } q ^ { 2 } + q _ { \parallel } p ^ { 2 } \right) \right.
{ \begin{array} { r l } { { F ^ { \alpha \beta } } _ { ; \beta } } & { = 0 } \\ { F _ { [ \alpha \beta ; \gamma ] } } & { = { \frac { 1 } { 3 } } \left( F _ { \alpha \beta ; \gamma } + F _ { \beta \gamma ; \alpha } + F _ { \gamma \alpha ; \beta } \right) = { \frac { 1 } { 3 } } \left( F _ { \alpha \beta , \gamma } + F _ { \beta \gamma , \alpha } + F _ { \gamma \alpha , \beta } \right) = 0 . } \end{array} }
F _ { r { \tilde { \theta } } } ^ { 1 } = - \, \frac { 1 } { 2 \lambda } \, { \dot { a } } ~ ~ , ~ ~ F _ { r { \tilde { \varphi } } } ^ { 2 } = \frac { 1 } { 2 \lambda } \, { \sin { \widetilde { \theta } } \, { \dot { a } } } ~ ~ , ~ ~ F _ { { \tilde { \theta } } { \tilde { \varphi } } } ^ { 3 } = \frac { 1 } { 2 \lambda } \, { \sin { \widetilde { \theta } } } \left( 1 - a ^ { 2 } \right) ~ ~ .
( \mathfrak { n } ^ { + } \otimes \mathbb C [ t ] ) v _ { \pmb { \xi } } = 0 , \quad \, ( h _ { \alpha _ { i } } \otimes t ^ { s } ) v _ { \pmb { \xi } } = \lambda ( h _ { \alpha _ { i } } ) \delta _ { s , 0 } v _ { \pmb { \xi } } , \quad \, ( x _ { \alpha _ { i } } ^ { - } \otimes 1 ) ^ { \lambda ( h _ { \alpha _ { i } } ) + 1 } v _ { \pmb { \xi } } = 0 , \quad \, \mathrm { f o r } \, i = 1 , 2 ,
\begin{array} { r l r } { \sum _ { i } P _ { i , \nu } d Q _ { i , \nu } } & { { } = } & { \frac { 1 } { 2 } \left[ ( P _ { k } - P _ { - k } ) d ( Q _ { k } - Q _ { - k } ) - ( Q _ { k } + Q _ { - k } ) d ( P _ { k } + P _ { - k } ) \right] = } \end{array}
a _ { \mu } ^ { \mathrm { S U S Y } } = { \frac { \tan \beta } { 4 8 \pi } } { \frac { m _ { \mu } ^ { 2 } } { M _ { \mathrm { S U S Y } } ^ { 2 } } } ( 5 \alpha _ { 2 } + \alpha _ { 1 } )
\Phi = { \left\{ \begin{array} { l l } { \tan { \frac { \pi \alpha } { 2 } } } & { { \mathrm { i f ~ } } \alpha \neq 1 } \\ { - { \frac { 2 } { \pi } } \log | c \, t | } & { { \mathrm { i f ~ } } \alpha = 1 } \end{array} \right. }
+ { \frac { 1 } { 2 m ^ { 2 } } } \Pi _ { \psi } ^ { 2 } + A _ { 0 } \Pi _ { \psi } + { \frac { m ^ { 2 } } { 2 } } ( ( \partial _ { i } \psi ) ^ { 2 } + 2 A _ { i } \partial _ { i } \psi ) .
H ^ { 2 } ( \mathrm { ~ \boldmath ~ q ~ } ^ { ( 1 ) } , \mathrm { ~ \boldmath ~ q ~ } ^ { ( 2 ) } ) = \frac { 1 } { 2 } \sum _ { j } \left( \sqrt { q _ { j } ^ { ( 1 ) } } - \sqrt { q _ { j } ^ { ( 2 ) } } \right) ^ { 2 } ,
{ \mathcal { K } } ( \hat { M } _ { f } , \hat { M } _ { b } ) = \mathrm { c o n s t } \; \int { \mathcal { D } } { \mathcal { F } } { \mathcal { D } } { \mathcal { B } } \exp \left\{ - { \mathcal { W _ { F } } } - { \mathcal { W _ { B } } } - { \mathcal { W _ { F B } } } \right\}
\begin{array} { r l } { h _ { 0 0 } ^ { \mathrm { P D } } \big | _ { \vec { x } = ( x , 0 , 0 ) } } & { = \frac { 1 - \cos ( x \omega _ { g } c _ { \vartheta } ) } { \omega _ { g } ^ { 2 } c _ { \vartheta } ^ { 2 } } \, ( \partial _ { 0 } ) ^ { 2 } h _ { 1 1 } ^ { \mathrm { T T } } \big | _ { \vec { x } = ( 0 , 0 , 0 ) } \, , } \\ { h _ { 0 1 } ^ { \mathrm { P D } } \big | _ { \vec { x } = ( x , 0 , 0 ) } } & { = h _ { 1 1 } ^ { \mathrm { P D } } \big | _ { \vec { x } = ( x , 0 , 0 ) } = 0 \, . } \end{array}
{ \mathcal { M } } = { \frac { 1 } { \sqrt { Z } } } { \Big ( } \operatorname* { l i m } _ { x _ { 1 } ^ { 0 } \rightarrow - \infty } - \operatorname* { l i m } _ { x _ { 1 } ^ { 0 } \rightarrow + \infty } { \Big ) } \int \! \mathrm { d } ^ { 3 } x _ { 1 } \; \mathrm { e } ^ { i p _ { 1 } \cdot x _ { 1 } } \langle \beta \ \mathrm { o u t } | { \bar { \Psi } } ( x _ { 1 } ) \gamma ^ { 0 } u _ { { \textbf { p } } _ { 1 } } ^ { s _ { 1 } } | \alpha ^ { \prime } \ \mathrm { i n } \rangle ,
T ^ { 0 } ( x ) : = 1 2 0 0 \mathrm { e x p } \left( - \frac { ( x - l / 2 ) ^ { 2 } } { 2 0 0 m ^ { 2 } } \right) \quad \mathrm { ~ a ~ n ~ d ~ } \quad S ^ { 0 } \equiv 1 .
\begin{array} { r l } { \mathrm { P L ^ { C I } } } & { ( f , d ) [ \mathrm { d B } ] = \mathrm { F S P L } ( f , \mathrm { 1 \, m } ) [ \mathrm { d B } ] + 1 0 \eta \log _ { 1 0 } ( d ) } \\ & { + \mathrm { A T } [ \mathrm { d B } ] + \mathrm { O 2 I } [ \mathrm { d B } ] + \mathrm { F L } [ \mathrm { d B } ] + \chi _ { \sigma } ^ { C I } [ \mathrm { d B } ] } \end{array}
\left| \frac { \epsilon _ { 1 } \epsilon _ { 2 } } { \epsilon _ { + } ^ { 2 } } \frac { t r ( f h ^ { l } f ^ { \dag } \tilde { h } ^ { l } ) } { t r \{ ( f f ^ { \dag } ) ^ { 2 } \} } - \frac { h _ { D 1 } ^ { 2 } } { 2 f _ { 1 } ^ { 2 } } \right| < 1 0 ^ { - 8 } .
\begin{array} { r l } { \left\langle w , \frac { \delta H } { \delta u } - D u \right\rangle } & { { } = 0 , \, \forall w \in \mathbb { V } _ { h } ^ { 1 } , } \\ { \left\langle \phi , \frac { \delta H } { \delta D } - \frac { 1 } { 2 } | u | ^ { 2 } - g ( D + b ) \right\rangle } & { { } = 0 , \quad \forall \phi \in \mathbb { V } _ { h } ^ { 2 } , } \end{array}
\begin{array} { r } { | \psi _ { k } \rangle = \sum _ { x } ( \psi _ { k } ( x , A ) a _ { x } ^ { \dagger } + \psi _ { k } ( x , B ) b _ { x } ^ { \dagger } ) | \varphi , g \rangle + \psi _ { e } | \varphi , e \rangle , } \end{array}
\int _ { - \infty } ^ { \infty } \hat { G } _ { w } ^ { ( P ) } ( x , \omega + P \omega _ { m } ) \mathrm { e } ^ { - \mathrm { i } ( \kappa + P \kappa _ { m } ) x } \, \mathrm { d } x = \mathbf { \tilde { G } } ( \kappa + P \kappa _ { m } , \omega + P \omega _ { m } ) .
B = \left[ { \begin{array} { r r r r r r r r } { - 2 6 } & { - 3 } & { - 6 } & { 2 } & { 2 } & { - 1 } & { 0 } & { 0 } \\ { 0 } & { - 2 } & { - 4 } & { 1 } & { 1 } & { 0 } & { 0 } & { 0 } \\ { - 3 } & { 1 } & { 5 } & { - 1 } & { - 1 } & { 0 } & { 0 } & { 0 } \\ { - 3 } & { 1 } & { 2 } & { - 1 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 1 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { 0 } \end{array} } \right] .
D ( x , x ^ { \prime } ) = \sum _ { n e v e n } \int { \frac { d ^ { 4 } k } { ( 2 \pi ) ^ { 4 } } } \int _ { 0 } ^ { \infty } { \frac { q d q } { \pi } } { \frac { e ^ { i k _ { \mu } ( x ^ { \mu } - x ^ { \mu } ) } e ^ { i n ( \theta - \theta ^ { \prime } ) } } { k _ { 4 } ^ { 2 } + q ^ { 2 } } } J _ { n } ( q r ) J _ { n } ( q r ^ { \prime } ) .
\left\langle n \right\rangle = { \frac { 1 } { 4 8 } } { \left( \frac { m k _ { B } } { { \hbar ^ { 2 } \pi } } \right) ^ { 3 / 2 } } { \frac { T _ { F } ^ { 3 } } { T ^ { 3 / 2 } } }
\theta _ { \chi } \left( { \frac { a z + b } { c z + d } } \right) = \chi ( d ) \left( { \frac { - 1 } { d } } \right) ^ { \nu } \left( { \frac { \theta _ { 1 } \left( { \frac { a z + b } { c z + d } } \right) } { \theta _ { 1 } ( z ) } } \right) ^ { 1 + 2 \nu } \theta _ { \chi } ( z )
\begin{array} { r l r l } { f _ { 2 n - 1 } ^ { ( n ) } } & { = f _ { 2 n - 1 } ^ { ( n - 1 ) } + c _ { 2 n - 1 } , } & { f _ { 2 n + 1 } ^ { ( n ) } } & { = \left\{ \begin{array} { l l } { c _ { 5 } + f _ { 3 } ^ { ( 1 ) } \{ c _ { 3 } \} - \frac { 1 } { 2 } c _ { 3 } \{ c _ { 3 } \} , } & { n = 2 ; } \\ { c _ { 2 n + 1 } + f _ { 3 } \{ c _ { 2 n - 1 } \} , } & { n > 2 ; } \end{array} \right. } \end{array}
\begin{array} { r l } { \delta \mathcal { B } _ { 1 2 , \widehat { \boldsymbol { \sigma } } } \left( c _ { 1 } ^ { 2 } + c _ { 2 } ^ { 2 } \right) = } & { { } 2 \overline { { \alpha } } ( \overline { { \alpha } } - 1 ) ( \mathbf { c } _ { 1 2 } \cdot \widehat { \boldsymbol { \sigma } } ) ^ { 2 } + 2 \overline { { \beta } } ( \overline { { \beta } } - 1 ) } \\ { \delta \mathcal { B } _ { 1 2 , \widehat { \boldsymbol { \sigma } } } \left( w _ { 1 } ^ { 2 } + w _ { 2 } ^ { 2 } \right) = } & { { } \frac { 2 \overline { { \beta } } ^ { 2 } } { \kappa \theta } \left( \widehat { \boldsymbol { \sigma } } \times \mathbf { c } _ { 1 2 } \right) ^ { 2 } + 8 \frac { \overline { { \beta } } } { \kappa } \left( \frac { \overline { { \beta } } } { \kappa } - 1 \right) } \end{array}
\begin{array} { r l } { { L ( m _ { \pi ^ { 0 } \eta } ^ { 2 } ) } } & { { = \ \frac { 1 } { 2 ( a - b ) } - \frac { 2 } { ( a - b ) ^ { 2 } } \left[ f \left( \frac { 1 } { b } \right) - f \left( \frac { 1 } { a } \right) \right] } } \\ { { } } & { { + \ \frac { a } { ( a - b ) ^ { 2 } } \left[ g \left( \frac { 1 } { b } \right) - g \left( \frac { 1 } { a } \right) \right] \ , } } \end{array}
d { \tilde { \rho } } _ { K } \left( X \right) = \sum _ { k = 1 } ^ { 2 K } { \overbrace { { \tilde { \rho } } _ { 1 / 2 } \left( 1 \right) \otimes \cdots \otimes { \tilde { \rho } } _ { 1 / 2 } \left( 1 \right) } ^ { k - 1 } \otimes d { \tilde { \rho } } _ { 1 / 2 } \left( X \right) \otimes \overbrace { { \tilde { \rho } } _ { 1 / 2 } \left( 1 \right) \otimes \cdots \otimes { \tilde { \rho } } _ { 1 / 2 } \left( 1 \right) } ^ { 2 K - k } } \ ,
\left\langle \left[ \partial _ { \mu y } A _ { \nu } ^ { m } ( y ) \right] \left[ g \epsilon ^ { 3 m n } a ^ { \mu } ( x ) A ^ { n \nu } ( x ) \right] \right\rangle \Bigr | _ { y = x } = - 8 i g a ^ { \mu } ( x ) \partial _ { \mu y } \mathcal P ( y , x ) \Bigr | _ { y = x } .
\begin{array} { r } { \langle x _ { s } \rangle = \frac { \alpha _ { g } \alpha _ { l } [ 2 + ( \alpha _ { g } + \alpha _ { l } ) s + e ^ { ( \alpha _ { g } + \alpha _ { l } ) s } ( ( \alpha _ { g } + \alpha _ { l } ) s - 2 ) ] } { s ( \alpha _ { g } + \alpha _ { l } ) ^ { 2 } [ \alpha _ { g } + \alpha _ { l } e ^ { ( \alpha _ { g } + \alpha _ { l } ) s } ] } . } \end{array}
S _ { \operatorname* { m i n } } = - \rho W _ { 0 } \left[ - \frac { 1 } { \rho } \exp \left( - \frac { H _ { \mathrm { S I R } } } { \rho } \right) \right] \, , \quad S _ { \operatorname* { m a x } } = - \rho W _ { - 1 } \left[ - \frac { 1 } { \rho } \exp \left( - \frac { H _ { \mathrm { S I R } } } { \rho } \right) \right] \, .
\mathrm { S S P } ( \mathbf { y } _ { i } , \mathbf { \hat { y } _ { i } } ) = \mathrm { S S P } _ { i } = \frac { \sqrt { \int | F _ { \mathbf { y } _ { i } } ( k ) - F _ { \mathbf { \hat { y } } _ { i } } ( k ) | ^ { 2 } d k } } { \sqrt { \int | F _ { \mathbf { y } _ { i } } ( k ) | ^ { 2 } d k } + \sqrt { \int | F _ { \mathbf { \hat { y } } _ { i } } ( k ) | ^ { 2 } d k } } \in [ 0 , 1 ] ,
\frac { 1 } { q } \varepsilon _ { S S l } \log \left( 1 + \frac { 1 } { \varepsilon _ { S S l } } \cdot \frac { C ^ { * } } { q } \right) \sim \frac 1 q \left[ \varepsilon _ { S S l } \log \frac { 1 } { \varepsilon _ { S S l } } + \varepsilon _ { S S l } \log \frac { C ^ { * } } { q } + \cdots \right] .