harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Fraction\;P\!\!-\!wave={\frac{(\overline{{p}}p\rightarrow\pi^{+}\pi^{-})_{P}+(\overline{{p}}p\rightarrow\pi^{0}\pi^{0})_{P}}{(\overline{{p}}p\rightarrow\pi^{+}\pi^{-})_{S\&P}+(\overline{{p}}p\rightarrow\pi^{0}\pi^{0})_{P}}}.
J_{i}=\intd^{2}\sigma\sqrt{g}\left[4\xi_{i}^{2}\mathscr{M}_{i}^{(0)}-{\frac{1}{4}}\xi_{i}^{4}\mathscr{M}_{i}^{(2)}g^{ab}(\partial_{a}t_{\mu\nu})(\partial_{b}t_{\mu\nu})\right]+O(\xi_{i}^{6}),
\begin{array}{lll}{T_{\alpha{\dot{\alpha}}}}&{=}&{-2D_{\alpha}G{\bar{D}}_{\dot{\alpha}}G{\mathcalF}^{\prime\prime}+G[D_{\alpha},{\bar{D}}_{\dot{\alpha}}]{\mathcalF}^{\prime}\,}\\{K_{\alpha{\dot{\alpha}}}}&{=}&{2G\partial_{\alpha{\dot{\alpha}}}{\mathcalF}^{\prime}=-2\partial_{\alpha{\dot{\alpha}}}({\mathcalF}-G{\mathcalF}^{\prime})\,}\\{J}&{=}&{0\.}\\\end{array}
\begin{array}{ll}{\Gamma^{(1)}\left[A_{\mu},V\right]}&{=\left.\displaystyle{\frac{d}{ds}}\right|_{s=0}\left(\displaystyle{\frac12}\displaystyle{\frac{1}{\Gamma\left(s\right)}}\displaystyle{\int_{0}^{\infty}}\mathrm{d}tt^{s-1}\mathrm{Str}\;\mathrm{e}^{-Ht}\right)}\\{{}}&{\equiv\frac12\zeta^{\prime}\left(0\right).}\\\end{array}
\left(1-{\frac{r_{0}}{r}}\right){\frac{r}{r_{0}}}{\frac{d}{dr}}\left(r^{2}{\frac{df_{\ell}}{dr}}\right)+\left(2\ell+(2\ell+1+2ikr_{0}){\frac{r_{0}}{r}}\right){\frac{r^{2}}{r_{0}}}\;{\frac{df_{\ell}}{dr}}+\ell(-\ell+2ikr_{0})\;f_{\ell}=0
=(2\pi)^{\frac{d}{2}}\e^{\eta\xi}\int_{0}^{\infty}{\frac{\mathrm{d}t}{(2t)^{\frac{d-2r}{2}}(2t+1)^{r}}}e^{-{\frac{2\eta\xi}{2t+1}}-{\frac{y^{2}}{4t}}}.
\sigma_{\mathrm{string}}^{(\mathrm{open})}(D)=A(D)\beta_{H}^{-\frac{(D+1)}{2}}\:r_{S}^{D-2}\:\left(\frac{c^{2}\beta_{S}}{b}\right)^{-\frac{(D-3)}{2}}\:\cdot\:\lbrace\frac{\beta_{H}}{\beta_{H}-\beta_{S}}\:e^{-(\beta_{H}-\beta_{S})c^{2}m_{0}}\:-\:E_{i}\left(-(\beta_{H}-\beta_{S})c^{2}m_{0}\right)\rbrace
U(q,a)\=\\frac{g^{2}}2\left(|q|^{2}-|\tilde{q}|^{2}\right)^{2}+2g^{2}\left|\tilde{q}q+\frac{i}{2}\,\eta+\frac{\mu_{D}}{\sqrt2}\,a\right|^{2}+2|a|^{2}\left(|q|^{2}+|\tilde{q}|^{2}\right)\.
h\rightarrowh+\frac{\Gamma(1/2)}{(4\pi)^{3/2}}R[\int_{B^{3}}[\frac{\lambda^{2}}{4}\hat{\phi}^{4}-\frac{\lambda}{6}\Box\hat{\phi}^{2}](\frac{\lambda}{2}\hat{\phi}^{2})^{-1/2}dv+\int_{S^{2}}\frac{4}{315R^{3}}(\frac{\lambda}{2}\hat{\phi}^{2})^{-1/2}ds].
V\Bigl(V(\psi\otimes\bar{v},z-\zeta)\;(\chi\otimes\bar{w}),\zeta\Bigr)\;(\varphi\otimes\bar{u})=\sum_{\bar{v}^{\prime},\bar{w}^{\prime},\bar{u}^{\prime}}{\Phi\;}_{\bar{v}^{\prime}\;\bar{w}^{\prime}\;\bar{u}^{\prime}}^{\bar{v}\;\bar{w}\;\bar{u}}\;\;V(\psi\otimes\bar{v}^{\prime},z)\;V(\chi\otimes\bar{w}^{\prime},\zeta)\;(\varphi\otimes\bar{u}^{\prime}),
L=L_{m}+L_{ext},\\\\L_{m}=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}(m^{2}-\xiR)\varphi^{2}-\frac{\lambda}{24}\varphi^{4}+a_{5}\Box\varphi^{2},
{\calG}_{0}(x,y)=\frac1{2\pii}\left(\begin{array}{cc}{0}&{\frac1{X-Y}\left(\fracXY\right)^{k+1}}\\{\frac1{X^{*}-Y^{*}}\left(\frac{Y^{*}}{X^{*}}\right)^{k+1}}&{0}\\\end{array}\right),
N:\z^{\alpha}\toz^{\alpha\,\prime}=\left(\frac{\partialz^{\prime}}{\partialz}\right)^{1/2}N^{-1}{}_{\beta}{}^{\beta}\qquad\bar{z}^{\alpha}\rightarrow\bar{z}^{\alpha\,\prime}=\left(\frac{\partial\bar{z}^{\prime}}{\partial\bar{z}}\right)^{1/2}N^{-1}{}_{\beta}{}^{\alpha}\bar{z}^{\beta}\;,
\Psi_{j}\left({\bfr},t\right)=\frac{\mathrm{e}^{i{\bfk}_{j}\cdot{\bfr}-iE_{{\bfk}_{j}}t}}{\sqrt{2\vertE_{{\bfk}_{j}}\vert\Omega}}\,,\quad\Phi\left({\bfr},t\right)=\frac{g}{\mu_{0}^{2}}\frac{N}{2\vertE_{{\bfk}_{j}}\vert\Omega}\,,\quad\Omega\rightarrow\infty\,,
S_{\mathrm{YM}}[A]=\pm\frac{8\pi^{2}}{g^{2}}K[A]+\frac{1}{8\pi^{2}}\intd^{4}x\left(F_{\mu\nu}^{a}\mp\tilde{F}_{\mu\nu}^{a}\right)\left(F_{a}^{\mu\nu}\mp\tilde{F}_{a}^{\mu\nu}\right)\geq\frac{8\pi^{2}}{g^{2}}\left|K[A]\right|\,.
D_{{\mu}{\nu}}^{p}(q)=\frac{i}{q^{2}+i{\epsilon}}\left(-g_{{\mu}{\nu}}+\frac{n_{\mu}q_{\nu}+n_{\nu}q_{\mu}}{q_{-}}-n^{2}\frac{q_{\mu}q_{\nu}}{q_{-}^{2}}\right)-{\delta}_{{\mu}+}{\delta}_{{\nu}+}\frac{i}{q_{-}^{2}},
\int_{0}^{1}dt\delta_{\vec{u}}(t)<\mathrm{Tr}(H_{12}(\gamma)H_{34}(\gamma))>=-{\frac{i\pi}{k}}\mathrm{Tr}(H_{12}(\gamma)H_{34}(\gamma))+{\frac{2i\pi}{k}}\mathrm{Tr}(H_{12}(\gamma))\mathrm{Tr}(H_{34}(\gamma))
\frac{g_{1}^{2}}{4\pim_{0}^{2}}=\alpha\left(\sigma_{1}\right)\,\frac{\mu_{0}}{\sigma_{1}m_{0}}\sqrt{1-\frac{1}{4}\left(\frac{\mu_{0}}{\sigma_{1}m_{0}}\right)^{2}}=\alpha_{0}+\frac{\alpha_{1}}{\sqrt{2}}\frac{\mu_{0}}{m_{0}}+O\left(\left(\frac{\mu_{0}}{m_{0}}\right)^{2}\right)\,,
R_{r_{g}}^{(z)}(r)=B_{\mathrm{in}}\left(\Omega_{\beta}^{(+)}\frac{1}{r^{1-\beta}}+i\Omega_{\beta}^{(-)}\frac{1}{r^{\beta}}\right)+B_{\mathrm{out}}\left(\Omega_{\beta}^{(+)}\frac{1}{r^{1-\beta}}-i\Omega_{\beta}^{(-)}\frac{1}{r^{\beta}}\right),
\begin{array}{lllll}{\mathrm{for~any~\psi~,}}&{\mathrm{at~}\phi=+1}&{~:~}&{(i^{k},1,\pm\sqrt{\psi},z_{4},z_{5},1,0)}&{,(k=0,...,3)~,}\\{}&{\mathrm{or~}\phi=-1}&{~:~}&{(i^{k}e^{i\pi/4},1,\pm\sqrt{\psi},z_{4},z_{5},1,0)}&{,(k=0,...,3)~,}\\{\mathrm{and~for~any~\phi~,}}&{\mathrm{at~}\psi=\pm\sqrt{\phi^{2}-1}}&{~:~}&{(\phi^{1/4}i^{k},1,0,z_{4},z_{5},1,0)}&{,(k=0,...,3)~,}\\\end{array}
{\calL}=-\frac{1}{4}\varepsilonF_{\mu\nu}F^{\mu\nu}=\frac{1}{2}\varepsilon({\bfE}^{2}-{\bfB}^{2})=\frac{1}{2}({\bfE}\cdot{\bfD}-{\bfH}\cdot{\bfB})=-\frac{1}{2}\mu({\bfH}^{2}-{\bfD}^{2}).
\mathrm{(f)}+\mathrm{(g)}=-\frac{d_{\Gamma}g^{2}}{2x_{ij}^{4}}\sum_{k}\frac{x_{jk}^{\mu}x_{ik}^{\nu}+x_{jk}^{\nu}x_{ik}^{\mu}-\delta^{\mu\nu}x_{jk}\cdotx_{ik}}{x_{jk}^{2}\,x_{ik}^{2}},
e^{3\varphi}=\left({\frac{r-r_{H}-2\eta_{Q}{\bfQ}_{ik\,\infty}}{r-r_{H}+2\eta_{P}{\bfP}_{ij\,\infty}}}\right)^{\frac{1}{2}}=\left({\frac{r-2|{\bfP}_{ij\,\infty}|}{r-2|{\bfQ}_{ik\,\infty}|}}\right)^{\frac{1}{2}},
\Omega_{+2}^{+2i}(z^{M})+\Omega_{-2}^{-2i}(z^{M})+4i\left(\psi_{+2q}^{1+}(z^{M})\gamma_{q\dot{q}}^{i}\psi_{-2\dot{q}}^{1-}(z^{M})-\psi_{-2q}^{1+}(z^{M})\gamma_{q\dot{q}}^{i}\psi_{+2\dot{q}}^{1-}(z^{M})+1\leftrightarrow2\right)=0.
\intd^{2}\sigma\,\sqrt{\hat{g}}\biggl\{-c_{1}\hat{g}_{ab}\partial_{a}t\partial_{b}t+c_{2}e^{\varphi}+c_{3}\hat{R}+\frac{25}{96\pi}(\hat{g}^{ab}\partial_{a}\varphi\partialb\varphi+2\hat{R}\varphi)\biggr\}\.
(\Box-m^{2})\,\Phi(\rho)\,e^{i\vec{k}\cdot\vec{x}}=0\quad\Rightarrow\quad[\rho^{2}\partial_{\rho}^{2}+(1-d)\,\rho\partial_{\rho}-\vec{k}^{2}\,\rho^{2}-m^{2}]\Phi(\rho)=0.
\hat{\calS}_{ij}^{\alpha\beta}\equiv\epsilon_{~~\mu\nu}^{\alpha\beta}\bigg(\delta_{ij}\hat{\theta}^{\mu}\hat{P}^{\nu}+{\itg_{2}}\big(\tilde{F}_{(ij)}^{\mu\nu}+{\sqrt2}\hat{\theta}_{5}F_{(ij)}^{\mu\nu}\big)\bigg)+{\frac{i\delta_{ij}}{2\kappa}}\big(e_{A}-{\sqrt2}e_{B}\hat{\theta}_{5}\big)\sigma^{\alpha\beta}\,,
{\calS}_{WZW}\,=\,\frac{\kappa}{4\pi}\int_{\Sigma}d\sigma^{2}Tr(g^{-1}\partialg\,g^{-1}\bar{\partial}g)-i\frac{\kappa}{6\pi}\int_{B}d\sigma^{3}\epsilon^{abc}Tr(g^{-1}\partial_{a}g\wedgeg^{-1}\partial_{b}g\wedgeg^{-1}\partial_{c}g)\,,
H={\frac{1}{2}}\sum_{j=1}^{r}p_{j}^{2}+{\frac{1}{2}}\sum_{j=1}^{r}\left[\left({\frac{\partialW_{0}}{\partialq_{j}}}\right)^{2}+{\frac{\partial^{2}W_{0}}{\partialq_{j}^{2}}}\right]-2a{\mathcalM}\sum_{j=1}^{r}q_{j}^{2},
\begin{array}{rcl}{F_{01}}&{=}&{R(r,u)^{-4}\left[Q(r^{2}-(N+au)^{2})\,-\,2Pr(N+au)\right]\;,}\\{}&{}&{}\\{F_{23}}&{=}&{-R(r,u)^{-4}\left\{P(r^{2}-(N+au)^{2})\,+\,2Qr(N+au)\right\}\;,}\\\end{array}
\begin{array}{rcl}{}&{}&{{\calK}_{ij}^{(2)\prime}-2(2\pi\alpha^{\prime})\frac{(k.h)}{k^{2}}{\calK}_{[i}^{(1)\prime}{\calK}_{j]\sigma}^{(2)\prime}=-{\calG}_{ij}^{(2)}}\\{}&{}&{}\\{}&{}&{{\calK}_{i}^{(1)\prime}{\calK}_{j}^{(1)\prime}+e^{2\phi}{\calK}_{i\sigma}^{(2)\prime}{\calK}_{j\sigma}^{(2)\prime}=e^{\phi}{\calK}_{i}^{T}M{\calK}_{j}\,.}\\\end{array}
\frac{\partial\Phi_{n,l}^{*}(z^{-1})}{\partialt_{-1}}=-\frac{S_{n,l}^{*}}{S_{n-1,l}^{*}}\frac{h_{n,l}}{h_{n-1,l}}(\Phi_{n,l}^{*}(z^{-1})-z^{-1}\Phi_{n-1,l}^{*}),
J(\Theta_{1}^{\mu},\Theta_{1}^{\nu},\Theta_{2}^{\sigma})=\frac{1}{M}(g^{\sigma\nu}P^{\mu}-g^{\sigma\mu}P^{\nu})\Psi_{1}+(g^{\sigma\nu}\Theta_{2}^{\mu}-g^{\sigma\mu}\Theta_{2}^{\nu})\Psi_{1}.
\Pi_{\mu\nu\rho\sigma}^{\mathrm{\mathrm{Gvac}}}(p)\sim\kappa^{2}\Lambda_{\mathrm{Gvac}}^{4}N_{\mu\nu\rho\sigma}(p^{2})\sim\frac{\Lambda_{\mathrm{Gvac}}^{4}}{M_{\mathrm{PL}}^{2}}N_{\mu\nu\rho\sigma}(p^{2}),
a_{-n_{1}}^{\lambda_{1}}...a_{-n_{p}}^{\lambda_{p}}\epsilon_{-r_{1}}^{\rho_{1}}...\epsilon_{-r_{q}}^{\rho_{q}}\phi_{j}^{j}(0)\vert0\rangle{}~,{}~{}~{}~n_{i}>0{}~,{}~r_{k}\geq0{}~.
M^{\prime}={\calU}^{T}\left(\begin{matrix}{U_{5}}&{0}\\{0}&{U_{21}}\\\end{matrix}\right){\calU}M{\calU}^{T}\left(\begin{matrix}{U_{5}^{T}}&{0}\\{0}&{U_{21}^{T}}\\\end{matrix}\right){\calU},
\{\Pi_{i}(x),\theta(y)\}^{*}=3D{\frac{\partial_{i}}{\nabla=^{2}}}\delta(x-y);~~\{\Pi_{i}(x),\Pi_{0}(y)\}^{*}=3D-m^{2}{\frac{\partial_{i}}{={\nabla^{2}}}}\delta(x-y).
\left\{\Phi,\Psi\right\}\left(M\right)=\left\langle\left\langle{\calR}\left(\begin{array}{c}{\nabla\Phi}\\{\nabla^{\prime}\Phi}\\\end{array}\right),\left(\begin{array}{c}{\nabla\Psi}\\{\nabla^{\prime}\Psi}\\\end{array}\right)\right\rangle\right\rangle,
\frac{\delta\Sigma}{\delta\bar{\varphi}^{a}}-\partial_{\mu}[\sqrt{g}g^{\mu\nu}\frac{\delta\Sigma}{\delta\tau^{a\nu}}]-\partial_{\mu}[M^{\mu\nu}\frac{\delta\Sigma}{\delta\Omega^{a\nu}}]=\partial_{\mu}[s(\sqrt{g}g^{\mu\nu})\psi_{\nu}^{a}]+\partial_{\mu}(L^{\mu\nu}A_{\nu}^{a})-\partial_{\mu}(M^{\mu\nu}\psi_{\nu}^{a}).
Z_{0}=Tre^{-\beta(H_{D}+\frac{VB^{2}}{2})}=e^{-\beta\frac{VB^{2}}{2}}(1+e^{-\betaE_{0}})^{N_{d}}\;\prod_{n=1}^{\infty}(1+e^{-\betaE_{n}})^{2N_{d}}\;e^{\betaE_{n}N_{d}}\;
R^{\pm}(\zeta)=\left(\begin{array}{cccc}{a^{\pm}(\zeta)}&{}&{}&{d^{\pm}(\zeta)}\\{}&{b^{\pm}(\zeta)}&{c^{\pm}(\zeta)}&{}\\{}&{c^{\pm}(\zeta)}&{b^{\pm}(\zeta)}&{}\\{d^{\pm}(\zeta)}&{}&{}&{a^{\pm}(\zeta)}\\\end{array}\right).
\tilde{\Psi}_{l\zetaj}^{{\vec{k}}{\vec{m}}}\left(y_{\alpha}+L_{2}\right)=\left(e^{i\pi/\kappa}\right)^{N_{A}-1}e^{-i2\pij/\kappa}\tilde{\Psi}_{l\zeta,j}^{{\vec{k}}{\vec{m}}}\left(y_{\alpha}\right)
{\hat{K}}_{3}\equiv{\frac{1}{2}}\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+1\right)\,\quad{\hat{K}}_{+}\equiva_{1}^{\dagger}a_{2}^{\dagger}\,\quad{\hat{K}}_{-}\equiv-a_{1}a_{2}
\{J,H_{c}\}=\frac{1}{k}\intd^{2}\vec{x}d^{2}\vec{y}d^{2}\vec{z}\{G(\vec{x}-\vec{y})G(\vec{y}-\vec{z})-G(\vec{x}-\vec{z})G(\vec{y}-\vec{z})\}f_{abc}A_{2}^{a}(\vec{x})J_{2}^{b}(\vec{y})J_{\circ}^{c}(\vec{z})
M_{p-scalar\,\pm}^{2}~=~M_{\pm}^{2}~-~{\frac{3}{4}}\,\alpha\gamma\,\mu\,\widetilde{m}_{\lambda}\Bigg(1\pm\sqrt{1+x}~\Bigg)\Bigg(4~\pm~{\frac{1}{\sqrt{1+x}}}~\Bigg),
Z_{-}=\sum_{h=0}^{\infty}\int_{Met_{h}}{\calD}ge^{-S(g)+i{\frac{\Theta}{2\pi}}\int_{\Sigma_{h}}R\sqrtg}=\sum_{h=0}^{\infty}(-1)^{1-h}\int_{Met_{h}}{\calD}ge^{-S(g)},\quad\Theta={\frac{\pi}{2}}.
\frac{d}{d\Omega}P(t)\equiv|\vec{L}(t)|^{2}\sim|\left(\frac{e^{2}}{4\pic}\right)^{1/2}\left[(1+\frac{3\lambda}{2\zeta})\frac{1}{\zeta^{3}}\;\vec{n}\times(\vec{\beta}\times\dot{\vec{\beta}})\;\right]_{\mathrm{ret}}|^{2}\;,
\frac{1}{\gamma}(\partial_{i}\dot{A}_{i}^{a}+\epsilon^{abc}A_{i}^{b}\dot{A}_{i}^{c})+\frac{k}{8\pi}\epsilon^{ij}F_{ij}^{a}+\frac{1}{2}\mathrm{Tr}(\partial_{t}\phi[T^{a},\phi])=0.
K_{F}[\psi]+I[j^{\mu}]+\intd^{3}xs_{\mu}j^{\mu}\leftrightarrowK_{B}[A]+I[\epsilon^{\mu\nu\rho}\partial_{\nu}A_{\rho}]+\intd^{3}xs_{\mu}\epsilon^{\mu\nu\rho}\partial_{\nu}A_{\rho}\;.
\langlej_{m,n}^{0}\rangle=C{\frac{e}{16}}\langle[(\psi_{1}^{\dagger}(1+T))_{1}+(\psi_{1}^{\dagger}(1+T))_{0}][((1+T^{\dagger})\psi_{1})_{1}+((1+T^{\dagger})\psi_{1})_{0}]\rangle,
L=\frac{1}{2}(D_{t}A)_{ai}(D_{t}A)_{ai}+\frac{i}{2}\left({\bar{\Psi}}_{\alpha}\gamma_{0}\dot{\Psi}_{\alpha}-\dot{{\bar{\Psi}}}_{\alpha}\gamma_{0}{\Psi}_{\alpha}\right)-m\bar{\Psi}_{\alpha}\Psi_{\alpha}-g\rho_{a}\,Y_{a}+gj_{ia}A_{ai}-V(A)~,
\begin{array}{l}{\psi_{0}^{0}=\displaystyle\frac{1}{2}D_{0}^{0},\quad\psi_{1}^{0}=\displaystyle\frac{1}{2}t_{f}\partial_{r}^{f}D_{0}^{0},\quad\psi_{1}^{1}=D_{r}^{r},\quad\psi_{2}^{2}=\psi_{3}^{3}=0,}\\{\omega_{1}^{1}=\omega_{2}^{2}=\omega_{3}^{3}=1,\quad\omega_{1}^{0}=t_{f}\partial_{r}^{f}D_{0}^{0}.}\\\end{array}
\left(\begin{matrix}{0}&{0}&{0}&{1}\\{0}&{0}&{1}&{0}\\{0}&{1}&{0}&{0}\\{1}&{0}&{0}&{0}\\\end{matrix}\right);\left(\begin{matrix}{0}&{1}&{0}&{0}\\{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{1}&{0}\\\end{matrix}\right);\left(\begin{matrix}{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\\end{matrix}\right)
\Gamma[g]=\frac{k\psi^{2}}{24\pi}\int_{{\cal{B}}}d^{3}X\epsilon^{\alpha\beta\gamma}tr(\tilde{g}^{-1}\partial_{\alpha}\tilde{g}\tilde{g}^{-1}\partial_{\beta}\tilde{g}\tilde{g}^{-1}\partial_{\gamma}\tilde{g})
b_{n}^{i}=A_{n}a_{n}^{i}-B_{n}^{*}\tilde{a}_{n}^{i^{\dagger}},\qquad\tilde{b}_{n}^{i}=A_{n}\tilde{a}_{n}^{i}-B_{n}^{*}a_{n}^{i^{\dagger}}.
e_{\bot}(x_{\bot})=g\nabla_{\bot}\intdy^{-}dy_{\bot}d(x_{\bot}-y_{\bot})\left\{f^{3ab}A_{\bot}^{a}(y_{\bot},y^{-})\Pi_{\bot}^{b}(y_{\bot},y^{-})+\rho_{m}^{3}(y_{\bot},y^{-})\right\}\frac{\tau^{3}}{2}\,.
\begin{array}{lcl}{w_{3}}&{=}&{u_{3}-2\thetau_{2}^{\prime}}\\{w_{4}}&{=}&{u_{4}-\frac{9}{100}u_{2}^{2}-\thetau_{3}^{\prime}+\frac{{4\theta^{2}}}{5}u_{2}^{\prime\prime}}\\\end{array}
\intd^{4}x\mathrm{Tr}{\calR}_{\mu\nu}H_{3}(\Box){\calR}^{\mu\nu}=\intd^{4}x\left[\frac{1}{2}RH_{3}(\Box)R-2R_{\mu\nu}H_{3}(\Box)R^{\mu\nu}+{\calO}({\calR}^{3})\right].
S=-\frac{1}{16\piG}\int_{\calM}d^{n+2}x\sqrt{-g}\left(R-\frac{n(n+1)}{l^{2}}\right)+\frac{1}{8\piG}\int_{\partial\calM^{-}}^{\partial\calM^{+}}d^{n+1}x\sqrt{h}K.
l_{p}^{-2}ds^{2}=\hat{H}^{-1/3}\left[-\hat{f}dt^{2}+\sum_{i=1}^{5}(dx^{i})^{2}+\hat{H}\Big(d\hat{z}^{2}+\hat{f}^{-1}d\hat{r}^{2}+\hat{r}^{2}d\Omega_{3}^{2}\Big)\right]
i\Pi_{\zeta\mu\nu}^{\underline{{a}}\underline{{b}}}(q)=-\frac{Ng^{2}}{2}\delta_{\underline{{a}}\underline{{b}}}\mu^{2\epsilon}\int\frac{d^{n}\ell}{(2\pi)^{n}}\frac{(2\ell+q)_{\mu}(2\ell+q)_{\nu}}{\ell^{2}(\ell+q)^{2}}
\langlet,\tau\vertt^{\prime},0\rangle=\delta(t-t^{\prime})-i\int_{0}^{\tau}d\tau^{\prime}\intdt^{\prime\prime}\langlet,\tau^{\prime}\vertt^{\prime\prime},0\rangle\theta(t^{\prime\prime}-t^{\prime})A(t^{\prime})e^{-im(t^{\prime\prime}-t^{\prime})}\;.
A=(A^{\prime}+B^{\prime})/{\sqrt{2}}\,,\;\;\;B=(A^{\prime}-B^{\prime})/{\sqrt{2}}\,,\;\;\;\psi=(i\Gamma_{5})^{\frac{1}{2}}\psi^{\prime}=\frac{1+i\Gamma_{5}}{\sqrt{2}}\psi^{\prime}\,,
{\calF}_{IIA}=-\frac{1}{6}C_{ABC}t^{A}t^{B}t^{C}-\frac{\chi\zeta(3)}{2(2\pi)^{3}}\sum_{d_{1}\dots,d_{n}}n_{d_{1},\dots,d_{n}}{\calL}_{i_{3}}[e^{2\pii[\sum_{k}d_{k}t^{k}]}],
V(\varphi,\chi)=\frac{1}{2}m^{2}\varphi^{2}+\frac{1}{2}\mu^{2}\chi^{2}+\frac{\lambda_{\varphi}}{4}\varphi^{4}+\frac{\lambda_{\chi}}{4}\chi^{4}-\frac{\lambda_{\varphi\chi}}{2}\varphi^{2}\chi^{2}.
{\calZ}_{GU}={\displaystyle\int}{\calD}(A^{\mu},\pi_{\mu},\phi,\pi_{\phi},\mu)~exp\left(i{\displaystyle\int}dxdt\left[\pi_{0}{\dot{A}^{0}}+\pi_{1}{\dot{A}^{1}}+\pi_{\phi}{\dot{\phi}}-{\cal{\widetildeH}}_{GU}-\mu\chi\right]\right),
\betaM_{q}=\frac{1}{2\pil_{s}^{2}}\int_{0}^{\pi}dt\int_{1}^{\rho_{\mathrm{probe}}}d\rho\sqrt{g_{tt}g_{\rho\rho}}=\frac{1}{2\pil_{s}^{2}}r_{+}^{2}\pi(\rho_{\mathrm{probe}}-1)=\sqrt{\frac{\eta}{2}}(\rho_{\mathrm{probe}}-1),
W_{\mathrm{total}}=c\frac{\Lambda^{21/5}}{(\psi\psiH)^{2/5}}\F\left(\frac{\lambda\psi\psiH}{c\Lambda^{21/5}/(\psi\psiH)^{2/5}},\frac{MH^{2}/2}{c\Lambda^{21/5}/(\psi\psiH)^{2/5}}\right),
\overline{{R}}(z_{1}-z_{2})t_{d}^{a}(z_{1})\otimest_{c}^{d}(z_{2})=\sum_{b}\overline{{W}}\left(\left.\begin{array}{cc}{a}&{b}\\{d}&{c}\\\end{array}\right|z_{1}-z_{2}\right)t_{c}^{b}(z_{1})\otimest_{b}^{a}(z_{2}).
\frac{a(\lambda_{j})}{d(\lambda_{j})}\prod_{k=1\atop{k\nej}}^{N+1}\frac{f(\lambda_{j},\lambda_{k})}{f(\lambda_{k},\lambda_{j})}=1,\qquad\mathrm{or}\qquad\frac{a(\lambda_{j})}{d(\lambda_{j})}\prod_{k=1}^{N+1}\frac{h(\lambda_{j},\lambda_{k})}{h(\lambda_{k},\lambda_{j})}=(-1)^{N}
\sum_{\epsilon_{l}<0}u_{\epsilon_{l}l}\left(r\right)u_{\epsilon_{l}l}\left(r^{\prime}\right)+\int_{0}^{\infty}\frac{\mathrm{d}k}{2\pi}\,u_{kl}\left(r\right)u_{kl}\left(r^{\prime}\right)=\delta\left(r-r^{\prime}\right)\,,
\begin{array}{ccc}{\theta^{i}=\left(\begin{array}{cc}{\bar{\theta}_{a}}&{0}\\{0}&{\theta^{a}{}^{t}}\\\end{array}\right)~~~}&{\Longrightarrow}&{~~~\bar{\theta}_{i}=\left(\begin{array}{cc}{0}&{\theta^{a}}\\{-\bar{\theta}_{a}^{t}}&{0}\\\end{array}\right)}\\\end{array}
{\calL}_{HEC}^{(1)}(a,B)=\frac{1}{8\pi^{2}}\int_{0}^{\infty}\frac{ds}{s^{3}}e^{-sm^{2}}\biggl[\frac{\sqrt{\pis}}{2a}+\sum_{n=1}^{\infty}e^{-(an)^{2}/s}\biggr]\,seB\,{\calM}(seB)\;.
E_{\dot{q}}^{-}=(E^{++}-{\frac{1}{2}}\Psi_{-}^{I}d\Psi_{-}^{I})\psi_{++\dot{q}}^{~~-}=(E^{++}-E^{--}{\frac{1}{2}}\Psi_{-}^{I}{\calD}_{--}\Psi_{-}^{I})\psi_{++\dot{q}}^{~~-}
L_{\mathrm{super}}=p\dot{\omega}_{\theta}+iZ_{\alpha\beta}\theta^{\alpha}\dot{\theta}^{\beta}+i\bar{Z}_{\dot{\alpha}\dot{\beta}}\bar{\theta}^{\dot{\alpha}}\dot{\bar{\theta}}^{\dot{\beta}}+iZ_{\alpha\dot{\beta}}(\theta^{\alpha}\dot{\bar{\theta}}^{\dot{\beta}}-\dot{\theta}^{\alpha}\bar{\theta}^{\dot{\beta}})-\frac{e}{2}(p^{2}+m^{2})\,.
E_{ADM}=\frac{1}{16\piG_{10}}\oint_{\infty}d\Sigma^{\alpha}\lbrace\partial_{\beta}h_{\alpha\beta}-\partial_{\alpha}h_{\beta\beta}+{}^{\circ}g^{ij}({}^{\circ}D_{i}h_{\alphaj})-{}^{\circ}g^{ij}\partial_{\alpha}h_{ij}\rbrace
\begin{array}{ccccccc}{}&{}&{}&{1}&{}&{}&{}\\{}&{}&{0\!}&{}&{\!0}&{}&{}\\{}&{0\!}&{}&{h^{1,1}}&{}&{\!0}&{}\\{1\!}&{}&{h^{1,2}\!}&{}&{\!h^{1,2}}&{}&{\!1}\\{}&{0\!}&{}&{h^{1,1}}&{}&{\!0}&{}\\{}&{}&{0\!}&{}&{\!0}&{}&{}\\{}&{}&{}&{1}&{}&{}&{}\\\end{array}\,
\theta_{m,k}\:(\tau)\;\theta_{m^{\prime},k^{\prime}}\:(\tau)=\sum_{\ell=1}^{k+k^{\prime}}\;\theta_{mk^{\prime}-m^{\prime}k+2\ellkk^{\prime},kk^{\prime}(k+k^{\prime})}\:(\tau)\;\theta_{m+m^{\prime}+2\ellk,k+k^{\prime}}\:(\tau),
X_{\mu\nu}^{L}=\sum_{a}\sum_{s=0}^{\infty}D^{s}[x_{a\mu}\delta_{\nu}^{i}-x_{a\nu}\delta_{\mu}^{i}-v_{a}^{i}(x_{a\mu}\sigma_{a\nu}-x_{a\nu}\sigma_{a\mu}]\frac{\partial}{\partialx_{a}^{i(s)}},
\begin{array}{c}{\\Omega_{\mu}=\frac14\gamma_{;\mu}^{\nu}\,\gamma_{\nu}=-\frac14\Omega_{\muab}\Sigma_{ab}}\\{\Omega_{\muab}=\partial_{\mu}g_{\alphaa}g^{\alphab}-\Gamma_{\mu\beta}^{\alpha}g_{\alphaa}g^{\betab}}\\{\\Sigma_{ab}=\frac12(\gamma_{a}\gamma_{b}-\gamma_{b}\gamma_{a})\qquad\Sigma_{ab}^{2}=1\qquad\gamma^{a}\Sigma_{ab}=0}\\\end{array}
(M_{\mathrm{bh}}^{\mathrm{dyon}})^{2}=|Z^{\mathrm{local}}(t,\bar{t},q,p)|^{2}=\left|e^{\frac{K(t,\bar{t})}{2}}[X^{\Lambda}(t)q_{\Lambda}-F_{\Lambda}(t)\,p^{\Lambda}]\right|^{2}\,
u_{\underline{{m}}}^{(0)}\delta_{AB}=v_{A}^{~\underline{{\mu}}}\sigma_{\underline{{m}}~\underline{{\mu}}\underline{{\nu}}}v_{B}^{~\underline{{\nu}}},\qquadu_{\underline{{m}}}^{i}\Gamma_{AB}^{i}=v_{A}^{~\underline{{\mu}}}\sigma_{\underline{{m}}\underline{{\mu}}\underline{{\nu}}}v_{B}^{~\underline{{\mu}}}.
\langle\,\Lambda\,\vert\Lambda\,\rangle\,=1\,,\,\,\langle\,\Phi\vertX\Psi\,\rangle\,=\,\langle\,{^tX}\Phi\vert\Psi\,\rangle\,,\,\,\langle\,\Phi\vert\Psi\,\rangle=\langle\,\Psi\vert\Phi\,\rangle\,.
{\calL}=\partial_{\mu}V^{*}\partial^{\mu}V+\partial_{\mu}W^{*}\partial^{\mu}W+h(V^{*}+V)+\frac{1}{2\piN}\epsilon^{\mu\nu\lambda}w^{*}\partial_{\mu}w\partial_{\nu}(v^{*}\partial_{\lambda}v)
{\bfM}_{\Lambda}^{RL}=\frac{1}{2}\theta_{\Lambda}\Lambda-2\,,\quad{\bfM}_{\Lambda}^{Ri}=-\frac{1}{2\sqrt{2}}\hat{\beta}\theta_{\Lambda}\gamma^{i}\theta_{\Lambda}\,,\quad{\bfM}_{\Lambda}^{Li}=\frac{1}{2\sqrt{2}\hat{\beta}}\Lambda\gamma^{i}\Lambda\,,
\left(\begin{matrix}{m_{u}+m_{d}}&{{\frac{m_{u}-m_{d}}{\sqrt3}}}&{m_{u}-m_{d}}\\{{\frac{m_{u}-m_{d}}{\sqrt3}}}&{{\frac{m_{u}+m_{d}+4m_{s}}{3}}}&{{\frac{m_{u}+m_{d}-2m_{s}}{\sqrt3}}}\\{m_{u}-m_{d}}&{{\frac{m_{u}+m_{d}-2m_{s}}{\sqrt3}}}&{m_{a}}\\\end{matrix}\right)
G(x,0)=\frac{{\displaystyle\int}{\calD}S{\calD}{\bar{\psi}}{\calD}\psi~{\bar{\psi}}(x)~e^{-2i\gamma^{5}S(x)}~\psi(x)~{\bar{\psi}}(0)~{\displaystylee}^{-2i\gamma^{5}S(0)}~\psi(0)\,\,\,e^{iS_{eff}}}{{\displaystyle\int}{\calD}S{\calD}{\bar{\psi}}{\calD}\psi\,\,\,{\displaystylee}^{iS_{eff}}},
Z=\int\prod_{n=2}^{\calN}D[t_{n}]D[\bar{t}_{n}]D[{z^{\prime}}_{n}^{c}]D[\bar{z^{\prime}}_{n}^{c}]D[\sigma_{R}]~D[F^{0}]~\delta(\chi^{1})\delta(\chi^{2})|\mathrm{Det}\{\chi^{\mu},H_{\nu}\}|\delta(\frac{H}{\sqrt{h}})e^{iS}.
\deltaA_{m}^{\alpha}={\frac{\varphi(x)}{u^{2}}}{\calL}^{\alpha\beta}{\calF}_{mn}^{\beta}u^{n},\qquad\delta\Lambda_{mn}={\frac{\varphi(x)}{(u^{2})^{2}}}{\calF}_{m}^{\alphar}u_{r}{\calF}_{n}^{\betas}u_{s}{\calL}^{\alpha\beta}.
\frac{d^{2}f}{dx^{2}}+\frac{(a^{2}-6a+4-(a-2)^{2}x^{a/2}}{4x(1-x^{a/2})}\frac{df}{dx}+\frac{\lambda^{2}}{x^{3-a/2}(1-x^{a/2})^{2}}f-\frac{l(l+a)}{4x^{2}(1-x^{a/2})}f=0\,.
h_{2}^{2}=\partial_{2}X\partial_{2}Y~,~~h_{1}^{2}=\frac{\partial_{2}Y}{\partial_{2}Z}~,~~h_{3}^{2}=\frac{\partial_{2}X}{\partial_{2}Z}~,~~A_{1}=\frac{\partial_{1}X}{\partial_{2}X}~,~~A_{3}=\frac{\partial_{3}Y}{\partial_{2}Y}.
{\calZ}=\int{\calD}{\bar{\Psi}}\;{\calD}\Psi\;{\calD}A_{\mu}\;{\calD}A_{0}^{\pi\pi}\;e^{i\intd^{3}x\;{\calL}({\bar{\Psi}},\Psi,A_{\mu},A_{0}^{\pi\pi})}
\frac{1}{2r}\prod_{n=1}^{\infty}\left(1-r^{2n}\right)^{-4}\sum_{\alpha=2,3,4}e_{\alpha}\Theta_{\alpha}(-\mathrm{i}\epsilon|\mathrm{i}\tau)\left[\Theta_{\alpha}(0|\mathrm{i}\tau)\right]^{3}~,
{\calB}_{a}(\tau,\bar{\tau})=\frac{2}{|\eta(\tau)|^{4}}\sum_{{even}\;s}\frac{(-1)^{s_{1}+s_{2}}}{2\pii}\frac{dZ_{\psi}({\bfs},\bar{\tau})}{d\tau}\timestr_{s_{1}}\left[Q_{a}^{2}(-1)^{s_{2}F}q^{H}\bar{q}^{\bar{H}}\right]_{int}.
\mathcal{E}\left(x\right)=\int\mathrm{d}^{3}k\sqrt{\frac{2\omega}{\left(2\pi\right)^{3}}}\left\{\begin{array}{c}{\left[c^{1}e_{1}+c^{3}\left(e_{3}+e_{4}\right)\right]\mathrm{e}^{-ikx}+}\\{\left[c^{*2}e_{1}+c^{*4}\left(e_{3}+e_{4}\right)\right]\mathrm{e}^{ikx}}\\\end{array}\right\},\quad\omega\equiv\sqrt{\overrightarrow{k}^{2}},
{\widehatB_{i}}^{2}|_{\theta=0,\pi}={\frac{1}{2}}F_{(i)}^{2}|_{r\rightarrow\infty,\;\theta=0,\pi}=B_{i}^{2}{\frac{e^{\sigma_{i}^{0}}}{(\lambda_{i}|_{\theta=0,\pi})^{3}}},
\tilde{T}_{\tilde{\sigma}^{+}\tilde{\sigma}^{+}}=\left(\frac{d\sigma^{+}}{d\tilde{\sigma}^{+}}\right)^{2}T_{\sigma^{+}\sigma^{+}}\left(\sigma^{+}\right)+\frac{c}{12}S\left(\tilde{\sigma}^{+},\sigma^{+}\right)
N_{\alphaA}^{(\beta)\:\dagger}\;:=\;-i\delta_{\alpha\beta}\:q_{A}\:-\:i\Gamma_{\alpha\beta}^{j}\:\frac{\partial}{\partialx_{jA}}\;=\;F_{\beta}^{-1}\:M_{\alphaA}^{(\beta)\:\dagger}\:F_{\beta}\;=\;-N_{\alphaA}^{(\beta)}\;\;\;.
\left|\Psi\right\rangle=\psi_{0}\left|0\right\rangle+\psi_{1}\left|1\right\rangle+\psi_{2}\left|2\right\rangle+....,\\quad\left|n\right\rangle=\frac{a^{*n}}{\sqrt{n!}}\left|0\right\rangle,\quada\left|0\right\rangle=0
=\frac{M_{i}+\eta_{i}}{\sqrt{2}}\left(\begin{array}{c}{\psi_{lR}}\\{\psi_{l}+\psi_{u}}\\\end{array}\right)=\frac{M_{i}+\eta_{i}}{\sqrt{2}}\left(\begin{array}{c}{\psi_{1R}}\\{\sqrt{2}\psi_{1L}+\psi_{1R}}\\\end{array}\right).
\mathrm{Tr}\left(J_{i}^{(1)}J_{j}^{(1)}J_{k}^{(1)}\right)\epsilon_{ijk}=\mathrm{Tr}\left(J_{i}^{(2)}J_{j}^{(2)}J_{k}^{(2)}\right)\epsilon_{ijk}=iNC_{2}^{(2)}=\frac{i}{4}N(N^{2}-1).
\begin{array}{cl}{}&{A_{\lambda}^{(m)}(x|z^{(n)}|\cdots|z^{(1)})=\displaystyle\sum_{k=1}^{n}\prod_{j=1}^{m}(x-z_{j}^{(k)}\tau^{2k-n-1})\times}\\{}&{\times\displaystyle\frac{1}{x}f_{\lambda}^{((n-1)m)}(x\tau^{n+1-2k}|z^{(n)}\tau,\cdots,z^{(k+1)}\tau\stackrel{k}{\hat{~}}z^{(k-1)}\tau^{-1},\cdots,z^{(1)}\tau^{-1}).}\\\end{array}
S_{\mathrm{GM}}(W)\\rightarrow\S_{\mathrm{GM}}(W)\+\{\frac{i}{2}}\intd^{4}xd^{2}\theta\\widetildeW^{\alpha}W_{\alpha}\-\{\frac{i}{2}}\intd^{4}xd^{2}\bar{\theta}\\widetilde{\overline{{W}}}_{\dot{\alpha}}\overline{W}^{\dot{\alpha}}\.
Q_{2}=-\frac{1}{4\pi}\,\xi\,\left[\int\frac{\mathrm{d}\phi}{({\calW}^{\prime})^{3}}\right]^{-1/2}\int\mathrm{d}\phi\,\frac{{\calW}^{\prime\prime\prime}}{({\calW}^{\prime})^{2}}\,,
\left\{-\frac{\gamma}{8N^{2}}\intd\theta\left(\rho(\theta)\left(\frac{\partial}{\partial\theta}\frac{\delta}{\delta\rho(\theta)}\right)^{2}-N^{4}\frac{\pi^{2}}{3}\rho^{3}(\theta)\right)+V[\rho]\right\}~\tilde{\psi}[\rho]~=~E~\tilde{\psi}[\rho]\;,
\overline{{N}}_{mn}^{rs}\approx\frac{1}{2\pi}\frac{(-1)^{r(m+1)+s(n+1)}}{\alpha_{s}\omega_{rm}+\alpha_{r}\omega_{sn}}\sqrt{\frac{|\alpha_{r}\alpha_{s}|(\omega_{rm}+\mu\alpha_{r})(\omega_{sn}+\mu\alpha_{s})}{\omega_{rm}\omega_{sn}}}s_{rm}s_{sn},
L_{B}=-{\frac{1}{2\cdot3!}}\,H^{\,2}{}_{\mu\nu\rho}(B)-{\frac{m}{4}}\,\epsilon^{\mu\nu\rho\sigma}\,B_{\mu\nu}\,\partial_{\,[\,\rho}\,V_{\sigma\,]}-{\frac{1}{4}}\,F_{\mu\nu}(\,V\,)\,F^{\mu\nu}(\,V\,)
1+S=\left(\begin{array}{cc}{\frac{1}{2V_{oo}}\sqrt{1+V_{oo}}\left[\sqrt{1+V_{oo}}-\sqrt{1-3V_{oo}}\right]}&{0}\\{0}&{\frac{1}{2V_{ee}}\sqrt{1+3V_{ee}}\left[\sqrt{1+3V_{ee}}-\sqrt{(1-V_{ee})}\right]}\\\end{array}\right)\,,