harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R_{1,2}^{\prime\prime}+{\frac{1}{\rho}}R_{1,2}^{\prime}-{\frac{1}{\rho^{2}}}\left(l_{1,2}-{\frac{\phi\rho^{2}}{a^{2}}}\right)^{2}R_{1,2}+p_{1,2}^{2}R_{1,2}=0\,,
\partial_{\mu}=\sum_{r=1}^{n}\partial_{\mu}^{r+1}u\frac{\partial}{\partial(\partial_{\mu}^{r}u)}+\sum_{r=1}^{m}\partial_{\mu}^{r+1}\bar{u}\frac{\partial}{\partial(\partial_{\mu}^{r}\bar{u})}+\partial_{\mu}u\frac{\partial}{\partialu}+\partial_{\mu}\bar{u}\frac{\partial}{\partial\bar{u}}.
S_{WZ}=\intd^{4}x\,\biggl\{\frac{1}{2}\left[\hat{\Phi}^{\dagger}\hat{\Phi}\right]_{D}+\left[\mu\hat{\Phi}+\frac{m}{2}\hat{\Phi}^{2}+\frac{\lambda}{3!}\hat{\Phi}^{3}\,\mathrm{+~h.c.}\right]_{F}\biggr\}
L_{0}^{\beta}\left(N\right)=\frac{1}{2}\beta_{0}^{2}+\sum_{e\geq2}^{2N}\beta_{-e}^{x}\cdot\beta_{e}^{x}+\sum_{o\geq1}^{2N-1}\beta_{-o}^{p}\cdot\beta_{o}^{p}.
\sum_{l_{2}=1}^{*}{\frac{(-1)^{l_{2}}l_{2}^{2}}{(l_{1}^{2}-l_{2}^{2})^{3}}}=2\sum_{l_{2}=2,4,..}^{*}{\frac{l_{2}^{2}}{(l_{1}^{2}-l_{2}^{2})^{3}}}-\sum_{l_{2}=1}^{*}{\frac{l_{2}^{2}}{(l_{1}^{2}-l_{2}^{2})^{3}}}.
D=[Q,K]=\frac{1}{\pi}\int_{0}^{\pi}d\sigmac^{0}(\sigma)\frac{1}{2}\left[4\pi^{2}\alpha^{\prime}2\epsilonP_{i}P_{j}\eta^{ij}-2\epsilon\frac{1}{\alpha^{\prime}}X^{i}X^{j}\eta_{ij}\right]\.
S_{\sigma,s}^{(1)}(t,t^{\prime})=Q_{\tilde{\Gamma}_{s}}(t_{s})^{-2}\left[G_{b}^{(1)}(z_{s},z_{s}^{\prime})\theta_{s}^{\prime}+\theta_{s}G_{(\sigma)}^{(1)}(z_{s},z_{s}^{\prime})-\tilde{\varepsilon}_{s}^{\prime}\Sigma_{\sigma}(z_{s}^{\prime})\right]Q_{\tilde{\Gamma}_{s}}(t_{s}^{\prime})^{3}
\Gamma=-{\frac{1}{2(4\pi)^{d/2}}}\int_{0}^{\infty}{\frac{ds}{s^{1+{\frac{d}{2}}}}}e^{-m^{2}s}\sum_{l\geq0}{\frac{(-s)^{l}}{l!}}\intd^{d}x\sqrtga_{l}(x)
\rho+p_{1}=-\frac{\ddot{a}_{2}}{a_{2}}-\frac{\ddot{a}_{3}}{a_{3}}+\frac{\dot{a}_{1}}{a_{1}}\frac{\dot{a}_{2}}{a_{2}}+\frac{\dot{a}_{1}}{a_{1}}\frac{\dot{a}_{3}}{a_{3}}\ge0
\{\zeta_{A},u_{i}^{\pm}|x_{A}^{\mu}=x^{\mu}-2i\theta^{(i}\sigma^{\mu}{\bar{\theta}}^{j)}u_{(i}^{+}u_{j)}^{-},~\theta^{+}=\theta^{i}u_{i}^{+},~{\bar{\theta}}^{+}={\bar{\theta}}^{i}u_{i}^{+},~u_{i}^{\pm}\},
\left(\delta_{ij}M_{km}(u)\frac{\partial^{2}}{\partialu_{k}\partialu_{m}}+\delta_{ij}K_{m}(u)\frac{\partial}{\partialu_{m}}+L(u)_{ij}\right)\left(\begin{array}{c}{a_{j}}\\{a_{D}^{j}}\\\end{array}\right)=0.
\left({\vec{\Lambda}}^{(n)}\,\cdot\,{\vec{h}},-{\vec{\Lambda}}^{(m)}\,\cdot\,{\vec{h}}\right)\,=\,\left({\vec{W}}^{(p)}\,\cdot\,{\vec{h}}\right)\quad;\quad(n,m=1,...,28\,;\,p=1,...,56)
g_{\mu\nu}^{E}dx^{\mu}dx^{\nu}=-{\frac{1}{\left(H_{1}H_{2}H_{3}\right)^{\frac{2}{3}}}}fdt^{2}+\left(H_{1}H_{2}H_{3}\right)^{\frac{1}{3}}\left[f^{-1}dr^{2}+r^{2}d\Omega_{3}^{2}\right],
\frac{2\betaM^{2}}{(d-2)\Lambda}\,\sqrt{\rho_{0}\lambda_{1}}\,\left(\lambda_{0}\,+\,\lambda_{1}\right)\,=\,1\,{,}\quad\sqrt{\lambda_{0}\lambda_{1}}\,\left(\rho_{0}^{-1}\,-\,1\right)\,=\,\frac{(d-2)\pi}{6\beta^{2}M^{2}}\,\rho_{0}^{-1}\,{.}
{\calT}^{tt}=\rho_{\Phi}+\hat{\rho}{\frac{c_{\mathrm{ph}}}{c}},\\\\\\{\calT}^{ij}=\left(p_{\Phi}+\hat{p}{\frac{c_{\mathrm{ph}}}{c}}\right){\frac{1}{a^{2}}}\gamma^{ij}.
\langle{\calO}_{1}(x){\calO}_{1}(y)\rangle_{1}^{\mathrm{disc.}}=2\,\langle{\calO}_{\bf20}^{LM}(x){\calO}_{\bf20}^{NP}(y)\rangle_{0}\\langle{\calO}_{\bf20}^{LM}(x){\calO}_{\bf20}^{NP}(y)\rangle_{1}\,,
{\calH}={\frac{1}{2}}(\pi_{\phi}-eA_{1})^{2}+{\frac{1}{2}}\pi_{1}^{2}+{\frac{1}{2}}\phi^{2}+\pi_{1}A_{0}^{\prime}+e\phi^{\prime}A_{0}+{\frac{1}{2}}ae^{2}(A_{0}^{2}-A_{1}^{2}).
\eta_{+}^{A}=e^{-\frac{1}{2}k|y|}\left(\begin{array}{c}{\hat{\eta}_{R}}\\{-\hat{\eta}_{L}}\\\end{array}\right)^{A}\,\qquad\eta_{-}^{A}=\epsilon(y)e^{-\frac{1}{2}k|y|}\left(\begin{array}{c}{\hat{\eta}_{L}}\\{\hat{\eta}_{R}}\\\end{array}\right)^{A}\,
I_{A}=\frac{1}{\widetildeC_{A}}(-1)^{\frac{S-\widetilde{S}}{2}}\:\sum_{\nu_{0}=0}^{K-1}\:(-1)^{({\frac{d}{2}}+r)\nu_{0}}\prod_{j=1}^{r}\:n_{L_{j}\widetilde{L}_{j}}^{2\nu_{0}+M_{j}-\widetilde{M}_{j}}.
\nu_{p}(p)=\sum_{a_{1}=odd}^{2k-1}\left(3k+\frac{1}{2}-\frac{3a_{1}}{2}\right)+\sum_{a_{1}=even}^{2k}\left(3k+\frac{3a_{1}}{2}+1\right)=3k^{2}=\frac{1}{12}p^{2}
A=\frac{d^{l+m}}{dx^{l+m}}(1-x)^{l^{-}}(1+x)^{l^{+}}=\sum_{k=0}^{l+m}{\binom{l+m}{k}}\frac{d^{k}}{dx^{k}}(1-x)^{l^{-}}\frac{d^{l+m-k}}{dx^{l+m-k}}(1+x)^{l^{+}}.
Z_{N}(\tau,\bar{\tau})=\sum_{\scriptsize\begin{array}{c}{N_{n},M_{n,m}\geq1}\\{\sumnN_{n}=N}\\{\summM_{n,m}=N_{n}}\\\end{array}}\prod_{n,m\geq1}\left(\frac{1}{m}Z(n,m|\tau,\bar{\tau})\right)^{M_{n,m}}.
S_{T}[\overline{{\psi}},\psi]=\frac{1}{2}\;g\;\sum_{b,b^{\prime}=1}^{N}\intd^{2}xj_{\mu}^{(b)}(x)\Bigl(\delta_{\mu\nu}-\lambda^{\prime}\partial_{\mu}\partial_{\nu}\Big)^{-1}j^{(b^{\prime})}(x)\;.
\zeta(s)=\int{\frac{d^{4}p}{(2\pi)^{4}}}{\frac{1}{[p^{2}+m^{2}]^{s}}}={\frac{1}{2}}{\frac{2\pi^{2}}{(2\pi)^{4}}}{\frac{m^{4-2s}}{(s-2)(s-1)}}
\partial_{t}g_{1}\,g_{1}^{-1}\,|_{\varphi=0}|_{r=-\bar{r}}^{r=\bar{r}}=0\qquad,\qquad\partial_{t}g_{2}\,g_{2}^{-1}\,|_{\varphi=0}|_{r=-\bar{r}}^{r=\bar{r}}=0\qquad,\qquad
\overline{{V_{A}^{a}}}=V_{\dot{A}a},\overline{{V^{Aa}}}=-V_{a}^{\dot{A}},\overline{{V_{Aa}}}=-V_{\dot{A}}^{a},\overline{{V_{a}^{A}}}=V^{\dot{A}a},\overline{{\varepsilon_{ab}}}=\varepsilon^{ab},\overline{{\chi^{Aa}\psi^{Bb}}}=-\overline{{\chi^{Aa}}}\\overline{{\psi^{Bb}}}
6e^{2\sigma(\phi)}\left({\calB}\sigma_{i}\sigma_{j}+\frac{1}{2}\left({\calB}_{i}\sigma_{j}+\sigma_{i}{\calB}_{j}\right)\right)=A(\phi)_{ij}\;,\;\;\;{\calB}(\phi)e^{2\sigma(\phi)}=B(\phi)\;,\;\;\;\frac{\lambda(\phi)}{{\calB}(\phi)}=\frac{\Lambda(\phi)}{B(\phi)}
S=-2^{k}(0.606)^{2k+1}1.2T_{p}\int_{R^{1,8-2k}}d^{p+1}x\left[1+\frac{(2\pi\alpha^{\prime})^{2}}{4}(F_{\mu\nu}F^{\mu\nu}+2\partial_{\mu}X^{i}\partial^{\mu}X^{i}+2i\overline{{\theta}}\Gamma^{\mu}\partial_{\mu}\theta)\right]
G_{{\bfa},k}(x,x^{\prime})={\frac{e^{\imathk|x-x^{\prime}|}}{4\pi|x-x^{\prime}|}}+{\frac{1}{{\frac{1}{\bfa}}-{\frac{\imathk}{4\pi}}}}~{\frac{e^{\imathk|x|}}{4\pi|x|}}~{\frac{e^{\imathk|x^{\prime}|}}{4\pi|x^{\prime}|}}~~.
\lbrack{G_{ri},N_{j}}\rbrack=-\frac{\imath}{2}\epsilon_{ijk}Q_{rk},\quad\lbrack{G_{ri},J_{j}}\rbrack=\frac{\imath}{2}\epsilon_{ijk}G_{rk},\quad\lbrack{Q_{ri},N_{j}}\rbrack=\frac{\imath}{2}\epsilon_{ijk}G_{rk}.
2^{r+3}\mathrm{Im}\left[\frac{(z_{1}z_{2})^{r}}{r!r!}\int_{0}^{\infty}e^{i\beta\mu\xi}(1-e^{-r\xi})\sum_{n=0}^{\infty}<n|\lambda^{r}|n+r>^{2}e^{-(n+1/2)\xi}\frac{r}{p^{2}-r^{2}}\right].
\tau_{+}=\left(\begin{array}{cc}{0}&{\sqrt2}\\{0}&{0}\\\end{array}\right)\quad\tau_{-}=\left(\begin{array}{cc}{0}&{0}\\{\sqrt2}&{0}\\\end{array}\right)\quad\tau_{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right).
\frac{\left[\left(\begin{array}{c}{\psi}\\{0}\\\end{array}\right),\widetilde{A}\left(\begin{array}{c}{\psi}\\{0}\\\end{array}\right)\right]_{{\calK}_{2}}}{\left[\left(\begin{array}{c}{\psi}\\{0}\\\end{array}\right),\left(\begin{array}{c}{\psi}\\{0}\\\end{array}\right)\right]_{{\calK}_{2}}}=\frac{\langle\psi\midA\psi\rangle}{\langle\psi\mid\psi\rangle},
B=\left(\begin{matrix}{0}&{\beta}&{0}&{0}&{0}&{0}&{0}\\{\beta}&{0}&{-\theta_{2}}&{0}&{0}&{0}&{0}\\{0}&{\theta_{2}}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{-\theta_{3}}&{0}&{0}\\{0}&{0}&{0}&{\theta_{3}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}\\\end{matrix}\right)~,
\Phi({\bfr})=U_{2}^{-1}({\bfr})\left(\begin{matrix}{t_{4}-{\frac{1}{2y}}}&{0}&{0}&{0}\\{0}&{t_{2}}&{0}&{0}\\{0}&{0}&{t_{2}}&{0}\\{0}&{0}&{0}&{t_{1}+{\frac{1}{2y}}}\\\end{matrix}\right)U_{2}({\bfr})+O(b/y^{2})
S(\gamma,X,A)=-\frac{1}{2}\int_{\SigmaxR}d^{3}\xi\sqrt{-\gamma}\left(\gamma^{ij}\partial_{i}X^{m}\partial_{j}X^{n}\eta_{mn}+\frac{1}{2}\gamma^{ij}\gamma^{kl}F_{ik}^{r}F_{jl}^{r}-1\right)
\kappa_{i}(\tau)=\frac{(\dot{n}_{i}\acute{x}_{i})}{\acute{x}_{i}^{2}(\tau)}=\frac{(n_{i}\dot{x}_{i}^{\prime})}{\dot{x}_{i}^{2}(\tau)}=\frac{A}{\dot{x}_{i}^{2}(\tau)},\,\,(i=1,3).
e^{-\Phi}\starH_{3}=-mU\epsilon_{7}+m^{-1}T_{ij}^{-1}\star{\calD}T_{jk}\wedge(\mu^{k}{\calD}\mu^{i})-{\frac{m^{-2}}{2}}T_{ik}^{-1}T_{jl}^{-1}\starF^{ij}\wedge{\calD}\mu^{k}\wedge{\calD}\mu^{l}
\begin{array}{rcccl}{\langle\psi_{s_{o}}(1/t)\vert\psi_{s}(1/t)\rangle}&{=}&{\langle\psi_{s_{o}}(t)\vert\psi_{\beta-s}(t)\rangle}&{=}&{\langle\psi_{1/2}(t)\vert\psi_{\beta-s+s_{o}-1/2}(t)\rangle=}\\{Z[a(\beta-s+s_{o}-1/2)+b]}&{=}&{Z(1-s^{\prime})}&{=}&{0.}\\\end{array}
L=\frac{1}{2}\langle\Phi|\Box-D_{1}\bar{D}_{1}-D_{2}\bar{D}_{2}-\frac{3}{4}P_{11}\Box\bar{P}_{11}+\frac{3}{4}(P_{11}\bar{D}_{1}^{2}+D_{1}^{2}\bar{P}_{11}+P_{11}D_{2}\bar{D}_{2}\bar{P}_{11})|\Phi\rangle\,,
a_{4}=\frac{Y^{4}}{4!}+\frac{M}{6}\tau_{3}Y^{3}-\frac{M^{2}}{12}(Y^{2}-Y\tau_{3}Y\tau_{3})-\frac{M^{3}}{3}\tau_{3}Y+\frac{1}{12}(Y+M\tau_{3})(\partialY)^{2}+\frac{1}{120}(\partial^{2}Y)^{2}.
\triangle(T)=-\frac{1728(j^{\prime})^{6}}{(48\pi^{2})^{3}j^{4}(j-j(i))^{3}}=\eta^{24}(T)=\frac{1}{(2\pii)^{6}}\frac{(j^{\prime})^{6}}{j^{4}(j-j(i))^{3}},
\frac{i}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\,\frac{e^{2}}{k^{2}\Big(1+(-1)^{n}k^{2n}/\Lambda^{2n}\Big)(k+p)^{2}}\to-\frac{1}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{e^{2}}{k^{2}\Big(1+k^{2n}/\Lambda^{2n}\Big)(k+p)^{2}}.
\langlek_{\mu}(x)k_{\nu}(y)\rangle=-\eta^{2}\delta_{\mu\nu}\int{\frac{d^{4}p}{(2\pi)^{4}}}\left(1-{\frac{\eta^{2}}{p^{2}}}\right)^{-1}e^{ip(x-y)}=-\eta^{2}\delta_{\mu\nu}\delta(x-y)+O(\eta^{4}),
H[\varphi]=P_{0}(e^{(0)})=\intd^{3}x\;\left(\frac{1}{2}({\partial_{0}\varphi})^{2}+\frac{1}{2}(\vec{\nabla}\varphi)^{2}+{\frac{m_{0}^{2}}{2}}\varphi^{2}+{\frac{g_{0}}{4!}}\varphi^{4}\right),
P_{Y}^{(6)}\underset{\mathrm{pairing}}{\rightarrow}P_{Y}^{(6)(i_{1}i_{2})...(i_{2n-1}i_{2n}),(j_{1}j_{2})...(j_{2n-1}j_{2n})}=\sum_{I}Q_{Y,i_{1}i_{2}...i_{2n}}^{T,I_{1}I_{2}...I_{n}}Q_{Y,j_{1}j_{2}...j_{2n}}^{I_{1}I_{2}...I_{n}}
{\calD}_{l}=\prod_{l_{i}>0}\left(\frac{\partial}{\partiala_{i}}\right)^{l_{i}}-\prod_{l_{i}<0}\left(\frac{\partial}{\partiala_{i}}\right)^{-l_{i}},\quad{\calZ}_{\alpha}=\sum_{i}\tilde{v}_{i,\alpha}a_{i}\frac{\partial}{\partiala_{i}},\quad{\calZ}_{0}=\sum_{i}a_{i}\frac{\partial}{\partiala_{i}}+1.
D_{i}\,\{\tilde{g}_{(4)j}^{\i}+\frac{l^{4}}{16}[\tilde{R}\tilde{R}_{\,j}^{i}-2\tilde{R}^{im}\tilde{R}_{mj}+\frac{1}{2}\delta_{\,j}^{i}(\tilde{R}^{mn}\tilde{R}_{mn}-\frac{5}{9}\tilde{R}^{2})]\}=0\,.
(A^{(V)})_{\mathrm{~}\hat{c}}^{\hat{b}}\equiv[\omega^{a}J_{a}^{(V)}+E^{a}\tilde{P}_{a}^{(V)}]_{\mathrm{~}\hat{c}}^{\hat{b}}=\left(\begin{array}{cccc}{0}&{\omega^{2}}&{-\omega^{1}}&{E^{0}}\\{\omega^{2}}&{0}&{-\omega^{0}}&{E^{1}}\\{-\omega^{1}}&{\omega^{0}}&{0}&{E^{2}}\\{-E^{0}}&{E^{1}}&{E^{2}}&{0}\\\end{array}\right).
M_{\mathrm{BPS}}=e^{-{\frac{\phi_{\infty}}{2}}}|g_{11\,\infty}^{\frac{1}{2}}\beta_{1}^{({2})}+g_{11\,\infty}^{-{\frac{1}{2}}}\beta_{1}^{(1)}|+e^{\frac{\phi_{\infty}}{2}}|g_{22\,\infty}^{{\frac{1}{2}}}\alpha_{2}^{(1)}+g_{22\,\infty}^{-{\frac{1}{2}}}\alpha_{2}^{(2)}|.
D_{\nu}D^{\nu}\psi+m^{2}\psi-\frac{i}{2}eF_{\mu\nu}S^{\mu}{}^{\nu}\psi-\frac{e}{2m}\left(\beta^{\mu}\beta^{\nu}\beta^{\alpha}+\beta^{\mu}\eta^{\nu\alpha}\right)D_{\nu}\left(F_{\alpha\mu}\psi\right)=0.
\begin{array}{lll}{P}&{=}&{p_{1}\tau_{1}+\cdots+p_{k}\tau_{k},}\\{Q}&{=}&{(p_{1}^{2}-p_{1})\tau_{1}+\cdots+(p_{k}^{2}-p_{k})\tau_{k},}\\{\rho}&{=}&{N_{1}+\cdots+N_{k}.}\\\end{array}
\left[-\frac{4\kappa\xi}{\left(1+\kappa\xi\Phi^{2}\right)}\left(\frac{1}{\kappa}\Lambda-\frac{1}{2}\mu^{2}\Phi^{2}+\frac{1}{4}\lambda\Phi^{4}\right)-\mu^{2}+\lambda\Phi^{2}\right]\Phi=0.
\frac{e^{3}}{|e|}\left[\prod_{a=1}^{2}\delta\left(\frac{F^{a}}{|F|}+\frac{e^{a}}{|e|}\right)+\prod_{a=1}^{2}\delta\left(\frac{F^{a}}{|F|}-\frac{e^{a}}{|e|}\right)\right],
Z=\int\prod_{x}\{\mathrm{d}v_{\mu}\,\mathrm{d}j^{\mu}\,\mathrm{d}j_{5}^{\mu}\,\mathrm{d}|h|^{2}\,\mathrm{d}\alpha\,K[j^{\mu},j_{5}^{\mu},|h|^{2}]\}\,e^{\mathrm{i}\int\mathrm{d}^{4}x\,{\calL}[v_{\mu},j^{\mu},j_{5}^{\mu},|h|,\alpha]},
{\frac{1}{2}}R_{ab,cd}^{(+)}\lambda_{+}{}^{a}\lambda_{+}{}^{b}\lambda_{-}{}^{c}\lambda_{-}{}^{d}={\frac{1}{4}}R_{ab,cd}^{(+)}\lambda_{+}{}^{a}\lambda_{+}{}^{b}\lambda_{-}{}^{c}\lambda_{-}{}^{d}+{\frac{1}{4}}R_{ab,cd}^{(-)}\lambda_{-}{}^{a}\lambda_{-}{}^{b}\lambda_{+}{}^{c}\lambda_{+}{}^{d}\.
g^{2}f_{s}^{2}\left(\begin{matrix}{2}&{-1}&{0}&{0}&{0}&{\dots}&{-1}\\{-1}&{2}&{-1}&{0}&{0}&{\dots}&{0}\\{0}&{-1}&{2}&{-1}&{0}&{\dots}&{0}\\{\vdots}&{\vdots}&{\ddots}&{\ddots}&{\ddots}&{\vdots}&{\vdots}\\{0}&{\dots}&{0}&{0}&{-1}&{2}&{-1}\\{-1}&{\dots}&{0}&{0}&{0}&{-1}&{2}\\\end{matrix}\right)
M_{\alphaA}^{(\beta)\:\dagger}\;=\;-i\delta_{\alpha\beta}\:q_{A}\:-\:i\Gamma_{\alpha\beta}^{j}\:\frac{\partial}{\partialx_{jA}}\:-\:\frac{1}{2}\:f_{ABC}\:x_{jB}\:x_{kC}\:\Gamma_{\alpha\beta}^{jk}
S^{BH}(G_{\tiny\mathrm{ren}},c_{\tiny\mathrm{ren}}^{i})=S^{BH}(G_{\tiny\mathrm{B}},c_{\tiny\mathrm{B}}^{i})+S_{\tiny\mathrm{div}}^{C}-{Q}_{\tiny\mathrm{div}}~~~.
\delta_{1}(\gamma^{\mu\nu\lambda\rho}\partial_{\nu\lambda}\psi_{\rho})=\gamma^{5}\widetilde{p}^{\mu\rho}p_{\rho\upsilon}\varepsilon_{\nu}\backsimPfaff(p)\varepsilon_{\mu}=\delta_{2}(Pfaff(p)\psi_{\mu})
R_{\mu\nu\rho\sigma}=g_{\mu\rho}R_{\nu\sigma}+g_{\nu\sigma}R_{\mu\rho}-g_{\nu\rho}R_{\mu\sigma}-g_{\mu\sigma}R_{\nu\rho}-\frac{R}{2}(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho}).
T_{\alpha\beta}[\psi]=(\frac{2}{3}\nabla_{\alpha}\psi\nabla_{\beta}\psi-\frac{1}{6}g_{\alpha\beta}\nabla_{\gamma}\psi\nabla^{\gamma}\psi)-\frac{1}{3}(\psi\nabla_{\alpha}\nabla_{\beta}\psi-g_{\alpha\beta}\psi\Box\psi)+\frac{1}{6}\psi^{2}G_{\alpha\beta}
W={\frac{1}{64\pi^{{\frac{1}{2}}}}}{\frac{\tau}{a}}(\Phi_{0}^{2}a^{2})^{2}Log({\frac{1}{M^{2}a^{2}}})[1+m^{2}a^{2}+...]-{\frac{c}{192(2\pi)^{\frac{1}{2}}}}({\frac{\tau}{a}}(\Phi_{0}^{2}a^{2})^{3})
{\calL}={\textstyle\frac12}~\partial_{\mu}n^{a}\partial^{\mu}n^{a}+{\textstyle\frac12}~i\,\overline{{\psi}}^{a}\gamma^{\mu}\partial_{\mu}\psi^{a}+{\textstyle\frac18}~g\left(\overline{{\psi}}^{a}\psi^{a}\right)^{2}~.
V_{0}=\textstyle{\left[-{\frac{1}{2}}\left(-{\frac{1}{2}}-{\frac{1}{2}}-{\frac{1}{2}}\right)^{3}||-{\frac{1}{2}}\left(-{\frac{1}{2}}-{\frac{1}{2}}-{\frac{1}{2}}\right)^{3}\right]}~.
D_{1}A^{\prime\prime}+D_{2}B^{\prime\prime}+{\frac{D_{1}}{2}}{A^{\prime}}^{2}+{\frac{D_{2}}{2}}{B^{\prime}}^{2}={\frac{4\kappa^{2}}{D-2}}\left(-\Lambda+{\frac{F^{2}}{4g^{2}}}\mathrm{e}^{-2B}\right)~,
\left(\frac{\partialt}{\partialT}\right)_{r,x^{i}}/\left(\frac{\partialt}{\partialr}\right)_{T,x^{i}}=-\left(\frac{\partialr}{\partialw}\right)_{t,x^{i}}/\left(\frac{\partialT}{\partialw}\right)_{t,x^{i}},
L_{eff}=\frac{1}{2}M[c_{1}^{2}\sum_{p_{1}=1}^{n_{1}}\vec{V}_{p_{1}}^{2}+c_{2}^{2}\sum_{p_{2}=1}^{n_{2}}\vec{V}_{p_{2}}^{2}]+2\pi\kappa\frac{\sigma_{1}\sigma_{2}}{q_{1}q_{2}}\sum_{p_{1}=1}^{n_{1}}\sum_{p_{2}=1}^{n_{2}}\frac{d}{dt}Arg(\vec{R}_{p_{1}}^{(1)}-\vec{R}_{p_{2}}^{(2)})\;\;.
l_{\dot{J}}^{\dot{I}}l_{\dot{L}}^{\dot{K}}=\delta_{\dot{J}}^{\dot{K}}l_{\dot{L}}^{\dot{I}}+\sum_{\dot{J}_{1}J_{2}=\dot{J}}\delta_{\dot{J}_{1}}^{\dot{K}}l_{\dot{L}J_{2}}^{\dot{I}}+\sum_{\dot{K}_{1}K_{2}=\dot{K}}\delta_{\dot{J}}^{\dot{K}_{1}}l_{\dot{L}}^{\dot{I}K_{2}}.
\rho_{\smallreflection}^{i}(k_{i})=-{\frac{k_{i}}{4\pi^{2}}}S\;\Im\left[\;\int_{0}^{\infty}{\frac{R^{i}(k_{i}+i\epsilon,k_{t})}{K(k_{i}+i\epsilon,k_{t})^{2}}}\;k_{t}\;dk_{t}\right].
{\mathcalM}_{l+\nu}^{(<)}(\tilde{\xi};\tilde{k})\!\equiv\!\left[{\mathcalL}_{l+\nu}^{(<)}(\tilde{\xi};\tilde{k})\right]^{2}+\left[\tilde{\xi}^{2}-\!(l+\nu)^{2}-\tilde{\xi}^{2}\check{U}(\tilde{\xi};\tilde{k})\right]
{\calA}_{\mu}^{\prime\prime}+\frac{\ddot{{\calA}}_{\mu}}{{\calL}^{2}}-\frac{2}{m_{H}^{2}~M^{2}}\Box{\calA}_{\mu}+(2H+F){\calA}_{\mu}^{\prime}-\biggl[\frac{P^{\prime\prime}}{P}+(2H+F)\frac{P^{\prime}}{P}\biggr]{\calA}_{\mu}=0.
\frac{A^{n+1}}{V^{n}}\ge\frac{4\,\pi}{(n-2)}\,\,\mathrm{Vol(M)}\,\frac{2^{\frac{1}{n-2}}(\frac{1}{n})^{\frac{n}{2}}(n-2)^{\frac{n+1}{2}}}{\left(1-(\frac{2}{n})^{\frac{n}{n-2}}\right)^{n}}
\Big\{\Big\{f,g\Big\},h\Big\}+\Big\{\Big\{h,f\Big\},g\Big\}+\Big\{\Big\{g,h\Big\},f\Big\}=-\varphi^{ijl}\Big(\overline{X}_{i}f\Big)\Big(\overline{X}_{j}g\Big)\Big(\overline{X}_{l}h\Big)\,.
N_{0,0}=ig_{0}^{2}\{\frac{m_{H}^{2}}{16m^{2}}[(\sum_{a=1}^{3}\phi_{a}^{2})^{2}+2H^{2}\sum_{a=1}^{3}\phi_{a}^{2}]+\lambdaH^{4}\}\delta(x_{1}-x_{2}),
\hat{S}=\intd^{10}\hat{x}\sqrt{|\hat{g}|}\,\left\{e^{-2\hat{\phi}}\left[\hat{R}-4(\partial\hat{\phi})^{2}\right]+{\textstyle\frac{(-1)^{7-p}}{2\cdot(8-p)!}}\left(\hat{G}^{(8-p)}\right)^{2}\right\}\,.
{\displaystyleV_{\underline{\alpha}}^{(1)}=V_{\underline{\alpha}}-\frac{i{\calP}_{\underline{{\alpha\dot{\beta}}}}\bar{V}^{\underline{{\dot{\beta}}}}}{(m-ieW/2)};\\,barV_{\underline{{\dot{\alpha}}}}^{(1)}=\bar{V}_{\underline{{\dot{\alpha}}}}-\frac{i{\calP}_{\underline{{\beta\dot{\alpha}}}}V^{\underline{\beta}}}{(m+ie\bar{W}/2)},}
\Phi_{k}\vert_{V_{\ell}}(\tau,\boldsymbol{z}):=\ell^{k-1}\sum_{\substack{ad=\ell\,a>0}}\sum_{b=0}^{d-1}d^{-k}\Phi_{k}\left(\frac{a\tau+b}{d},a\boldsymbol{z}\right)\,.
\begin{array}{lcr}{|X_{a-1}|^{2}+|X_{a}|^{2}-(|Y_{a-1}|^{2}+|Y_{a}|^{2})=R_{a}}\\{\sum_{j}q_{a}^{j}X_{j}\overline{{Y}}_{j}=X_{a-1}\overline{{Y}}_{a-1}+X_{a}\overline{{Y}}_{a}=0}\\{\sum_{j}q_{a}^{j}\overline{{X}}_{j}{Y}_{j}=\overline{{X}}_{a-1}{Y}_{a-1}+\overline{{X}}_{a}{Y}_{a}=0.}\\\end{array}
\stackrel{1}{M}_{\pm}\stackrel{2}{u}(x)\,=\,\stackrel{2}{u}(x)R^{\pm}\stackrel{1}{M}_{\pm}\,,\quad\stackrel{1}{\bar{M}}_{\pm}\!\!\!^{-1}R^{\pm}\stackrel{2}{\bar{u}}(x)\,=\,\stackrel{2}{\bar{u}}(x)\stackrel{1}{\bar{M}}_{\pm}\!\!\!^{-1}
S=\left(\begin{array}{ccccc}{\mp1}&{0}&{0}&{0}&{0}\\{0}&{1/2}&{1/2}&{1/2}&{1/2}\\{0}&{1/2}&{\pm1/2}&{-1/2}&{\mp1/2}\\{0}&{1/2}&{-1/2}&{1/2}&{-1/2}\\{0}&{1/2}&{\mp1/2}&{-1/2}&{\pm1/2}\\\end{array}\right)\,.
\partial_{\underline{i}}\epsilon_{I}-{\textstyle\frac{i}{4}}e^{2\phi}\bigl(\partial_{\underline{i}}a\bigr)\epsilon_{I}+{\textstyle\frac{1}{2}}\sqrt2e^{-U-\phi}\biggl(F_{0i}^{+(1)}\alpha_{IJ}+iF_{0i}^{+(2)}\beta_{IJ}\biggr)\gamma^{0}\epsilon^{J}=0\,.
\partial_{i}x^{j}=\tilde{J_{i}^{\alpha}}v_{\alpha}x^{j}=\partial_{i}(x^{j})+\underbrace{\tilde{J_{i}^{\alpha}}O_{\alpha}{}^{\beta}(x^{j})J_{\beta}{}^{k}}_{L_{i}{}^{k}(x^{j})}\partial_{k}.
\left(\begin{matrix}{i\hbar(\partial/\partialx_{0})}&{i\hbar\bbox{\sigma}\cdot\bbox{\nabla}}\\{-i\hbar\bbox{\sigma}\cdot\bbox{\nabla}}&{-i\hbar(\partial/\partialx_{0})}\\\end{matrix}\right)\left(\begin{matrix}{\psi}\\{\chi}\\\end{matrix}\right)=mc\left(\begin{matrix}{\psi}\\{\chi}\\\end{matrix}\right)\quad,
p=-\frac{L_{p}^{(3-d)}}{4\pi}\left\{\frac{(d-2)}{A}\left(k+\frac{A^{2}}{\ell^{2}}+\dot{A}^{2}\right)^{1/2}+\frac{\ddot{A}+A/\ell^{2}}{\sqrt{k+\dot{A}^{2}+A^{2}/\ell^{2}}}\right\},
\begin{array}{rcl}{i\Sigma_{ij}(p)}&{=}&{\displaystylei\lambda^{2}\delta_{ij}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}[U(k_{i},p)-T(k_{i})],}\\{i\Sigma_{00}(p)}&{=}&{\displaystylei\lambda^{2}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}[V(k_{i},p)-T(k_{i})],}\\\end{array}
\left[\frac{E(x^{2}b)E(x^{10}b)E(x^{12}b)E(x^{32}/b)E(x^{40}/b)E(x^{42}/b)}{E(x^{2}/b)E(x^{10}/b)E(x^{12}/b)E(x^{32}b)E(x^{40}b)E(x^{42}b)}\right]^{N}=b^{3N}.
g(s;y,z)=\frac{y}{\Gamma(s)}\int_{0}^{\infty}\dt\t^{s-1}\mathrm{e}^{-zt}\left\{2\frac{\mathrm{e}^{-2t}+\mathrm{e}^{-2yt}}{(1-\mathrm{e}^{-2t})(1-\mathrm{e}^{-2yt})}+1\right\},
\{\eta_{MN}\}=\left(\begin{array}{cc}{\eta_{\alpha\beta}}&{\eta_{\mu\beta}}\\{\eta_{\alpha\nu}}&{\eta_{\mu\nu}}\\\end{array}\right)=\left(\begin{array}{cc}{g_{\alpha\beta}+g_{\rho\sigma}\,A_{\alpha}^{\rho}\,A_{\beta}^{\sigma}}&{g_{\mu\rho}\,A^{\rho\beta}}\\{A_{\alpha}^{\rho}\,g_{\rho\nu}}&{g_{\mu\nu}}\\\end{array}\right),
b_{m}={\frac{\alpha}{2}}\left\{m(m+\lambda+\frac{1}{2})\right\}^{-\frac{1}{2}}b_{m-1}=\frac{(\frac{\alpha}{2})^{m}}{\{m!\Gamma(m+\lambda+\frac{3}{2})\}^{\frac{1}{2}}}b_{0}
i\frac{\partial}{\partialt}G(\bar{\xi}^{\prime\prime},\xi^{\prime};t)=\sum_{m=1}^{N}\frac{\mu_{m}}{\sqrt{2m(m+1)}}\left(\left[(m+1)\xi_{m}^{*\prime\prime}\frac{\partial}{\partial\xi_{m}^{*\prime\prime}}+\left(\sum_{k=m+1}^{N}\xi_{k}^{*\prime\prime}\frac{\partial}{\partial\xi_{k}^{*\prime\prime}}\right)\right]-2J\right)G(\bar{\xi}^{\prime\prime},\xi^{\prime};t)
{\tilde{\psi}}({\bf{k}}_{n}r)=exp(-i{\mathrm{~}}\sum_{{\bf{q}}_{m}}T_{-{\bf{q}}_{m}}({\bf{k}}_{n})X_{{\bf{q}}_{m}r})\mathrm{~}exp(i{\mathrm{~}}\sum_{{\bf{q}}_{m}}T_{{\bf{q}}_{m}}({\bf{k}}_{n})\phi([\rho];{\bf{q}}_{m}r)){\mathrm{~}}(N_{r}^{0}+\sum_{{\bf{q}}_{m}}\delta\rho({\bf{q}}_{m}r)T_{{\bf{q}}_{m}}({\bf{k}}_{n}))^{\frac{1}{2}}\delta_{{\bf{k}}_{n},{\bf{0}}}
\{J^{\alpha\beta}(\varphi),~Q^{\gamma\delta}(\varphi^{\prime})\}=\Omega_{\gamma\alpha;\beta\delta}^{-1}(\varphi,\varphi^{\prime})=\frac{1}{2}\left(\delta^{\beta\gamma}Q^{\alpha\delta}-\delta^{\alpha\delta}Q^{\gamma\beta}\right)\delta(\varphi-\varphi^{\prime}),
\widetilde{S}_{0}^{(D_{0})}(k)=-\frac{1+i\left(2\omega_{_\mathrm{(gs)}}^{-1}/\pi-Q_{_\mathrm{(gs)}}\right)}{1-i\left(2\omega_{_\mathrm{(gs)}}^{-1}/\pi-Q_{_\mathrm{(gs)}}\right)}\;,
{\cal{L}}=(\partial_{\mu}+igB_{\mu})\varphi^{+}(\partial^{\mu}-igB^{\mu})\varphi+\mu^{2}\varphi^{+}\varphi-\lambda(\varphi^{+}\varphi)^{2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu},
S^{(D9)}=\mathrm{Tr}\,\int_{PP_{6}\timesR^{4}}\frac{1}{2}F\wedge*F+\frac{1}{4!}C^{(2)}\wedgeF\wedgeF\wedgeF\wedgeF+\frac{1}{2!}C^{(6)}\wedgeF\wedgeF
\bar{Q}_{i}=\intd^{3}x\:\biggl\{\varepsilon_{ijk}\left(F_{0j}-\tilde{F}_{0j}\right)\chi_{k}+\psi_{0}D_{i}\lambda+\psi_{i}D_{0}\lambda-\left(F_{0i}-\tilde{F}_{0i}\right)\eta+\varepsilon_{ijk}\psi_{j}D_{k}\lambda+{\frac{1}{2}}\left[\phi,\lambda\right]\chi_{i}\biggl\}.
E_{4}(q)=\left(2\frac{\eta(q^{2})^{2}}{\eta(q)}\right)^{8}+\left(\frac{\eta(q)^{2}}{\eta(q^{2})}\right)^{8}=1+240\sum_{n=1}^{\infty}n^{3}\frac{q^{n}}{1-q^{n}}.
\Biggl\langle\bar{b}_{-1}\vert0\!\!>,\Bigl(b_{-\!m_{1}\!-\!m_{2}}\bar{b}_{-m_{3}}\vert0\!\!>\!\!+b_{-\!m_{2}\!-\!m_{3}}\bar{b}_{-m_{1}}\vert0\!\!>\!\!+b_{-\!m_{3}\!-\!m_{1}}\bar{b}_{-m_{2}}\vert0\!\!>\Bigr)_{0}b_{-1}\vert0\!\!>\Biggr\rangle\.
\left[{\frac{dn_{s}(\omega|\mu)}{d\omega}}\right]_{\tiny\mathrm{div}}={\frac{1}{(4\pi)^{2}\kappa}}\int_{\Sigma}\left[2b+a\left({\frac{\omega^{2}}{\kappa^{2}}}{\calP}+2\left(\frac16-\xi\right)R\right)\right]~~~,
[b_{i}({\bf{q}}),a_{{\bf{k}}}^{\dagger}({\bf{q}})]=(\frac{\Lambda_{{\bf{k}}}(-{\bf{q}})}{{\tilde{\omega}}_{i}({\bf{q}})-\frac{{\bf{k.q}}}{m}})g_{i}({\bf{q}})=[a_{{\bf{k}}}({\bf{q}}),b_{i}^{\dagger}({\bf{q}})]
L:=\frac12\int_{s_{0}}^{s_{1}}[\frac12Tr(D_{1}A_{2}-D_{2}A_{1})^{*}(D_{1}A_{2}-D_{2}A_{1})+(D_{1}Z)^{*}(D_{1}Z)+(D_{2}Z)^{*}(D_{2}Z)]ds
H_{\left(s\right)\alpha}^{\prime}\equivP_{\left(s\right)\alpha}+H_{\left(s\right)\alpha}\left(t_{\left(u\right)\alpha};q_{\left(u\right)a};p_{\left(u\right)a}=\frac{\partialS}{\partialq_{\left(u\right)a}}\right)=0.
\Lambda\left(\varepsilon,u,v\right)=\frac{-2\varepsilon_{+^{\prime}-^{\prime}}+\left(\varepsilon_{+^{\prime}+}u+\varepsilon_{+^{\prime}-}v\right)-\left(\varepsilon_{-^{\prime}-}v+\varepsilon_{-^{\prime}+}u\right)\left(-2+uv\right)}{1-uv}.
{\left(\rho_{ab}^{(2)}\right)}_{ij}^{-1}=\left(\begin{array}{cccc}{0}&{-\delta_{ab}D_{ij}}&{0}&{\partial_{i}^{x}\nabla^{-2}}\\{\delta_{ab}D_{ij}}&{0}&{\partial_{i}^{x}\nabla^{-2}}&{0}\\{0}&{-\partial_{j}^{x}\nabla^{-2}}&{0}&{\nabla^{-2}}\\{-\partial_{j}^{x}\nabla^{-2}}&{0}&{-\nabla^{-2}}&{0}\\\end{array}\right)\delta(\vec{x}-\vec{y}),
\prod_{k=0}^{n-1}\left[\,1\,+\,\frac{(\xi\cdot\partial)^{2}}{(k+1)(2k+1)}\\frac{\partial_{\xi}\cdot\partial_{\xi}}{\Box}\\,-\,\frac{\xi\cdot\partial}{k+1}\\frac{\partial_{\xi}\cdot\\partial}{\Box}\,\\right]\,\hat{\calF}(\hat{\Phi})\,=\,0\,
\Gamma\lbrackA\rbrack=\!-2\!\int_{0}^{\infty}{\frac{dT}{T}}\mathrm{e}^{-m^{2}T}\int{\calD}X\mathrm{exp}\left\{-\int_{0}^{T}\!\!\!\!d\tau\int\!\!d\theta\,\left[\!-{\frac{1}{4}}X\,D^{3}X\!-\!ieDX^{\mu}A_{\mu}(X)\right]\right\}
\left[{\hbar}^{2}\frac{{\partial}^{2}}{{\partial}x_{1}^{2}}\pmi\frac{e{\hbar}}{c}{\calE}^{ext}+\left(\frac{eA_{0}^{ext}-E}{c}\right)^{2}\right]u_{1(2)}=m^{2}c^{2}u_{1(2)},
q=\frac{1}{8\pi^{2}a^{4}}\sum_{l=1}^{\infty}(2l+1)\sum_{\lambda=0,1}\int_{0}^{\infty}{z^{3}\psi_{\mu}(z/a)\frac{f_{\lambdal}^{(q)}(z,zx)}{\Omega_{1\lambdal}(z)}dz},\quadq=\varepsilon,p_{\perp}.
J_{\mu\nu}=2i\big\{e_{[\mu}{}^{\alpha\dot{\alpha}}e_{\nu]}{}^{\beta}{}_{\dot{\alpha}}\widehatA_{\alpha}^{(1)}\star\widehatA_{\beta}^{(1)}+\mathrm{h.c.}\big\}_{z=0}+L_{\mu\nu}+\mathrm{~W~-terms},
J_{m}(z)\equiv\Big(\frac{z}{2}\Big)^{m}\sum_{n=0}^{\infty}\frac{(-1)^{n}(z/2)^{2n}}{n!(m+n)!}\quad,\quadJ_{-m}(z)\equiv(-1)^{m}J_{m}(z)\quad\mathrm{for}\quadm\geq0.
\begin{array}{ccccccc}{\nabla_{i}\Phi}&{=}&{(\partial_{i}+\frac{1}{2}p\partial_{i}{\calK})\Phi}&{;}&{\nabla_{\bar{\imath}}\Phi}&{=}&{(\partial_{\bar{\imath}}-\frac{1}{2}p\partial_{\bar{\imath}}{\calK})\Phi}\\\end{array}