harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{ccccccccccl}{D2:}&{1}&{2}&{\_}&{\_}&{\_}&{\_}&{\_}&{\_}&{\_}&{\quad\mathrm{probe}\nonumber}\\{F1:}&{\_}&{\_}&{\_}&{4}&{\_}&{\_}&{\_}&{\_}&{\_}&{\quad\mathrm{soliton}\nonumber}\\{D2:}&{\_}&{\_}&{3}&{4}&{\_}&{\_}&{\_}&{\_}&{\_}&{\quad\mathrm{soliton.}}\\\end{array}
V_{n}^{i_{1}...i_{n}}=\frac{1}{n}(-)^{\sum_{s=1}^{n-1}(\alpha_{s}+1)\sum_{r=1}^{n}\alpha_{r}}(-)^{\sum_{r,s=1}^{n}(i_{r}\alpha_{s})}V_{\alpha_{1}...\alpha_{n}}^{i_{1}...i_{n}}\left[\Phi\right]c^{\alpha_{1}}...c^{\alpha_{n}}\mathrm{~,}\tag{65}
\chi_{k}^{n}(\tau)={\frac{1}{(\eta(\tau))^{3}}}\cdot2(k+2)\sum_{m=-\infty}^{\infty}\left(m+{\frac{n+1}{2(k+2)}}\right)e^{2\pii(k+2)\tau\left(m+{\frac{n+1}{2(k+2)}}\right)^{2}}.
R^{(d+1)}=R^{(d)}-D_{\alpha}^{(d)}D^{(d)\alpha}\psi_{1}-\frac{1}{2}D_{\alpha}^{(d)}\psi_{1}D^{(d)\alpha}\psi_{1}-\frac{1}{4}e^{\psi_{1}}g^{(d)\alpha\beta}g^{(d)\gamma\delta}F_{\alpha\gamma}F_{\beta\delta},
\sum_{i,j,k,l=1}^{h}U_{i}U_{j}U_{k}U_{l}\widehat\Theta_{ijkl}[n]+\sum_{i,j=1}^{h}\left({\frac{3}{4}}V_{i}V_{j}-U_{i}W_{j}\right)\widehat\Theta_{ij}[n]+d\widehat\Theta[n]=0,\qquadn\in{\bfZ}_{2}^{h}.
\left({\mathcalL}_{m}^{(r,s)}(u),\tilde{\mathcalL}_{n}^{(\alpha,\beta)}(v)\right)=\delta_{\alpha+s+1,0}\delta_{\beta+r+1,0}{\widehatRes}\left[{\mathcalL}_{m}^{(r,s)}(u)\star\tilde{\mathcalL}_{n}^{(\alpha,\beta)}(v)\right],
(1-\delta_{IJ}\delta_{bb^{\prime}})\{{\frac{\alpha^{(I)}\alpha^{(J)}}{2}}+s-{\frac{d_{I}d_{J}}{D-2}}+\sum_{i=1}^{N}r_{i}\Lambda_{bi}^{(I)}\Lambda_{b^{\prime}i}^{(J)}\}=0,
\frac{\partial^{2}\tilde{\phi}}{\partial\tilde{t}^{2}}-\frac{\partial^{2}\tilde{\phi}}{\partial\tilde{x}^{2}}+\tilde{\eta}_{m}\tilde{\phi}^{2}\frac{\partial\tilde{\phi}}{\partial\tilde{t}}=-\tilde{\phi}+\tilde{\alpha}\tilde{\phi}^{2}-\tilde{\lambda}\tilde{\phi}^{3}+\tilde{\xi}_{m}\tilde{\phi}.
\begin{array}{rclrcl}{F^{(1)1}}&{=}&{{\textstyle\frac{1}{2}}F\,,}&{F^{(1)2}}&{=}&{{\textstyle\frac{1}{2}}{}^{\star}F\,,}\\{}&{}&{}&{}&{}&{}\\{F^{(2)}{}_{1}}&{=}&{\mp{\textstyle\frac{1}{2}}F\,,}&{F^{(2)}{}_{2}}&{=}&{\mp{\textstyle\frac{1}{2}}{}^{\star}F\,,}\\\end{array}
{\frac{I^{2}}{c^{2}}}\left(1-{\frac{c^{2}B}{I^{3/2}}}\varrho\right)\ddot{\Phi}+{\frac{3I}{c^{2}}}{\frac{\dot{a}_{0}}{a_{0}}}\dot{\Phi}\left(1+{\frac{B}{\sqrt{I}}}\wp\right)+V^{\prime}(\Phi)=0.
{\frac{\omega^{\prime}}{T_{L}^{\prime}}}={\frac{\omega+k_{4}}{T_{L}}}\,\quad{\frac{\omega^{\prime}}{T_{R}^{\prime}}}={\frac{\omega-k_{4}}{T_{R}}}\,\quad{\frac{\omega^{\prime}}{T_{H}^{\prime}}}={\frac{\omega-\phik_{4}}{T_{H}}}\,
\sqrt{detM_{1}}={\frac{1}{\sqrt{2}}}\biggl(l\mu_{q}-{\frac{i\hbar}{l}}\mu_{p}\biggr)\equivg_{1},\quadM_{1}^{-1}M_{2}={\frac{1}{\sqrt{2}g_{1}}}\left(l\mu_{q}+{\frac{i\hbar}{l}}\mu_{p}\right){\bfI_{2}}\equivg_{2}{\bfI_{2}}\\,
\left.\frac{p^{\mu}\Gamma_{A^{\mu}\varphi_{1}\varphi_{2}}(p,p_{1},p_{2})}{p_{1}^{2}-p_{2}^{2}}\right|_{p^{2}=0\atopp_{1}^{2}=p_{2}^{2}=m_{H}^{2}}=ez
\mathcal{H}_{ext}=\frac{1}{\epsilon}p_{i}^{b}b_{i}-\frac{1}{\epsilon}(\bar{q}_{i}-a_{i})\bar{p}_{i}-\bar{q}_{i}G_{i}-\bar{q}_{i}c_{j}\mathcal{C}_{ijk}p_{k}^{c}+p_{i}^{c}b_{i}\;.
(\kappa_{4})^{4}U_{OM}\simeq2.3\Big(\frac{|h|\beta}{d^{1/3}}\Big)^{2}\frac{r_{OM}}{V_{1}^{7/9}}e^{-0.9\frac{\betaV_{1}^{4/3}}{r_{OM}d^{1/3}}}\;.
\leq\;\prod_{a=1}^{N}\left(8^{[1-\frac{g}{\pi+gN}]}\right)^{2n_{a}}\;\int_{\Lambda}\prod_{b=1}^{N}\prod_{i=1}^{n_{b}}d^{2}x_{i}^{(b)}d^{2}y_{i}^{(b)}\;{\calF}\Big(\{x,y\}\Big)\;,
(M_{0,\pi})_{A}^{\;B}=\frac{1}{2}\alpha_{0,\pi}(\sigma_{1})_{A}^{\;B}\,\quad(\bar{M}_{0,\pi})_{A}^{\;B}=\frac{1}{2}\alpha_{0,\pi}\mathrm{i}(\sigma_{2})_{A}^{\;B}\,
\left[{\calH}-\zeta^{\,0}\right]_{\mathrm{quad}}=\frac{1}{2}\left\{\dot{\chi}^{2}+[(\partial_{\,z}+F^{\prime})\chi]^{2}+i\psi_{2}(\partial_{\,z}-F^{\prime})\psi_{1}+i\psi_{1}(\partial_{\,z}+F^{\prime})\psi_{2}\right\}\,,
{\frac{\Gamma(\lambda)}{{4{\pi}^{\lambda+1}}{\Bigl({[x(\tau_{a})-x(\tau_{b})]}^{2}\Bigr)}^{\lambda}}}=\int_{0}^{\infty}d\bar{T}{(4\pi\bar{T})}^{-{\frac{D}{2}}}\mathrm{exp}\Biggl[-{\frac{{\Bigl(x(\tau_{a})-x(\tau_{b})\Bigr)}^{2}}{4\bar{T}}}\Biggr]\,.
S^{\theta}\;=\;\intd^{3}x\left[{i\bar{\widehat{\psi}}}\star{\widehatD}_{0}{\widehat\psi}\,-\,\frac{1}{2m}(\overline{{{\widehatD}_{i}{\widehat\psi}}})\star{\widehatD}_{i}{\widehat\psi}\right]
\partial_{T}^{3}f=-{\frac{2i}{\pi}}\frac{j_{T}(T)}{j(T)-j(U)}\left\{\frac{j(U)}{j(T)}\right\}\left\{\frac{j_{T}(T)}{j_{U}(U)}\right\}\left\{\frac{j(U)-j(i)}{j(T)-j(i)}\right\}\,
L_{eff}=\int\;d^{2}x\;\mid\partial_{t}\phi\mid^{2}\;+\;\deltaL_{eff}=\pic_{1}^{2}\int\;rdr\;F^{2}g^{2}(\dot{\lambda}_{1}^{2}+\dot{\lambda}_{2}^{2})\;+\;\deltaL_{eff}\;\;.
\lambda=\frac{\mathrm{d}x}{y}={R}\,\frac{\mathrm{d}a}{\mathrm{d}u}\,{\calW}={R}\left(\frac{\mathrm{d}a}{\mathrm{d}u}\mathrm{d}B-\frac{\mathrm{d}a_{D}}{\mathrm{d}u}\mathrm{d}A\right)\,
\|c_{jl}\|^{-1}=\frac{1}{D^{(N)}}\;\mathrm{diag}\left(\sqrt{D_{11}^{(N-1)}},\ldots,\sqrt{D_{NN}^{(N-1)}}\right)\;\|\widetilde{c}_{jl}\|\;\mathrm{diag}\left(\sqrt{D_{11}^{(N-1)}},\ldots,\sqrt{D_{NN}^{(N-1)}}\right),
\begin{matrix}{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{1}&{0}&{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{1}&{0}&{0}&{1}\\\end{matrix}
\begin{aligned}{b}&{{}=Wa_{-}W^{\dagger},}&{b^{\dagger}}&{{}=Wa_{-}^{\dagger}W^{\dagger}}\\{c}&{{}=Wa_{+}W^{\dagger},}&{c^{\dagger}}&{{}=Wa_{+}^{\dagger}W^{\dagger}}\\\end{aligned}
{\calL}=(\partial^{\mu}\bar{\phi})(\partial_{\mu}\phi)+\psi^{\alpha}i\partial_{\alpha\dot{\alpha}}\bar{\psi}^{\dot{\alpha}}+\bar{F}F+\left\{F\,{\calW}^{\prime}(\phi)-\frac{1}{2}{\calW}^{\prime\prime}(\phi)\psi^{2}+\mathrm{H.c.}\right\}\,.
\intd\vec{y}\,f^{(2)}(\vec{x},\vec{y})\,f^{(2)^{-1}}(\vec{y},\vec{z})=\left(\begin{array}{cccccc}{\delta_{ij}^{mn}}&{0}&{0}&{0}&{0}&{0}\\{0}&{\delta_{ij}^{mn}}&{0}&{0}&{0}&{0}\\{0}&{0}&{\delta_{i}^{m}}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{\delta_{i}^{m}}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}\\\end{array}\right)\delta(\vec{x}-\vec{z})
\begin{array}{rcl}{p^{\prime}}&{=}&{P(p)}\\{}&{=}&{p^{8}+8p^{7}(1-p)+28p^{6}(1-p)^{2}+56p^{5}(1-p)^{3}+\frac{525}{8}p^{4}(1-p)^{4}+\frac{35}{2}p^{3}(1-p)^{5}}\\{}&{=}&{\frac{105}{8}p^{8}-55p^{7}+\frac{315}{4}p^{6}-\frac{63}{2}p^{5}-\frac{175}{8}p^{4}+\frac{35}{2}p^{3}\,\,\,\,\,.\,\,\,\,\,}\\\end{array}
\ddot{x}=-2xy^{2}+{\frac{(p_{\alpha}^{2}+p_{\beta}^{2})x(x^{4}+10x^{2}y^{2}+5y^{4})-2p_{\alpha}p_{\beta}y(y^{4}+10x^{2}y^{2}+5x^{4})}{\left(x^{2}-y^{2}\right)^{4}}}
S=M^{6}\intd^{8}x\,R_{8}+M^{6}\;L^{2}\intd^{8}x\,R_{8}^{2}+\cdots+M^{6}r_{c}^{4}\intd^{4}x\,R_{4}+\lambda_{2}\intd^{4}x\,R_{4}^{2}+\cdots\,.
\begin{array}{cccccccc}{q=}&{1}&{2}&{3}&{4}&{5}&{6}&{7}\\{}&{c_{i}}&{d_{i}}&{e_{i}}&{f_{i}}\\{}&{}&{}&{}&{Yc_{i}}&{Yd_{i}}&{Ye_{i}}&{Yf_{i}\quad.}\\\end{array}
L_{D}^{\ast}=\int_{X}d^{3}x\varepsilon^{ijk}\bigg\{-{\frac{3}{4a}}(a\partial_{i}B_{jAB}+bG_{ijAB})[{\bfR}^{-1}]_{kn}^{ABCD}\varepsilon^{lmn}(a\partial_{l}B_{mCD}+bG_{lmCD})+c\alpha_{i}^{AB}G_{jkAB}\bigg\},
M^{2}=\pi\left(\begin{matrix}{\vec{q}^{(e)\,\prime}}&{\vec{q}_{(m)}^{\,\prime}}\\\end{matrix}\right)S\left(\begin{matrix}{\eta}&{0}\\{0}&{\eta}\\\end{matrix}\right)S^{T}\left(\begin{matrix}{\vec{q}^{(e)\,\prime}}\\{\vec{q}_{(m)}^{\,\prime}}\\\end{matrix}\right).
K=-{\frac{2}{l}}\left[1+{\frac{l^{2}}{2}}\varphi^{\prime\prime}e^{2\varphi}+{\frac{l^{4}}{8}}(\varphi^{\prime})^{2}(2\varphi^{\prime\prime}+(\varphi^{\prime})^{2})e^{4\varphi}+...\right]~~,
(\Phi,H_{\mathrm{coul}}\Psi)=\int_{\Lambda}dA^{tr}\sigma\intd^{3}x\,2^{-1}[(E_{i}^{a}\Phi)^{*}E_{i}^{a}\Psi+\Phi^{*}B_{i}^{a}B_{i}^{a}\Psi]\;\mathrm{,}
e_{\mu}^{a}=\left(\begin{array}{ccc}{N}&{0}&{0}\\{0}&{N^{-1}}&{0}\\{rN^{\phi}}&{0}&{r}\\\end{array}\right),~~~e_{a}^{\mu}=\left(\begin{array}{ccc}{N^{-1}}&{0}&{0}\\{0}&{N}&{0}\\{-N^{-1}N^{\phi}}&{0}&{r^{-1}}\\\end{array}\right)
(-4\partial_{+}\partial_{-}\phi+4\partial_{+}\phi\partial_{-}\phi+2\partial_{+}\partial_{-}\rho)e^{-2\phi}+\left(\lambda^{2}-\frac{\beta}{16}(f^{2}-\frac{\mu^{2}}{\beta})^{2}\right)e^{2\rho-2\phi}=0,
S_{N}=g^{1/3}\sum_{n=0}^{N}\sum_{k=0}^{N-n}\sum_{l=0}^{k}c_{n}{\frac{\Gamma(3n/2+k-1/2)}{\Gamma(3n/2-1/2)(k-l)!}}{\frac{1}{y_{N}^{3n+2l-1}}}{\frac{(-1)^{l}}{l!}}(g^{-2/3})^{l}.
\left\langle\xi_{m}\cdot\cdot\cdot\xi_{1}\mid\eta_{1}\cdot\cdot\cdot\eta_{n}\right\rangle=\delta_{mn}\sum_{\sigma}\left\langle\xi_{1}\mid\eta_{\sigma(1)}\right\rangle...\left\langle\xi_{n}\mid\eta_{\sigma(n)}\right\rangle
S_{E}=\frac{1}{16\piG_{4}}\int_{M}d^{4}x\sqrt{\hat{g}}(\frac{1}{8}e^{-\Phi}\bar{F}^{2}+\frac{1}{6}e^{-2\Phi}H^{2})-\frac{1}{8\piG_{4}}\int_{\partial\Sigma}({\mathcalK}-{\mathcalK}_{0})\sqrt{c}.
p_{\mathrm{int}}^{2}={\frac{F}{2\sigma\mu_{\parallel}}}\left[{\frac{4\mu_{\parallel}}{F}}-1\pm\sqrt{1+{\frac{4\mu_{\parallel}}{F}}}\right]\qquad\quad\mathrm{or}\qquad\quadp_{\mathrm{int}}^{2}=0
U_{k_{1}\,\,k_{2}}^{\,\,l_{1}\,\,l_{2}}\left(\begin{array}{ccc}{\lambda}&{\mu}&{\rho}\\{k_{1}}&{k_{2}}&{m}\\\end{array}\right)\left(\begin{array}{ccc}{\lambda}&{\mu}&{\rho^{\prime}}\\{l_{1}}&{l_{2}}&{m^{\prime}}\\\end{array}\right)^{\ast}
\int\prod_{I}[dC_{I}^{\mu}]\delta(\partial_{\mu}C_{I}^{\mu})=\int\prod_{I}[dC_{I}^{\mu}][dH_{I}^{\mu}]\delta(C_{I}^{\mu}-\frac{1}{2\piq_{I}}\varepsilon^{\mu\nu\lambda}\partial_{\nu}H_{\lambda}^{I})\;\;
m_{1}^{2}+m_{2}^{2}=\gamma^{4}/\eta^{2},\quadm_{1}^{2}m_{2}^{2}=\gamma^{4},\quadm_{1}^{2}-m_{2}^{2}=(\gamma^{4}/\eta^{2})\sqrt{1-4\eta^{4}/\gamma^{4}}.
{\check{c}}_{\gamma,\sigma\dot{\beta}}^{\enspace\enspace\enspace\,\sigma\dot{\sigma}}=\bar{c_{a}}^{\sigma\dot{\sigma}}E_{\gamma}{c_{\sigma\dot{\beta}}}^{a}+{c_{\sigma\dot{\beta}}}^{a}\bar{c_{b}}^{\sigma\dot{\sigma}}{c_{\gammaa}}^{b}-{c_{\gamma\dot{\beta}}}^{\dot{\sigma}}
S=\int_{0}^{1}\left\{-\frac{1}{2e}\left[\dot{x}^{\mu}-i(\psi^{\mu}-\frac{2i\alpha}{3}\epsilon^{\mu\nu\rho\varsigma}\psi_{\nu}\psi_{\rho}\psi_{\varsigma})\chi\right]^{2}-i\psi_{\mu}\dot{\psi}^{\mu}\right\}d\tau\;.
\langle\psi^{\dagger}({\bf{0}},0)\psi({\bf{x}},t)\rangle=\rho\mathrm{~}\langle\mathrm{~}e^{i\mathrm{~}\Pi({\bf{0}},0)}e^{-i\mathrm{~}\Pi({\bf{x}},t)}\mathrm{~}\rangle
S[A_{0}+a_{0},a_{j},{\bar{\psi}},\psi]=\intd^{3}x\left[\frac{1}{4}F_{\mu\nu}(A+a)F_{\mu\nu}(A+a)+\sum_{b}{\bar{\psi}}_{b}(\not\!\partial+ie_{N}(\not\!\!A+\not\!a))\psi_{b}\right]\;.
\hat{S}=\intd^{11}\hat{x}\sqrt{|\hat{g}|}\left[\hat{R}-{\textstyle\frac{1}{2\cdot4!}}\hat{G}{}^{2}-{\textstyle\frac{1}{(144)^{2}}}{\textstyle\frac{1}{\sqrt{|\hat{g}|}}}\,\hat{\epsilon}\hat{G}\hat{G}\hat{C}\right]\,.
\omega_{F}^{2}\left(\begin{array}{c}{U}\\{V}\\\end{array}\right)=\left(\begin{array}{cc}{D_{F}D_{F}^{\dagger}}&{0}\\{0}&{D_{F}^{\dagger}D_{F}}\\\end{array}\right)\left(\begin{array}{c}{U}\\{V}\\\end{array}\right).
|F(p)|e^{-\epsilon|p|}\leq\int_{0}^{1}\frac{\mathrm{d}L}{L}L^{-\sigma}|G(Lp)|e^{-\epsilon|p|}\leq\int_{0}^{1}\frac{\mathrm{d}L}{L}L^{-\sigma}e^{-(1-L)\epsilon|p|}\|G\|_{\infty,\epsilon}
\langle{\mathrm{tr}}\,\lambda\lambda\rangle_{k}=\frac{32\pi^{2}}{3}\langle\phi^{3}\rangle_{k}=\frac{32\pi^{2}}{3}\left(\frac{3m}{4}\right)^{N_{f}/N_{c}}{\mathrme}^{-2\piikN_{f}/N_{c}}.
\frac{d}{dt}(\sum_{n}P_{n}z_{n}^{2})=-2c_{0}\sum_{n}P_{n}z_{n}-c_{1}\sum_{n}P_{n}z_{n}^{2}+\sum_{n}\frac{\partial\beta_{n}}{\partial\mu}z_{n}^{2}+\sum_{B}\frac{\partial\beta_{B}}{\partial\mu}z_{B}^{2}.
{\calL}=\bar{\psi}\gamma^{\mu}\left(i\nabla_{\mu}+A\omega_{\mu}+a_{\mu}-gH_{\mu}\right)\psi+g^{\mu\nu}H_{\mu}H_{\nu}-\epsilon^{\mu\nu}\rho(\partial_{\mu}a_{\nu}-\frac{B}{2}\epsilon_{ab}e_{\mu}^{a}e_{\nu}^{b}).
\begin{aligned}{}&{{}(K_{ij}f)(z_{1},\dots,z_{i},\dots,z_{j},\dots,z_{N})=f(z_{1},\dots,z_{j},\dots,z_{i},\dots,z_{N})\,,}\\{}&{{}(K_{i}f)(z_{1},\dots,z_{i},\dots,z_{N})=f(z_{1},\dots,-z_{i},\dots,z_{N})\,,}\\\end{aligned}
\partial^{\mu}\partial_{\mu}f_{kl}(z_{\alpha\beta})=\sum_{\alpha,\beta,\gamma,\delta}\left(\frac{\partialf_{kl}}{\partialz_{\alpha\beta}}\partial^{\mu}\partial_{\mu}z_{\alpha\beta}+\frac{\partial^{2}f_{kl}}{\partialz_{\alpha\beta}\partialz_{\gamma\delta}}\partial_{\mu}z_{\alpha\beta}\partial^{\mu}z_{\gamma\delta}\right)=0
W=C(N,\eta)\lambda^{\frac{N}{N-3\eta}}X^{-{\frac{N-2\eta}{N-3\eta}}}\Lambda^{\frac{4N-11\eta}{N-3\eta}}+N(\lambdaX)^{2-{\frac{4\eta}{N}}}(\Lambda^{\prime})^{1+{\frac{4\eta}{N}}}~,
U_{{\bf{k}},{\bf{q}}}^{i}=f_{{\bf{k}},{\bf{q}}}^{*}({\bf{x}})[a_{{\bf{k}}}({\bf{q}}),b_{i}^{\dagger}({\bf{q}})]+f_{{\bf{k}},-{\bf{q}}}({\bf{x}})[a_{{\bf{k}}}(-{\bf{q}}),b_{i}({\bf{q}})]
x_{r}(t)=a\,{\frac{\mathrm{cn}(at,k)}{\mathrm{sn}(at,k)}},\quady_{r}(t)=-a^{2}{\frac{\mathrm{dn}(at,k)}{\mathrm{sn}^{2}(at,k)}},\quadz_{r}(t)=-b^{2}\wp(bt),
G_{\underbrace{{\scriptstyle{2\dots2}}}_{m}\underbrace{{\scriptstyle{1\dots1}}}_{n-m}}(1,\dots,n)=(-i)^{n-1}\langle{\widetildeT}[\hat{\phi}(1)\cdots\hat{\phi}(m)]T[\hat{\phi}(m+1)\cdots\hat{\phi}(n)]\rangle\,.
\bar{a}_{i}^{b}e_{b}^{\mu}D_{\muL}|a\rangle=0\,,\quad(a_{i}^{b}\bar{a}_{i}^{b}-h_{i})|a\rangle=0\,,\quad\bar{a}_{i}^{b}\bar{a}_{i}^{b}|a\rangle=0\,,\quad\varepsilon^{ij}a_{i}^{b}\bar{a}_{j}^{b}|a\rangle=0\,,
\partial_{t}U_{k}(\rho)=\frac{1}{2}\int\frac{d^{d}q}{(2\pi)^{d}}\,\partial_{t}R_{k}\,\left(\frac{N-1}{Z_{k}q^{2}+R_{k}+U_{k}^{\prime}(\rho)}+\frac{1}{Z_{k}q^{2}+R_{k}+U_{k}^{\prime}+2\rhoU_{k}^{\prime\prime}(\rho)}\right),
R_{s^{\prime},s}^{\alpha^{\prime},\beta^{\prime};\alpha,\beta}(z^{\prime},z)=\int_{-\infty}^{\infty}\Phi_{s^{\prime},z^{\prime}}^{\ast\;\alpha^{\prime},\beta^{\prime}}(\eta)\Phi_{s,z}^{\alpha,\beta}(\eta)\;d\eta\;,
e^{I}{}_{i}(\Phi)D_{\gammac}D_{\alphaa}\Phi^{i}-{\frac{1}{4}}(\gamma^{I}{}_{J})_{a}{}^{b}e^{J}{}_{j}(\Phi)D_{\gammac}D_{\alphab}\Phi^{j}=f^{I}{}_{\gamma\alphaca},
\left[D_{\mu},D_{\nu}\right]\left(\begin{array}{c}{A}\\{B}\\\end{array}\right)=\left(\begin{array}{c}{\left[A,F_{\mu\nu}\right]}\\{\left[B,F_{\mu\nu}\right]+\left[A,\tilde{F}_{\mu\nu}\right]}\\\end{array}\right),
G_{\perp}^{\prime}=\nabla_{\perp}\Pi_{\perp}^{\prime}+gf^{abc}\frac{\tau^{a}}{2}A^{\prime}\,_{\perp}^{b}\left({\Pi^{\prime}}\,_{\perp}^{c}-\frac{1}{L}{e}\,_{\bot}^{c}\right)+g\rho_{m}\,.
{\bar{S}}_{s}^{rs}\equivS_{\phi}^{rs}=J_{\phi}^{rs}{|}_{{\vec{P}}_{\phi}=0}=\intd^{3}\sigma\Big(\sigma^{r}\Big[\pi_{\phi^{*}}\partial^{s}\phi^{*}+\pi_{\phi}\partial^{s}\phi\Big](\tau,\vec{\sigma})-(r\leftrightarrows)\Big).
\deltaS=\int_{M}d^{2}X\mathrm{~}\left(\frac{\partial{\calL}}{\partial\Lambda_{A}}\mathrm{~}\delta\Lambda_{A}+\frac{\partial{\calL}}{\partial\partial_{\alpha}\Lambda_{A}}\mathrm{~}\delta\left(\partial_{\alpha}\Lambda_{A}\right)\right)\mathrm{.}
T_{nk}(x|g)=\left(\begin{array}{cc}{\Lambda_{\mu\nu}(x|g)}&{0}\\{0}&{-\varepsilon}\\\end{array}\right)\;,\;\;\Lambda_{\mu\nu}(x|g)=\eta_{\mu\nu}\varepsilon-\frac{e_{0}}{2}\varepsilongF_{\mu\nu}(x)\varepsilon\;.
a,b=\sqrt{({\calF}^{2}+{\calG}^{2})^{\frac{1}{2}}\pm{\calF}}\,\,;\,\,\,{\calF}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,\,\,{\calG}=-\frac{1}{4}F_{\mu\nu}^{*}F^{\mu\nu}.
\begin{array}{c}{{}[P_{\mu},K_{\nu}]=2i(M_{\mu\nu}+\eta_{\mu\nu}D)\,,}\\{{}}\\{{}[P_{\mu},S^{a}]=-\gamma_{\mu}Q^{a}\,,}\\{{}}\\{{}[K_{\mu},Q^{a}]=-\gamma_{\mu}S^{a}\,,}\\{{}}\\{{}\{Q^{a\alpha},\bar{S}_{b\beta}\}=-i\delta^{a}{}_{b}(2\delta^{\alpha}{}_{\beta}D+(\gamma^{[\mu}\gamma^{\nu]})^{\alpha}{}_{\beta}M_{\mu\nu})+2i\delta^{\alpha}{}_{\beta}A^{a}{}_{b}\,.}\\\end{array}
\qquad\qquad=\quad\left((2\sigma)^{2}\left(\delta_{\alpha\beta}\hat{x}_{\gamma}+\delta_{\alpha\gamma}\hat{x}_{\beta}+\delta_{\beta\gamma}\hat{x}_{\alpha}\right)+(2\sigma)^{3}\hat{x}_{\alpha}\hat{x}_{\beta}\hat{x}_{\gamma}\right)\frac{e^{-\frac{(x-y)^{2}}{4\sigma(n+1)}}}{(4\pi\sigma(n+1))^{2}}
\Gamma_{KZ}(E,E^{\prime})|r\!>_{E^{\prime}}={[_{E^{\prime}}<\!S_{z,1/z}|\Gamma_{KZ}(E,E^{\prime})]}\left(\frac{}{}\Gamma_{KZ}(E,E^{\prime})|p\!>_{E^{\prime}}\Gamma_{KZ}(E,E^{\prime})|p\!>_{E^{\prime}}\right).
{\calE}=\frac{1}{-g^{tt}}\left\{-g^{t\phi}p_{\phi}+\left[\left(g^{t\phi}p_{\phi}\right)^{2}+(-g^{tt})\left(\mu^{2}+g^{\phi\phi}p_{\phi}^{2}+g^{rr}p_{r}^{2}+g^{\theta\theta}p_{\theta}^{2}\right)\right]^{1/2}\right\}-qA_{t}.
V_{II}(\beta,\phi)=\mu^{D}\sqrt{a}\sum_{s=1}^{\infty}\frac{(-1)^{s+1}}{2s}g^{s}\phi^{2s}\frac{\pi^{\frac{d}{2}}}{\Gamma(s)}\Gamma(s-\frac{d}{2})\sum_{n=-\infty}^{\infty}\frac{1}{(an^{2}+c^{2})^{s-\frac{d}{2}}}.
\langlec|\mathrm{e}^{-\pi\ellH}|c\rangle_{\mathrm{NS-NS}}\sim\sum_{m,k}\left[1+(-1)^{m}\right]\left[1+(-1)^{k}\right]\mathrm{e}^{-{\frac{\pi\alpha^{\prime}\ell}{2R^{2}}}(k-\gammam)^{2}}
\sigma_{i}=2^{1/2}\left(\begin{array}{cc}{e_{i}^{3}}&{e_{i}^{2}}\\{e_{i}^{1}}&{-e_{i}^{4}}\\\end{array}\right);\qquad\bar{\sigma}_{i}=2^{1/2}\left(\begin{array}{cc}{-e_{i}^{4}}&{-e_{i}^{2}}\\{-e_{i}^{1}}&{e_{i}^{3}}\\\end{array}\right).
\langleg^{l}\rangle^{(m)}=\omega^{l(m+1)}\displaystyle\frac{(q^{2};q^{2})_{\infty}^{2}}{(t^{2};t^{2})_{\infty}^{2}}\displaystyle\frac{(t^{2}\omega^{l};t^{2})_{\infty}(t^{2}\omega^{-l};t^{2})_{\infty}}{(q^{2}\omega^{l};q^{2})_{\infty}(q^{2}\omega^{-l};q^{2})_{\infty}}.
E_{1}(x)\left|\Omega\right\rangle=\overline{\Psi}(y)\Psi(y^{\prime})E_{1}(x)\left|0\right\rangle+q\left({1-\frac{{e^{2}}}{\pi}\frac{1}{{\partial_{1}^{2}}}}\right)^{-1}\int_{y}^{y^{\prime}}{dz_{1}\delta(x_{1}-z_{1}})\left|\Omega\right\rangle.
Z_{\lambda}^{(1)}=\delta_{\mathrm{T}}-\delta_{\mathrm{L}}=\left[\left(\frac{13}6-\frac\lambda2\right)+\lambda\xi(1-\xi)\right]\frac{(g\mu^{-\epsilon})^{2}}{(4\pi)^{2}}\frac{C_{2}(G)}{\epsilon}.
\psi_{+}=\frac{1}{2\partial_{-}}(\delta^{IJ}+\Gamma^{IJ})\partial_{J}\psi_{I}~,~~~\rho_{+}=\frac{1}{2\partial_{-}}(\Gamma^{+}\Gamma^{I}\partial_{I}\psi_{+}+\frac{1}{4}\Gamma^{I}J_{I})~.
S_{|P_{T_{l}}|}^{WZ}(\eta)\doteq\frac{\kappa}{4\pi\sqrt{-1}}\int_{V_{M}}\widehat{h}^{\ast}\chi_{SU(2)}-\frac{\kappa}{4\pi\sqrt{-1}}\sum_{j=1}^{N_{0}}\int_{U_{\rho^{2}(j)}}\left.\widehat{h}\right|_{U_{\rho^{2}(j)}}^{\ast}\omega_{j},
F\Big(m_{N},H-\sum_{\ell=1}^{N-1}{U}_{\ell}\Big){\calA}_{k}^{\dagger}{\calA}_{\ell}=F\Big(m_{N},H-\sum_{\ell=1}^{N-1}{U}_{\ell}\,+1\Big){\calA}_{\ell}{\calA}_{k}^{\dagger}.
V(x,\lambda)=\frac{1}{2}\frac{L_{0}(x)+L_{1}(x)}{1-\lambda}+\,\frac{1}{2}\frac{L_{0}(x)-L_{1}(x)}{1+\lambda}=\,\frac{\lambdaL_{1}(x)+L_{0}(x)}{1-{\lambda}^{2}}
g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=\lambda(r)\mathrm{d}t^{2}-\lambda^{-1}(r)\mathrm{d}r^{2}-R(r)(\mathrm{d}\theta^{2}+\mathrm{sin}^{2}\theta\mathrm{d}\phi^{2}),
\frac{\delta^{2}W}{\deltax_{\mu}(s)\deltax_{\mu}(s^{\prime})}\mathop{=}_{s\tos^{\prime}}\frac{1}{|s-s^{\prime}|^{4}}(C_{1}+C_{2})(1-\dot{\phi}^{2})+\frac{1}{|s-s^{\prime}|^{2}}\left(\frac{(C_{1}+C_{2})}{12}\dot{\phi}\stackrel{{\,...}}{\phi}+\frac{C_{1}}{4}\ddot{\phi}^{2}\right)+O(\phi^{4})\;.
C^{\prime\prime}+\biggl[3{\calH}+2\frac{{\calH}^{\prime}}{\calH}-2\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}\biggr]C^{\prime}+3\biggl[3{\calH}^{\prime}-2{\calH}\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}\biggr]C-\partial_{\alpha}\partial^{\alpha}C=0.
A_{\mu}^{\nu}=diag\left(A_{0}^{0},-{\cal{T}}+A_{0}^{0}+B_{0}^{0},{\cal{T}}-A_{0}^{0}-\frac{B_{0}^{0}}2,{\cal{T}}-A_{0}^{0}-\frac{B_{0}^{0}}2\right)\,
{\mathsfW}_{-+}^{(210)}=\frac{1}{4!}h_{ab}^{^{(210)}}<\widehat{\Psi}_{(-)a}^{*}|B\Gamma_{[\mu}\Gamma_{\nu}\Gamma_{\rho}\Gamma_{\lambda]}|\widehat{\Psi}_{(+)b}>\Phi_{\mu\nu\rho\lambda}
{\calL}_{d=4}^{SUSY}=\frac{1}{4e^{2}}\int\left(W^{2}d^{2}\theta+\bar{W}^{2}d^{2}\bar{\theta}\,\right)+\frac{1}{e^{2}}\sum_{s,t=0}^{\infty}b_{st}^{(\beta)}\intd^{4}\theta~W^{2}\bar{W}^{2}~X^{s}Y^{t}
{\calH}^{\rho_{1}}{}_{r_{1}\rho_{2}r_{2}\cdots\rho_{n}r_{n}}(\widetilde{z_{21}},\widetilde{z_{31}},\cdots,\widetilde{z_{n1}})={\calH}^{\rho_{1}}{}_{r_{1}\rho_{2}r_{2}\cdots\rho_{n}r_{n}}(\widetilde{z_{21}^{\prime\prime}},\widetilde{z_{31}^{\prime\prime}},\cdots,\widetilde{z_{n1}^{\prime\prime}})\,.
d\epsilon_{ab}=\left(\begin{array}{cccc}{0}&{-dv_{1}}&{-dv_{2}}&{-dv_{3}}\\{dv_{1}}&{0}&{-d\theta_{12}}&{d\theta_{31}}\\{dv_{2}}&{d\theta_{12}}&{0}&{-d\theta_{23}}\\{dv_{3}}&{-d\theta_{31}}&{d\theta_{23}}&{0}\\\end{array}\right)
\epsilon(r,\theta,\phi,\psi)=e^{-{\frac{i}{2}}\Gamma_{012}\theta}e^{{\\frac{1}{2}}\Gamma_{23}\phi}e^{-{\frac{i}{2}}\Gamma_{014}\psi}\Big({\\frac{1}{2r}}-itr({\frac{3}{(Z_{e})_{\mathrm{cr}}})}^{\frac{3}{2}}\Gamma_{1}(1+i\Gamma_{0})\Big)\varepsilon_{0},
f_{\hat{\alpha}\hat{\beta}}^{-1}=\left(\begin{array}{cccc}{O}&{I}&{\vec{0}}&{\vec{0}}\\{-I}&{O}&{\vec{0}}&{\vec{0}}\\{\vec{0}^{T}}&{\vec{0}^{T}}&{0}&{1}\\{\vec{0}^{T}}&{\vec{0}^{T}}&{-1}&{0}\\\end{array}\right),
\eta(s,t)=(s_{11}^{2}+s_{12}^{2})\,\int_{0}^{\infty}\int_{0}^{\infty}ds_{11}^{\prime}ds_{12}^{\prime}\,\frac{s_{11}^{\prime}s_{12}^{\prime}}{s_{11}^{\prime2}+s_{12}^{\prime2}}\,\gamma(s_{11},s_{11}^{\prime},t)\gamma(s_{12},s_{12}^{\prime},t)\.
\times\left(1-\frac{p^{\prime}q}{pp^{\prime}}\right)\delta_{+}\left(q^{2}-\mu^{2}\right)\theta\left(q_{0}\right)\delta_{+}\left[\left(p+p^{\prime}-q\right)^{2}-s^{\prime}\right]\theta\left(p_{0}+p_{0}^{\prime}-q_{0}\right)d^{4}qds^{\prime},
\Omega_{O}^{(1)2}[(g_{1}+\frac{3}{2}g_{2})M^{\prime}(q_{s})+\frac{1}{4}M^{\prime\prime}(q_{s})]+(g_{1}+\frac{1}{2}g_{2})V^{\prime\prime\prime}(q_{s})+\frac{1}{8}V^{\prime\prime\prime\prime}(q_{s})<0,
H_{{\mathbfi}{\mathbfj}{\mathbfk}{\mathbfl}}-{\bar{H}}_{{\mathbfi}{\mathbfj}{\mathbfk}{\mathbfl}}=-\frac{3i}{2}\eta_{{\mathbfm}({\mathbfi}}[\theta^{\mathbfm}{}_{{\mathbfj}}{\bar{\theta}}_{{\mathbfk}{\mathbfl})}+{\bar{\theta}}^{\mathbfm}{}_{{\mathbfj}}\theta_{{\mathbfk}{\mathbfl})}].
D_{\phi}=D_{\phi}^{(2)}+\frac{\lambda^{1}}{N^{2}}\sum_{n=3}^{\infty}\frac{1}{n2^{n}}\frac{V_{n}}{N^{n}}Tr_{p}(6\Sigma_{\phi}^{(1)}(p)N^{2/3})^{n}.
q^{{\frac{1}{5}}}\chi_{-\frac{1}{5}}(q)=1+q+q^{2}+q^{3}+2\,q^{4}+2\,q^{5}+3\,q^{6}+3\,q^{7}+4\,q^{8}+5\,q^{9}+6\,q^{10}+7\,q^{11}+9\,q^{12}+O(q^{13})
R_{\PQL}^{M}\varphi_{,T}^{P}\varphi_{,T}^{Q}\psi^{L}=\kappa\left[\varphi_{,T}^{M}\left(\psi_{N}\varphi_{,T}^{N}\right)-\psi^{M}\left(\varphi_{N,T}\varphi_{,T}^{N}\right)\right]=0
2\pi\alpha^{\prime}{\calL}_{b}={\frac{1}{2}}\left[(\partial_{a}\vec{r})^{2}+(\partial_{a}\omega)^{2}+(\partial_{a}\vec{y})^{2}-\mu^{2}\vec{y}^{2}\right]+2\,(1+e^{2U_{0}})^{-1/2}\mu\,y_{2}\partial_{\sigma}y_{1}\.
P_{R}^{-}={\frac{g^{2}L}{4\pi^{2}}}\left[\sum_{N=-\infty}^{\infty}\!\!\!\!{}^{\stackrel{\prime}{}}\thinspace\thinspace{\frac{1}{N^{2}}}C_{N}^{3}C_{-N}^{3}+\sum_{n=-\infty}^{\infty}{\frac{1}{(n-D_{0}^{3})^{2}}}\left\{C_{n}^{+},C_{-n}^{-}\right\}\right]\;.
\frac{\partialS(L(\stackrel{o}p_{s},p_{s}))}{\partial{p_{s}}_{\mu}}=\frac{\partiale^{A(p_{s})}}{\partial{p_{s}}_{\mu}}=\int_{0}^{1}dxe^{xA(p_{s})}\frac{\partialA(p_{s})}{\partial{p_{s}}_{\mu}}e^{-xA(p_{s})}e^{A(p_{s})}.
S_{ji}^{({\widehatG}/H)}(\theta,q)~=~\left(x^{-\rho_{H}\cdoth}\otimesx^{-\rho_{H}\cdoth}\right)~S_{ji}^{({\widehatG})}~\left(x^{\rho_{H}\cdoth}\otimesx^{\rho_{H}\cdoth}\right)\,
{\frac{N}{2}}\biggl(\sum_{i=1}^{N-1}2\theta_{i}\delta_{ik}+\sum_{1\leqi<j\leqN-1}(\theta_{j}\delta_{ik}+\theta_{i}\delta_{jk})\biggr)=\pi\sum_{i=1}^{N-1}(N-i)\delta_{ik},\quadk=1,2,\cdots,N-1.
L_{s}=\;\frac{q^{2}}{1+q^{2}}\;e^{-1}\;(\dot{x}_{\mu})^{2}+\;\frac{e}{1+q^{2}}\;m^{2}+b\;\dot{e}+\frac{b^{2}}{1+q^{2}}+\dot{\bar{c}}\;\dot{c}.
TrfK(t,x,x)=(4\pit)^{-3/2}tr(\int_{B^{3}}dx\sqrt{g}\sum_{k=0}^{\infty}\frac{(-t)^{k}}{k!}(fa_{k})+\int_{S^{2}}d\theta\sqrt{\gamma}\sum_{k=0}^{\infty}t^{\frac{k+1`}{2}}c_{\frac{k+1}{2}}(f))
\frac{D}{D\tau}g_{\mu\nu}=\dot{x}^{\sigma}\frac{\partial}{\partialx^{\sigma}}g_{\mu\nu}=-\frac{1}{2}\left[\frac{\partialf_{\mu}}{\partial\dot{x}^{\nu}}-\frac{\partialf_{\nu}}{\partial\dot{x}^{\mu}}\right]
I=-\frac{1}{4\pi\alpha^{\prime}}\intd^{2}\sigma\sqrt{h}(h^{\alpha\beta}g_{\mu\nu}+\epsilon^{\alpha\beta}B_{\mu\nu})\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}+\frac{1}{4\pi}\intd^{2}\sigma\sqrt{h}(\frac{1}{2}h^{\alpha\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi+\PhiR^{(2)})
\deltaS_{g}=\frac{i}{48}[\int(H-{\bar{H}})F^{{\mathbfi}{\mathbfj}}F_{{\mathbfi}{\mathbfj}}+\frac{3}{10}(H-{\bar{H}})_{{\mathbfi}{\mathbfj}{\mathbfk}{\mathbfl}}F^{{\mathbfi}{\mathbfj}}F^{{\mathbfk}{\mathbfl}}\;]
\intd^{4}x(\epsilon_{\mu\nu\rho\sigma}\partial^{\nu}B^{\rho\sigma})^{2}=\intd^{4}x2(-B_{\mu\nu}\partial^{2}B_{\mu\nu}-B_{\mu\nu}\partial_{\nu}\partial_{\lambda}B_{\lambda\mu}+B_{\nu\mu}\partial_{\mu}\partial_{\lambda}B_{\lambda\nu})
\partial_{n}X^{i}=\left({\frac{\partialz}{\partial\tau}}\right)\partialX^{i}-\left({\frac{\partial\bar{z}}{\partial\tau}}\right)\bar{\partial}X^{i}=\left({\frac{\partialz}{\partial\tau}}\right)\partialY^{i}+\left({\frac{\partial\bar{z}}{\partial\tau}}\right)\bar{\partial}Y^{i}=\partial_{\tau}Y^{i},