harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbf{E}^{\prime}(\mathbf{x})=\frac{1}{4\pi}\int\rho_{e}^{\prime}(\mathbf{x}^{\prime})\frac{\mathbf{r}}{r^{3}}\,d^{3}x^{\prime}\,-\frac{1}{4\pi}\int\mathbf{J}_{m}^{\prime}(\mathbf{x}^{\prime})\,{\bf\times}\,\frac{\mathbf{r}}{r^{3}}\,d^{3}x^{\prime},
4D:\quaddet\left(\begin{array}{cc}{i\sigma_{2}\otimesz}&{\sigma^{\mu}P_{\mu}\otimes\gamma^{m}v_{m}}\\{\sigma^{T\mu}P_{\mu}\otimes\gamma^{Tm}v_{m}}&{i\sigma_{2}\otimesz^{*}}\\\end{array}\right)=0
\sum_{L=0}^{M}\frac{(\rho_{1})_{L}(\rho_{2})_{L}(aq/\rho_{1}\rho_{2})^{L}}{(aq/\rho_{1})_{L}(aq/\rho_{2})_{L}}\frac{\alpha_{L}}{(q)_{M-L}(aq)_{M+L}}=\sum_{L=0}^{M}\frac{(\rho_{1})_{L}(\rho_{2})_{L}(aq/\rho_{1}\rho_{2})^{L}(aq/\rho_{1}\rho_{2})_{M-L}}{(aq/\rho_{1})_{M}(aq/\rho_{2})_{M}(q)_{M-L}}\beta_{L},
\begin{array}{ccccccc}{\nabla_{i}\Phi}&{=}&{(\partial_{i}+{\frac{1}{2}}p\partial_{i}{\calK})\Phi}&{;}&{\nabla_{i^{*}}\Phi}&{=}&{(\partial_{i^{*}}-{\frac{1}{2}}p\partial_{i^{*}}{\calK})\Phi}\\\end{array}
\nu=e^{\frac{2\,\pi}{n}\sqrt{-1}},\,\,\,\,\,\,\,\,\,\theta_{1}=1,\,\,\,\,\,\,\,\,\,\theta_{\kappa}=\left(\begin{array}{c}{\frac{\gamma+s\kappa}{n}-1}\\{\kappa-1}\\\end{array}\right),
\Omega_{\epsilon}^{\frac{D-2}{2}}{P_{\epsilon}}\Omega_{\epsilon}^{\frac{2-D}{2}}=\Omega_{\epsilon}^{-2}P=-{\frac{1}{\Omega_{\epsilon}^{2}a^{2}}}\left(z^{D-2}\partial_{z}z^{2-D}\partial_{z}+\sum_{i=1}^{D-1}\partial_{i}^{2}\right).
\hat{H}_{0}=\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{q}^{2}=\frac{1}{2}\hbar\omega[a^{\dagger}a+aa^{\dagger}]=\hbar\omega\left[a^{\dagger}a+\frac{1}{2}\right]\.
u(\mathbf{p,}i)\simeq\left(1+\frac{\mathbf{p}^{2}}{8m^{2}}\right)\left[\begin{array}{c}{1}\\{\frac{(\overrightarrow{\mathbf{\sigma}}\mathbf{\cdotp})}{2m}}\\\end{array}\right]\varphi_{i}=\left[\begin{array}{c}{\left(1+\frac{\mathbf{p}^{2}}{8m^{2}}\right)}\\{\frac{(\overrightarrow{\mathbf{\sigma}}\mathbf{\cdotp})}{2m}}\\\end{array}\right]\varphi_{i}
\begin{array}{rcl}{a_{i,\kappa}}&{=}&{\frac{\kappa}{n-2}+\frac{n-i}{n}-\frac{1}{n(n-2)}\,\,\,(i=1,2,\dots,n),}\\{b_{i,\kappa}}&{=}&{\frac{\kappa}{n-2}+\frac{3-i}{2}-\frac{1}{2(n-2)}\,\,\,(i=1,2),}\\{b_{i,\kappa}}&{=}&{\frac{\kappa}{n-2}+\frac{i-2}{n-2}+\frac{\delta_{i}}{n-2}\,\,\,(i=3,\dots,n-1),}\\\end{array}
B=\left(\begin{matrix}{0}&{-\theta_{1}}&{0}&{0}&{0}&{0}&{0}&{0}\\{\theta_{1}}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{-\theta_{2}}&{0}&{0}&{0}&{0}\\{0}&{0}&{\theta_{2}}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{-\theta_{3}}&{0}&{0}\\{0}&{0}&{0}&{0}&{\theta_{3}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{-\theta_{4}}\\{0}&{0}&{0}&{0}&{0}&{0}&{\theta_{4}}&{0}\\\end{matrix}\right)~,
\theta^{\mu\nu}=2\pi\alpha^{\prime}\left(\begin{array}{cccc}{0}&{-\frac{e}{g(1-e^{2})}}&{0}&{0}\\{\frac{e}{g(1-e^{2})}}&{0}&{0}&{0}\\{0}&{0}&{0}&{-\frac{gb}{1+g^{2}b^{2}}}\\{0}&{0}&{\frac{gb}{1+g^{2}b^{2}}}&{0}\\\end{array}\right),
S_{\pm}(x,y)=\langleq_{\alpha}^{a}(x){\bar{q}}_{\beta}^{b}(y)\rangle=\intd^{4}z\,\frac{\left[\Psi_{z}^{\pm}(x){\bar{\Psi}}_{z}^{\pm}(y)\right]_{\alpha\beta}^{ab}}{m^{*}}
{\widehat{\tilde{\calQ}}}{}_{\alpha}^{I}=-\frac{i}{m}p_{\alpha}{}^{\beta}\widehat{\calQ}_{\beta}^{I}=\left.i\widehat{K}\cdot\widehat{\calQ}_{\alpha}^{I}\right|_{\mathrm{q}\to2-\mathrm{q}}\,,
a^{2}=\left({\calF}^{2}+{\calJ}^{2}\right)^{\frac{1}{2}}+{\calF},b^{2}=\left({\calF}^{2}+{\calJ}^{2}\right)^{\frac{1}{2}}-{\calF};{\calF}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu},{\calJ}=-\frac{1}{4}F_{\mu\nu}^{*}F^{\mu\nu}\;\;.
\frac{d\,{\mathcalA}^{\{{\mathcalN}\}}}{dt}={\mathcalA}^{\{{\mathcalN}\}}\sum_{j}\sum_{\zeta_{j}}\left[\frac{d\,{\mathcalN}_{(\zeta_{j})}^{(\alpha_{j})}}{dt}\psi({\mathcalN}_{(\zeta_{j})}^{(\alpha_{j})}+1)-\sum_{\nu_{\alpha_{j}}}\frac{d\,{\mathcalN}_{\nu_{\alpha_{j}};(\zeta_{j})}^{(\alpha_{j})}}{dt}\psi({\mathcalN}_{\nu_{\alpha};(\zeta_{j})}^{(\alpha_{j})}+1)\right]
q(\hat{A}_{4}=0)=\ointK^{\alpha}\,dS_{\alpha}=-\left.\int\frac{\,d^{3}\vec{x}}{16\pi^{2}}\epsilon^{ijk}\,\mathrm{tr\,}\left(\hat{A}_{i}\,\hat{F}_{jk}+\frac{2i}3\hat{A}_{i}\hat{A}_{j}\hat{A}_{k}\right)\right|_{x_{4}=-\infty}^{x_{4}=+\infty}.
\langle\vec{x}_{b}|\frac{1}{{\hat{\vec{p}}}^{2}+\hat{V}(r)-E_{r}}|\vec{x}_{a}\rangle=i\int_{0}^{\infty}d\tau_{f}\int{\calD}\vec{p}{\calD}\vec{x}exp\{i\int_{0}^{\tau_{f}}d\tau[\vec{p}\dot{\vec{x}}-f(\vec{x}(\tau))H]\}
\frac{1}{i\partial_{-}-eA_{-}}=\frac{1}{i\partial_{-}}+\left(\frac{1}{i\partial_{-}}eA_{-}\right)\ast\frac{1}{i\partial_{-}-eA_{-}}=\frac{1}{i\partial_{-}}+\frac{1}{i\partial_{-}-eA_{-}}\ast\left(eA_{-}\frac{1}{i\partial_{-}}\right).
{\bfa}_{\;\;\mu}^{\nu}=\left(\begin{array}{ccccccc}{p_{1}}&{0}&{p_{3}}&{0}&{0}\\{-2p_{1}}&{p_{2}}&{0}&{0}&{0}\\{p_{1}}&{-2p_{2}}&{p_{3}}&{0}&{0}\\{0}&{p_{2}}&{-2p_{3}}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\\end{array}\right).
{\bfA}_{\pm}^{(1,0)}=\frac{i}{\sqrt{2}}\,\big(\,\pm{\bfB}_{+}^{(1,0)}+{\bfB}_{-}^{(1,0)}\,\big)\,,\qquad\{\bfA}_{\pm}^{(0,1)}=\frac{i}{\sqrt{2}}\,\big(-{\bfB}_{+}^{(0,1)}\pm{\bfB}_{-}^{(0,1)}\,\big)\,,
{H^{lf}\equivP^{-}}=\intdx\,\Bigl[\omega(\lambda\omega^{2}-m^{2})\varphi+{\frac{1}{2}}(3\lambda\omega^{2}-m^{2})\varphi^{2}+\lambda\omega\varphi^{3}+{\frac{\lambda}{4}}\varphi^{4}\Bigr],
F_{2k}(r)={\frac{r}{(r^{2}-n^{2})^{k}}}\int^{r}\left[{\frac{(s^{2}-n^{2})^{k}}{s^{2}}}+{\frac{2k+1}{l^{2}}}{\frac{(s^{2}-n^{2})^{k+1}}{s^{2}}}\right]\,ds.
X_{1}\equivX_{\uparrow}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}{1}\\{0}\\{1}\\{0}\\\end{array}\right]\;,\quadX_{-1}\equivX_{\downarrow}=\frac{1}{\sqrt{2}}\left[\begin{array}{r}{0}\\{1}\\{0}\\{-1}\\\end{array}\right]\;.
V(x)=\frac{\Gamma\left(\frac{d-2-\eta_{A}}{2}\right)}{2^{\eta_{A}}(4\pi)^{d/2}\Gamma\left(\frac{2+\eta_{A}}{2}\right)}\left[\left(\frac{|x|}{a}\right)^{2-d+\eta_{A}}-1\right].
{\calA}_{4}=~_{41i}\langleV_{3}|_{j23}\langleV_{3}|Y_{-2}^{(i)}\frac{b_{o}^{(i)}}{L^{(i)}}|V_{2}\rangle_{ij}|\tilde{A}_{4}\rangle_{4}|\tilde{A}_{1}\rangle_{1}|\tilde{A}_{2}\rangle_{2}|\tilde{A}_{3}\rangle_{3},
W^{\prime}=\frac{\Lambda^{\prime(3N-N_{f})/(N-N_{f})}}{(\mathrm{det}\tilde{Q}^{\primei}Q^{\primej})^{1/(N-N_{f})}}=\frac{\Lambda^{(3N-N_{f})/(N-N_{f})}(Z_{Q}Z_{\tilde{Q}})^{N_{f}/2(N-N_{f})}}{(\mathrm{det}\tilde{Q}^{i}Q^{j})^{1/(N-N_{f})}(Z_{Q}Z_{\tilde{Q}})^{N_{f}/2(N-N_{f})}}=W.
\Big[\underbrace{\left(8-4\cdot3\right)\cdot3}_{\stackrel{\scriptstyle\mathrm{vertex\,correction}}{{\scriptstyle\mathrm{+self\,energy}}}}-\underbrace{4\cdot1}_{{\scriptstyle\mathrm{vertex}}}-\underbrace{4\cdot({\scriptstyle-}4)}_{{\scriptstyle\mathrm{nonplanar}}}\Big]\,\,Y_{R}^{(e)}\,\mathrm{Tr}\,T^{A}T^{B}\,=0.
changed\intd^{25-p}q_{\bot}\,\left(\frac{1+W_{0}^{2}}{k_{0}W_{0}^{2}}\right)^{q_{\bot}^{2}}\alpha_{-1}\cdot\tilde{\alpha}_{-1}\tilde{c}_{1}(c_{0}+\tilde{c}_{0})c_{1}|q_{\bot}\rangle.
(M_{\hat{a}\hat{b}})_{\mathrm{~}\hat{d}}^{\hat{c}}=\delta_{\hat{a}}^{\hat{c}}\eta_{\hat{b}\hat{d}}-\delta_{\hat{b}}^{\hat{c}}\eta_{\hat{a}\hat{d}}=-(M_{\hat{b}\hat{a}})_{\mathrm{~}\hat{d}}^{\hat{c}},
\frac{8\pi}{3}\rho_{eff}=\frac{1}{4}[(f^{-1}(T)+EHe^{\Phi_{0}})^{2}-(1+\frac{\ell^{2}e^{2\Phi_{0}}}{2}H)]\frac{Q^{2}e^{2\Phi_{0}}}{r^{10}}H^{-\frac{5}{2}}
E_{C}^{(2)\;reg}=-\frac{M_{0}}{4\sqrt{\alpha}}-\frac{M_{0}^{2}R}{8\pi\alpha}\Gamma(-1)-\frac{M_{0}}{2\pi\sqrt{\alpha}}\sum_{n=1}^{\infty}n^{-1}K_{1}\left(\frac{2nM_{0}R}{\sqrt{\alpha}}\right).
F(r)={\frac{1}{3l^{2}(r^{2}-n^{2})^{2}}}\left[3r^{6}+(l^{2}-15n^{2})r^{4}-3n^{2}(2l^{2}-15n^{2})r^{2}-6mrl^{2}-3n^{4}(l^{2}-5n^{2})\right]
\hat{\phi}(t,r)=\frac{1}{2\pi}\int_{0}^{\infty}\frac{d\omega}{\sqrt{\omega}}\,\left[b_{\omega}e^{-i\omegat}\psi_{\omega}(t,r)+b_{\omega}^{\dagger}e^{i\omegat}\psi_{\omega}^{\ast}(t,r)\right].
{\frac{\sqrt{2}-1}{\sqrt{2}}}=\sum_{k=0}^{\infty}\left[(\sum_{n=0}^{\infty}(-1)^{n}X_{2n,2k+1})X_{2k+1,0}\right]\neq\sum_{n=0}^{\infty}\left[(-1)^{n}\sum_{k=0}^{\infty}\left(X_{2n,2k+1}X_{2k+1,0}\right)\right]=1\,.
\lambda_{L}^{(l)}=\bar{\lambda}_{L}\biggr[1+\sum_{j=1}^{l-1}\biggr(\bar{\lambda}_{L}\frac{2}{\pi}\sum_{i=0}^{L}\frac{k^{2(L-i)}\Lambda^{2i+1}}{2i+1}\biggr)^{j}\biggr].
\omega^{-1}(\eta(t))\tilde{E}_{\mu}[\eta|t]\omega(\eta(t))=-\frac{2}{\bar{N}}\epsilon_{\mu\nu\rho\sigma}\dot{\eta}^{\nu}(t)\int\delta\xidsE^{\rho}[\xi|s]\dot{\xi}^{\sigma}(s)\dot{\xi}^{-2}(s)\delta(\xi(s)-\eta(t)),
u=\Phi(\chi_{x},\chi_{xx},\chi_{xxx})\equiv\frac{\chi_{xxx}}{\chi_{x}}-\frac{\chi_{xx}^{\;2}}{\chi_{x}^{2}}+2\alpha\chi_{xx}+\alpha^{2}\chi_{x}^{2}+2\alpha\varepsilon+\frac{\varepsilon^{2}}{\chi_{x}^{2}}
\psi_{R}\{p^{\mu}\}\,=\,\left[\begin{array}{c}{\phi_{R}(p^{\mu})}\\{0}\\\end{array}\right]\,,\,\psi_{L}\{p^{\mu}\}\,=\,\left[\begin{array}{c}{0}\\{\phi_{L}(p^{\mu})}\\\end{array}\right]\quad,
\delta\varphi_{0}^{(1)}={\calN}\dot{\varphi}_{0}(y)\int_{y^{(+)}}^{y}{dy^{\prime}}\left({\frac{2\kappa}{3a^{2}}}-{\frac{(m_{0}^{(1)})^{2}}{a^{4}(y^{\prime})\dot{\varphi}_{0}^{2}(y^{\prime})}}\right).
\tilde{\Omega}_{5}={\frac{4}{5!R^{2}}}\varepsilon_{\hat{m}_{0}^{\prime}\hat{m}_{1}^{\prime}\dots\hat{m}_{5}^{\prime}}X^{\hat{m}_{0}^{\prime}}dX^{\hat{m}_{1}^{\prime}}\dotsdX^{\hat{m}_{5}^{\prime}}\.
c\frac{d}{dt}\left((1-\frac{1}{c^{2}}|\frac{d{\bfx}_{1}}{dt}|^{2})^{-1/2}\right)=\frac{1}{c}\sum_{i=1}^{3}\frac{\partialA_{0}({\bfx}_{1})}{\partialx_{1}^{i}}\frac{dx_{1}^{i}}{dt},
\Xi_{1}(\theta,\varphi)=\binom{C_{nl}^{1}(\theta,\varphi)}{C_{nl}^{2}(\theta,\varphi)},\quad\Xi_{2}(\theta,\varphi)=\binom{C_{nl}^{1}(\theta,\varphi)}{-C_{nl}^{2}(\theta,\varphi)}
w(b,\bar{b})=C\oint\prod_{j=1}^{N+1}{\frac{dz_{j}}{z_{j}-z_{j+1}}}\delta\left(b^{\mu}(\tau)-\sum_{j=1}^{N}k_{j}^{\mu}\delta(\tau-\tau_{j})\right)\delta\left(\bar{b}^{\mu}(\tau)-\sum_{j=1}^{N}k_{j}^{\mu}\delta(\tau-\tau_{j})\right)
\alpha_{(\cdot),(\cdot)}^{-1,1}=A_{(\cdot),(\cdot)}^{-1,1}+\frac{1}{N^{2}}A_{(\cdot),(\cdot)}^{0,0}-\frac{1}{N}B_{(\cdot),(\cdot)}^{0,0}-\frac{1}{N}B_{(\cdot),(\cdot)}^{-1,1}.
{\mathrm{}^{*}\mathcal{L}}_{\lambda}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\beta^{2}\left(\frac{\lambda}{4}+\frac{1}{\lambda}\right)+\frac{\lambda}{24m^{2}}H_{\mu\nu\rho}H^{\mu\nu\rho}-\frac{\chi}{2\theta}\varepsilon^{\mu\nu\rho\lambda}B_{\nu\rho}\partial_{\mu}A_{\lambda}\;,
F~=~f_{+}(x)\left(\begin{matrix}{1}&{0}\\{0}&{1}\\\end{matrix}\right)~+~f_{-}(x)\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right)~=~\left(\begin{matrix}{f_{1}(x)}&{0}\\{0}&{f_{2}(x)}\\\end{matrix}\right)~,
a_{\beta,2}={\frac{\pi}{3\alpha}}(1-\alpha^{2})\int_{\Sigma}\left[\frac16R+{\frac{1+\alpha^{2}}{60\alpha^{2}}}(2R_{\mu\nu\lambda\rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}-R_{\mu\nu}n_{i}^{\mu}n_{i}^{\nu})\right]~~~.
\langle{\calO}(\hat{\Phi},\hat{\Pi})\rangle=\langle{\calO}(\Phi,\Pi)\rangle\equiv{\frac{1}{Z}}\int{\calD}\Phi(x){\frac{{\calD}\Pi(x)}{2\pi}}\;{\calO}(\Phi,\Pi)\;W[\Phi,\Pi;t]\;,
\left(\frac{\delta\rho}{\rho}\right)_{\Lambda}\sim\left(\frac{t_{\Lambda}}{t_{EQ}}\right)^{2/3}\left(\frac{\delta\rho}{\rho}\right)_{0}\sim\left(\frac{R\,T_{EQ}^{2}}{M_{pl}}\right)^{2/3}\left(\frac{\delta\rho}{\rho}\right)_{0}\equiv(1+z)_{EQ}\left(\frac{\delta\rho}{\rho_{0}}\right)\.
H_{l.c}=-p_{+}={\frac{p_{\alpha}^{2}}{4p_{-}}}+{\frac{1}{2\alpha^{\prime}p_{-}}}{\calH},~~{\calH}={\calH}_{0}+{\calH}_{R}+{\calH}_{L}+N_{R}^{0}+N_{L}^{0},
\frac{{\delta}{\calH}[N]}{{\delta}A_{ai}^{(n)}}=\frac{1}{2}{\epsilon}_{abc}\partial_{b}\left(\frac{N}{{\calH}+1}((-1)^{n}B_{ci}^{(n)}-{\epsilon}_{cde}D_{di}^{(n)}{\calH}_{e}^{M})\right)
\frac{d}{dt}\biggl|_{t=0}Z(R(S),\langle\cdot\,,\cdot\rangle^{A(t)})=\biggl(\sum_{k=0}^{N}\frac{1}{2}(-1)^{k}Tr(B_{k})\biggr)\,Z(R(S),\langle\cdot\,,\cdot\rangle)\,.
C_{m}^{d}(\sigma_{m})=(-1)^{l(\sigma_{m})}\frac{d^{l(\sigma_{m})}}{\prod_{j=1}^{l(\sigma_{m})}d_{j}\prod_{i=1}^{m}mul(i,\sigma_{m})!}=(-1)^{l(\sigma_{m})}\frac{d^{l(\sigma_{m})}\cdotN(\sigma_{m})}{\prod_{j=1}^{l(\sigma_{m})}d_{j}\cdotl(\sigma_{m})!},
\Psi_{s}^{\pm}(x_{\perp})=N\left(\begin{array}{c}{\left[1+\left(p^{3}+s\widetilde{M}\right)/M\right]g_{0}\left(\varphi\right)\psi^{\pm}\left(r\right)}\\{\left[-1+\left(p^{3}+s\widetilde{M}\right)/M\right]g_{0}\left(\varphi\right)\psi^{\pm}\left(r\right)}\\\end{array}\right)\;,
K=\left(\begin{array}{ccccc}{\frac{\pi}{\pi\!+\!gN}\frac{1}{-\triangle\!+\!\frac{e^{2}N}{\pi\!+\!gn}}}&{}&{}&{}&{}\\{}&{\frac{1}{-\triangle}}&{}&{}&{}\\{}&{}&{.}&{}&{}\\{}&{}&{}&{.}&{}\\{}&{}&{}&{}&{\frac{1}{-\triangle}}\\\end{array}\right)\;.
A_{0}^{(0)}\sim-DA_{1}^{(1)}\sim\frac{D}{D-1}A_{2}^{(2)}\sim\frac{2^{D}D(\zeta_{c}-\zeta)}{(D-1)^{2}\Gamma(D)}\Gamma^{4}\left(\frac{D+1}{2}\right),\quadm=0.
{\tilde{\cal{H}}^{(2)}}=2\eta^{1}\eta^{1}-\eta^{1}\eta^{1^{\prime\prime}}-\eta^{1}\eta^{2^{\prime}}-\eta^{2}\eta^{2}+\eta^{2}\eta^{1^{\prime}}-\eta^{2}\eta^{4}+\frac{1}{2}\eta^{3}\eta^{3}.
a_{i}=\sum_{\begin{array}{c}{k_{1}\alpha_{1}+\dots+k_{m}\alpha_{m}=i}\\{\alpha_{1}>\alpha_{2}>\dots>\alpha_{m}}\\\end{array}}\frac{(-1)^{k_{1}+\dots+k_{m}}s_{\alpha_{1}}^{k_{1}}\dotss_{\alpha_{m}}^{k_{m}}}{k_{1}!\alpha_{1}^{k_{1}}k_{2}!\alpha_{2}^{k_{2}}\dotsk_{m}\alpha_{s}^{k_{m}}!}
\Theta_{\mu\nu}=-\frac{1}{48\pi}\left(\partial_{\mu}\varphi\partial_{v}\varphi-\frac{1}{2}\gamma_{\mu\nu}\gamma^{\alpha\beta}\partial_{\alpha}\varphi\partial_{\beta}\varphi\right)-\frac{1}{24\pi}\left(d_{\mu}d_{\nu}\varphi-\frac{1}{2}\gamma_{\mu\nu}\gamma^{\alpha\beta}d_{\alpha}d_{\beta}\varphi\right)
ds_{M2}^{2}\approx{\frac{r^{4}}{R^{4}}}(-dt^{2}+dx_{1}^{2}+dx_{2}^{2})\,+R^{2}\,\frac{dr^{2}}{r^{2}}+R^{2}\,\frac{\Lambda}{6}\,ds_{X_{7}}^{2}.
A_{\alpha}=\left(\begin{matrix}{A_{\alpha}^{1}}&{0}&{0}&{0}\\{0}&{A_{\alpha}^{2}}&{0}&{0}\\{0}&{0}&{A_{\mu}^{3}}&{0}\\{0}&{0}&{0}&{A_{\alpha}^{4}}\\\end{matrix}\right)
Z_{T}\sime^{\pi^{2}\ell^{2}/2G\beta}e^{-\frac{\pi^{2}\ell^{2}}{\beta^{2}}\left(\frac{\partialE_{r}}{\partialr}\right)B+\frac{r^{2}}{\ell^{2}}\left(\frac{\partialE_{r}}{\partialr}\right)B}
\left(\sum_{k,m}M_{km}^{(p)}(u)\frac{\partial^{2}}{\partialu_{k}\partialu_{m}}+\sum_{m\ne1}K_{m}^{(p)}(u)\frac{\partial}{\partialu_{m}}+L^{(p)}(u)\right)\left(\begin{array}{c}{a_{i}}\\{a_{D}^{i}}\\\end{array}\right)=0,
\left(\begin{matrix}{x}\\{x^{\prime}}\\\end{matrix}\right)=-\left(\begin{matrix}{\frac{s^{\prime}}{s}}&{1}\\{-k^{2}+\left(\frac{s^{\prime}}{s}\right)^{2}}&{\frac{s^{\prime}}{s}}\\\end{matrix}\right)\left(\begin{matrix}{y}\\{y^{\prime}}\\\end{matrix}\right)~.
L_{4}^{g}=48\kappa^{2}\left[2B_{\mu\nu\alpha\beta}B^{\mu\alpha\nu\beta}-B_{\mu\nu\alpha\beta}B^{\mu\nu\alpha\beta}+6B_{\mu\rho\alpha}^{\\\\\rho}B_{\\\\,sigma}^{\mu\sigma\alpha}-\frac{15}{49}\:(B_{\\~\mu\nu}^{\mu\nu})^{2}+P_{\mu\nu\alpha\beta}P^{\mu\nu\alpha\beta}\right].
N^{NS}=\left(\begin{matrix}{a+b}&{0}&{0}&{0}&{0}&{0}&{a+b-1}&{0}&{0}\\{0}&{a}&{b}&{0}&{0}&{0}&{0}&{a-1}&{b}\\{0}&{b}&{a}&{0}&{0}&{0}&{0}&{b}&{a-1}\\{0}&{0}&{0}&{1}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}&{0}&{0}&{0}\\{a+b-1}&{0}&{0}&{0}&{0}&{0}&{a+b}&{0}&{0}\\{0}&{a-1}&{b}&{0}&{0}&{0}&{0}&{a}&{b}\\{0}&{b}&{a-1}&{0}&{0}&{0}&{0}&{b}&{a}\\\end{matrix}\right)
\begin{array}{rcl}{M_{a,b}(\mu,\nu;2\lambda;q)}&{=}&{q^{\lambda(2(a+b)\lambda+a-b)}\displaystyle\left[\begin{array}{c}{\mu+\nu}\\{\nu-(a+b)\lambda}\\\end{array}\right]_{q},}\\{M_{a,b}(\mu,\nu;2\lambda-1;q)}&{=}&{q^{(2\lambda-1)((a+b)\lambda-a)}\displaystyle\left[\begin{array}{c}{\mu+\nu}\\{\nu-(a+b)\lambda+a}\\\end{array}\right]_{q}.}\\\end{array}
\widetilde{V}_{1}=\left(\begin{array}{ll}{1}&{-\widehat{\theta}}\\{0}&{1}\\\end{array}\right),\widetilde{V}_{2}=\left(\begin{array}{ll}{1}&{0}\\{\Phi}&{1}\\\end{array}\right),\widetilde{V}_{3}=\left(\begin{array}{ll}{A}&{0}\\{0}&{(A^{t})^{-1}}\\\end{array}\right),\widetilde{V}_{4}=\left(\begin{array}{ll}{1}&{\theta}\\{0}&{1}\\\end{array}\right).
{\calB}^{i}=\frac{1}{2}\\varepsilon^{abc}{\calF}_{jk}=\frac{1}{2gv^{3}}\\varepsilon^{ijk}\varepsilon^{abc}\phi^{a}\partial_{j}\phi^{b}\partial_{k}\phi^{c}+\varepsilon^{ijk}\partial_{j}A_{k}\.
S_{D}={\frac{1}{\sqrt2\cdot2^{2}}}\int_{u_{0}}^{u}d\tilde{u}(\taua-2\tilde{u}\partial_{\tilde{u}}a_{D})+S_{D}(u_{0}),\qquadS={\frac{1}{\sqrt2\cdot6}}(ua-2(u^{2}-\Lambda_{SW}^{4})a^{\prime}),
p^{2}\left(\frac{\lambda}{\lambda_{l}^{(\ast)}}-1\right)\left[1+O\left(\frac{1}{p}\right)\right]=\frac{p^{2}}{\lambda_{l}^{(\ast)}}\,z_{2}^{4/\epsilon}\,\left[1+O\left(\frac{1}{p}\right)\right]\;,
H_{n,k}=-n^{-(k+\frac{2}{n})}\,\prod_{s=0}^{k}\,(s\,n+1\,)\sum_{m=0}^{\lfloor\frac{k}{2}\rfloor}\frac{1}{m!\,(k-2\,m)!\,n^{m}}\,\prod_{l=0}^{m}\,{\frac{1}{l\,n+1}}\,v^{mn+1}u^{k-2m}
\begin{array}{rcl}{\mathbb{L}_{k_{(I)}}\psi}&{=}&{k_{(I)}{}^{\mu}\partial_{\mu}\psi-k_{(I)}{}^{\mu}\vartheta^{i}{}_{\mu}\Gamma_{s}(M_{i})\psi-\Gamma_{\mathrm{Adj}}(u^{-1})^{i}{}_{I}\Gamma_{s}(M_{i})\psi}\\{}&{}&{}\\{}&{=}&{\mathcal{L}_{k_{(I)}}\psi+W_{I}^{i}\Gamma_{s}(M_{i})\psi\,.}\\\end{array}
{\calA}={\calA}_{0}+{\calA}^{\mathrm{int}},~~~~~~{\calA}_{0}=\int_{t_{a}}^{t_{b}}dt\left[\frac{M}{2}\dot{x}^{2}-\frac{\Omega^{2}}{2}x^{2}\right],~~~~{\calA}^{\mathrm{int}}=\int_{t_{a}}^{t_{b}}dt\,V_{\Omega}^{\mathrm{int}}\equiv\int_{t_{a}}^{t_{b}}dt\left[-\frac{(\omega^{2}-\Omega^{2})}{2}x^{2}-V^{\mathrm{int}}(x)\right],
\delta=\delta_{1}t+\delta_{-1/2}t^{-1/2}+t\int^{t}dt^{\prime}(t^{\prime})^{-5/2}\int^{t^{\prime}}\left(\frac{\alpha}{(t^{\prime\prime})^{1/2}}\delta\phi-\frac{4}{3}(t^{\prime\prime})^{3/2}\deltaE_{00}\right)
{\calL}=\mathrm{i}\varphi^{\dagger}\partial_{+}\varphi+\mathrm{i}\chi^{\dagger}\partial_{-}\chi+\frac{\mathrm{i}}{\sqrt{2}}\left(\varphi^{\dagger}\tilde{\partial}_{m}\chi+\chi^{\dagger}\partial_{m}\varphi\right)+{\calL}_{\mathrm{int}}(\varphi,\chi)
\phi^{i}(x,z)\phi^{j}(x,z^{\prime})|_{ct}=\frac{b^{ij}}{|z-z^{\prime}|^{2}}\delta(z-z^{\prime})+c^{ij}\delta^{2}(z-z^{\prime})+\delta^{(2)}(z-z^{\prime})\\Gamma_{k}^{ij}\phi^{k}(x,z)
\omega_{0}^{2}-k_{n}^{2}\left[{\frac{1}{g}}+{\frac{1}{2}}k_{n}^{D-4}\int^{\Lambda/k_{n}}{\frac{d^{D}q}{(2\pi)^{D}}}\,{\frac{1}{q^{2}(q+1)^{2}}}\right]=0.
\begin{array}{ll}{{\calL}_{Y}=}&{-\frac{ie}{\sqrt{2}}E_{I}\left[\sigma_{I}\bar{\chi}_{I}({\bar{\tilde{\lambda}}}-\frac{\alpha}{\sqrt{\beta}}{\bar{\tilde{\xi}}})-\sigma_{I}\chi_{I}(\tilde{\lambda}-\frac{\alpha}{\sqrt{\beta}}\tilde{\xi})\right]}\\{}&{-\frac{iq}{\sqrt{2\beta}}Q_{i}\left[\phi_{i}\bar{\psi}_{i}{\bar{\tilde{\xi}}}-\phi_{i}\psi_{i}\tilde{\xi}\right].}\\\end{array}
f_{ij}^{(0)}(x,y)=\left(\begin{array}{ccccc}{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{0}&{-1}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\\end{array}\right)\delta^{2}(x-y).
J(w^{\prime}x^{4}w_{2n}|\zeta^{\prime},x^{2}\zeta_{2n})=\{v_{2n}-u_{2n}+\frac{1}{2}-k\}\prod_{j=1}^{2n-1}[v_{2n}-u_{j}+\frac{3}{2}]\prod_{a\inA\atopa\neq2n}\frac{[u_{2n}-v_{a}+\frac{3}{2}]}{[v_{a}-v_{2n}-1]},
\hat{e}_{\hat{\mu}}{}^{\hat{a}}\,=\,\left(\begin{array}{cc}{K^{-1/7}\e_{\mu}{}^{a}}&{K^{1/2}\v_{m}{}^{i}A_{\mu}^{(m)}}\\{0}&{K^{1/2}\v_{m}^{i}}\\\end{array}\right)\;,\;
\Vertl_{jk}\Vert_{0\lej,k\le2}=\left(\begin{array}{ccc}{-\frac{1}{3}}&{\frac{2}{3}}&{\frac{1}{3}}\\{\frac{2}{3}}&{-\frac{1}{3}}&{-\frac{2}{3}}\\{\frac{1}{3}}&{-\frac{2}{3}}&{-\frac{1}{3}}\\\end{array}\right).
I=\intd^{4}x\left[1-\frac{1}{2}v(1+\frac{1}{2}e^{-\phi}F^{mn}F_{mn}-\frac{1}{16}e^{-2\phi}({\cal{F}}^{mn}F_{mn})^{2})-\frac{1}{2v}+\frac{1}{4}\psi{\cal{F}}^{mn}F_{mn}\right].
L_{2j+1}^{\epsilon_{1}}Z_{21}^{-1}L_{1j}^{\epsilon_{2}}=\tilde{Z}_{21}^{-1}L_{1j}^{\epsilon_{2}}\tilde{Z}_{21}L_{2j+1}^{\epsilon_{1}}\tilde{Z}_{21}^{-1}\ll{zlzl1+}
W(x,x^{\prime})=W(x)-\frac{1}{2}W_{;\alpha}(x)\sigma^{;\alpha}+\frac{1}{2}W_{\alpha\beta}(x)\sigma^{;\alpha}\sigma^{;\beta}+\frac{1}{4}\{\frac{1}{6}W_{;\alpha\beta\gamma}(x)-W_{\alpha\beta;\gamma}(x)\}\sigma^{;\alpha}\sigma^{;\beta}\sigma^{;\gamma}+0(\sigma^{2})
{\tilde{e}}_{x}^{a}=-{\frac{1}{g_{xx}}}e_{x}^{a}\,\qquad{\tilde{e}}_{\alpha}^{a}=e_{\alpha}^{a}-{\frac{1}{g_{xx}}}\bigl(g_{x\alpha}+B_{x\alpha}\bigr)e_{x}^{a}\.
\nonumber(i)^{s}\frac{d}{ds}\zeta_{H}(s;iz)+(i)^{-s}\frac{d}{ds}\zeta_{H}(s;-iz)=(2\pi)^{s}(\frac{d}{ds}\frac{1}{\Gamma(s)})\sum_{n=1}^{+\infty}\frac{e^{-2n\pi{z}}}{n^{1-s}}+\frac{(2\pi)^{s}}{\Gamma(s)}\sum_{n=1}^{+\infty}\frac{e^{-2n\pi{z}}\mathrm{ln}(2n\pi)}{n^{1-s}}+
a_{i}^{(1)}(x)=\epsilon_{ij}\partial_{x^{j}}\int\!d\vec{y}{G}_{\mathrm{cyl}}(\vec{x},\vec{y})b^{(1)}(x)+\partial_{x^{i}}\int\!d\vec{y}{G}_{\mathrm{cyl}}(\vec{x},\vec{y})\partial_{y^{j}}a_{j}^{(1)}(y)\,.
\frac{\ddot{b}}{b}+\frac{\dot{b}^{2}}{b^{2}}+3\frac{\dot{a}}{a}\frac{\dot{b}}{b}=\dot{H}_{b}+2H_{b}^{2}+3HH_{b}=\frac{V(b)}{b^{2}}-\frac{V^{\prime}(b)}{4b}\equiv-\frac{B^{\prime}(b)}{b}
\sum_{D}\gamma_{ij}^{D}(k_{1},p_{1})\gamma_{i^{\prime}j^{\prime}}^{D}(k_{2},p_{2})=2g^{2}(K^{(0)}c_{ij}c_{i^{\prime}j^{\prime}}+K^{(1)}c_{ij}^{k}c_{i^{\prime}j^{\prime}}^{k}+K^{(2)}c_{ij}^{kl}c_{i^{\prime}j^{\prime}}^{kl})
{\calA}\cdot\Psi_{0}=\int_{\Sigma}\left([-f+i(Cf)]\varphi-(Bf)\frac{\delta}{\delta\varphi}\right)\Psi_{0}+\int_{\Sigma}\left(i(Dg)\varphi-(Ag+ig)\frac{\delta}{\delta\varphi}\right)\Psi_{0}=0
\pi^{j}=-\frac\kappa2\varepsilon^{ij}A_{i}-i\fracg2\varepsilon^{ij}\left[\phi^{*}(D_{i}\phi)-\phi(D_{i}\phi)^{*}\right]-\frac{g^{2}}2\partial^{j}A_{0}\left|\phi\right|^{2}+\frac{g^{2}}4(\partial_{0}A_{0})\left|\phi\right|^{2}
\begin{array}{l}{<((\phi_{\tau},f_{1})-E[(\phi_{\tau},f_{1})])((\phi_{\tau},f_{2})-E[(\phi_{\tau},f_{2})])....}\\{((\phi_{\tau},f_{2n})-E[(\phi_{\tau},f_{2n})])>=\hbar^{2n}\sum_{pairs}\prod_{(j,k)}\frac{1}{2i}(f_{j},{\calA}^{-1}f_{k})}\\\end{array}
\frac{B^{3}}{|B|}\left[\prod_{a=1}^{2}\delta\left(\frac{F^{a}}{|F|}+\frac{B^{a}}{|B|}\right)+\prod_{a=1}^{2}\delta\left(\frac{F^{a}}{|F|}-\frac{B^{a}}{|B|}\right)\right],
\frac{i}{2}\int_{0}^{\infty}\frac{I_{i\omega}(k\xi_{1})I_{-i\omega}(k\xi_{2})-I_{-i\omega}(k\xi_{1})I_{i\omega}(k\xi_{2})\,}{I_{i\omega}(k\xi_{2})K_{i\omega}(k\xi_{1})-I_{i\omega}(k\xi_{1})K_{i\omega}(k\xi_{2})}
A^{\mu}(x)=(2\pi)^{-3/2}\sum_{\lambda=0}^{3}\int\frac{d^{3}k}{\sqrt{2\omega}}\Bigl(\epsilon_{\lambda}^{\mu}({\bf{k}})a_{\lambda}({\bf{k}})e^{-ikx}\pm(\epsilon_{\lambda}^{\mu}({\bf{k}})a_{\lambda}^{+}({\bf{k}})e^{+ikx}\Bigr)\quad,\quad\omega=\sqrt{{\bf{k}}^{2}+m^{2}}
x_{H}=\omega_{4}\,M\left(\sqrt{1+\frac{9\,Q^{2}}{16\,M^{2}}}-1\right)\;,\;\;x_{H}=\frac{9\,\omega_{4}\,Q^{2}}{32\,M}\;,\;\;x_{H}=\omega_{4}\,M\left(1-\sqrt{1-\frac{9\,Q^{2}}{16\,M^{2}}}\right)\,\,,
[\nabla_{\mu},\nabla_{\nu}]\Psi^{\mu}=-R_{\mu\nu}^{~~~\mu\tau}(\Gamma)\Psi_{\tau}+\frac{1}{4}R_{\mu\nu}^{~~mn}(\omega)\gamma_{\,mn}\Psi^{\mu}=\Lambda\Psi_{\nu}+\frac{\Lambda}{6}\gamma_{\mu\nu}\Psi^{\mu}\,.
\begin{array}{c}{{\bfz}_{i}{\bf{\bar{z}}}_{j}-c^{2}\delta_{ij}\approx0,}\\{{\bfz}_{i}{\bfz}_{j}\approx0,}\\{{\bfp}{\bfz}_{i}\approx0,}\\{{\bfp}^{2}\approx0.}\\\end{array}
A_{0}^{(0)}\simA_{0}^{(i)}\sim-2m\xiA_{0}^{(1)}\sim\pi(1/4-\zeta)\Gamma\left(\frac{D-1}{2}\right)(m\xi)^{(D+1)/2}e^{-2m\xi},\quadm\xi\gg1,\quadi=2,3,\cdots,D.
\left\{{\frac{\partial^{2}}{\partial\rho^{2}}}+{\frac{5}{\rho}}{\frac{\partial}{\partial\rho}}+{\frac{(\tilde{\omega}\tilde{R}_{\frac{\pi}{2}})^{4}}{\rho^{4}}}-{\frac{2}{\rho^{2}}}\right\}\hat{\phi}_{I}(\rho)=0,
S_{1}=\frac{\sqrt{(y^{2}+x_{0}^{2})^{3}+(y^{2}+x_{0}^{2})^{2}I_{2}+(y^{2}+x_{0}^{2})(I_{2}^{2}-I_{4})/2+I_{3}^{2}}}{y^{2}+x_{0}^{2}}\=||(X,\nabla_{\!\!X}\Big(\frac{I_{3}}{R}\Big))||\,,
U=-\frac{1}{R^{\prime}}\frac{d}{dz}\frac{R{R^{\prime}}^{2}R^{\prime\prime}}{(1+{R^{\prime}}^{2})^{3/2}}-\frac{d}{dz}\frac{R^{\prime}}{\sqrt{1+{R^{\prime}}^{2}}}+\frac{\sqrt{1+{R^{\prime}}^{2}}}{R}.
\frac{\delta^{2}\mathcal{A}}{\deltax_{\mu}(s)\deltax_{\mu}(s^{\prime})}=\frac{\delta^{2}\mathcal{A}}{\deltax_{1}(s)\deltax_{1}(s^{\prime})}+\frac{\delta^{2}\mathcal{A}}{\delta\vec{\phi}(s)\delta\vec{\phi}(s^{\prime})}\;.
M^{2}(\vec{v}^{\pm})=\frac{m^{2}}{2}\cdot\left(\begin{array}{cccc}{4}&{0}&{\dots}&{0}\\{0}&{\sigma_{2}^{2}}&{\dots}&{0}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{0}&{0}&{\dots}&{\sigma_{N}^{2}}\\\end{array}\right)
\begin{array}{ccccccc}{}&{}&{}&{1}&{}&{}&{}\\{}&{}&{0}&{}&{0}&{}&{}\\{}&{0}&{}&{h^{11}}&{}&{0}&{}\\{1}&{}&{h^{12}}&{}&{h^{21}}&{}&{1}\\{}&{0}&{}&{h^{22}}&{}&{0}&{}\\{}&{}&{0}&{}&{0}&{}&{}\\{}&{}&{}&{1}&{}&{}&{}\\\end{array}\;\qquad\mathrm{with}\qquad\begin{array}{l}{h^{11}=h^{22}}\\{h^{12}=h^{21}\.}\\\end{array}
{\calL}=\sum_{i,j=1}^{n}\left\{G^{i\bar{j}}\,\partial_{\mu}\phi_{i}\,\partial^{\mu}\bar{\phi}_{\bar{j}}-\left[G^{-1}\right]_{i\bar{j}}\frac{\partial{\calW}}{\partial\phi_{i}}\,\frac{\partial\bar{\calW}}{\partial\bar{\phi}_{\bar{j}}}\right\}+\mathrm{fermions}
K(z,u_{a})=\biggl[{\frac{3}{{G_{Bab}}^{2}}}-{\frac{12}{G_{Bab}}}\biggr]+\biggl[-{\frac{1}{2}}{\frac{1}{G_{Bab}^{2}}}+{\frac{1}{G_{Bab}}}+2\biggr]z^{2}+O(z^{4})
\frac{1}{\sqrt{\{n_{1}\}!\{n_{2}\}!...\{n_{M}\}!}}\phi_{M,\kappa_{M}}^{\daggern_{M}}\phi_{M-1,\kappa_{M-1}}^{\daggern_{M-1}}...\phi_{1,\kappa_{1}}^{\daggern_{1}}|0\rangle.