harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\frac{1}{4}}\lambda^{\prime}\gamma^{03}\varepsilon-{\frac{1}{4}}\lambda\partial_{r}\varphi\gamma^{03}\varepsilon+{\frac{i}{12}}{\frac{e_{a}^{i}e_{b}^{j}P_{ij}}{Re^{-{\frac{\varphi}{2}}}}}\lambda^{\frac{1}{2}}\gamma^{120}\gamma^{ab}\varepsilon-{\frac{i}{6}}{\frac{e_{i}^{a}e_{j}^{b}Q^{ij}}{Re^{\frac{\varphi}{2}}}}\lambda^{\frac{1}{2}}\gamma^{3}\gamma^{ab}\varepsilon=0,
{\mathbbE}=\frac{1}{2\pi}\int_{0}^{2\pi}d\xi\,e^{-i\xi\Phi}=1-\Phi^{2}=1-f_{1}^{\dagger}f_{1}-f_{2}^{\dagger}f_{2}+2f_{1}^{\dagger}f_{1}f_{2}^{\dagger}f_{2}
{\calI}=\frac{1}{4\pi}\int\left(\frac{(|R_{z}|^{2}-|R_{{\bar{z}}}|^{2})(1+|z|^{2})^{2}}{(1+|R|^{2})^{2}}\right)^{2}\frac{2i~\mathrm{d}z\mathrm{d}{\bar{z}}}{(1+|z|^{2})^{2}}.
{\bar{D}}^{{\mathbfm}}{}_{{\mathbfj}}D_{{\mathbfk}{\mathbfl}}D^{({\mathbfi}{\mathbfj}}X^{{\mathbfk}{\mathbfl})}=\frac{5}{18}\varepsilon^{{\mathbfm}{\mathbfi}}{\bar{D}}_{{\mathbfj}{\mathbfk}}D^{4}X^{{\mathbfj}{\mathbfk}}+\frac{5}{9}{\bar{D}}^{({\mathbfm}}{}_{{\mathbfj}}D^{4}X^{{\mathbfi}){\mathbfj}}.
\begin{array}{l}{{\calX}^{+k}\vertn,z\rangle_{+k}=q^{2n}z\vertn,z\rangle_{+k}~,}\\{{\calD}_{+k}\vertn,z\rangle_{+k}=q^{1-2n}\lambda^{-1}z^{-1}\vertn-1,z\rangle_{+k}~,}\\{\Lambda_{+k}\vertn,z\rangle_{+k}=\vertn-1,z\rangle_{+k}~,}\\\end{array}
\sum_{M_{1}+M_{2}=M}\bigl(J_{1},M_{1};J_{2},M_{2}\vertJ_{1},J_{2};J,M\bigr)\bigl(J_{1},M_{1};J_{2},M_{2}\vertJ_{1},J_{2};{\overline{J}},M)=\delta_{J,\,{\overline{J}}}
E_{1s}^{KG}=mc^{2}{\left[\frac{1}{2}+\sqrt{\frac{1}{4}-z^{2}{\alpha}^{2}}\right]}^{\frac{1}{2}}\stackrel{z\rightarrow\frac{1}{2\alpha}}{\longrightarrow}\frac{1}{\sqrt{2}}mc^{2}
{\calL}_{3}\equiv{\calL}_{4}\left(\varphi,\partial_{\mu}\varphi,\partial_{\mu\nu}\varphi;B_{\rho},F^{(1)},F^{(2)},\tilde{C}^{(1)},C^{(2)};B_{\rho\sigma},D^{(2)},\tilde{G}^{(2)},G^{(3)}\right),
\int_{0}^{\infty}d\rho{\frac{R^{3}(3\rho^{2}+x_{0}^{2})x_{0}^{5}\rho^{3}}{(\rho^{2}+x_{0}^{2})^{7}}}\sim{\frac{R^{3}}{x_{0}^{3}}}\rightarrow0,
V_{\mu}=\left(\begin{array}{c}{q_{1}+\bar{q}_{1}}\\{i(q_{1}-\bar{q}_{1})}\\{q_{2}+\bar{q}_{2}}\\{i(q_{2}-\bar{q}_{2})}\\{q_{3}+\bar{q}_{3}}\\{i(q_{3}-\bar{q}_{3})}\\{q_{4}+\bar{q}_{4}}\\{i(q_{4}-\bar{q}_{4})}\\{q_{5}+\bar{q}_{5}}\\{i(q_{5}-\bar{q}_{5})}\\{q_{6}+\bar{q}_{6}}\\{i(q_{6}-\bar{q}_{6})}\\\end{array}\right)
S_{Ricci\,\mathrm{eff}}=2\int_{0}^{\tilde{y}}e^{-A(y)}dy\intd^{4}x\sqrt{\tilde{g}}\tilde{R}^{(4)}=2e^{A(\tilde{y})}\int_{0}^{\tilde{y}}e^{-A(y)}dy\intd^{4}x\sqrt{g^{ind}}R^{(4)}.
S[q(\tau)]=\int_{0}^{T}[\frac{1}{2}M(q)\dot{q}^{2}+V(q)+\deltaV(q)]d\tau+\frac{1}{2}\int_{-\infty}^{\infty}d\tau^{\prime}\int_{0}^{T}d\tau\gamma(\tau-\tau^{\prime})\{q(\tau)-q(\tau^{\prime})\}^{2},
\left<{\calO}\left[j_{\mu}^{M}\right]\right>_{j_{\mu}^{M}}=\prod_{i=1}^{N}\intd^{4}y^{i}{\calD}z^{i}\mu\left[z^{i}\right]\sum_{q_{a}=\pm1}^{}{\calO}\left[j_{\mu}^{M{\,}(N)}\right].
\sum_{i=1}^{N-1}\bigl(v_{i}-y_{i}^{2}\bigr)=\sum_{i=1}^{N-1}{\frac{\displaystyle\prod_{j=1}^{N-1}(v_{j}-y_{i}^{2})}{\displaystyle\prod_{j=1,j\neqi}^{N-1}(y_{j}^{2}-y_{i}^{2})}}\,\cdotp
\int_{0}^{2\pi}\frac{d\phi_{1}}{2\pi}f(\phi_{1},\ldots,\phi_{N_{f}},\theta)\sum_{\nu=-\infty}^{\infty}e^{i\nu(\theta-\phi_{1}\ldots-\phi_{N_{f}})}=f(\phi_{2}+\ldots+\phi_{N_{f}}-\theta,\phi_{2},\ldots,\phi_{N_{f}},\theta)\;,
u=H_{0}=\frac{rs}{\sqrt{(1+r^{2})(1+s^{2})}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,v=\tilde{H}_{-1}=\sqrt{(1+r^{2})(1+s^{2})}
\intd^{4}x\psi_{[\omega,\frac{l+1}{2},\frac{l}{2},mm^{\prime}]}^{\dagger}(x)\psi_{k_{x},+}(x)=(2\pi)\delta(\omega-k^{0})\cdotf_{\frac{l+1}{2},\frac{l}{2},mm^{\prime}}.
E_{F}=\int_{x_{1}}^{x_{2}}\left[{\frac{1}{2}}(\nabla\phi)^{2}+{\frac{1}{2}}(\nabla\psi_{1})^{2}+{\frac{1}{2}}(\nabla\psi_{2})^{2}+V(\phi,\psi_{1},\psi_{2})\right]\,dx\.
\int\mathrm{d}^{n}x\,\frac{\delta^{R}\Gamma}{\delta\Phi_{c}^{a}(x)}\,\frac{\delta^{L}\Gamma}{\deltaK_{a}(x)}+\sum_{r\geq2}\frac1{r!}\,\frac{\partial^{R}\Gamma}{\partial\xi^{B}}\,f_{A_{1}\cdotsA_{r}}^{B}\xi^{A_{r}}\cdots\xi^{A_{1}}=0.
{\frac{\phi^{\prime\prime}}{A}}+{\frac{1}{A}}\left(-{\frac{1}{2}}{\frac{A^{\prime}}{A}}+2{\frac{B^{\prime}}{B}}+{\frac{2}{r}}\right)\phi^{\prime}-{\frac{2}{r^{2}}}\phi-\lambda\phi(\phi^{2}-\eta^{2})=0\,.
\intdx^{-}\psi_{a}^{\dagger}\psi_{a}=\intdx^{-}\lambda(x^{-})=\int{\frac{dx^{-}}{2}}\sqrt{\partial_{-}^{2}f^{i}\partial_{-}^{2}f^{i}}=\int_{-\infty}^{\infty}{\frac{ds}{2\partial_{-}s}}\sqrt{(\partial_{-}s)^{4}\kappa^{2}(s)+(\partial_{-}^{2}s)^{2}}=2.
\begin{array}{rccccc}{\mathrm{\underline{{sector}}}}&{{}}&{\mathrm{\underline{{state}}}}&{{}}&{{}}&{\mathrm{\underline{{~{\cal~R}~}}}}\\{\mathrm{NS}}&{{}}&{\psi_{-{\frac{1}{2}}}^{i}|0\rangle}&{i=2,\cdots,9}&{{}}&{-}\\{\mathrm{R}}&{{}}&{|-,2s_{1},\cdots,2s_{4}\rangle}&{8s_{1}s_{2}s_{3}s_{4}=-1}&{{}}&{-}\\\end{array}
\frac{\deltaX}{X_{\mathrm{sr}}}\simeq-\frac{1}{12}\frac{\dot{\varepsilon}}{\varepsilon^{3/2}}\simeq-\frac{\sigma}{3}\frac{\overline{{K}}^{\prime}}{\overline{{K}}^{3/2}}\simeq+\frac{2\sigma}{3}\left(\frac{1}{\sqrt{\overline{{K}}}}\right)^{\prime},
D(x,x^{\prime})=\int{\frac{d^{4}k}{(2\pi)^{4}}}\int{\frac{d^{2}{\vec{q}}}{(2\pi)^{2}}}{\frac{e^{ik_{\mu}(x^{\mu}-x^{\mu})}}{k_{4}^{2}+q^{2}}}\left(e^{i{\vec{q}}\cdot({\vec{x}}-{\vec{x}}^{\prime})}+e^{i{\vec{q}}\cdot({\vec{x}}+{\vec{x}}^{\prime})}\right),
\delta^{(k)}e^{ighB_{0}}=\sum_{N=0}^{\infty}{\frac{(igh)^{N}}{N!}}\sum_{l=0}^{N-1}B_{0}^{l}{\frac{(igh)^{k}}{h}}{\frac{{\calB}_{k}}{k!}}{\underbrace{[\cdots[\delta\omega_{m+1n}-\delta\omega_{mn},B_{0}],\cdots,B_{0}]}_{\mathrm{~k~nested~commutators}}}B_{0}^{N-l-1},\quadk\ne1,
<\chi\mid\chi>=\S_{\begin{array}{l}{{\scriptstylen_{1},...,n_{N}=0}}\\{{\scriptstylem_{1},...,m_{M}=0}}\\{{\scriptstyleq_{1},...,q_{Q}=0}}\\{{\scriptstylet_{1},...,t_{T}=0}}\\\end{array}}^{1}{\left|c^{\prime}({\{n_{r}\}}_{1}^{N};{\{m_{r}\}}_{1}^{M};{\{q_{r}\}}_{1}^{Q};{\{t_{r}\}}_{1}^{T})\right|}^{2}.
S[A_{\mu},a_{\mu}]=\intd^{2}x[\frac{-1}{4\pi}(-\partial\cdotA+\epsilon^{\mu\nu}\partial_{\mu}a_{\nu}-\partial\cdota)\Box^{-1}(-\partial\cdotA+\epsilon^{\mu\nu}\partial_{\mu}a_{\nu}-\partial\cdota)+\phi\partial\cdotA]
\frac{\sigma^{\prime}}{\sigma}=r\Phi^{2}+\frac{2f^{2}e^{-2\Phi}}{r}+\frac{4\beta}{r}\Big((\frac{\Phi^{\prime}(w-1)e^{-2\Phi}}{\sigma})^{\prime}\sigma-\frac{W^{\prime}\Phi^{\prime}}{\sigma^{2}}e^{-2\Phi}\Big)\;,
X\equiv<\!0|\chi^{2}|0\!>=\intd^{3}xd^{3}y{\frac{F(\mathbf{x})F(\mathbf{y})+\mathbf{\nabla}G(\mathbf{x})\!\cdot\!\mathbf{\nabla}G(\mathbf{y})}{4\pi^{2}|\mathbf{x}-\mathbf{y}|^{2}}}
\left.-\epsilon_{n}^{s}\rho_{n}r_{n}\left(r_{n-1}/r_{n}\right)^{\lambda}\frac{(E-V)^{2}}{2Mc^{2}}+\epsilon_{n}^{s}\rho_{n}r_{n}\left(r_{n-1}/r_{n}\right)^{\lambda}\frac{Mc^{2}}2\right].
\zeta_{\mu_{1}}^{(1)}\zeta_{\mu_{2}}^{(2)}\tilde{\zeta}_{\nu_{2}}^{(2)}\tilde{\zeta}_{\nu_{3}}^{(3)}k_{\rho_{2}}^{2}k_{\rho_{4}}^{4}\langle\psi^{\mu_{1}}(z_{1}){\psi}^{\mu_{2}}(z_{2})\rangle\langle\bar{\partial}X^{\nu_{2}}(\bar{z}_{2})\bar{\partial}X^{\nu_{3}}(\bar{z}_{3})\rangle\langle{\psi}^{\rho_{2}}({z}_{2}){\psi}^{\rho_{4}}(z_{4})\rangle.
6p^{2}x^{2}\mapsto{\mathfrakp}^{2}{\mathfrakx}^{2}+{\mathfrakx}^{2}{\mathfrakp}^{2}+{\mathfrakp}{\mathfrakx}{\mathfrakp}{\mathfrakx}+{\mathfrakp}{\mathfrakx}^{2}{\mathfrakp}+{\mathfrakx}{\mathfrakp}{\mathfrakx}{\mathfrakp}+{\mathfrakx}{\mathfrakp}^{2}{\mathfrakx}~.
\begin{array}{l}{|X_{1}|^{2}+|X_{3}|^{2}-|X_{2}|^{2}-|X_{0}|^{2}=\xi_{1}}\\{|X_{2}|^{2}+|X_{4}|^{2}-|X_{3}|^{2}-|X_{0}|^{2}=\xi_{2}}\\{|X_{3}|^{2}+|X_{5}|^{2}-|X_{4}|^{2}-|X_{0}|^{2}=\xi_{3}}\\\end{array}
\bar{D}_{1}VZ_{2}=Z_{2}V\bar{D}_{1}+2P_{-}\quad,\quadP_{-}=\frac{1}{2}\left(\begin{array}{crrc}{0}&{0}&{0}&{0}\\{0}&{1}&{-1}&{0}\\{0}&{-1}&{1}&{0}\\{0}&{0}&{0}&{0}\\\end{array}\right)
l_{4}=\left(\frac{N_{2}^{(1)}\bar{N}_{2}^{(1)}N_{5}^{(1)}\bar{N}_{5}^{(1)}}{N_{2}^{(2)}\bar{N}_{2}^{(2)}N_{5}^{(2)}\bar{N}_{5}^{(2)}}\right)^{\frac14}l_{1},
{\hat{\gamma}}_{(\lambda,\mu,\alpha)}^{r_{i}}\eta_{r_{j}}^{\lambda\mu}=-\eta_{r_{j}}^{\lambda\mu}{\hat{\gamma}}_{(\lambda,\mu,\alpha)}^{r_{i}},\quad{\hat{\gamma}}_{(\lambda,\mu,\alpha)}^{r_{j}}\eta_{r_{i}}^{\lambda\mu}=\eta_{r_{i}}^{\lambda\mu}{\hat{\gamma}}_{(\lambda,\mu,\alpha)}^{r_{j}}.
t_{reg}^{jj}(s)|_{F=0}=t_{reg}^{rr}(s)|_{F=0}=r^{2}t_{reg}^{\varphi\varphi}(s)|_{F=0}=\frac{m^{d-2s}}{2(4\pi)^{\fracd2}}\frac{\Gamma(s-\fracd2)}{\Gamma(s+1)}\,.
\delta(\zeta_{\alpha})_{\betak}={\frac{e}{2}}{\calU}_{a}^{\dot{B}\gammal}(\sigma^{a})_{\alpha\dot{B}}\epsilon_{\beta\gamma}\delta_{kl}={\frac{e}{2}}{\calU}_{a}^{\dot{B}\gammak}(\sigma^{a})_{\alpha\dot{B}}\epsilon_{\beta\gamma}.
\begin{array}{ll}{g\longrightarrow}&{hgh^{-1}}\\{{\bfA}\longrightarrow}&{h^{-1}({\bfA}+\partial)h}\\{\overline{{{\bfA}}}\longrightarrow}&{h^{-1}(\overline{{{\bfA}}}+\overline{{\partial}})h}\\\end{array}
\Psi^{(+)}({\bfr})\stackrel{\scriptscriptstyle(r\rightarrow\infty)}{\sim}e^{i{\bfk}\cdot{\bfr}}+f_{k}^{(D)}(\Omega^{({D})})\,e^{i\gamma_{{D}}}\,\frac{e^{ikr}}{r^{({D}-1)/2}}\;,
\deltaU={\frac{1}{2\chi_{T}}}[(\delta\phi_{T})^{2}+(\delta\phi_{L})^{2}-{\frac{(\delta\vec{\phi}^{2})^{2}}{4\bar{\phi}^{2}}}]+{\frac{1}{2\chi_{L}}}[{\frac{(\delta\vec{\phi}^{2})^{2}}{4\bar{\phi}^{2}}}]
M=\mu\left(\begin{array}{cccccc}{0}&{-a_{1}}&{0}&{0}&{0}&{0}\\{a_{1}}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{-a_{2}}&{0}&{0}\\{0}&{0}&{a_{2}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{-a_{3}}\\{0}&{0}&{0}&{0}&{a_{3}}&{0}\\\end{array}\right)
\widetilde{\calJ}^{\nu}=(\eta^{\mu\nu}\bot_{\sigma\rho}+2\eta^{\nu}{_{[\sigma}}\eta_{\rho]}{^{\mu}})[\acute{\xi}^{\sigma}\widetilde{\nabla}_{\mu}\xi^{\rho}-(\widetilde{\nabla}_{\mu}\acute{\xi}^{\rho})\xi^{\sigma}].
\frac{\mathrm{PP}}{kq}\:(kq)^{j}\left(\frac{q^{2}}{4}\right)^{r}\cdots\:{\mathcal{B}}_{q}\:\cdotsT^{(n+j+r)}(k)\:\left[g(\vec{k}+\frac{\vec{q}}{2})-g(\vec{k}+\frac{\vec{q}}{2})\right]\;,
S=\intd^{2}z(-\partial\psi\overline{{\partial}}\psi+\partials\overline{{\partial}}s+\partial\rho\overline{{\partial}}\rho{G}_{\rho\rho}+\partial\lambda\overline{{\partial}}\lambda{G}_{\lambda\lambda}+(\partial\rho\overline{{\partial}}\lambda-\partial\lambda\overline{{\partial}}\rho)B)
\langle\Omega|T\left(a_{3}\left(x_{\perp}\right)a_{3}\left(0\right)\right)|\Omega\rangle\sim\Bigl(\frac{\ell}{g^{2}L}\Bigr)^{x_{\perp}/\ell}\!\!\rightarrow\delta^{3}\left(x_{\perp}\right)\.
{\calZ}_{\chi=0,L}=\int(DX^{i})~e^{-i\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}\partialX^{i}{\overline{\partial}}X^{j}\eta_{ij}-i\frac{1}{2\pi\alpha^{\prime}}\int_{\partial\Sigma}\left(\epsilong_{i}^{C}+g_{i}^{D}\frac{1}{\epsilon}\right)\partial_{n}X^{i}}
\delta(p+q)=(2\pi)^{\,4}\,\tilde{C}_{\alpha}^{\,\alpha_{0}}(q)\,delta_{\Phi_{c}(p)}\delta_{\Phi_{c}(q)}\,\Gamma_{0}^{\,\alpha,\,\alpha_{0}}(\Phi_{c})\qquad\mathrm{for~}\r=0\quad,
\omega^{1}\,_{2}=d\rho,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\omega^{1}~_{3}=\frac{\ell~e^{2\rho}}{2~r_{ex}}~du+\frac{r_{ex}}{\ell}~dv,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\omega^{2}~_{3}=-\frac{\ell~e^{2\rho}}{2~r_{ex}}du.
\hat{H}=\frac{1}{2m}\,\hat{g}^{-1/4}\,\hat{p}_{i}\,\hat{g}^{ij}\,\hat{g}^{1/2}\,\hat{p}_{j}\,\hat{g}^{-1/4}+\frac{\lambda\hbar^{2}}{m}\,\hat{R}\;,
B=-\frac{1}{6}e^{\hat{R}}\wedgee^{\hat{S}}\wedgee^{\hat{U}}C_{\hat{R}\hat{S}\hat{U}}+\frac{1}{2}e^{\hat{R}}\wedgee^{\hat{S}}\wedge\bar{\theta}\Gamma_{\hat{R}\hat{S}}D\theta+{\calO}(\theta^{4})\,,
S~=~\intd^{d+1}x\sqrt{g}e^{-\Phi}\big(R+\partial_{\mu}\Phi\partial^{\mu}\Phi-{\frac{1}{12}}H_{\mu\nu\lambda}H^{\mu\nu\lambda}-{\frac{\alpha^{\prime}}{4}}F^{N}{}_{\mu\nu}F^{N}{}^{\mu\nu}+2\Lambda\big).
\frac{\partial}{\partialx_{\mu}}(\left|x-a\right|+\left|x-b\right|+\left|x-c\right|)=\frac{(x-a)_{\mu}}{\left|x-a\right|}+\frac{(x-b)_{\mu}}{\left|x-b\right|}+\frac{(x-c)_{\mu}}{\left|x-c\right|}
\check{R}(z_{1}-z_{2})\phi_{ad}(z_{1})\otimes\phi_{dc}(z_{2})=\sum_{b}W_{z_{1}-z_{2}}\left(\begin{array}{cc}{a}&{b}\\{d}&{c}\\\end{array}\right)\phi_{ab}(z_{2})\otimes\phi_{bc}(z_{1}).
\begin{aligned}{\Phi^{j}(g_{o}\cdotX)}&{{}=\Omega(X)\Phi^{j}(X)\left[\Omega(X)\right]^{-1}\quadj=1,\dots,6}\\{A(g_{o}\cdotX)}&{{}=\Omega(X)\,A(X)\left[\Omega(X)\right]^{-1}-i\,d\Omega(X)\left[\Omega(X)\right]^{-1}}\\\end{aligned}
\epsilon_{1}^{i}=e^{{\frac{1}{2}}A}\tilde{\eta}_{+}^{i}\,\qquad\epsilon_{2}^{i}=\left(e^{-{\frac{1}{2}}A}-2\kappa^{D-2}W(\phi_{0})e^{{\frac{1}{2}}A}x^{\mu}\Gamma_{\underline{\mu}}\right)\tilde{\eta}_{-}^{i}\.
{\calI}=\sum_{i=1}^{6}\int\frac{d^{2}\tau}{\tau_{2}^{3/2}}{\bar{C}}_{l}(\bar{\tau})\partial_{\bar{\tau}}(\tau_{2}^{1/2}\sum_{p_{L},p_{R}\in\Gamma_{l}}e^{\pii\tau|p_{L}|^{2}}e^{-\pii{\bar{\tau}}p_{R}^{2}}).
f_{1-loop}=-\frac{i}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\Bigg(\frac{1}{k^{2}(k+p)^{2}}-\sum_{i}c_{i}\frac{1}{(k^{2}-M_{i}^{2})\Big((k+p)^{2}-M_{i}^{2}\Big)}\Bigg).
\frac{\partial{\mathsfW}^{^{(16\times\overline{{16}})}}}{\partial\Phi_{\calX}}=0;~~~\frac{\partial{\mathsfW}^{^{(16\times\overline{{16}})}}}{\partial\Phi_{\mu\nu{\calY}}}=0;~~~\frac{\partial{\mathsfW}^{^{(16\times\overline{{16}})}}}{\partial\Phi_{\mu\nu\rho\lambda{\calZ}}}=0.
E_{\mathrm{in}}^{(\mathrm{surf})}=-\frac{(4\xi-1)A_{1}B}{2a}\sum_{l=0}^{\infty}D_{l}\sum_{k=1}^{\infty}\frac{\lambda_{\nu,k}^{2}}{\sqrt{\lambda_{\nu,k}^{2}+m^{2}a^{2}}[A^{2}+B^{2}(\lambda_{\nu,k}^{2}-\nu^{2})]},
\frac{\partial^{p-2}}{\partialx_{i_{1}}^{p-2}}\frac{\partial^{p-3}}{\partialx_{i_{2}}^{p-3}}\dots\frac{\partial}{\partialx_{i_{p-2}}}F_{r_{1},r_{2},\dots,r_{n};\theta_{1},\theta_{2},\dots,\theta_{n}}^{p,g;\iota_{1},\dots,\iota_{n}}\biggr|_{x_{i_{1}}=x_{i_{2}}=\dots=x_{i_{p-1}}}=0.
\deltax^{\mu}=\dot{x}^{\mu}\xi\;,\;\;\deltae=\frac{d}{d\tau}(e\xi)\;,\;\;\delta\psi^{a}=\dot{\psi}^{a}\xi\;\;,\delta\chi=\frac{d}{d\tau}(\chi\xi)\;,\;\;\delta\kappa=\frac{d}{d\tau}(\kappa\xi)\;,
W(p,q)=W_{4}(p,q;0)=\left(\begin{array}{cccc}{1+\frac{p^{2}+q^{2}}{2}}&{p}&{q}&{-\frac{p^{2}+q^{2}}{2}}\\{p}&{1}&{0}&{-p}\\{q}&{0}&{1}&{-q}\\{\frac{p^{2}+q^{2}}{2}}&{p}&{q}&{1-\frac{p^{2}+q^{2}}{2}}\\\end{array}\right)
\hat{\varphi}(\theta)=\frac{1}{2\pi}\,\sum_{n\ne0}\,\sqrt{\frac{\pi}{|n|}}\,(\hat{a}_{n}\,e^{i\,n\,\theta}+\hat{a}_{n}^{\dagger}\,e^{-i\,n\,\theta}),\;\;\hat{\pi}(\theta)=\frac{i}{2\pi}\,\sum_{n\ne0}\,\sqrt{\pi|n|}\,(-\hat{a}_{n}\,e^{i\,n\,\theta}+\hat{a}_{n}^{\dagger}\,e^{-i\,n\,\theta}).
{\frac{J_{b}^{G/K_{b}}}{J_{f}^{G/H_{f}}}}\,{\frac{{\calV}(G/K_{b})}{{\calV}(G/H_{f})}}={\frac{J_{b}^{G/K_{b}}}{J_{f}^{G/H_{f}}}}\,{\calV}(H_{f}/K_{b})
{\hat{V}}={\frac{1}{2}}{\sum_{\sigma,\sigma^{\prime}}}\sum_{X_{1},X_{2},X_{3},X_{4}}V_{X_{1}X_{2}X_{3}X_{4}}c_{X_{1}\sigma}^{\dagger}c_{X_{2}\sigma^{\prime}}^{\dagger}c_{X_{4}\sigma^{\prime}}c_{X_{3}\sigma}
(W_{\alpha\dot{\alpha}}^{\mathrm{inst}})^{\dot{f}\dot{g}}=2i\frac{\rho^{2}}{(x^{2}+\rho^{2})^{2}}\left(x_{\alpha}^{\dot{g}}\delta_{\dot{\alpha}}^{\dot{f}}+x_{\alpha}^{\dot{f}}\delta_{\dot{\alpha}}^{\dot{g}}\right)\,.
\mathrm{index}(\mathrm{bulk})\,=\,\frac{1}{24.8\pi^{2}}\int_{{\calM}}\,\Big[R(e)^{mn}\wedgeR(e)_{nm}-\frac{1}{2}F(A)\wedgeF(A)-2\,\sqrt{g}\,D_{\mu}{\calK}^{\mu}\Big]\,.
\delta\biggl(k^{0}-{\frac{1}{2}}\sqrt{{\frac{\epsilon\eta^{2}}{i}}}-\vec{\eta}\cdot\vec{k}\biggr)\biggl[k_{\mu}k_{\nu}D_{1}+\eta_{\mu}k_{\nu}D_{2}+\eta_{\nu}k_{\mu}D_{3}\biggr]+\mu\leftrightarrow\nu;k\rightarrow-k
\hat{Q}^{+}=\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{Q_{0,1}^{+}}&{0}&{0}&{0}\\{0}&{Q_{1,2}^{+}}&{0}&{0}\\{0}&{0}&{Q_{2,3}^{+}}&{0}\\\end{array}\right),\qquad\hat{Q}^{-}=\left(\hat{Q}^{+}\right)^{\dagger}=\left(\begin{array}{cccc}{0}&{Q_{1,0}^{-}}&{0}&{0}\\{0}&{0}&{Q_{2,1}^{-}}&{0}\\{0}&{0}&{0}&{Q_{3,2}^{-}}\\{0}&{0}&{0}&{0}\\\end{array}\right),
J^{(2)}(4-2\varepsilon;1,1)=\mathrm{i}\pi^{2-\varepsilon}\Gamma(\varepsilon)\frac{m_{0}^{1-2\varepsilon}}{\sqrt{\Lambda^{(2)}}}\;\left\{\Omega_{1}^{(2;4-2\varepsilon)}+\Omega_{2}^{(2;4-2\varepsilon)}\right\},
L_{1}=e^{\alpha\phi/M_{p}}{-\frac{1}{\kappa}R(\omega,e)+\frac{1}{2}g^{\mu\nu}\phi,_{\mu}\phi,_{\nu}+\frac{i}{2}\overline{{\Psi}}\left[\gamma^{a}e_{a}^{\mu}(\overrightarrow{\partial}_{\mu}+\frac{1}{2}\omega_{\mu}^{cd}\sigma_{cd})-(\overleftarrow{\partial}_{\mu}-\frac{1}{2}\omega_{\mu}^{cd}\sigma_{cd})\gamma^{a}e_{a}^{\mu}\right]\Psi-m\overline{{\Psi}}\Psie^{\frac{1}{2}\alpha\phi/M_{p}}}
d(z,\lambda,\mu)~=~(s-r)(z,\lambda,\mu)~=~{\frac{2\lambda\mu}{1\!-\!\lambda\mu}}\left({\frac{C}{\lambda\!-\!\mu}}\,-\,{\frac{2\,j(z)}{\lambda(1\!-\!\mu^{2})}}\right)~~~,
\left(c^{-1}\right)^{AB}={\frac{1}{2\gamma\Phi\sqrtg}}\left(\begin{matrix}{{\frac{2\gamma\Phif}{\Omega^{2}}}+A_{\lambda}A^{\lambda}}&{-A^{\nu}}\\{-A^{\mu}}&{g^{\mu\nu}}\\\end{matrix}\right).
\dot{\epsilon}_{\delta\sigma}=\epsilon_{\delta\sigma}\left[3{\frac{\dot{a}}{a}}(1+w)+2{\frac{\dot{c}}{c}}{\frac{1+\epsilon_{\delta\sigma}}{1+\epsilon}}+\left({\frac{\dot{G}_{4}}{G_{4}}}-4{\frac{\dot{c}}{c}}\right)(1+\epsilon_{\delta\sigma})+{\frac{3Cc\dot{c}}{4\piG_{4}a_{0}^{4}\varrho_{m}}}\right].
\frac{\partial\tau}{\partialy}=l\dot{\alpha}e^{-\alpha}\,\\frac{\partialz}{\partialy}=-l\alpha^{\prime}e^{-\alpha}\,\\frac{\partial\tau}{\partialt}=l\alpha^{\prime}e^{-\alpha}\,\\frac{\partialz}{\partialt}=-l\dot{\alpha}e^{-\alpha}
\dot{R}^{2}=\frac{\kappa^{2}}{6}\rho_{B}R^{2}+\frac{\kappa^{4}}{36}\left[\frac{M}{R^{2}}+\frac{\rho_{\phi0}R_{0}^{3(1+\omega)}}{R^{3\omega+2}}\right]^{2}.
\hat{B}_{L}^{(\tau)}({\bfx})=\left(\begin{array}{cc}{\delta({\bfx})}&{\;\;f_{equ}^{(\tau)}({\bfx})}\\{\delta({\bfx})}&{\;\;\delta({\bfx})+f_{equ}^{(\tau)}({\bfx})}\\\end{array}\right),\;\;\;\;\hat{B}_{R}^{(\tau)}({\bfx})=\left(\begin{array}{cc}{\delta({\bfx})+f_{equ}^{(\tau)}({\bfx})}&{\;\;f_{equ}^{(\tau)}({\bfx})}\\{\delta({\bfx})}&{\;\;\delta({\bfx})}\\\end{array}\right)
\begin{array}{l}{S_{1}=\intdx(-\partial_{\nu\kappa}h_{\alpha\beta}\partial_{\nu\kappa}h_{\alpha\beta}+4\partial_{\beta\kappa}h_{\alpha\beta}\partial_{\nu\kappa}h_{\alpha\nu}-\partial_{\beta\kappa}h_{\alpha\beta}\partial_{\nu\alpha}h_{\kappa\nu}}\\{+4\partial_{\alpha\beta}h_{\alpha\kappa}\partial_{\beta\kappa}h+\partial_{\beta\kappa}h\partial_{\beta\kappa}h)}\\\end{array}
F_{mn}\rightarrowF_{AB}^{\!~~~~ab}=\mathrm{~\frac{1}{4}~}\epsilon_{AB}\epsilon_{CD}F^{CDab}+\mathrm{~\frac{1}{4}~}\Omega_{ABCD}\Omega^{abcd}F_{\!~~~~cd}^{CD}+\mathrm{~\frac{1}{2}~}\epsilon^{ab}\epsilon^{cd}F_{\!~~~~cd}^{AB},
\theta_{\mathrm{ev}}(\tau,z)=\sum_{n\in{\bfZ}}q^{\frac{1}{4}(2n)^{2}}\zeta^{2n}\,,\qquad\theta_{\mathrm{od}}(\tau,z)=\sum_{n\in{\bfZ}}q^{\frac{1}{4}(2n+1)^{2}}\zeta^{2n+1}
{\bfv}_{0}=\frac{\sqrt2}l\left[\begin{array}{c}{\,1}\\{\,\,a^{2}\,}\\\end{array}\right],\,\,\,{\bfv}_{i}=\frac{{\sqrt2}a}l\left[\begin{array}{c}{1/J_{2}(j_{i})}\\{-1\,}\\\end{array}\right]\.
S_{eff}=i\{[\int\frac{\vec{j}(x)\cdot\vec{j}(y)}{2N}{\calG}[x,y;\chi]-V[\rho]+\frac{1}{2}\rho\chi+S\chi+J{\rho}+\frac{i}{2}Tr\mathrm{Ln}{\calG}^{-1}[x,y,\chi]\}\biggr]
a_{2,2}=\int_{{\calM}_{3}}h^{1/2}d^{3}x~\mathrm{Tr}\left[-{\frac{1}{32}}F^{2}-{\frac{1}{12}}F^{jk}F_{jk}\right]=-{\frac{1}{24}}\int_{{\calM}_{3}}h^{1/2}d^{3}x~F^{jk}F_{jk}
0=\partial_{y}\left[\frac{\kappa}{2}\left(\frac{\partial_{x}\Phi^{(0)}}{\partial_{y}\Phi^{(0)}}\right)^{2}+\partial_{y}\Phi^{(0)}\right]-\kappa\partial_{x}\left[\frac{\partial_{x}\Phi^{(0)}}{\partial_{y}\Phi^{(0)}}\right].
c_{D_{8}}(-,-)=\left[\begin{matrix}{c(\tau,\tau)}&{c(g,\tau)}&{c(\taug,\tau)}\\{c(\tau,g)}&{c(g,g)}&{c(\taug,g)}\\{c(\tau,\taug)}&{c(g,\taug)}&{c(\taug,\taug)}\\\end{matrix}\right]=\left[\begin{matrix}{+}&{-}&{-}\\{+}&{+}&{+}\\{+}&{-}&{-}\\\end{matrix}\right]
{\calS}\left[\vec{j}_{\mu}^{\calN},\vec{A}_{\mu}\right]=\frac{1}{2}\intd^{4}xd^{4}y\vec{j}_{\mu}^{\calN}(x)D_{0}(x-y)\vec{j}_{\mu}^{\calN}(y)+\frac{i\thetag^{2}}{8\pi^{2}}\intd^{4}x\vec{A}_{\mu}\vec{j}_{\mu}^{\calN},
\widetilde{j}_{\mu}^{a,\mathrm{reg}}=\widetilde{\psi}_{\varepsilon}^{+}T^{a}\gamma_{\mu}\widetilde{\psi}_{\varepsilon}=\psi_{\varepsilon}^{+}\Omega_{+}T^{a}\gamma_{\mu}\Omega_{-}\psi_{\varepsilon}.
\begin{array}{ll}{\displaystyle(p^{2}+m^{2})F^{\mathrm{phys}}(p,\bar{z},\bar{\theta},\bar{\chi})=0}\\{\displaystyle[(p,{\bfJ})-m{\bfP}_{4}-ms]F^{\mathrm{phys}}(p,\bar{z},\bar{\theta},\bar{\chi})=0\;.}\\\end{array},
B=\left(\begin{array}{ccc|cccc}{}&{}&{}&{0}&{\ldots}&{0}&{g_{1}}\\{}&{h}&{}&{\vdots}&{}&{\vdots}&{\vdots}\\{}&{}&{}&{0}&{\ldots}&{0}&{g_{m}}\\\hline{f_{1}}&{\ldots}&{f_{m}}&{0}&{\ldots}&{0}&{e_{1}}\\{0}&{\ldots}&{0}&{1}&{\ddots}&{}&{e_{2}}\\{\vdots}&{}&{\vdots}&{}&{\ddots}&{\ddots}&{\vdots}\\{0}&{\ldots}&{0}&{0}&{\ldots}&{1}&{e_{n-m}}\\\end{array}\right).
\frac{i\theta^{im}}{2\pi}\frac{i\theta^{jn}}{2\pi}\int_{0}^{1}d\tau^{\prime}~e^{-2\pii\tau^{\prime}}l^{m^{\prime}}F_{mm^{\prime}}(x+l\tau^{\prime})\int_{0}^{1}d\tau^{\prime\prime}~e^{2\pii\tau^{\prime\prime}}l^{n^{\prime}}F_{nn^{\prime}}(x+l\tau^{\prime\prime}).
\Gamma_{P}^{Q^{\prime}Q^{\prime\prime}}(-i\Omega_{m},\stackrel{\rightharpoonup}{p})[\delta_{Q^{\prime\prime}Q}-N_{Q^{\prime\prime}5}(-i\Omega_{m},\stackrel{\rightharpoonup}{p})g_{Q^{\prime\prime}Q}^{\prime}]=g_{Q^{\prime}Q}^{\prime}/2,
T_{N_{0}}(\{\lambda^{\prime}\},\{\rho^{\prime}\};\{\lambda\},\{\rho\})=\delta_{\lambda_{1},\lambda_{1}^{\prime}}...\delta_{\lambda_{p},\lambda_{p}^{\prime}}\delta_{\rho_{1},\rho_{1}^{\prime}}...\delta_{\rho_{s},\rho_{s}^{\prime}}(-1)^{\nu(p,s)}{}~,
[a_{m}^{\dagger},\;\tilde{\psi}_{\dot{\alpha}}]=[a_{m}^{\dagger},\;\tilde{\psi}_{\dot{\alpha}}^{\dagger}]=[\tilde{a}_{m}^{\dagger},\;\psi_{\dot{\alpha}}]=[\tilde{a}_{m}^{\dagger},\;\psi_{\dot{\alpha}}^{\dagger}]=0
{\calL}_{eff}=\sum_{n=0}^{\infty}\left(\psi_{n}^{(L)*}i\partial_{+}\psi_{n}^{(L)}+\psi_{n}^{(R)*}i\partial_{-}\psi_{n}^{(R)}\right)+\sum_{n_{1},n_{2},m_{1},m_{2}=0}^{\infty}g_{0}\left(\begin{array}{cc}{n_{1}}&{n_{2}}\\{m_{1}}&{m_{2}}\\\end{array}\right)\psi_{n_{1}}^{(L)*}\psi_{n_{2}}^{(R)}\psi_{m_{1}}^{(R)*}\psi_{m_{2}}^{(L)}.
\stackrel{\wedge}{\psi}^{\dagger}=\sigma_{-}=\left(\begin{array}{cc}{0}&{0}\\{1}&{0}\\\end{array}\right),\qquad\stackrel{\wedge}{\psi}=\sigma_{+}=\left(\begin{array}{cc}{0}&{1}\\{0}&{0}\\\end{array}\right)
32M_{\mu}^{(2)}=\frac{\tilde{A}^{2}\mu^{2}}{48\pi\alpha\omega^{2}}\left(\frac{1-\omega}{1+\omega}\right)\left((1+\omega)^{2}-4g\right)e^{2\omega\delta}\left(\frac{\Lambda}{\mu}\right)^{2\omega}.
m^{2}\geq4\piv^{2}(n_{e},n_{m})\,\frac{1}{\mathrm{Im}\,\tau}\left(\begin{array}{cc}{1}&{-\mathrm{Re}\,\tau}\\{-\mathrm{Re}\,\tau}&{|\tau|^{2}}\\\end{array}\right)\left(\begin{array}{c}{n_{e}}\\{n_{m}}\\\end{array}\right)
\alpha^{-1}=\left(\begin{array}{ccc}{\omega_{2k}^{-2}}&{0}&{0}\\{0}&{\omega_{2k}^{1}}&{0}\\{0}&{0}&{\omega_{2k}^{1}}\\\end{array}\right)~~~~~~~~\beta=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{\omega_{2d}}&{0}\\{0}&{0}&{\omega_{2d}^{-1}}\\\end{array}\right)~~~~~~~\gamma=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{0}&{i}\\{0}&{i}&{0}\\\end{array}\right)
\beta^{0}=\left(\begin{array}{ccccc}{0}&{0}&{0}&{0}&{1}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{1}&{0}&{0}&{0}&{0}\\\end{array}\right);\\beta^{1}=\left(\begin{array}{ccccc}{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}&{0}\\\end{array}\right);
R_{\beta\mu\nu}^{\alpha}=\partial_{\mu}\Gamma_{\nu\beta}^{\alpha}-\partial_{\nu}\Gamma_{\mu\beta}^{\alpha}+\Gamma_{\mu\gamma}^{\alpha}\Gamma_{\nu\beta}^{\gamma}-\Gamma_{\nu\gamma}^{\alpha}\Gamma_{\mu\beta}^{\gamma}
\left(1+{\frac{16}{9}}{\frac{z_{0}^{2}}{l^{2}}}\right)+\left({\frac{32}{9}}\omegaz_{0}^{2}-l^{2}\right)\varepsilon^{2}+\left({\frac{16}{9}}l^{2}z_{0}^{2}\omega^{2}\right)\varepsilon^{4}>0\,,
I_{21}(y,a)|_{m=0}=\left(\frac{B_{0}}{8\pi^{2}y^{2}}+\frac{B_{1}}{4\piya}+\frac{1}{2\pi}\sum_{n=2}^{\infty}\frac{B_{n}}{n!}(\frac{2\piy}{a})^{n-1}\right).
\left(\begin{matrix}{\cdot}&{\cdot}&{0.407663}&{0.405138}&{0.386709}&{0.396082}&{}\\{\cdot}&{0.362383^{*}}&{0.405248}&{0.408042^{*}}&{0.392824}&{}&{}\\{0.414164}&{0.395111}&{0.392686}&{0.395422}&{}&{}&{}\\{0.377688}&{0.392075}&{0.394118}&{}&{}&{}&{}\\{0.4843^{*}}&{0.403122}&{}&{}&{}&{}&{}\\{0.406194}&{}&{}&{}&{}&{}&{}\\\end{matrix}\right)
{\frac{1}{2}}\gamma_{b}^{a}\left(\xi_{ij}^{b}+\xi_{ji}^{b}-\partial_{i}\rho_{j}^{b}-\partial_{j}\rho_{i}^{b}-\nu_{ic}^{b}\rho_{j}^{c}-\nu_{jc}^{b}\rho_{i}^{c}\right)=0\,\,\,.
i\,\mathbf{G}_{\mu\nu,\lambda\sigma}=\left[\widehat{\mathbf{Q}}_{\mu\nu},\widehat{\mathbf{Q}}_{\lambda\sigma}\right]_{\star}-i\,{F_{\mu\nu,\lambda\sigma}}^{\alpha\beta}\,\widehat{\mathbf{Q}}_{\alpha\beta}.
{\Phi}^{(I)}=-\frac{s\,e^{2}F(1-F)}{12{\pi}^{2}|m|}\int_{1}^{\infty}\frac{dv}{v{\sqrt{v-1}}}\frac{(1+F)(1+A{v}^{F})-(2-F)(1+A^{-1}v^{1-F})}{A{v}^{F}+2+A^{-1}v^{1-F}}.
E_{\mathrm{total}}=NM_{\mathrm{D}2l}+\sum_{i=1}^{l}N_{i}M_{\mathrm{D}2}^{\left(i\right)}+\sum_{i,j=1}^{l}N_{ij}M_{\mathrm{D}4}^{\left(ij\right)}+\ldots+N_{i_{1}\ldotsi_{l}}M_{\mathrm{D}0}.