harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}{\Psi(x)=R({\bfB}_{f})\Psi_{f}(x_{f}),\quad\bar{\Psi}(x)=\bar{\Psi}_{f}(x_{f})\widetilde{R}^{+}({\bfB}_{f}),}\\{g^{\mu}(x)\nabla_{\mu}\Psi(x)=S(B_{f})R({\bfB}_{f})\gamma^{l}D_{l}\Psi_{f}(x_{f}),}\\{\left(\nabla_{\mu}\bar{\Psi}(x)\right)g^{\mu}(x)=S(B_{f})\left(D_{l}\bar{\Psi}_{f}(x_{f})\right)\gamma^{l}\widetilde{R}^{+}({\bfB}_{f}),}\\\end{array}
M=\left(\begin{array}{cc}{\frac{\partial^{2}\mathrm{ln}Z}{\partial\alpha^{2}}|_{(\alpha_{0},\beta_{0})}}&{\frac{\partial^{2}\mathrm{ln}Z}{\partial\alpha\partial\beta}|_{(\alpha_{0},\beta_{0})}}\\{\frac{\partial^{2}\mathrm{ln}Z}{\partial\beta\partial\alpha}|_{(\alpha_{0},\beta_{0})}}&{\frac{\partial^{2}\mathrm{ln}Z}{\partial\beta^{2}}|_{(\alpha_{0},\beta_{0})}}\\\end{array}\right)\
\hat{\phi}_{1}^{\prime}=\frac{\phi}{\sqrt{2\mathrm{Tr}\phi^{2}}},\\\hat{\phi}_{2}^{\prime}=\frac{1}{\sqrt{2[-2(\mathrm{Tr}\hat{\phi}_{1}^{\prime}{}^{3})^{2}+\mathrm{Tr}\hat{\phi}_{1}^{\prime}{}^{4}]}}[-2(\mathrm{Tr}\hat{\phi}_{1}^{\prime}{}^{3})\hat{\phi}_{1}^{\prime}+\hat{\phi}_{1}^{\prime}{}^{2}]
W_{\mathrm{YM/\Phi}}^{(\mathrm{q})}\toS_{\mathrm{NG}}={\frac{1}{8\pig^{2}\,\sqrtV}}\int_{\Sigma}d^{2}\sigma\left[\,{\frac{1}{\Phi(\sigma)}}\left\{\,X_{\mu}(\sigma)\,X_{\nu}(\sigma)\,\right\}_{\mathrm{PB}}^{2}+g^{2}\,\Phi(\sigma)\,\right]
S_{eff}=NS_{0}(\sigma^{0})+S_{1}(\sigma^{0})+{\frac{i}{2}}B_{2}+{\frac{1}{2}}A_{2}+\sum_{n=1}^{\infty}{\frac{1}{\sqrt{N^{n}}}}\big[-i{\frac{(-1)^{n+1}}{(n+2)}}B_{n+2}+iL^{d}\delta^{d}(0){\frac{(-1)^{n+1}}{n}}B_{n}+{\frac{1}{(n+2)!}}A_{n+2}\big]
\nabla^{\mu}\nabla_{\mu}=-{\frac{1}{B}}(\xi^{\mu}\nabla_{\mu})^{2}+{\frac{1}{2B}}B^{,\nu}\nabla_{\nu}+h^{\mu\nu}\nabla_{\mu}\nabla_{\nu}=-{\frac{1}{B}}\partial_{t}^{2}-{\frac{1}{2B}}B_{,i}h^{ij}D_{j}+h^{ij}D_{i}D_{j}~~~,
\Pi_{\prime\prime}^{\mu\nu,\alpha\beta}\left(k\right)=\Pi^{\mu\nu,\alpha\beta}\left(k\right)+\frac\kappa2\left(\displaystyle\frac{2b-2c-d}{a+4b}\right)\left(\eta^{\mu\alpha}\Gamma^{\nu\beta}+\eta^{\mu\beta}\Gamma^{\nu\alpha}+\eta^{\nu\alpha}\Gamma^{\mu\beta}+\eta^{\nu\beta}\Gamma^{\mu\alpha}\right),
\langle\,{\bf\cdot}\midH-E\mid{\bf\cdot}\,\rangle\mid\psi(R,\Lambda_{3})\,\rangle+\!\int\!d^{2}\lambda(d\Omega)\sigmau_{R}^{\ast}(\lambda_{1},\lambda_{2})\langle\,0\midH\mid\Psi_{Q}(x,y)\,\rangle=0
S=-\frac{1}{2}\int^{\Lambda}\frac{d^{D}k}{(2\pi)^{D}}[(1-au\theta^{2})k^{2}+bk^{4}+m^{2}]\phi(-k)\phi(k)-\frac{1}{4!}\int^{\Lambda}u(4321)\phi(4)\phi(3)\phi(2)\phi(1),
m_{f,modified}^{(late)}\simeq\upsilon\left(\frac{m_{1}}{m_{2}}\right)^{2}\left[2h\left(\frac{m_{1}}{m_{2}}\right)^{4}e^{-[3(\beta-\frac{1}{2}\alpha)-\gamma]\phi/M_{p}}+\tilde{h}e^{-(\beta-\frac{1}{2}\alpha-\tilde{\gamma})\phi/M_{p}}\right]\qquadas\qquad\phi>M_{p}.
\begin{array}{rcl}{E_{\mathrm{vib-rot}}(v,J)}&{=}&{\displaystylehc\omega_{e}\left(v+{\frac{1}{2}}\right)-hc\omega_{e}x_{e}\left(v+{\frac{1}{2}}\right)^{2}+hc\omega_{e}y_{e}\left(v+{\frac{1}{2}}\right)^{3}}\\{}&{}&{\displaystyle-hc\alpha_{e}(v+\frac{1}{2})J(J+1)-hc\beta_{e}(v+\frac{1}{2})J^{2}(J+1)^{2}}\\{}&{}&{+hcB_{e}J(J+1)-hcD_{e}J^{2}(J+1)^{2}+\cdots~.}\\\end{array}
F^{M_{1}\cdotsM_{n}}={\frac{1}{\sqrt{-g}}}\,e^{-a\phi}\,{\frac{1}{(D-n)!}}\,\epsilon^{M_{1}\cdotsM_{n}N_{1}\cdotsN_{D-n}}\,\tilde{F}_{N_{1}\cdotsN_{D-n}}.
\widetilde\Pi^{\prime}=\left(\begin{matrix}{\widetildeF_{0}^{\prime}}\\{\widetildeF_{1}^{\prime}}\\{\widetildeX^{0}}\\{\widetildeX^{1}}\\\end{matrix}\right)=\left(\begin{matrix}{\widetildeX_{0}}\\{\widetildeF_{1}}\\{-\widetildeF_{0}}\\{\widetildeX^{1}}\\\end{matrix}\right)\.
p_{\Phi}={\frac{\deltaS}{\delta\dot{\Phi}}}=m\dot{\Phi}\left[\sqrt{(\dot{x}^{0})^{2}-\dot{\Phi}^{2}}\right]^{-1}\equivm\dot{\Phi}\left[\sqrt{-\left(\dot{x}^{2}+\dot{\varphi}^{2}+2\dot{x}^{1}\dot{\varphi}\right)}\right]^{-1}\quad\mathrm{(no~~trace!)},
\epsilon^{\mu\alpha\beta\ldots}~\mathrm{Tr}(\psi[A_{\alpha},A_{\beta}][A_{\mu},Y^{-1}])\rightarrow\epsilon^{\mu\alpha\beta\ldots}~\intd^{2}\sigma~\psi~\{X_{\mu},1/\sqrtg\}_{PB}\{X_{\alpha},X_{\beta}\}_{PB}=0.
{\calI}_{8}=\frac{1}{8}\left(\mathrm{tr}R^{2}-\mathrm{tr}F_{N_{+}}^{2}+\mathrm{tr}F_{N_{-}}^{2}\right)\left(\mathrm{tr}F_{D_{+}}^{2}-\mathrm{tr}F_{D_{-}}^{2}\right)\,\nonumber
\qquad\qquad\quad\qquad=\frac1{A(S^{d-1})(d-1)!}\int_{M_{i}}\in^{A_{1}\cdotsA_{d}}\in_{a_{1}\cdotsa_{d}}\partial_{A_{1}}n^{a_{1}}\partial_{A_{2}}n^{a_{2}}\cdots\partial_{A_{d}}n^{a_{d}}d^{d}v.
\Psi^{\mu\,i}=\varepsilon^{\mu\nu\rho\sigma}D_{\nu}\Sigma_{\rho\sigma}^{\\i}=\varepsilon^{\mu\nu\rho\sigma}\left(\partial_{\nu}\Sigma_{\rho\sigma}^{\\i}+2i\varepsilon_{\jk}^{i}\omega_{\nu}^{\j}\Sigma_{\rho\sigma}^{\\k}\right)=0.
\begin{array}{lll}{\delta=\frac{3L}{L+4},}&{\,\,\alpha=\frac{2(L-2)}{3L}}&{\quad\mathrm{regime~1},}\\{\delta=\frac{3(L+2)}{L-2},}&{\,\,\alpha=\frac{2(L+4)}{3(L+2)}}&{\quad\mathrm{regime~2}.}\\\end{array}
z(1-z)\partial_{z}^{2}g(z)+(1-z)(2\alpha+1)\partial_{z}g(z)+\left[\frac{1}{z}\left(\alpha^{2}+\frac{\omega^{2}\ell^{4}}{4r_{H}^{2}}\right)-\left(\alpha^{2}+\frac{\omega^{2}\ell^{4}}{4r_{H}^{2}}\right)\right]g(z)=0,
\mathrm{Res}_{w_{2n}=xz_{2n}}\frac{dw_{2n}}{w_{2n}}\psi\left(\frac{w_{2n}}{z_{2n}}\right)=\frac{1}{(x^{2};x^{2})_{\infty}(x^{2r};x^{2r})_{\infty}}.
Z_{T}\sime^{\pi^{2}\ell^{2}/2G\beta}e^{\left(\frac{\pi^{2}\ell^{2}}{\beta^{2}r}-\frac{r}{\ell^{2}}\right)BE_{r}}e^{\left(\frac{\pi^{2}\ell^{2}}{\beta^{2}r}-\frac{3}{\ell^{2}}\right)B}\.
\sigma^{2}=\left(\frac1{2m}\varepsilon_{\mu\nu\alpha\beta}p_{\nu}J_{\alpha\beta}\right)^{2}=\frac1{m^{2}}\left(J_{\mu\nu}^{2}p^{2}-J_{\mu\sigma}J_{\nu\sigma}p_{\mu}p_{\nu}\right).
\frac{1}{-p^{2}-i0}\to\frac{1}{-p^{2}-i0}+\frac{1}{-p^{2}-i0}\,g(p^{2})\,\frac{1}{-p^{2}-i0}=\frac{1}{-p^{2}-i0-g(p^{2})}+O(\kappa^{4})\,,
\left<{\calO}\left[j_{\mu}^{\mathrm{gas}}\right]\right>_{\mathrm{gas}}=\prod_{i=1}^{N}\intd^{4}y^{i}Dz^{i}\mu\left[z^{i}\right]\sum_{q_{a}=\pm1}^{}{\calO}\left[j_{\mu}^{\mathrm{gas}}\right].
\tau(\zeta)=\frac{1}{\zeta}\frac{\Theta_{q^{4}}(q\zeta^{2})}{\Theta_{q^{4}}(q\zeta^{-2})},\qquad\Theta_{p}(z)=(z;p)_{\infty}(pz^{-1};p)_{\infty}(p;p)_{\infty},
D_{\mu}\,\varepsilon_{a}\;=\;\nabla_{\mu}\,\varepsilon_{a}\;+\;\frac{1}{2}\,g_{1}\,a_{\mu}(\Gamma_{45})_{a}^{\,b}\,\varepsilon_{b}\;+\;\frac{1}{2}\,g_{2}\,A_{\mu}^{I}(\Gamma_{I45})_{a}^{\,b}\,\varepsilon_{b},
H(\vec{\phi})=\displaystyle{\int}d^{d}x\left[\frac{1}{2}\sum_{i=1}^{N}{\phi}_{i}(-{\nabla}^{2}){\phi}_{i}+\frac{u}{4}(\sum_{i=1}^{N}{\phi}_{i}^{2})^{2}+\frac{v}{4}\sum_{i=1}^{N}\phi_{i}^{4}\right]\,\,.
\omega=\sum_{i=1}^{n}\frac{du_{i}}{u_{i}}\wedged\beta_{i}-\sum_{i<j}\frac{d\beta_{i}\wedged\beta_{j}}{\beta_{i}-\beta_{j}}+\sum_{i=1}^{m}\frac{dv_{i}}{v_{i}}\wedged\kappa_{i}-\sum_{i<j}\frac{d\kappa_{i}\wedged\kappa_{j}}{\kappa_{i}-\kappa_{j}}.
{\calL}+{\calL}_{ct}={\frac{1}{2}}(\partial_{\mu}\phi)^{2}-{\frac{1}{2}}M^{2}\phi^{2}-{\frac{\lambda}{4!}}\mu^{2\epsilon}\phi^{4}+{\frac{1}{2}}A(\partial_{\mu}\phi)^{2}-{\frac{1}{2}}M^{2}B\phi^{2}-{\frac{\lambda}{4!}}\mu^{2\epsilon}C\phi^{4}\,-{\frac{1}{2}}(m^{2}-M^{2})(1+B)\phi^{2}
Q_{~\mu\nu}^{a}(e,\tilde{w})\equiv-\frac{1}{2}\left(\partial_{\mu}e_{~\nu}^{a}-\partial_{\nu}e_{~\mu}^{a}+\tilde{w}_{~b\mu}^{a}e_{~\nu}^{b}-\tilde{w}_{~b\nu}^{a}e_{~\mu}^{b}\right)
W_{\mathrm{SUSY-breaking}}={\frac{e^{{\frac{24\pi^{2}}{b_{0}}}S}\lbrackh(T_{i})A\rbrack^{3(b_{0}-b_{1})/b_{0}}}{\prod_{i=1}^{3}\lbrack\eta(T_{i})\rbrack^{6({b^{\prime}}_{a}^{i}-\delta_{GS}^{i})/b_{0}}}}{}.
\oint_{C}{\bfE}\,d{\bfl}=-\frac{1}{c}\frac{d}{dt}\int_{S}{\bfB}\,d{\bfS}=-\frac{1}{c}\left\{\frac{\partial}{\partialt}+({\bfv}\cdot\nabla)\right\}\int_{S}{\bfB}\,d{\bfS}
\mathcal{L}_{B}=\frac{1}{2g^{2}}\widetilde{\mathrm{Tr}}\left(\left(\partial_{\lbrack\mu}A_{\nu]}+[A_{\mu},A_{\nu}]\right)^{2}\right)+\frac{1}{g^{2}}\widetilde{\mathrm{Tr}}\left(\left(\partial_{\mu}\varphi-i\partial_{5}A_{\mu}+[A_{\mu},\varphi]\right)^{2}\right)
{\calF}_{mn}^{\alpha,a}={\calL}^{\alpha\beta}L^{ab}F_{mn}^{\beta,b}-({\calL}^{T}{\cal{ML}})^{\alpha\beta}(L^{T}ML)^{ab}F_{mn}^{*\beta,b}\equiv{\frac{\sqrt{-g}}{2}}(-{\calL}^{T}{\calM})^{\alpha\beta}(L^{T}M)^{ab}\epsilon_{mnpq}{\calF}^{\beta,b~pq},
K=\frac{R}{2}=\frac{1}{(X^{2}+Y^{2}+Z^{2})^{2}}\cdot\left(X^{\prime\prime}Y^{\prime\prime}Z^{2}+X^{\prime\prime}Z^{\prime\prime}Y^{2}+Y^{\prime\prime}Z^{\prime\prime}X^{2}\right)
\langle\Phi_{1}^{(\lambda_{1},\mu_{1})}(z_{1},\bar{z}_{1})\ldots\Phi_{N-1}^{(\lambda_{N-1},\mu_{N-1})}(z_{N-1},\bar{z_{N-1}})\tilde{\Phi}_{N}^{(\lambda_{N},\mu_{N})}(z_{N},\bar{z_{N}})\rangle.
S=\left(\begin{matrix}{ae^{i\alpha}P_{-}}&{0}&{ice^{i\alpha}P_{-}JP_{+}}\\{0}&{e^{i\phi}P_{0}}&{0}\\{ide^{i\alpha}P_{+}JP_{-}}&{0}&{be^{i\alpha}P_{+}}\\\end{matrix}\right)
S_{\mathrm{int}}(\phi_{\pm})=\intd^{4}x[-V_{\Lambda}(\phi_{\pm})+\frac{1}{2}Z_{\Lambda}(\phi_{\pm})\dot{\phi}_{\pm}^{2}-\frac{1}{2}Y_{\Lambda}(\phi_{\pm})({\bf\nabla}\phi_{\pm})^{2}+\ldots].
L=\sum\overline{{\psi_{i}}}\left(\triangle_{\mu}\gamma_{\mu}\otimes\mathrm{1}+\frac{1}{2}1\otimes\gamma_{1}\Box_{\mu}+g(\sigma+i\gamma_{3}\pi)\otimes(1+\gamma_{3})/2\right)\psi_{i}+{\bf\sigma}^{2}/2+{\bf\pi}^{2}/2,
\int\frac{d^{3}\tilde{u}}{(2\pi)^{3}}\frac{\tilde{u}^{2}\tilde{u}_{0}^{2}}{[\tilde{u}^{2}+\tilde{u}_{0}^{2}+\tilde{M}^{2}]^{\frac{3}{2}}}\sim\frac{\tilde{u}_{0}^{2}}{[\tilde{u}_{0}^{2}+\tilde{M}^{2}]^{\frac{3}{2}}}
{\frac{{\bfq}_{1}}{\sqrt{T_{0}}}}={\bfu}_{1}+{\bfu}_{2},\qquad{\frac{{\bfq}_{2}}{\sqrt{T_{0}}}}={\bfu}_{1}-{\bfu}_{2},\qquad{\frac{{\bfp}_{1}}{\sqrt{T_{0}}}}={\bfv}_{1}+{\bfv}_{2},\qquad{\frac{{\bfp}_{2}}{\sqrt{T_{0}}}}={\bfv}_{1}-{\bfv}_{2},
L_{\mathrm{T}}=\mathrm{}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{m^{2}}{2}\left(A_{\mu}-\frac{1}{m}\partial_{\mu}\theta\right)^{2}+b\left(\partial^{\mu}A_{\mu}+\frac{\alpha}{2}b\right)+i\bar{c}\Boxc.
\mathrm{if}~~F\in{\calA}~~,~~F=\left(\begin{matrix}{f_{1}}&{0}\\{0}&{f_{2}}\\\end{matrix}\right)~~,~~\mathrm{then}~{\tilde{F}}=\left(\begin{matrix}{f_{2}}&{0}\\{0}&{f_{1}}\\\end{matrix}\right).
Ref_{\pm}(s,t)=\frac{(s)^{k}}{\pi}P\!\int_{s_{0}}^{+\infty}ds^{\prime}\frac{Imf_{\pm}(s^{\prime},t)}{(s^{\prime})^{k}}\left[\frac{1}{s^{\prime}-s}\pm\frac{(-1)^{k}}{s^{\prime}+s}\right].
I_{2}^{(2)}=\intd^{D}k~d^{D}l\frac{1}{k^{2}(q-l)^{2}l^{2}}T_{q}\left[\frac{1}{(k-q)^{2}}\right]T_{l}\left[\frac{1}{[(k-l)^{2}-M^{2}]}\right],
j_{i_{1}\ldotsi_{s}}=\sum_{k=0}^{s}{\frac{(-1)^{k}}{k!\left(k+{\frac{D-4}{2}}\right)!(s-k)!\left(s-k+{\frac{D-4}{2}}\right)!}}\partial_{i_{1}}\cdots\partial_{i_{k}}\phi^{*}\;\partial_{i_{k+1}}\cdots\partial_{i_{s}}\phi-\mathrm{traces}
\Sigma_{0}{}_{\mu\nu}^{k}\Sigma_{0}^{k\tau\rho}=2P_{\mu\nu}{}^{\rho\tau}\equiv2(\delta_{\mu}^{[\tau}\delta_{\nu}^{\rho]}-{\frac{1}{2}}\epsilon_{\mu\nu}{}^{\tau\rho})\Sigma_{0}{}_{\mu}^{k\rho}\Sigma_{0}{}_{\rho}^{l\nu}=\epsilon_{kln}\Sigma_{0\mu}^{n\nu}.
\begin{array}{rclcrcl}{\eta(u_{p}\wedgew_{q})}&{=}&{(\etau_{p})\wedge(\etaw_{q}),}&{\quad}&{\eta(u_{p}\veew_{q})}&{=}&{(\etau_{p})\vee(\etaw_{q}),}\\{\zeta(u_{p}\wedgew_{q})}&{=}&{(\zetaw_{q})\wedge(\zetau_{p}),}&{\quad}&{\zeta(u_{p}\veew_{q})}&{=}&{(\zetaw_{q})\vee(\zetau_{p}),}\\\end{array}
D_{\;\;ij}^{kl}=\frac12\left(\delta_{\;\;i}^{\left[k\right.}\delta_{\;\;j}^{\left.l\right]}-\frac1{\triangle}\delta_{\;\;m}^{\left[l\right.}\partial_{}^{\left.k\right]}\delta_{\;\;\left[j\right.}^{m}\partial_{\left.i\right]}^{}\right).
ds^{2}=-dt^{2}+[R_{0}^{2}-k(t-t_{0})^{2}]d\Omega_{k}^{2}+\frac{(t-t_{0})^{2}}{[R_{0}^{2}-k(t-t_{0})^{2}]}[dx^{5}+\kappaA_{0}(t)dt]^{2},
\Pi^{\mathrm{(non\mathrm{-}planar)}}(p)=\frac{2\pi^{\frac{n}{2}}}{(2\pi)^{n}}\int_{0}^{1}dx\,\frac{1}{(M^{2})^{2-\frac{n}{2}}}\,\left(\frac{|\bar{\theta}|M}{2}\right)^{2-\frac{n}{2}}\,K_{2-\frac{n}{2}}(|\bar{\theta}|M)
P^{+}=\int_{-\infty}^{\infty}dx^{-}\mathrm{Tr}\biggl[2\partial_{-}\phi_{-}\cdot\partial_{-}\phi_{+}+{\frac{\mathrm{i}}{2}}\sum_{h}\left(u_{h}^{\dagger}\cdot\partial_{-}u_{h}-\partial_{-}u_{h}^{\dagger}\cdotu_{h}\right)\biggr]
g(\lambda)\left(1-{\frac{z}{\lambda^{2}}}\right)^{-1/2}=\left(\sum_{m\geq1}c_{m}\lambda^{2m}\right)\left(\sum_{m\geq0}{\frac{(2m)!}{m!^{2}4^{m}}}{\frac{z^{m}}{\lambda^{2m}}}\right)=h(\lambda)+2{\frac{N}{\beta}}+O(\lambda^{-1}).
E^{\mathrm{(vol)}}=E^{\mathrm{(tot)}}-E^{\mathrm{(surf)}}=\sum_{m=1}^{2}E^{\mathrm{(s)(vol)}}(\beta_{m})+a\left(\varepsilon_{c}^{(1)}+4D(\xi-\xi_{c})\varepsilon_{c}^{(2)}\right),
{\calR}_{2}={\calA}^{4}=\left(\begin{matrix}{3}&{-2}&{4}&{0}&{0}&{-4}\\{0}&{-1}&{0}&{0}&{0}&{0}\\{0}&{0}&{-3}&{4}&{0}&{0}\\{0}&{0}&{-2}&{3}&{0}&{0}\\{0}&{0}&{1}&{-2}&{-1}&{0}\\{2}&{-1}&{0}&{4}&{0}&{-3}\\\end{matrix}\right)
ds_{n+3}^{2}=2dx^{+}dx^{-}+\frac{2n+4}{(n+4)^{2}}\frac{1}{{x^{-}}^{2}}\Big(n(y^{0})^{2}-\sum_{i=1}^{n}{y^{i}}^{2}\Big)(dx^{-})^{2}+\sum_{i=0}^{n}{dy^{i}}^{2}
C_{2}=\sum_{i,j=1}^{m}E_{ij}E_{ji}-\sum_{\bar{k},\bar{l}=m+1}^{m+n}E_{\bar{k}\bar{l}}E_{\bar{l}\bar{k}}+\sum_{i=1}^{m}\sum_{\bar{k}=m+1}^{m+n}(E_{\bar{k}i}E_{i\bar{k}}-E_{i\bar{k}}E_{\bar{k}i}).
\begin{array}{l}{\mathrm{vol}\;X=\int_{X}d\sigma_{0}\wedged\sigma_{1}\wedged\sigma_{2}=}\\{=\intd\sigma_{0}d\sigma_{1}d\sigma_{2}\sqrt{(\overline{{w^{3,0}}},w^{3,0})+3(w,(I\otimesI\otimes1).w)}=}\\{=\intd\sigma_{0}d\sigma_{1}d\sigma_{2}\sqrt{(\mathrm{Re}\;\Omega(w))^{2}+(\mathrm{Im}\;\Omega(w))^{2}+3(w,(I\otimesI\otimes1).w)}=}\\{=\int\mathrm{Re}\;\Omega+\ldots}\\\end{array}
\psi_{\alpha}^{i}(x^{0},x^{1},x^{2},x^{3})=\sum_{n}\left(\begin{matrix}{b_{n}(x^{0},x^{3})\psi_{n1}^{i}(x^{1},x^{2})}\\{c_{n}(x^{0},x^{3})\psi_{n2}^{i}(x^{1},x^{2})}\\\end{matrix}\right)
\chi_{\dot{q}\dot{p}}^{1}=\chi^{1}\delta_{\dot{q}\dot{p}}+\chi^{1ij}\tilde{\gamma}_{\dot{q}\dot{p}}^{ij}+\chi_{\dot{q}\dot{p}}^{1ijkl},\\chi_{qp}^{2}=\chi^{2}\delta_{qp}+\chi^{2ij}\gamma_{qp}^{ij}+\chi^{2ijkl}\gamma_{qp}^{ijkl}.
f=f_{\mathrm{pert}}+4\piVe^{-\phi}\sum_{m\ne0}\sum_{\mathcal{N}}\frac{|m|}{\|\mathcal{N}\|}K_{1}\left(2\pi|m|e^{-\phi}\|\mathcal{N}\|\right)e^{2\piim\langle\mathcal{R},\mathcal{N}\rangle}
W_{q}^{2}=q^{2}+V^{\prime\prime}+{\frac{5}{16}}\left[\left({\frac{V^{\prime\prime\prime}}{q^{2}+V^{\prime\prime}}}\right)^{2}-{\frac{V^{\prime\prime\prime\prime}}{4(q^{2}+V^{\prime\prime})}}\right]\dot{\phi}^{2}-{\frac{V^{\prime\prime\prime}}{4(q^{2}+V^{\prime\prime})}}\ddot{\phi}+\ldots
{\calV}=\frac{1}{4}\int_{\mathbb{F}}\frac{dv_{1}\,dv_{2}}{v_{2}}\sum_{\alpha\inP/mQ^{\vee}}\left(E_{2}(\rho)-\frac{3}{\piv_{2}}\right)^{-k/2}h_{\alpha}(\rho)\Theta_{\alpha,m}(\rho,\bar{\rho},[\omega])\,,
I[\,\phi\,]\,=\,{\frac{1}{2}}\intd^{n+1}x\,\partial_{\mu}\Big((x^{0})^{-n+1}\,\phi\partial_{\mu}\phi\Big)\,-\,{\frac{1}{2}}\intd^{n+1}x\,\phi\,\partial_{\mu}\Big((x^{0})^{-n+1}\,\partial_{\mu}\phi\Big)\,.
\{Q_{(\alphai)},Q_{(\betaj)}\}=-i\delta_{ij}(\mathcal{C}\gamma^{a})_{\alpha\beta}P_{a}+{\textstyle\frac{i}{R_{2}}}\mathcal{C}_{\alpha\beta}\epsilon_{ij}M_{1}+{\textstyle\frac{1}{R_{2}}}(\mathcal{C}\gamma_{5})_{\alpha\beta}\epsilon_{ij}M_{2}\,.
\Phi=\left(\begin{array}{cc}{-HC}&{0}\\{0}&{-HC}\\\end{array}\right),\,\,\,\,\,\,{\calK}=\left(\begin{array}{cc}{K^{11}}&{K^{12}}\\{K^{21}}&{K^{22}}\\\end{array}\right),\,\,\,\,\,\,\chi^{\mu}=\left(\begin{array}{c}{K^{13}\psi^{3\mu\dagger}}\\{K^{23}\psi^{3\mu\dagger}}\\\end{array}\right).
(\partial_{u_{2}}^{2}+\ldots+\partial_{u_{d}}^{2})Q(u_{2},\ldots,u_{d})=\partial_{u_{1}}P_{f}(u_{1},\ldots,u_{d})+\int_{0}^{u_{1}}dv(\partial_{u_{2}}^{2}+\ldots+\partial_{u_{d}}^{2})P_{f}(v,u_{2},\ldots,u_{d})
F_{n}^{\mathcal{O}|\mu_{1}\ldots\mu_{n}}(\theta_{1},\theta_{2}\ldots,\theta_{n}):=\langle0|\mathcal{O}(0)|Z_{\mu_{1}}^{\dagger}(\theta_{1})Z_{\mu_{2}}^{\dagger}(\theta_{2})\ldotsZ_{\mu_{n}}^{\dagger}(\theta_{n})\rangle.
S_{dual}=\intd^{2}x\left[{\frac12}({\partial}_{\mu}{\rho})({\partial}^{\mu}{\rho})+e^{\prime}A^{\mu}({\epsilon}_{{\mu}{\nu}}-r^{\prime}g_{{\mu}{\nu}}){\partial}^{\nu}{\rho}+{\frac12}{e^{\prime}}^{2}a^{\prime}A_{\mu}A^{\mu}-{\frac14}F_{{\mu}{\nu}}F^{{\mu}{\nu}}\right],
\begin{array}{rcl}{\displaystyle\sum_{k}K^{*}(z)_{k}^{j}\phi^{*k}(z)|\mathrm{vac}\rangle_{i}}&{=}&{\nu^{(i)}(-z-\frac{n}{2}w)\phi^{*j}(-z)|\mathrm{vac}\rangle_{i},}\\{{}_{i}\langle\mathrm{vac}|\displaystyle\sum_{k}\phi_{k}^{*}(z)K^{*}(z)_{j}^{k}}&{=}&{\nu^{(i)}(-z-\frac{n}{2}w){}_{i}\langle\mathrm{vac}|\phi_{j}^{*}(-z).}\\\end{array}
\langle0|\varphi^{2}(x)|0\rangle=\langle0^{(\alpha)}|\varphi^{2}(x)|0^{(\alpha)}\rangle-\frac{\sigma^{1-D}}{\pinS_{D}r^{n}}\sum_{l=0}^{\infty}D_{l}\int_{m}^{\infty}dz\frac{z\Omega_{\alpha\nu_{l}}(az,bz)}{\sqrt{z^{2}-m^{2}}}G_{\nu_{l}}^{(\alpha)2}(\alphaz,rz),
M_{W^{+}}^{2}={\frac{v^{2}}{4}}g^{2}~,~M_{W^{-}}^{2}={\frac{v^{2}}{4}}g^{2}~,~M_{Z}^{2}={\frac{v^{2}}{4}}~(g^{\prime2}+g^{2})~,~M_{A}^{2}=0\.
\mathrm{(5a)}+\mathrm{(5b)}+\mathrm{(5c)}=\frac{N}{96\pim}\int_{x}\left[\mathrm{tr}\left\{[\tilde{A}_{\mu},\tilde{A}_{\nu}][\tilde{A}^{\mu},\tilde{A}^{\nu}]\right\}+O(\frac{\partial^{2}}{m^{2}})\right].
K_{I}^{\prime}(\theta)=\ell^{\prime}\;\;\frac{\textrm{sn}\left[\frac{\textbf{K}}{i\pi}\left(\theta-i\frac{\pi}{2}\right)\right]}{\textrm{cn}\left[\frac{\textbf{K}}{i\pi}\left(\theta-i\frac{\pi}{2}\right)\right]\textrm{dn}\left[\frac{\textbf{K}}{i\pi}\left(\theta-i\frac{\pi}{2}\right)\right]}\,\,\,.
g^{\mu\nu}g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}\deltag_{\mu\nu}-\nabla^{\mu}\nabla^{\nu}\deltag_{\mu\nu}=\nabla_{\mu}(g^{\alpha\beta}\nabla^{\mu}\deltag_{\alpha\beta}-g^{\mu\alpha}\nabla^{\beta}\deltag_{\alpha\beta})
\begin{array}{ll}{Z_{13}=}&{\frac{1}{s}(\sum_{l=1}^{g(s)}e^{-2\pii(l-1)/ns}+e^{2\pii/n}\sum_{l=g(s)+1}^{s}e^{-2\pii(l-1)/ns})}\\{}&{\times(\sum_{p=0}^{p_{max}}e^{-2\piip/n}\sum_{u(p)>j\geql(p)}e^{2\pii(j-1)/M}),}\\\end{array}
\delta_{ij}-{\frac{k_{i}k_{j}}{|{\bfk}|^{2}}}+{\frac{\bar{c}}{c}}{\frac{k_{i}k_{j}}{|{\bfk}|^{2}}}=\delta_{ij}+{\frac{2\nu}{1-{\frac{d}{2}}-\nu}}{\frac{k_{i}k_{j}}{|{\bfk}|^{2}}}.
X^{i}(\sigma,\tau)=X_{t}^{i}+\frac{1}{2}i\sum_{m=0}\left(\frac{\alpha_{m+\frac{1}{n}}^{i}}{m+\frac{1}{n}}e^{-i(m+\frac{1}{n})(\tau-\sigma)}+\frac{{\tilde{\alpha}}_{m-\frac{1}{n}}^{i}}{m-\frac{1}{n}}e^{-i(m-\frac{1}{n})(\tau+\sigma)}\right).
xy^{2}=\left[x\prod_{a=1}^{n_{c}}(x-\phi_{a}^{2})^{2}+2\Lambda^{2n_{c}+2-n_{f}}m_{1}\cdotsm_{n_{f}}\right]^{2}-4\Lambda^{2(2n_{c}+2-n_{f})}\prod_{i=1}^{n_{f}}(x+m_{i}^{2}),\quadUSp(2n_{c}).
b_{\underline{{n}}\sigma}^{\dagger}=\frac{\pi/L_{\perp}}{\sqrt{8\pi^{3}n}}b_{\underline{{p}}\sigma}^{\dagger}\,,\;\;a_{\underline{{m}}}^{\dagger}=\frac{\pi/L_{\perp}}{\sqrt{8\pi^{3}m}}a_{\underline{{q}}}^{\dagger}\,,\;\;
M(\lambda,L,m,d)=\lambda^{2}\intd^{d-1}r\intd^{d-1}r^{\prime}\int_{0}^{L}dz\int_{0}^{L}dz^{\prime}F(\vec{r},\vec{r}^{\prime},z,z^{\prime})(G_{0}^{(2)}(\vec{r}-\vec{{r}^{\prime}},z,z^{\prime}))^{2},
S_{4}^{\prime}(\phi_{s})=\biggl<S_{4}(\phi_{s},\phi_{f})\biggr>_{0f}+\frac{1}{2}\biggl(\biggl<S_{4}^{2}(\phi_{s},\phi_{f})\biggr>_{0f}-\biggl<S_{4}(\phi_{s},\phi_{f})\biggr>_{0f}^{2}\biggr)+\cdots.
y_{\ell}^{\vphantom+}(r)=\sqrt{r}\left(C_{\ell}^{-}\left(\frac{R}{r}\right)^{\left((\ell+1)^{2}-M^{2}N\right)^{1/2}}+C_{\ell}^{+}\left(\frac{r}{R}\right)^{\left((\ell+1)^{2}-M^{2}N\right)^{1/2}}\right)\,\\M^{2}\leq\frac{(\ell+1)^{2}}{N}.
\deltam^{2}=\frac{1}{24}\frac{1}{\beta^{2}}\left(4\tilde{\lambda}{\bfI}-\tilde{g}^{2}\gamma_{S}\right)+\frac{\tilde{\lambda}\tilde{m}^{2}{\bfI}}{4\pi^{2}}\left(\frac{1}{\varepsilon}+1-\gamma-ln{\frac{2\beta\mu}{\sqrt{\pi}}}\right),
2n^{\mu}n^{\nu}\frac{(p^{2}-q^{2})(p^{2}-r^{2})}{q\cdotn\;r\cdotn}=-2n^{\mu}n^{\nu}\frac{(p^{2}-q^{2})(p^{2}-r^{2})}{p\cdotn}\left[\frac{1}{q\cdotn}-\frac{1}{r\cdotn}\right].
\deltaS=\frac{1}{3}\intd^{2}xd_{ijk}(\partial_{+}\varphi^{i}\partial_{+}\varphi^{j}\partial_{+}\varphi^{k}\partial_{-}\lambda^{++}+\partial_{-}\varphi^{i}\partial_{-}\varphi^{j}\partial_{-}\varphi^{k}\partial_{+}\lambda^{--}).
S=\intd^{D}x\sqrt{|g|}\left(\frac{1}{2}M_{D}^{n+2}R-\frac{\Lambda_{D}}{M_{D}^{n+2}}+(-1)^{p}\frac{1}{4}F_{\mu_{1}\dots\mu_{p+1}}F^{\mu_{1}\dots\mu_{p+1}}\right)~.
\rho_{m_{0}\rightarrowm_{1}+\cdots+m_{N}}^{D}=\sum_{k_{i}}\frac{\delta_{\sum_{i}k_{i},0}}{(2\piR)^{N-1}}\rho_{m_{0}\rightarrowm_{1k_{1}}+\cdots+m_{Nk_{N}}}^{D-1},
\{F,G\}_{\alpha}=-\intd^{2}\theta\,\Bigr\{\theta^{2}\frac{\partial^{2}\deltaF}{\partial\theta^{2}\delta\Phi^{A}(\theta)}\,\frac{\deltaG}{\delta\bar{\Phi}_{B}(\theta)}(\sigma_{\alpha})_{B}^{~~\!A}+(-1)^{\epsilon(F)\epsilon(G)}(F\leftrightarrowG)\Bigr\}.
{\calL}_{class}^{\prime}(\varphi_{1},\varphi_{2})=-\varphi_{1}\Box\varphi_{2}+m^{2}\varphi_{1}\varphi_{2}+{\frac{\lambda}{4}}\varphi_{1}^{2}\varphi_{2}^{2}+J_{1}(\varphi_{1}+\alpha\varphi_{1}^{2})+J_{2}\varphi_{2}.
{\mathcalL}_{\mathrm{eff}}=\partial_{\mu}V\,\partial^{\mu}V^{*}-\lambda\bigl(V\,V^{*}-{\frac{g^{2}}{8\pi^{2}}}\bigr)^{2}+{\frac{m_{ph}^{2}}{4}}\left\{V^{2}+(V^{*})^{2}\right\}.
\intd^{10}\hat{x}\sqrt{|\hat{\jmath}|}\,\left[{\textstyle\frac{1}{2\cdot5!}}\left(\hat{G}_{(5)}\right)^{2}+{\textstyle\frac{1}{2\cdot5!}}\left(\hat{\overline{{G}}}_{(5)}\right)^{2}\right]\,.
{\calM}_{-1}^{+}={\calM}_{\infty}^{-1}{\calM}_{0}^{-1}=\left(\begin{array}{rrr}{0}&{1}&{0}\\{1}&{0}&{0}\\{-3}&{3}&{1}\\\end{array}\right)\;\qquad{\calM}_{-1}^{-}={\calM}_{0}^{-1}{\calM}_{\infty}^{-1}=\left(\begin{array}{rrr}{-1}&{-1}&{-1}\\{6}&{4}&{3}\\{-6}&{-3}&{-2}\\\end{array}\right)\.
M_{ADM}=\frac{1}{\sqrt{2}\kappa}\left[l_{1}l_{2}\hat{e}(N_{2}+\bar{N}_{2})+l_{1}V_{4}\hat{e}_{m}(N_{5}+\bar{N}_{5})+\frac{2\sqrt{2}\kappa\pi}{l_{1}}(n+\bar{n})\right]
y_{12}^{ij}=-{\frac{1}{m}}F_{~b}^{i}\epsilon^{jbc}S_{c}+{\frac{F^{il}\Lambda_{0l}p_{k}\epsilon^{jk}p_{0}(p.S)}{2m^{2}p_{0}\Lambda_{00}(p^{k}\Lambda_{0k}}}+{\frac{S_{0}\epsilon^{rl}F^{0r}S^{l}}{p_{0}(p.S)}}((p.\Lambda)g^{ij}-p^{i}\Lambda^{0j})
\begin{array}{c}{\omega=\mathrm{d}{\bfp}\wedge\mathrm{d}{\bfx}+\sum_{i=1}^{N}\mathrm{d}{\bfp}_{i}\wedge\mathrm{d}{\bfe}_{i}}\\{{\calH}={\tilde{s}}\left[{\bf\pi}{\bfe}_{1}+\sum_{i=1}^{N}{k}_{i-1}{\phi}_{i-1.i}+\frac{{k}_{N}}{2c}({\Phi}_{N.N}-c^{2})+\sum_{i,j=1}^{N}d_{ij}({\bfe}_{i}{\bfe}_{j}-\delta_{ij})\right],}\\\end{array}
\intd^{3}\zeta-\sqrt{-g(Z(\zeta))}=\int-h^{-1}\sqrt{1-hv^{2}}\Big[1+\frac{1}{2}h^{2/3}\big[2h^{-1/3}\bar{\theta}\Gamma^{\tilde{1}}\partial_{1}\theta+2h^{-1/3}\bar{\theta}\Gamma^{\tilde{2}}\partial_{2}\theta
X^{\prime\prime}+d{\calH}X^{\prime}-\partial_{\alpha}\partial^{\alpha}X-\frac{\partial^{2}V}{\partial\varphi^{2}}a^{2}X+\varphi^{\prime}[(d+1)\Psi^{\prime}+\Xi^{\prime}]+2(\varphi^{\prime\prime}+d{\calH}\varphi^{\prime})\Xi=0.
{\calA}_{0}(t,x)=\int_{0}^{1}{d\alpha}x^{1}\partial_{1}^{\alphax}\left({-\frac{{J^{0}(\alphax)}}{{\partial_{\alphax}^{2}-\lambda^{2}}}}\right)-4\pim\Sigma\int_{0}^{1}{d\alpha}x^{1}\frac{{\partial_{1}^{\alphax}(-J^{0}(\alphax))}}{{(\partial^{2}-\lambda^{2})_{\alphax}\partial_{\alphax}^{2}}},
\partial_{z_{+}}^{3}v^{+}(z_{+})=z_{-}^{2}\partial_{z_{-}}\bigg(z_{-}^{2}\partial_{z_{-}}\bigg(z_{-}^{2}\partial_{z_{-}}\bigg(z_{-}^{-2}v^{-}(z_{-})\bigg)\bigg)\bigg)=0,
A_{\epsilon}=A(x+\epsilonx)=A(x)+\epsilonx^{M}\partial_{M}A(x)+\sum_{n=1}^{\infty}{\frac{\epsilon^{n+1}}{(n+1)!}}x^{M_{1}}...x^{M_{n+1}}\partial_{M_{1}\dotsM_{n+1}}^{n+1}A(x).
{\calH}_{ch}=\intd^{d}x\Bigl\{{\frac{1}{2}}\sum_{a}\Bigl[(\partial_{\mu}\phi_{a})^{2}+r\phi_{a}^{2}\Bigr]+{\frac{1}{4!}}u\Bigl(\sum_{a}\phi_{a}^{2}\Bigr)^{2}+{\frac{1}{4!}}v\sum_{a,b}\Bigl[(\phi_{a}\cdot\phi_{b})^{2}-\phi_{a}^{2}\phi_{b}^{2}\Bigr]\Bigr\},
{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\;i}\left[\frac{1}{2}\frac{\partial\delta\Gamma^{(1)}}{\partialw^{i}}\right]=-{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\;i}\left[\deltaB_{i}^{w(2)}\right]+{e}_{\alpha}^{\;\;\mu}{e}_{\hat{0}}^{\;\;\nu}X_{\mu\nu}^{w}.
\deltaA=-\delta\bar{\theta}\Gamma_{11}\Gamma_{m}\theta\Pi^{m}+{\frac{1}{2}}\delta\bar{\theta}\Gamma_{11}\Gamma_{m}\theta\bar{\theta}\Gamma^{m}d\theta-{\frac{1}{2}}\delta\bar{\theta}\Gamma^{m}\theta\bar{\theta}\Gamma_{11}\Gamma_{m}d\theta.
\kappa^{2}g^{2}\alpha^{\prime}\left(-2\cdot(\frac{(32-5)\cdot2^{7/2}}{3^{5}})(-\frac{5\cdot2^{7/2}}{3^{5}})\right)\left(A_{i}^{2}A_{j}^{2}\right)=\kappa^{2}g^{2}\alpha^{\prime}\left(\frac{17280}{59049}\right)\left(2A_{i}A_{i}A_{j}A_{j}\right).
\zeta_{A}(s)=\frac{1}{(4\pi)^{3/2}\Gamma(s)}\sum_{k=0,1/2,1,...}^{\infty}\Gamma(s+k-3/2)(\int_{B^{3}}a_{k}dv+\int_{S^{2}}c_{k}ds)V^{3/2-k-s}
\begin{array}{l}{\theta_{m,24}({\frac{\tau}{2}},{\frac{z}{6}})\stackrel{z\toz+\tau}{\longrightarrow}q^{-1/3}y^{-2/3}~\theta_{m+8,24}({\frac{\tau}{2}},{\frac{z}{6}}),}\\\end{array}
(stu)_{\mathrm{fix}}\,|q_{1}|=\left({\frac{u}{st}}\right)_{\mathrm{fix}}|p_{2}|=\left({\frac{s}{tu}}\right)_{\mathrm{fix}}|q_{3}|=\left({\frac{t}{su}}\right)_{\mathrm{fix}}|p_{4}|\.