harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
N_{z}\simeqp_{z}\frac{\partial}{\partialE}+\left(E+\frac{{\tilde{L}}_{p}}{2}{\vec{p}}^{2}-{\tilde{L}}_{p}E^{2}\right)\frac{\partial}{\partialp_{z}}-{\tilde{L}}_{p}p_{z}\left(p_{j}\frac{\partial}{\partialp_{j}}\right)~.
\left\{d-{\textstyle\frac{1}{4}}\omega^{a}{}_{b}\gamma_{a}{}^{b}-{\textstyle\frac{1}{8\sqrt{3}}}(\gamma^{bc}\mathcal{F}_{bc}\gamma_{a}+2\gamma^{b}\mathcal{F}_{ba})e^{a}\right\}\kappa=0\,.
{\calA}\propto\intd^{J-1}\theta\int_{[p]}dz\ointdw\frac{1}{w^{n}}\eta_{\mu\nu}\langle\V_{-1}^{\mu}(0)\V_{-1}^{\nu}(\infty)\V_{+1/2}[u_{a}](w)\V_{-1/2}[\lambda_{a}](z)\\rangle
\left({\begin{array}{cc}{\frac{2\mu\tilde{\lambda}_{e\,1}+(2\mu-1)(2\mu^{2}-4\mu+3)}{(2\mu-3)(\mu-2)}}&{\frac{(2\mu-1)^{2}(2\mu-3)(\mu-3)}{(\mu-1)}}\\{(\mu^{2}-1)}&{\frac{4\mu\tilde{\lambda}_{e\,1}}{(2\mu-1)}+\frac{(2\mu-3)(\mu-2)(\mu-3)(\mu+1)(2\mu-1)}{(\mu-1)}}\\\end{array}}\right)
\partial_{a_{1}}\ldots\partial_{a_{n}}\delta_{n}\phi=\epsilon_{a_{1}}^{\bar{a}_{1}}\ldots\epsilon_{a_{n}}^{\bar{a}_{n}}\bar{\partial}_{\bar{a}_{1}}\ldots\bar{\partial}_{\bar{a}_{n}}(\rho_{0}^{\bar{c}}\bar{\partial}_{\bar{c}}\phi)\;+\;{\calO}(\phi^{2}).
\left(\begin{array}{ccccccc}{\hat{n}_{25}}&{\hat{n}_{26}}&{\hat{n}_{27}}&{\hat{n}_{28}}&{\hat{n}_{29}}&{\hat{n}_{30}}&{\hat{n}_{31}}\\{1}&{0}&{0}&{1}&{0}&{1}&{0}\\{0}&{1}&{0}&{0}&{1}&{0}&{1}\\{1}&{0}&{1}&{0}&{0}&{1}&{0}\\{0}&{1}&{0}&{1}&{0}&{0}&{1}\\{1}&{0}&{1}&{0}&{1}&{0}&{0}\\{0}&{1}&{0}&{1}&{0}&{1}&{0}\\{0}&{0}&{1}&{0}&{1}&{0}&{1}\\\end{array}\right).
\sum_{n=-\infty}^{\infty}\Big[\omega\Big(x+i(n+2)\fracm2\Big)-\omega\Big(x+in\fracm2\Big)\Big]^{2}=2\sum_{n=-\infty}^{\infty}\mathrm{Tr}\Big[\frac{W^{\prime}(\hat{\Phi})}{x-\hat{\Phi}+in\fracm2}\Big]\.
\deltag_{\mu\nu}(x)=f^{\star}[\overline{{\nabla}}_{\mu}\bar{\xi}_{\nu}+\overline{{\nabla}}_{\nu}\bar{\xi}_{\mu}](x)+\left[f^{\star}{\frac{\partial\bar{g}_{\mu\nu}(l)}{dl_{i}}}\deltal_{i}\right](x)=\nabla_{\mu}\xi_{\nu}(x)+\nabla_{\nu}\xi{_\mu}(x)+{\frac{\partialg_{\mu\nu}(x,l,f)}{\partiall_{i}}}\deltal_{i}\:.
D_{\alphaa}=\frac\partial{\partial\theta^{\alphaa}}+\frac12\sigma_{\alpha\dot{\beta}}^{\mu}\gamma_{a\dot{b}}^{m}\bar{\theta}^{\dot{\beta}\dot{b}}\partial_{\mu}\tilde{\partial}_{m},\quad\bar{D}_{\dot{\beta}\dot{b}}=\frac\partial{\partial\bar{\theta}^{\dot{\beta}\dot{b}}}\_+\frac12\theta^{\alphaa}\sigma_{\alpha\dot{\beta}}^{\mu}\gamma_{a\dot{b}}^{m}\partial_{\mu}\tilde{\partial}_{m}
<~...~>_{R_{\tau}}^{bc}\prod_{i=1}^{N-3}d\tau_{i}=(z_{1}-z_{2})(z_{1}-z_{N})(z_{2}-z_{N})\prod_{r=1}^{N}(\frac{d\rho}{dz}e^{-\rho})_{z_{r}}^{-1}\prod_{i=1}^{N-3}dz_{i},
\left(\begin{array}{cc}{a}&{b}\\{-b}&{a}\\\end{array}\right)={\calL}\left(\begin{array}{cc}{a-{\mathfrakt}b}&{0}\\{0}&{a+{\mathfrakt}b}\\\end{array}\right){\calL}^{-1}\,\,\,\,\,\,a^{2}+b^{2}\equiv\,1\,{\mathrmm\mathrmo\mathrmd}\,p^{n}
\widehat{\bfA}_{\perp}|{\bfA}_{\perp}^{\prime}\rangle={\bfA}_{\perp}^{\prime}|{\bfA}_{\perp}^{\prime}\rangle\qquad\widehat{\bbox\pi}_{\perp}|\bbox\pi_{\perp}^{\prime}\rangle=\bbox\pi_{\perp}^{\prime}|\bbox\pi_{\perp}^{\prime}\rangle,
G_{\alpha}{}^{\hat{\alpha}}=\frac{1}{\sqrt{2}}\left(\begin{matrix}{1}&{\gamma_{5}}\\{1}&{-\gamma_{5}}\\\end{matrix}\right),\qquadH^{\alpha\hat{\alpha}}=\frac{1}{\sqrt{2}}\left(\begin{matrix}{1}&{-\gamma_{5}}\\{1}&{\gamma_{5}}\\\end{matrix}\right),
\left(\begin{array}{c}{{x^{\prime}}^{0}}\\{{x^{\prime}}^{1}}\\{\lambda^{\prime}}\\\end{array}\right)=\left(\begin{array}{ccc}{\frac{p^{0}}{mc}}&{\frac{p}{mc}}&{a^{0}}\\{\frac{p}{mc}}&{\frac{p^{0}}{mc}}&{a^{1}}\\{-k\frac{p^{0}a^{0}-pa^{1}}{\Lambdamc}}&{kK}&{\Lambda}\\\end{array}\right)\left(\begin{array}{c}{x^{0}}\\{x^{1}}\\{\lambda}\\\end{array}\right)\,,
\begin{array}{rcl}{V_{e}/m^{2}}&{=}&{\displaystyle+[D(k_{1}+p)D_{m}(k_{1})+D(k_{1})D_{m}(k_{1}+p)]}\\{}&{}&{\displaystyle\times[D(k_{2}+p)D_{m}(k_{2})+D(k_{2})D_{m}(k_{2}+p)]K_{12}}\\{}&{}&{\displaystyle+D_{m}(k_{2})[D(k_{2}+p)-D(k_{2}-p)]D_{m}(k_{1}+k_{2})D(k_{1}+k_{2}+p)K_{1},}\\\end{array}
H=\frac{g^{2}}{2V}\mathrm{Tr}P^{i}g_{ij}P^{j}+\frac{V}{4g^{2}}\mathrm{Tr}(F_{ij}+\phi_{ij}\cdot{\bf1})g^{ik}g^{il}(F_{kl}+\phi_{kl}\cdot{\bf1})\,.
\lbrackF_{_{(1)}}]^{-1}=\left(\begin{array}{ccc}{f^{-1}}&{-f^{-1}M_{_{(1)}}R_{_{(1)}}^{-1}}&{0}\\{-R_{_{(1)}}^{-1}M_{_{(1)}}^{^T}f^{-1}}&{R_{_{(1)}}^{-1}\omega_{_{(1)}}R_{_{(1)}}^{-1}}&{-R_{_{(1)}}}\\{0}&{R_{_{(1)}}}&{0}\\\end{array}\right)\;,
\begin{array}{rcl}{iT^{\mathrm{1PI}}}&{=}&{\displaystylei\lambda^{2}v\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}T(k_{i}),}\\{T(k_{i})}&{=}&{\displaystyleT_{a}+T_{b}+T_{c},}\\\end{array}
\pi_{\mu\nu}=-\frac{1}{4}\tau_{\mu\nu}\tau_{\nu}^{\alpha}+\frac{1}{12}\tau\tau_{\mu\nu}+\frac{1}{8}q_{\mu\nu}\tau_{\alpha\beta}\tau^{\alpha\beta}-\frac{1}{24}q_{\mu\nu}\tau^{2}
\begin{array}{rl}{\displaystyle\int_{0}^{\infty}\sum_{n=1}^{-\pin^{2}x}x^{s/2-1}dx}&{=}\\{}&{}\\{}&{\displaystyle=\int_{0}^{\infty}x^{s/2-1}\omega(x)dx}\\{}&{}\\{}&{\displaystyle={\frac{1}{s(s-1)}}+\int_{1}^{\infty}[x^{s/2-1}+x^{(1-s)/2-1}]\omega(x)dx}\\{}&{}\\{}&{\displaystyle=Z(s)=Z(1-s).}\\\end{array}
+\frac{1}{2\mathrm{L}}\sum_{m\in\calZ}(\hat{\Pi}_{1,\pm}^{a}(m+n)\hat{\Pi}_{1,\pm}^{a}(-m)+\frac{e_{\pm}^{2}\mathrm{L}^{2}}{2\pi}A_{1,\pm}^{a}(m+n)A_{1,\pm}^{a}(-m)){\vdots},\,
\partial_{\pm2}\alpha_{AB}=-\frac14\alpha_{AC}\Omega_{\pm2}^{(i)(j)}\gamma_{CB}^{(i)(j)},\\,partial_{\pm2}\beta_{\dot{A}\dot{B}}=-\frac14\beta_{\dot{A}\dot{C}}\Omega_{\pm2}^{(i)(j)}\tilde{\gamma}_{\dot{C}\dot{B}}^{(i)(j)}.
\left({J_{H}^{-1}}^{\prime}\right)^{\frac{1}{2}}\partial_{u}{\frac{1}{{J_{H}^{-1}}^{\prime}}}\partial_{u}\left({J_{H}^{-1}}^{\prime}\right)^{\frac{1}{2}}\cdot\psi=\left(\partial^{2}+{\frac{T}{2}}\right)\cdot\psi=0,
\ddot{x}^{2}\big[\acute{x}{}^{2}x\!\!\dot{}\,\acute{}\,{}^{a}-(\acute{x}x\!\!\dot{}\,\acute{}\,)\acute{x}{}^{a}\big]-\acute{x}^{2}(\ddot{x}x\!\!\dot{}\,\dot{}\,\acute{}\,)\dot{x}{}^{a}=0.
ds^{2}~=~~(1-{\frac{2GM}{r}}+{\frac{GQ^{2}}{r^{2}}})dt^{2}{}~-~(1-{\frac{2GM}{r}}+{\frac{GQ^{2}}{r^{2}}})^{-1}dr^{2}~-~r^{2}d\Omega^{2}~,
\sum_{l,p}\sum_{r=0}^{\infty}r^{-27/4}(r+pl)^{-27/4}e^{4\pi\left(\sqrt{r}+\sqrt{r+pl}\right)-2\pi\tau_{2}(2r+pl)-\pi\alpha^{\prime}\tau_{2}\left(\frac{l^{2}}{R^{2}}+\frac{R^{2}p^{2}}{\alpha^{2}}\right)+\beta{B}\frac{Rp}{\alpha^{\prime}}+\beta{A}\frac{l}{R}}
[\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}]\theta_{L}=\frac{\mathrm{i}}{e}({\overline{\epsilon}}_{1R}\Gamma_{\mu}\epsilon_{2R}\Gamma^{\mu}({\dot{\theta}}_{R}-\frac{1}{e}\mathrm{D}x_{\nu}\Gamma^{\nu}\psi_{L}),
\begin{array}{rcl}{p_{q}}&{=}&{ap_{rr}+ap_{qq}+2dp_{qr}}\\{p_{qq}}&{=}&{\beta+ap_{qrr}+ap_{qqq}+2dp_{qqr}}\\{p_{r}}&{=}&{bp_{rr}+bp_{qq}+2ep_{qr}}\\{p_{qr}}&{=}&{\gamma+bp_{qrr}+bp_{qqq}+2ep_{qqr}}\\{p_{qr}}&{=}&{\gamma+2dp_{qrr}+ap_{rrr}+ap_{qqr}}\\{p_{rr}}&{=}&{\alpha+2ep_{qrr}+bp_{rrr}+bp_{qqr}\.}\\\end{array}
{\bf{S}}=\frac{1}{2}\intd^{D}X\,\sqrt{-G}e^{-2\Phi}\biggl\{-\frac{(D-26)}{3}+{\bf{R}}-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+4\nabla_{\mu}\Phi\nabla^{\mu}\Phi\biggr\}.
S_{i}=\left(\begin{matrix}{1}&{}&{}&{}\\{}&{}&{1}&{}\\{}&{1}&{}&{}\\{}&{}&{}&{\mathbb{I}_{d-3}}\\\end{matrix}\right)\,\quadT=\left(\begin{matrix}{1}&{}&{}&{}\\{-1}&{}&{-1}&{}\\{-1}&{-1}&{}&{}\\{}&{}&{}&{\mathbb{I}_{d-3}}\\\end{matrix}\right)
\begin{aligned}{V(\phi,\chi,\varphi)}&{{}=\frac{1}{2}e^{4\varphi/\sqrt{7}}\mathrm{Tr}(C^{2}+C\mathcal{M}^{-1}C^{T}\mathcal{M})\,,}\\{}&{{}=8(\frac{\deltaW}{\delta\phi})^{2}+8\,e^{-2\phi}(\frac{\deltaW}{\delta\chi})^{2}+8(\frac{\deltaW}{\delta\varphi})^{2}-{\textstyle{{\frac{32}{7}}}}W^{2}\,,}\\\end{aligned}
\frac{\Biggl(\displaystyle{\prod_{k=1}^{2n}}\chi\Bigl(w_{\beta},\frac{n}{2N}(n-2k+\frac{1}{2n})+\frac{1-(-1)^{n+1}}{4}\Bigr)\Biggr)^{2M}}{\displaystyle{\prod_{\alpha=1}^{M}}\Biggl(\chi\Bigl(w_{\beta}-w_{\alpha},0\Bigr)\Bigl(\displaystyle{\prod_{j=1}^{2n-1}}\chi\Bigl(w_{\beta}-w_{\alpha},\frac{n}{N}j\Bigr)\Bigr)^{2}\chi\Bigl(w_{\beta}-w_{\alpha},\frac{2n^{2}}{N}\Bigr)\Biggr)}=1,
\Omega=\left(\begin{array}{ccccc}{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{DV}&{-V}&{\Phi_{+}}&{0}&{1}\\{0}&{0}&{0}&{0}&{0}\\\end{array}\right),\,\,\bar{\Omega}=\left(\begin{array}{ccccc}{0}&{0}&{0}&{0}&{0}\\{-V}&{0}&{0}&{1}&{0}\\{\Phi_{-}}&{0}&{0}&{0}&{0}\\{-\bar{D}V}&{0}&{0}&{0}&{0}\\{\lambda}&{0}&{0}&{0}&{0}\\\end{array}\right),
\Biggl(\phi_{1}(x),{\binom{\phi_{2}(x)}{\phi_{3}(x)}},{\binom{\phi_{4}(x)}{\phi_{5}(x)}}\Biggr)\rightarrow\Biggl({\frac{1}{\phi_{1}(x)}},{\binom{\phi_{4}(x)}{\phi_{5}(x)}},{\binom{\phi_{2}(x)}{\phi_{3}(x)}}\Biggr)
T_{(\alpha\beta)\mu}=\left(\eta_{\mu\alpha}A_{\beta}-\eta_{\mu\beta}A_{\alpha}\right)+\left(\partial_{\alpha}B_{\mu\beta}-\partial_{\beta}B_{\mu\alpha}\right)-\frac{1}{6}\epsilon_{\alpha\beta\sigma\nu}\int(dy)\,f^{\sigma}(x-y)\Theta^{\mu\nu}(y)\.
C_{I,L}(N)g_{YM}^{2L-2}{\frac{\dot{U}^{I}}{U^{3L+2(I-2)}}}=C_{I,L}(N){\frac{\dot{U}^{2}}{g_{YM}^{2}}}\left({\frac{g_{YM}^{2}\dot{U}^{2}}{U^{7}}}\right)^{L}\left({\frac{\dot{U}}{U^{2}}}\right)^{I-2L-2}.
{\widetildeS}={\widehatM}_{P}^{D-d-2}\intd^{D-d}x~f(r)~\sqrt{-{\widehatG}}\left[{\widehatR}-{\widehat\Lambda}\right]+{M}_{P}^{D-2}\intd^{D}x~\sqrt{-{G}}~{R}~.
s^{\nu\left(N\right)}\,=\,\left(-1\right)^{N-1}2\,a\,\left(\partial_{\mu}\,f^{\mu\nu\left(N\right)}\,-\,\delta_{0}^{\nu}\partial_{\rho}f^{0\rho\left(N\right)}\right)\,-\,\theta\,\epsilon^{0\lambda\nu}\partial_{0}\,f_{\lambda}^{\left(N\right)},
\kappa=\displaystyle\frac{2\Gamma(h/2)}{\pi\Gamma(1-h/2)}\left(\displaystyle\frac{\sqrt{\pi}}{2\Gamma\left(\displaystyle\frac{1}{2-h}\right)\Gamma\left(\displaystyle\frac{h}{4-2h}\right)}\right)^{2-h}\,.
F_{R}(M)=\int\frac{db^{4}}{(2\pi)^{4}}\frac{1}{b^{2}+M^{2}+\Pi(b)}-\left[\int\frac{db^{4}}{(2\pi)^{4}}\frac{1}{b^{2}}-M^{2}\int\frac{db^{4}}{(2\pi)^{4}}\frac{1}{b^{4}}\right]
\left\{\mu\frac{\partial}{\partial\mu}+\left(\beta_{p}-d_{p}p\right)\,\frac{\partial}{\partialp}+\gamma_{i}\intd^{4}x\sqrt{-g}\;\Phi_{i}\,\frac{\delta}{\delta\Phi_{i}}\right\}\,\Gamma\left[g_{\alpha\beta},p,\Psi_{i},\mu\right]=0\,.
F_{\mu\nu}=\frac{-\vartheta_{\mu\nu}\left(-2\left(\partial_{0}\partial_{0}\phi\right)\phi+\left(\partial_{0}\phi\right)^{2}+\left(\partial_{1}\phi\right)^{2}+\left(\partial_{2}\phi\right)^{2}+\left(\partial_{3}\phi\right)^{2}\right)}{4\phi^{2}}\;.
ST=\frac{(d-3)\omega_{d-2}}{16\pi}2r_{0}^{\frac{2d-7}{d-3}}\left\{2\prod_{i=1}^{i=n}\left[r_{0}+\sqrt{r_{0}^{2}+q_{i}^{2}}\right]^{r_{i}}\right\}^{-\left(\frac{d-4}{d-3}\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}+{\textstyle\frac{10}{(5!)^{2}}}\hat{\epsilon}\left[\hat{G}_{(5)}\hat{\calH}\hat{C}_{(2)}-\hat{\overline{{G}}}_{(5)}\hat{\calH}\hat{\overline{{C}}}_{(2)}\right]\right\}\,,
M=\pm\;\left(\begin{array}{lll}{\lambda}&{\quad}&{0}\\{\;0}&{\quad}&{\lambda^{-1}}\\\end{array}\right)\\\\\\\mathrm{or}\\\\\M^{\prime}=\pm\;\left(\begin{array}{lll}{\;0}&{\quad}&{\lambda^{-1}}\\{\lambda}&{\quad}&{0}\\\end{array}\right)\,
\zeta(s)=\frac{L_{x}\,L_{y}}{c^{2s}}\int\frac{d^{2}{\bfk}}{(2\pi)^{2}}\left\{2\sum_{n=1}^{\infty}\left[{\bfk}^{2}+\left(\frac{n\pi}{a}\right)^{2}\right]^{-s}+\left({\bfk}^{2}+\mu^{2}\right)^{-s}\right\},
<\varphi,\beta|{\calN}_{Q}[V(\phi_{x})]|\varphi,\beta>=\int_{-\infty}^{\infty}{\frac{d\alpha}{2\sqrt{\pi}}}e^{-{\frac{\alpha^{2}}{4}}}V({\frac{\alpha}{2}}\sqrt{J_{1}(g)-I_{1}(Q^{2})}+\varphi_{x})
v_{2}^{(1)}=-2w_{1}^{(0)}v_{0}^{(2)}+\frac{1}{w_{1}^{(0)}}\left(v_{1}^{(1)}+w_{2}^{(0)}\right)\left(v_{1}^{(1)}+2w_{2}^{(0)}\right)-\frac{3}{2}w_{3}^{(0)}
\chi_{1,i}^{(2,2k+3)}(q)=\chi_{1,2k+3-i}^{(2,2k+3)}(q)=\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k}^{2}+n_{i}+\cdots+n_{k}}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-1}-n_{k}}(q)_{n_{k}}},
A_{n}^{*i}=\mathrm{e}^{-i{\frac{\pi}{4}}}\Omega_{-}^{-{\frac{1}{2}}}\bar{\psi}_{n}^{i\dot{1}},\qquadA_{n}^{i}=\mathrm{e}^{i{\frac{\pi}{4}}}\Omega_{-}^{{\frac{1}{2}}}\psi_{n2}^{i}.
\kappa=\frac{k}{m_{f}}\;\;;\;\;L^{2}=\frac{m_{i}^{2}}{m_{f}^{2}}=\frac{\left[T_{i}^{2}-T_{c}^{2}\right]}{\left[T_{c}^{2}-T_{f}^{2}\right]}\;\;;\;\;\tau=m_{f}t\;\;;\;\;\vec{x}=m_{f}\vec{r}
E_{1\,non-disp.}=(\varepsilon-1)^{2}\Bigl(-\frac{23}{128}\frac{V}{\pi^{2}\lambda^{4}}+\frac{23}{384}\frac{S}{\pi^{2}\lambda^{3}}+\frac{23}{1536}\frac{1}{\pia}\Bigr)\,\,+\,\mathrm{const}
d(d\omega)=\frac{1}{r!}\left(\frac{\partial^{2}}{\partialx^{\rho}x^{\nu}}\omega_{\mu_{1}\mu_{2}\cdots\mu_{r}}\right)dx^{\rho}\wedgedx^{\nu}\wedgedx^{\mu_{1}}\wedge\cdots\wedgedx^{\mu_{r}},
\lambda={\frac{2^{2/3}(-1+2\sqrt{7})(1+(-1+\sqrt{7})e^{2h}+(2+\sqrt{7})e^{4h})\Lambda(Z,T)}{81g_{c}e^{2h}(1+e^{2h})}}\epsilon^{-2/3}+{\calO}(\epsilon^{-1/3})\,
\left(\begin{array}{c|c}{-\bar{D}^{2}\delta_{\mu\sigma}+\bar{D}_{\mu}\bar{D}_{\sigma}-2\bar{F}_{\mu\sigma}}&{-i\bar{D}_{\mu}}\\{i\bar{D}_{\sigma}}&{\beta}\\\end{array}\right)\left(\begin{array}{cc}{G_{\sigma\nu}}&{\tilde{G}_{\sigma}}\\{G_{\nu}}&{G}\\\end{array}\right)=g^{2}\left(\begin{array}{cc}{\delta_{\mu\nu}}&{0}\\{0}&{1}\\\end{array}\right)
(D\theta)^{I}=\left(d+{\frac{1}{4}}(\omega^{ab}\gamma_{ab}+\omega^{a^{\prime}b^{\prime}}\gamma_{a^{\prime}b^{\prime}})\right)\theta^{I}-{\frac{1}{2}}i\epsilon^{IJ}(e^{a}\gamma_{a}+ie^{a^{\prime}}\gamma_{a^{\prime}})\theta^{J}
\begin{array}{rclrcl}{F_{1}}&{=}&{\bar{U}dV-Vd\bar{U},}&{F_{3}}&{=}&{\bar{U}H+V\bar{H},}\\{F_{5}}&{=}&{dA_{4}+\mathrm{i}(B\bar{H}-\bar{B}H),}&{H}&{=}&{dB,}\\\end{array}
L=\dot{q}_{i}^{a}p_{i}^{a}-\frac{1}{2}g^{2}p_{i}^{a}p_{i}^{a}-\frac{1}{2}\omega^{2}q_{i}^{a}q_{i}^{a}-\lambda\epsilon^{ab}p_{i}^{a}q_{i}^{b}=\dot{q}_{i}^{a}p_{i}^{a}-H_{0}-\lambda\phi\\\,
\left(v^{4}-\frac{u_{2}}{2}v^{2}-\left(\frac{u_{4}}{4}-\frac{u_{2}^{2}}{8}\right)\right)^{2}-\Lambda_{N=2}^{4}v^{2}(v^{2}-m_{1}^{2})=(v^{2}-p_{1}^{2})^{2}T(v^{2}).
W_{LL}=\sum_{k=2}^{N_{c}}\frac{g_{k}\mathrm{Tr}{\Phi_{cl}^{k}}}{k}\pm2{\Lambda_{L}}^{5/2}\sqrt{\lambdaM+m_{Q}}=\sum_{k=2}^{N_{c}}\frac{g_{k}\mathrm{Tr}{\Phi_{cl}^{k}}}{k}\pm2g_{N_{c}}{\Lambda}^{N_{c}-\frac{1}{2}}\sqrt{\lambdaM+m_{Q}}.
Q^{3}|\Phi^{\prime}\rangle=\intdy^{-}dy_{\bot}\left\{f^{3ab}A_{\bot}^{a}(y_{\bot},y^{-})\Pi_{\bot}^{b}(y_{\bot},y^{-})+\rho_{m}^{3}(y_{\bot},y^{-})\right\}|\Phi^{\prime}\rangle=0\,.
\overrightarrow{\xi}=\frac{1}{\textsf{P}_{0}}\biggl((ch(lm)P_{0}+sh(lm)m)\overrightarrow{\bigtriangledown}+m^{-1}sh(lm)\overrightarrow{P}(\overrightarrow{P}.\overrightarrow{\bigtriangledown})\biggr)
\dot{H}+H^{2}+{\frac{2}{3}}\sigma^{2}+{\textstyle{\frac{1}{6}}}\kappa^{2}(\rho+3p)-{\frac{1}{3}}\Lambda=-{\frac{1}{36}}\left({\frac{\widetilde{\kappa}}{\kappa}}\right)^{4}\left[{\kappa}^{4}\rho(2\rho+3p)+12{\calU}\right],
f^{2}g^{2}{\frac{\partial^{2}}{\partialr^{*2}}}t+f\{fg^{\prime\prime}+f^{\prime}g^{\prime}\}{\frac{\partial}{\partialr^{*}}}t-[\sqrt2ff^{\prime}-2f(1-f)]t-{\frac{\partial^{2}}{\partialt^{2}}}t=0,
{\calS_{D}^{\mp}}={\calS_{D}}+\int_{M}\left[\pm{\frac{i}{4}}\psi^{A}\epsilon_{ABCD}e^{C}\wedgee^{D}\wedgeT^{B}{\mp}{\frac{i}{2(3!)}}d(\psi^{A}\epsilon_{ABCD}e^{B}\wedgee^{C}\wedgee^{D})\right],
a(p)c(-p)\left\{\Gamma_{S}^{11}(p)+\left[\frac{d(-p)}{c(-p)}+\frac{b(p)}{a(p)}\right]\Gamma_{S}^{12}(p)+\frac{b(p)}{a(p)}\cdot\frac{d(-p)}{c(-p)}\Gamma_{S}^{22}(p)\right\}=0
L_{B}=-{\frac{1}{2\cdot3!}}\,H^{2}{}_{\mu\nu\rho}(B)+{\frac{m}{4!}}\,\epsilon^{\mu\nu\rho\sigma}\,B_{\mu\nu}\partial_{\,[\,\rho}\,A_{\sigma\,]}-{\frac{1}{4}}F^{2}{}_{\mu\nu}(A)
T_{\nu}^{ab}(\eta,z)=z\left\{\frac{\bar{J}_{\nu}^{(a)2}(z)}{\bar{J}_{\nu}^{(b)2}(\etaz)}\left[A_{b}^{2}+B_{b}^{2}(\eta^{2}z^{2}-\nu^{2})\right]-A_{a}^{2}-B_{a}^{2}(z^{2}-\nu^{2})\right\}^{-1}.
\chi_{2,2}^{Vir~(4)}(q)~\left[\chi_{2,1}^{Vir~(5)}(q)-\chi_{3,1}^{Vir~(5)}(q)\right]=\chi_{2,1}^{Vir~(4)}(q)~\left[\chi_{1,1}^{Vir~(5)}(q)-\chi_{4,1}^{Vir~(5)}(q)\right],
\left(\begin{array}{ccccccc}{0}&{a}&{a}&{0}&{0}&{...}&{0}\\{-a}&{0}&{0}&{b}&{c}&{...}&{d}\\{a}&{0}&{0}&{-b}&{-c}&{...}&{-d}\\{0}&{b}&{b}&{0}&{e}&{...}&{f}\\{0}&{c}&{c}&{-e}&{0}&{...}&{g}\\{...}&{...}&{...}&{...}&{...}&{...}&{...}\\{0}&{d}&{d}&{-f}&{-g}&{...}&{0}\\\end{array}\right)
D_{\alpha}^{I}=D_{\alpha}^{i}u_{i}^{I}\;,\quad\bar{D}_{J}^{\dot{\alpha}}=\bar{D}_{i}^{\dot{\alpha}}u_{J}^{i}\;,\quad1\leqI\leqp\;,\N-q+1\leqJ\leqN\;.
\{q_{0},p_{n}\}^{\ast}=\{q_{0},\Phi_{n}\}^{\ast}=-\frac{3\lambda[2q_{0}q_{-n}+\int_{-L/2}^{L/2}dx\varphi^{2}\;e^{-ik_{n}x}]}{[3\lambda(q_{0}/\sqrt{L})^{2}-m^{2}]L+3\lambda\int_{-L/2}^{L/2}dx\varphi^{2}}
\eta^{\sigma\rho}(\bot_{\nu}^{\mu}{\calR}_{\rho}{^{\nu}}{_{\sigma\lambda}}-\bot_{\lambda\nu}{\calR}_{\rho}{^{\nu}}{_{\sigma}}{^{\mu}})\acute{\xi}_{\mu}\xi^{\lambda}=\widetilde{\nabla}_{\nu}{\calJ}_{3}^{\nu}+(\widetilde{\nabla}^{\nu}\bot_{\lambda}^{\mu}+2K_{\lambda}{^{\nu\mu}})\widetilde{\nabla}_{\nu}(\acute{\xi}_{\mu}\xi^{\lambda}),
\deltaI_{\mathrm{F}}=-\int_{\Sigma}\;{\frac{d^{2}\xi}{2\pi\alpha^{\prime}}}\\deltaX^{\mu}\;\partial_{a}\partial^{a}X_{\mu}+\int_{\partial\Sigma}\;{\frac{d\xi^{a}}{2\pi\alpha^{\prime}}}\\deltaX^{\mu}\varepsilon_{ab}\partial^{b}X_{\mu}
W^{({D})}({\bfr})={\mathcalF}_{({D})}^{-1}\left\{\widetilde{W}({\bfk})\right\}=\int\frac{d^{{D}}k}{(2\pi)^{{D}}}\;e^{i{\bfk}\cdot{\bfr}}\,\widetilde{W}({\bfk})\;,
\begin{array}{lcl}{{\mathcalL}_{4}}&{=}&{p^{4}+u_{2}\starp^{2}+u_{3}\starp+u_{4}}\\{}&{=}&{p^{4}+u_{2}p^{2}+(u_{3}-2\thetau_{2}^{\prime})p+u_{4}-\thetau_{3}^{\prime}+\theta^{2}u_{2}^{\prime\prime}.}\\\end{array}
\lambda_{L}=\left(\textstyle{\frac{6}{\mu}E_{0}+\frac{1}{2}N,\,{-\frac{1}{2}j_{z}}\,,\,{-\frac{1}{2}(j_{u}+2j_{v}+j_{w})}\,,\,{-\frac{1}{2}(j_{u}+j_{w})}\,,\,{-\frac{1}{2}(j_{u}-j_{w})}}\right)\,.
u_{\underline{{m}}}^{(0)}\delta_{AB}=v_{\underline{{\mu}}}^{~A}\tilde{\sigma}_{\underline{{m}}}^{\underline{{\mu}}\underline{{\nu}}}v_{\underline{{\nu}}}^{~B},\qquadu_{\underline{{m}}}^{i}\Gamma_{AB}^{i}=v_{\underline{{\mu}}}^{~A}\tilde{\sigma}_{\underline{{m}}}^{\underline{{\mu}}\underline{{\nu}}}v_{\underline{{\mu}}}^{~B}.
g_{\mu\nu}=\left(\begin{array}{cc}{g_{00}}&{g_{0j}}\\{g_{i0}}&{g_{ij}}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{av_{j}}\\{av_{i}}&{{\bar{g}}_{ij}+av_{i}v_{j}}\\\end{array}\right),
p_{L}^{\primei}v_{L}^{i}-p_{R}^{\primei}v_{R}^{i}+p_{L}^{t}v_{L}^{t}-p_{R}^{t}v_{R}^{t}-\frac{1}{2}\ell((v_{L}^{i})^{2}-(v_{R}^{i})^{2}+(v_{L}^{t})^{2}-(v_{R}^{t})^{2})=0\\\bmod1.
Z(M,l)=\mathrm{e}^{i\phi_{\mathrm{fr}}}\sum_{\alpha_{1},\cdots,\alpha_{L}\in{\calF}_{l}}Z_{\alpha_{1},\cdots,\alpha_{L}}({\calL})\,{\calU}_{\alpha_{1}\rho}^{(p_{1},q_{1})}\cdots{\calU}_{\alpha_{L}\rho}^{(p_{L},q_{L})}.
\partial_{[m}G_{nkl]}+\frac{i}{2}e^{\phi}\partial_{[m}CG_{nkl]}=\frac{i}{2}e^{\phi}\partial_{[m}\tauG_{nkl]}^{\ast},\qquad\partial_{[m}F_{nklpq]}=\frac{5}{48}iG_{[mnk}G_{lpq]}^{\ast}
\langle{\calO}({\bfX}){\calO}({\bfY})\rangle={\frac{2(h_{L}+h_{R}-1)^{2}l^{2(h_{L}+h_{R})-3}}{\pi\left\vert{\bfX}-{\bfY}\right\vert^{2(h_{L}+h_{R})}}}.
\overline{{\Gamma}}(A)=-\int_{0}^{\infty}{\frac{ds}{s^{3}}}e^{-m^{2}s}\intd^{4}x\left[1-\left.\left(1+s{\frac{\partial}{\partials}}+{\frac{1}{2}}s^{2}{\frac{\partial^{2}}{\partials^{2}}}\right)\right|_{s=0}\right]\mathrm{tr}\left(s^{2}<xs|x>\right)
\partial_{J}^{I}u_{i}^{K}=\delta_{J}^{K}u_{i}^{I}-{\frac{1}{N}}\delta_{J}^{I}u_{i}^{K}\;,\qquad\partial_{J}^{I}u_{K}^{i}=-\delta_{K}^{I}u_{J}^{i}+{\frac{1}{N}}\delta_{J}^{I}u_{K}^{i}\;.
\begin{array}{rcl}{{\calH}}&{=}&{(\Lambda^{T})^{-1}\left(\begin{array}{c}{P_{(p_{1},p_{2})}}\\{0}\\\end{array}\right)\epsilon_{3}}\\{}&{=}&{\left(\begin{array}{c}{P^{(1)}}\\{P^{(2)}}\\\end{array}\right)\epsilon_{3},}\\\end{array}
S=\sum_{n=0}^{\infty}\intd^{2}x\[\dot{\varphi}_{(n)}{\pi}_{(n)}-\frac{(-1)^{n}}{2}({\pi}_{(n)}^{2}+\varphi_{(n)}^{2})-{\Lambda}_{(n+1)}{\Psi}_{(n+1)}]\,
S_{l.c.}\!=\!\frac{2i}{(\alpha^{\prime}){}^{1/2}}\!\int\!e\!\!\left[\!\left(\!1\!+\!\frac{2}{c(\alpha^{\prime}){}^{1/2}}\!D_{[+2]}x^{[+2]}\!\right)\!\!\tilde{D}_{[-2]}\!\theta^{-}\!\cdot\!\theta^{-}\!+\!\left(\!1\!+\!\frac{2}{c(\alpha^{\prime}){}^{1/2}}\!D_{[-2]}x^{[-2]}\!\right)\!\!D_{[+2]}\!\theta^{+}\!\cdot\!\theta^{+}\!\right]
Y^{m}=\left(\begin{array}{ccccc}{0}&{Y_{12}^{m}}&{Y_{13}^{m}}&{...}&{Y_{1K}}\\{Y_{12}^{\daggerm}}&{0}&{Y_{23}^{m}}&{...}&{Y_{2K}^{m}}\\{...}&{...}&{...}&{...}&{...}\\{Y_{1K}^{\daggerm}}&{Y_{2K}^{\daggerm}}&{...}&{Y_{K,K-1}^{\daggerm}}&{0}\\\end{array}\right)
\int\!\!d^{3}x\,\deltaA^{i}({\boldsymbol{x}};\phi)\deltaA^{i}({\boldsymbol{x}};\phi)=\int\!\!d^{3}x\,\deltaA^{j}({\boldsymbol{x}})M^{jk}(-i{\boldsymbol{\nabla}})\deltaA^{k}({\boldsymbol{x}}),
|T_{nn}(K)-T_{B}-T_{nn}^{(2)}(K)|\leqC_{1}(M)\int....\intd\vec{x}d\vec{x^{\prime}}\frac{\tilde{V}(|\vec{x}|)\tilde{V}(|\vec{x^{\prime}}|)}{|\vec{x^{\prime}}|}e^{+\frac{|n|}{R}|\vec{x}-\vec{x^{\prime}}|},
\sum_{i=1}^{r}\Lambda(H_{i})\Lambda^{\prime}(H^{i})=(\Lambda_{0},\Lambda_{0}^{\prime})\,,~~~~\forall\Lambda=(\Lambda_{0},\kappa,\sigma)\,,\,\Lambda^{\prime}=(\Lambda_{0}^{\prime},\kappa^{\prime},\sigma^{\prime})\in{\calH}^{*}
\begin{array}{c}{\left\{Q_{\alpha},Q_{\beta}\right\}=\left(S_{A}\right)_{\alpha\beta}}\\{\left(S_{A}\right)_{\alpha\beta}=\left(C\gamma^{m}\right)_{\alpha\beta}\,\,P_{m}+\left(C\gamma^{m_{1}m_{2}}\right)_{\alpha\beta}\,\,\,Z_{m_{1}m_{2}}+\left(C\gamma^{m_{1}\cdotsm_{5}}\right)_{\alpha\beta}\,\,\,X_{m_{1}\cdotsm_{5}}.}\\\end{array}
\Pi_{\tau}^{\underline{{m}}}=e_{\tau}^{++}u_{\underline{{m}}}^{--};\\\\\,Pi_{q}^{\underline{{m}}}=e_{q}^{++}u_{\underline{{m}}}^{--}\Longrightarrow\Pi_{\tau}^{\underline{{m}}}u_{\underline{{m}}}^{--}=0;\\\\\\Pi_{q}^{\underline{{m}}}u_{\underline{{m}}}^{--}=0
N_{gen.}(t,w)=a(t)\left[\frac{w}{2\pi}-\frac{1}{2\pi(1-\mu)}\frac{b(t)/a(t)}{2\pi(1-\mu)}+\frac{2}{e^{2(1-\mu)w}-1}\left(w-\frac{b(t)/a(t)}{2\pi(1-\mu)}\right)\right].
+C_{\xi}^{*k}C_{\eta}^{k}e_{n+1+k,M}\otimese_{kN}(\frac{q^{n+1}}{1-q^{2}})^{k}q^{k^{2}}\sqrt{\frac{[n+1]_{q^{2}}!}{[n+1+k]_{q^{2}}![k]_{q^{2}}!}}(1-q^{2(n+k+1)}))
{\frac{\partialL}{\partialq^{i}}}{\calF}\!L^{*}({\frac{\partialG}{\partialp_{i}}})+\dot{q}^{i}{\calF}\!L^{*}({\frac{\partialG}{\partialq^{i}}})+{\calF}\!L^{*}({\frac{\partialG}{\partialt}})=r^{\mu}\chi_{\mu}.
\omega=-\displaystyle\frac{1}{C_{K}}\left(R_{1}\,dR_{2}\wedgedL+R_{2}\,dL\wedgedR_{1}+\displaystyle\frac{(\lambda-8E)L+4\lambdaL^{3}}{(\lambda-8E)+8\lambdaL^{2}}dR_{1}\wedgedR_{2}\right)+\sum_{i=1}^{3}d\bar{C}_{i}\wedgedC^{i}~,
f^{\stackrel{\underbrace{\mathrm{\scriptsize~1\ldots~1}}}{m_{1}}\stackrel{\underbrace{\mathrm{\scriptsize~2\ldots~2}}}{m_{2}}\cdots\stackrel{\underbrace{\mathrm{\scriptsize~N-1\ldots~N-1}}}{m_{N-1}}}(u)\;,\qquad{m_{1}\geqm_{2}\geq\ldots\geqm_{N-1}}\;.
\tilde{n}^{A}=\left\{{\frac{c\,\dot{\calR}}{n^{2}\sqrt{1-{\frac{c^{2}}{n^{2}}}\dot{\calR}^{2}}}},0,0,0,0,{\frac{1}{c\sqrt{1-{\frac{c^{2}}{n^{2}}}\dot{\calR}^{2}}}}\right\}\,.
P^{y^{\prime}z^{\prime}}~T^{A,b}~(P^{y^{\prime}z^{\prime}})^{-1}=T^{A,b}~,~P^{y^{\prime}z^{\prime}}~T^{A,\hat{b}}~(P^{y^{\prime}z^{\prime}})^{-1}=-T^{A,\hat{b}}~.~\,
{\cal{M}}=\left(\begin{array}{cc}{S^{\mathrm{T}}S}&{S^{\mathrm{T}}R}\\{R^{\mathrm{T}}S}&{[(S^{\mathrm{T}}S)^{-1}+R^{\mathrm{T}}R]}\\\end{array}\right)\longrightarrow\left(\begin{array}{cc}{(S^{\mathrm{T}}S)_{AB}}&{{(S^{\mathrm{T}}R)_{A}}^{B}}\\{{(R^{\mathrm{T}}S)^{A}}_{B}}&{((S^{\mathrm{T}}S)^{-1})^{AB}+(R^{\mathrm{T}}R)^{AB}}\\\end{array}\right)
e_{i}^{(k)}=\frac{1}{h_{k}}\,\frac{\partialx_{i}}{\partialX_{k}}\,,\quadh_{k}=\left(\sum_{i=1}^{3}\left(\frac{\partialx_{i}}{\partialX_{k}}\right)^{2}\right)^{1/2},\quadh\equiv\prod_{i=1}^{3}h_{i},
\langleN\rangle_{T}\simk\pm\frac{4km}{T}e^{-\frac{m}{T}}\sum_{n=0}^{\infty}\frac{1}{(2n+2)!}\left(\frac{2k}{2^{\frac{1}{k}}\lambdaT}\right)^{2n}\beta\left(n+1+\frac{n}{k},n+1-\frac{n}{k}\right)
{\calJ}_{i}=\frac{1}{2}(z^{\alpha}\sigma_{\beta\alpha}^{i}\frac{\partial}{\partialz^{\beta}}-{\bar{z}}^{\alpha}\sigma_{\alpha\beta}^{i}\frac{\partial}{\partial{\bar{z}}^{\beta}}),\;\;{\calI}_{i}=-4\frac{\hbar^{2}}{2\mu}\sigma_{\alpha\beta}^{i}\frac{\partial}{\partialz^{\beta}}\frac{\partial}{\partial{\bar{z}}^{\alpha}}+\frac{\mu\omega^{2}}{2}\sigma_{\alpha\beta}^{i}z^{\alpha}{\bar{z}}^{\beta},
M_{Susy}=2.3M_{Pl}\sqrt{|h|}\bigg(\frac{1}{r_{v}}\Big(\frac{V_{1}}{d^{4}}\Big)^{1/3}e^{-\frac{1}{r_{v}}\big(\frac{3V_{1}^{4}}{2d}\big)^{1/3}}\bigg)^{1/4}\;.
X_{L}^{25}=\frac{X_{0}^{25}}{2}+\frac{a}{2}+\alpha^{\prime}P^{25}(\tau+\sigma)+i\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{l\neq0}\frac{1}{l}\alpha_{l}^{25}exp\bigg(-i2l(\tau+\sigma)\bigg)
(e^{-2\hat{\phi}})_{\mathrm{fix}}={\frac{|\hat{q}_{1}\hat{p}^{0}-\hat{q}_{0}|\hat{p}^{1}}{(\hat{p}^{1})^{2}+(\hat{p}^{0})^{2}}}\,\qquad(a)_{\mathrm{fix}}={\frac{-\hat{q}_{1}\hat{p}^{1}-\hat{q}_{0}\hat{p}^{0}}{(\hat{p}^{1})^{2}+(\hat{p}^{0})^{2}}}\.