harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{Tr}V_{1\nu}^{~\mu}(\nabla)=2\kappa^{2}\intd^{4}x\left[R_{\,\cdot\,\gamma\beta\sigma}^{\alpha}{\calE}^{\gamma\beta}-{\calE}^{\beta\gamma}\delta_{\sigma}^{\alpha}\nabla_{\beta}\nabla_{\gamma}\right](\Box\delta_{\alpha}^{\sigma}+R_{\alpha}^{\sigma})^{-2}\delta(x,y)|_{y=x}.
\Sigma_{1+2}=\intd^{3}x\frac{1}{2}\biggl\{A^{\mu}[M_{\mu\nu}]A^{\nu}-\varphi\square\varphi+\varphi\left[\epsilon_{\mu\alpha\nu}v^{\mu}\partial^{\alpha}\right]A^{\nu}+A^{\mu}\left[\epsilon_{\nu\alpha\mu}v^{\nu}\partial^{\alpha}\right]\varphi\biggr\},
\hat{H}_{c}^{r}\,=\,{\frac{1}{2}}\int\left[\left(-i{\frac{\delta}{\delta\phi_{1}}}\phi_{2}+i{\frac{\delta}{\delta\phi_{2}}}\phi_{1}\right)^{2}\,-\,{\frac{1}{\phi_{1}^{2}+\phi_{2}^{2}}}\left(\phi_{1}\partial_{i}\partial_{i}\phi_{1}+\phi_{2}\partial_{i}\partial_{i}\phi_{2}\right)\right]dx\,.
A^{g}~=~\int_{M_{g}}\langle\int\prod_{i=1}^{2g}d^{2}x_{i}V_{h}(x_{1})V_{h}(x_{2})\prod_{i=3}^{2g}V_{T}(x_{i})\prod_{k=1}^{3g-3}|(\mu_{k}b)|^{2}e^{\varphi}G^{-}(z_{k})e^{\widetilde\varphi}{\widetildeG}^{-}(\bar{z}_{k}^{\prime}){\rangle}_{g}~.
f=\left(\begin{matrix}{s_{1}}\\{s_{2}}\\{s_{3}}\\\end{matrix}\right),\qquadg=\left(\begin{matrix}{0}&{s_{3}}&{-s_{2}}\\{-s_{3}}&{0}&{s_{1}}\\{s_{2}}&{-s_{1}}&{0}\\\end{matrix}\right),\qquadh=\left(\begin{matrix}{s_{1}}&{s_{2}}&{s_{3}}\\\end{matrix}\right)
\langle\bar{\lambda}^{a}\lambda^{a}\rangle\\sim\{\frac{1}{2}}\left({\frac{N_{c}g^{2}(M^{2})}{8\pi^{2}}}\right)^{2}\,{\frac{32\pi^{2}}{g^{4}(M^{2})}}\,\Lambda_{1}^{3}\,
\Pi_{55}^{AA^{\prime}}(0)=-\frac{1}{8}g^{2}\delta^{AA^{\prime}}C_{2}({\calG})\mu^{4-d}\int\frac{d^{d}q}{{(2\pi)^{d}}}\sum_{l=-\infty}^{+\infty}\left(4(d-1)\frac{q^{2}-\frac{l^{2}}{R^{2}}}{\left(q^{2}+\frac{l^{2}}{R^{2}}\right)^{2}}\right).
2i{\frac{(12)(13)(23)}{x_{12}^{4}x_{34}^{4}}}\;\left[{\frac{c}{s}}+\beta_{0}{\frac{1+{\frac{\beta_{1}}{\beta_{0}}}U+{\frac{\beta_{2}}{\beta_{0}}}U^{2}}{1+{\frac{1+s-t}{s}}U+{\frac{1}{s}}U^{2}}}\right]\;X_{2}^{\mu}
\mathrm{F.T.~of~}\left[\tilde{\phi}(\left|{\bfr}\right|)e^{\pmi\ell\theta}\right]=\intd^{2}{\bfr}\tilde{\phi}(\left|{\bfr}\right|)e^{i{\bfr}.{\bfp}\pmi\ell\theta}=\phi(\left|{\bfp}\right|)e^{\pmi\ell\theta_{p}},
\theta_{m,20}(q)=\sum_{p=-\infty}^{+\infty}~\sum_{\ell=0}^{9}q^{20(10p+\ell)^{2}+(10p+\ell)m+\frac{m^{2}}{80}}=\sum_{\ell=0}^{9}\theta_{10m+400\ell,2000}(q),
(\psi_{i^{\prime}}^{a}\otimes\psi_{k^{\prime}}^{b},\psi_{i}^{a}\otimes\psi_{k}^{b}):=(\psi_{i^{\prime}}^{a},\psi_{i}^{a})\cdot(\psi_{k^{\prime}}^{b},\psi_{k}^{b})
(\not\!p_{2}-m)_{\rho^{\prime}\alpha}\bar{\eta}_{\rho\alpha}(p_{2},p_{3}-p_{2})(\not\!p_{3}-m)_{\rho\alpha^{\prime}}=i{\frac{\lambda_{0}^{2}}{4\pi^{2}}}\intdp_{1}\bar{\eta}_{\alpha\alpha}(p_{1},p_{3}-p_{2})\delta_{\alpha^{\prime}\rho^{\prime}}.
\delta_{\mu\nu}\partial_{\alpha}h_{\mu\alpha}(\partial^{2})^{\ell}\partial_{\beta}h_{\nu\beta}={\frac{1}{4}}(\epsilon_{\mu\nu\rho\sigma}\partial^{\nu}B^{\rho\sigma})(\partial^{2})^{\ell}(\epsilon_{\mu\alpha\beta\gamma}\partial^{\alpha}B^{\beta\gamma}).
\delta[Z_{D}^{\prime}(0)]=\gamma[\deltac_{0})]+\mathrm{Finite}_{\epsilon\to0}[\intd^{2}x\,2\delta\sigma(x)K_{D}(x,x;\epsilon)-\sum_{\nu:\,\lambda_{\nu}=0}\int_{D}d^{2}x\,2\delta\sigma(x)|\Psi_{\nu}(x)|^{2}]\,.
Z\left({\calA},G,p,t_{n,l}^{(A)}\right)=\sum_{\{r_{n,l}^{(A)}\}}Z\left({\calA},G,p,r_{n,l}^{(A)}\right)\prod_{A=1}^{p}\prod_{l=1}^{\infty}\prod_{n=-\infty}^{+\infty}\left[t_{n,l}^{(A)}\right]^{r_{n,l}^{(A)}}~.
(C_{ij}^{n}(q))=(C_{ji}^{n}(q))=\frac{n}{[n]_{q}}\,\frac{[2]_{q}^{2}}{q^{2n}+1+q^{-2n}}\,\left(\begin{array}{cc}{q^{n}+q^{-n}}&{(-1)^{n}}\\{(-1)^{n}}&{q^{n}+q^{-n}}\\\end{array}\right)\,;
\tilde{\pi}^{ij}=2\gamma^{1/2}\tilde{\gamma}^{-1/3}e^{-\frac{3}{2}(\phi-\tilde{\phi})}\left[\frac{\partialH}{\partial\tilde{h}_{ij}}-\frac{1}{3}\frac{\partialH}{\partial\tilde{h}_{kl}}\tilde{h}_{kl}\tilde{h}_{ij}\right]~~~,
{F_{\mu\nu,\lambda\sigma}}^{\alpha\beta}=\frac{1}{2}\delta_{(\mu}^{(\alpha}\,\,\varepsilon_{\nu)(\kappa}\,\,\delta_{\lambda)}^{\beta)}\qquad\mathrm{and}\qquad{F^{\mathrm{MN,KL}}}_{\mathrm{RS}}=\frac{1}{2}
{\calE}=\frac{\sqrt{1-\lambda_{9}^{2}}}{\lambda_{9}}\left(u-iv\right)=\frac{(1-\lambda_{9}^{2}){\calE}_{0}}{1-\lambda_{9}^{2}Re{\calE}_{0}},\;\;\;z=\frac{i}{\left(1-\lambda_{9}^{2}{\calE}_{0}\right)}.
\begin{array}{rcl}{v_{\mu}A_{\nu}^{a}=i\,\mathrm{sgn}(k){\epsilon}_{\nu\mu\rho}{\partial}^{\rho}c^{a}}&{,}&{v_{\mu}c^{a}=0\,,}\\{v_{\mu}{\bar{c}^{a}}=A_{\mu}^{a}}&{,}&{v_{\mu}B^{a}=-D_{\mu}c^{a}\,,}\\\end{array}
\mu_{11}=-{\frac{1}{6\pi^{2}}}{\frac{\pi^{D/2}}{2^{D}}}\left[{\frac{(D-3)(D-1)}{D}}\Gamma\left(2-{\frac{D}{2}}\right)\zeta(2-D)+\pi^{3/2-D}(D+1)\Gamma\left({\frac{D+1}{2}}\right)\zeta(D+1)\right]\,{\frac{\hbarL^{D-1}}{a^{D}}},
\sigma=\frac{1}{3!}ie^{2}\varepsilon^{\mu\nu\rho}\int\frac{d^{3}p}{(2\pi)^{3}}\mathrm{Tr}\left[\partial_{\mu}S^{-1}S\partial_{\nu}S^{-1}S\partial_{\rho}S^{-1}S\right]\,
\left(\hat{S}^{3}+\hat{S}+\frac{z}{2}\right)\left(\hat{S}^{3}-\hat{S}+\frac{z}{2}\right)=0~,\quad\hat{S}=\frac{S}{\hat{\Lambda}^{3}}~,\quadz=\hat{\Lambda}^{3}\frac{18\lambda^{2}}{m^{3}}~.
Z_{0(m)}^{(n)}(\{k_{s},u_{s},v_{s}\},L)=\frac{\Theta^{5}[l_{1},l_{2}](0|\omega^{(o)})}{\Theta^{5}[\{0\},\{1/2\}](0|\omega^{(o)})}\prod_{(k)}\prod_{m=1}^{\infty}\frac{(1-k^{m-1/2})^{10}}{(1-k^{m})^{10}}
{\calL}=-\frac{1}{2}[(\partial_{\mu}\bbox{\pi})^{2}+(\partial_{\mu}\tilde{\sigma})^{2}+m_{\pi}^{2}(\bbox{\pi}^{2}+3\tilde{\sigma}^{2})]-g_{3}(\tilde{\sigma}\bbox{\pi}\cdot\bbox{\pi}+\tilde{\sigma}^{3})-g_{4}[(\bbox{\pi}\cdot\bbox{\pi})^{2}+\tilde{\sigma}^{4}+2\bbox{\pi}^{2}\tilde{\sigma}^{2}],
\bigl\{T_{A}^{a},T_{B}^{(1)b}\bigr\}_{(A,\pi)}+\bigl\{T_{A}^{(1)a},T_{B}^{b}\bigr\}_{(A,\pi)}+\bigl\{T_{A}^{(1)a},T_{B}^{(2)b}\bigr\}_{(\eta)}+\bigl\{T_{A}^{(2)a},T_{B}^{(1)b}\bigr\}_{(\eta)}=0\,.
S_{0}=\sum_{l}{\frac{1}{2\sigma_{l}^{2}}}\phi_{l}\phi_{-l}+\sum_{l}{\frac{1}{2\tau_{l}^{2}}}\tilde{f}_{l}\tilde{f}_{-l}+\sum_{r}{\frac{1}{g_{r}}}\bar{\psi}_{r}\psi_{r}\,.
\partial_{\mu}q^{u}\to{D}_{\mu}q^{u}=\partial_{\mu}q^{u}-k_{I}^{u}A_{\mu}^{I}\,\qquad\partial_{\mu}t^{i}\to{D}_{\mu}t^{i}=\partial_{\mu}t^{i}-k_{I}^{i}A_{\mu}^{I}\.
{\calL}=\frac{1}{2}\left\{\partial_{\mu}\varphi_{i}\,\partial^{\mu}\varphi_{i}+i\bar{\psi}_{i}\gamma^{\mu}\partial_{\mu}\psi_{i}+f_{i}f_{i}+2f_{i}\,\frac{\partial{\calW}}{\partial\varphi_{i}}-\frac{\partial^{2}{\calW}}{\partial\varphi_{i}\partial\varphi_{j}}\,\bar{\psi}_{i}\psi_{j}\right\}\;.
{\frac{\ddot{\hat{a}_{0}}}{\hat{a}_{0}}}+{\frac{\ddot{\hat{c}}}{\hat{c}}}-2{\frac{\dot{\hat{c}}^{2}}{\hat{c}^{2}}}=-{\frac{4\pi\hat{G}_{4}}{3}}\left(\hat{\varrho}+3{\frac{\hat{\wp}}{\hat{c}^{2}}}\right)+{\frac{\hat{c}^{2}}{3}}\hat{\Lambda}_{\mathrm{eff}}-{\frac{\hat{\calC}\hat{c}^{2}}{\hat{a}_{0}^{4}}},
\{\frac{(d-1)!}{2}[Tr(h^{-1}\widehat{G})\,-\,Tr(\mathcal{F}F)]-\frac{(d-1)!}{2\alpha}[\mathrm{det}\,\left(h^{-1}+\mu\mathcal{F}\right)]^{\alpha}\}\mathrm{det}\,e\,d^{d}\xi
\{\varepsilon^{\mu\nu}\}\rightarrow\{\varepsilon^{\mu\nu}\}=\{\varepsilon^{\mu\nu}\}+\omega\left(\begin{array}{cccc}{0}&{f^{1}}&{f^{2}}&{f^{0}-f^{3}}\\{-f^{1}}&{0}&{0}&{-f^{1}}\\{-f^{2}}&{0}&{0}&{-f^{2}}\\{f^{3}-f^{0}}&{f^{1}}&{f^{2}}&{0}\\\end{array}\right)
\intd^{8}\eta^{a}\eta^{a_{1}}\cdots\eta^{a_{8}}=\epsilon^{a_{1}\cdotsa_{8}}\;,\quad\qquad\;\intd^{8}\dot{\eta}^{\dot{a}}\dot{\eta}^{\dot{a}_{1}}\cdots\dot{\eta}^{\dot{a}_{8}}=\epsilon^{\dot{a}_{1}\cdots\dot{a}_{8}}.
\mathrm{\nu}=\frac{\left(k_{2}k_{3}+k_{1}k_{3}+k_{1}k_{2}\right)-\left(l^{2}+m^{2}+t^{2}\right)}{k_{1}k_{2}k_{3}+2lmt-\left(k_{1}t^{2}+l^{2}k_{3}+m^{2}k_{2}\right)}
A_{i,j}^{\mathrm(int)}(s)=-R^{2s-1}\frac{\Gamma\left(s+\frac{i+j-1}{2}\right)}{\sqrt{\pi}\Gamma(s)\Gamma\left(\frac{i+j}{2}\right)}z^{-2s-(i+j-1)}f\left(s;j,\frac{i+j-1}2,z\right).
g_{i}({\bf{q}})=[\sum_{{\bf{k}}}\frac{n_{F}({\bf{k-q/2}})-n_{F}({\bf{k+q/2}})}{({\tilde{\omega}}_{i}({\bf{q}})-\frac{{\bf{k.q}}}{m})^{2}}]^{-\frac{1}{2}}
\hat{\calA}_{\alpha}=\left(\begin{array}{cc}{G_{(\alpha,\beta)}{}^{\gamma}}&{C_{(\alpha,\beta)}{}^{\bar{\gamma}}}\\{0}&{0}\\\end{array}\right)\;\qquad\hat{\calA}_{\bar{\alpha}}=\left(\begin{array}{cc}{0}&{0}\\{\bar{C}_{\bar{\alpha},\bar{\beta}}{}^{\gamma}}&{G_{\bar{\alpha},\bar{\beta}}{}^{\bar{\gamma}}}\\\end{array}\right)\,
\_{p}F_{q}\left(\begin{array}{ccccccc}{0}&{{\frac{1}{24}}}&{{\frac{2}{24}}}&{{\frac{3}{24}}}&{{\frac{4}{24}}}&{{\frac{5}{24}}}&{....\{\frac{23}{24}}}\\{1}&{1}&{{\frac{1}{2}}}&{{\frac{5}{8}}}&{...\-{\frac{2}{8}}}&{{\frac{1}{12}}}&{...\-{\frac{10}{12}}}\\\end{array}\right).
\begin{array}{c}{{\calR}_{+}=q^{-\frac12}\left(\begin{array}{cccc}{q}&{0}&{0}&{0}\\{0}&{1}&{q-q^{-1}}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{q}\\\end{array}\right),}\\{{\calR}_{-}=P\left({\calR}_{+}^{-1}\right),}\\\end{array}
E_{O}\,=\,\frac{l}{4}a_{3}\sum_{n=1}^{\infty}\left(\omega_{n2}^{2}\,-\,\omega_{n1}^{2}\right)\left(\omega_{n1}^{2}A_{n1}^{2}\,-\,\omega_{n2}^{2}A_{n2}^{2}\right).
F_{7}^{{\mathcal{O}}_{0,1}^{0,1}|++++++-}=\frac{-iH^{0,1}(\sigma_{1}^{-})^{3}/(\sigma_{6}^{+})^{1/2}}{8\prod_{1\leqi<j\leq6}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}).
\Omega^{\mu\nu}(\tau)=S^{\mu\nu}-\frac{1}{2}\epsilon_{\alpha}^{A}(p)\eta_{AB}\left(\frac{\partial\epsilon_{\beta}^{B}(p)}{\partialp_{\mu}}p^{\nu}-\frac{\partial\epsilon_{\beta}^{B}(p)}{\partialp_{\nu}}p^{\mu}\right)S^{\alpha\beta}.
W_{CDB}{}^{A}\equiv\varepsilon^{ij}f_{jC}^{X}f_{iD}^{Y}{\calR}_{XYB}{}^{A}=\frac12\varepsilon^{ij}f_{jC}^{X}f_{iD}^{Y}f_{kB}^{Z}f_{W}^{Ak}R_{XYZ}{}^{W}\,.
\langlem\rangle={\frac{\langle\mathrm{Tr}(e^{h}U-e^{-h}V)(e^{h}U+e^{-h}V)^{k-1}\rangle_{n}}{\langle\mathrm{Tr}(e^{h}U+e^{-h}V)^{k}\rangle_{n}}}\,
\langle(DX^{\rho})_{1}\bar{U}_{2}\rangle=\langle(DX^{\rho})_{1}\bar{X}^{\sigma}(\bar{Z}_{2})\rangle\bar{\partial}_{\sigma}\bar{U}_{2}=-\frac{1}{2}Q^{\rho\sigma}\theta_{1\bar{2}}Z_{1\bar{2}}^{\\-1}\bar{\partial}_{\sigma}\bar{U}_{2},
\lambda_{2N}(\alpha)={\frac{\sum_{m=0}^{N}\left({\frac{i\alpha}{2}}\right)^{2m}{\frac{(2N+2m)!}{(2m)!(2N-2m)!}}}{1+\sum_{m=0}^{N-1}\left({\frac{i\alpha}{2}}\right)^{2m+1}{\frac{(2N+2m+1)!}{(2m+1)!(2N-2m-1)!}}}}\times\lambda_{0}(\alpha).
P_{k}^{S}\approx\left.\frac{16}{9}\frac{G^{2}V^{2}}{c_{s}\dot{\phi}^{2}F_{,X}}\right|_{c_{s}k=aH}=\left.\frac{16}{9}\left(\frac{3}{8\pi}\right)^{2}\frac{H^{4}}{(c_{s}F_{,X})\dot{\phi}^{2}}\right|_{c_{s}k=aH}~,
L=\frac{l^{2}}{(x-y)^{2}}\left(+\frac{1}{G}(\frac{\mathrm{d}x}{\mathrm{d}\tau})^{2}+\frac{1}{F}(\frac{\mathrm{d}y}{\mathrm{d}\tau})^{2}+G(\frac{\mathrm{d}\varphi}{\mathrm{d}\tau})^{2}-F(\frac{\mathrm{d}t}{\mathrm{d}\tau})^{2}\right)
J_{E}\:\tilde{\psi}\;:=\;-if_{EAA^{\prime}}\left(x_{jA}\:\frac{\partial}{\partialx_{jA^{\prime}}}\:+\:z_{A}\:\partial_{A^{\prime}}\:+\:\overline{{z_{A}}}\:\overline{{\partial_{A^{\prime}}}}\:+\:\lambda_{\alphaA}\:\partial_{\lambda_{\alphaA^{\prime}}}\right)\tilde{\psi}\;=\;0
G_{b}^{(n)}(\zeta_{1},\cdots,\zeta_{2n})=\sum_{\varepsilon_{j}=\pm\atop\varepsilon_{1}+\cdots+\varepsilon_{2n}=0}v_{\varepsilon_{1}}\otimes\cdots\otimesv_{\varepsilon_{2n}}G_{b}^{(n)}(\zeta_{1},\cdots,\zeta_{2n})^{\varepsilon_{1}\cdots\varepsilon_{2n}}.
{\frac{\lambda_{1}}{\lambda_{2}}}={\frac{X_{1}A_{3}}{A_{3}X_{2}}},\quad{\frac{\lambda_{2}}{\lambda_{3}}}={\frac{X_{2}A_{1}}{A_{1}X_{3}}},\quad{\frac{\lambda_{3}}{\lambda_{1}}}={\frac{X_{3}A_{2}}{A_{2}X_{1}}}
\tilde{\chi}(m,\bar{m};\tau_{2})=\int\int_{F}\frac{d^{2}m^{\prime}}{(m_{2}^{\prime})^{2}}\llm,\bar{m};\tau_{2}|m^{\prime},\bar{m^{\prime}};\tau_{1}\gg\tilde{\chi}(m^{\prime},\bar{m^{\prime}};\tau_{1}).
\frac{F(1,\frac{i}{4}(k+k^{\prime}-2k^{\prime\prime}),\frac{i}{4}(k+k^{\prime}-2k^{\prime\prime})+1,e^{-4x_{0}})}{k+k^{\prime}-2k^{\prime\prime}}=\sum_{n=0}^{\infty}\frac{e^{-4nx_{0}}}{k+k^{\prime}-2k^{\prime\prime}-4ni}
\Pi_{\mu_{1}\mu_{2}}^{\mathrm{UV}}=\left(\frac{13}{3}-\xi\right)\frac{g^{2}C_{G}}{\left(4\pi\right)^{2}}\left(\frac{1}{2\epsilon}+\mathrm{ln}\left(M\right)\right)\left(\eta_{\mu_{1}\mu_{2}}k^{2}-k_{\mu_{1}}k_{\mu_{2}}\right),
E=-\sum_{l=1}^{\infty}(2l+1)\int_{-\infty}^{\infty}{\frac{d\omega}{2\pi}}e^{-i\omega\tau}\left\{\int_{0}^{a}r^{2}dr\,k^{\prime3}(A_{F}+A_{G})j_{l}^{2}(k^{\prime}r)+\int_{a}^{\infty}r^{2}dr\,k^{3}(B_{F}+B_{G})h_{l}^{2}(kr)\right\}.
S_{11}=\left\langle\frac{2\pi}{H}\right\rangle\left\langle\frac{2\pi}{3}\right\rangle\left\langle\frac{2\pi}{H}-\frac{\pi}{3}\right\rangle\left\langle\frac{\pi}{3}-\frac{4\pi}{H}\right\rangle\left\langle-\frac{4\pi}{H}\right\rangle,
f(x^{+},x^{-})=f(x_{0}^{+},x_{0}^{-})-\frac{1}{2}\int_{x_{0}^{+}}^{x^{+}}j_{+}^{X}(x^{+\prime},x_{0}^{-})dx^{+\prime}+\frac{1}{2}\int_{x_{0}^{-}}^{x^{-}}j_{-}^{X}(x^{+},x^{-\prime})dx^{-\prime}
\frac{k\mu}{2\pi}\widehat{L}\equiv-J^{-}(\theta)(L(\theta)-\partial_{\theta}J^{3}/2),\;\frac{k\nu}{2\pi}\widehat{\widetilde{L}}\equiv-\widetilde{J}^{+}(\theta)(\widetilde{L}(\theta)+\partial_{\theta}\widetilde{J}^{3}/2)
Le_{L}^{c}\hat{H}_{1}=\phi_{(2,\overline{{\tilde{u}+2}})}^{37_{3}}\phi_{(\tilde{u}+2,\bar{\tilde{u}})}^{7_{3}7_{3}}\phi_{(\tilde{u},\overline{{\tilde{u}+1}})}^{7_{3}7_{3}}\phi_{(\tilde{u}+1,\bar{1})}^{7_{3}3}\phi_{(1,\bar{\tilde{u}})}^{37_{3}}\phi_{(\tilde{u},\bar{2})}^{7_{3}3}M_{s}^{-3}
\begin{array}{l}{\frac{M}{A_{8-p}}=\frac{8-p}{2}r_{H}^{7-p}+\frac{7-p}{2}\alpha,}\\{\left(\frac{Q_{0}}{(7-p)A_{8-p}}\right)^{2}=\alpha\left(\alpha+r_{H}^{7-p}\right),}\\\end{array}
N^{-1}L_{aux~B}=-3RB^{2}+\left[-\frac{\kappa}{3R}\bar{\lambda}\lambda+\frac{6\sqrtk}{\kappa}R+\frac{3}{2\kappaR}G_{\bar{A}B}(\bar{\chi}^{\bar{A}}\chi^{B}+\phi^{B}\bar{\phi}^{\bar{A}})-\frac{6}{\kappa^{2}}R^{2}e^{\frac{\alphaG}{2}}\right]\,,
{\frac{1}{\tau}}{\frac{dE(\tau)}{d\tau}}={\frac{e}{2}}\left\langle\left[\bar{\Psi},\tilde{\gamma}^{\eta}\Psi\right]\right\rangle={\frac{e}{2\tau}}\left\langle\left[\Psi^{\dagger},\gamma^{0}\gamma^{3}\Psi\right]\right\rangle.
\frac{1}{2m}\left(-i\hbar\frac{\partial}{\partialx_{j}}-\hbarA_{j}^{a}\hat{T}_{a}\right)^{2}\psi+\frac{\hbar^{2}}{2Mr^{2}}{\hat{T}}^{2}\psi-\frac{e^{2}}{r}\psi=\epsilon\psi
T^{j}\!_{i.k}\equiv\nabla_{k}T^{j}\!_{i}=\partial_{k}T^{j}\!_{i}+\left\{\begin{array}{c}{j}\\{l\,k}\\\end{array}\right\}T^{l}\!_{i}-\left\{\begin{array}{c}{l}\\{i\,k}\\\end{array}\right\}T^{j}\!_{l}\;.
e^{ik(\theta)}=\frac{\vartheta_{4}\big(\,\frac{\theta}{2i\pi}-\frac{1}{4}\,,\,p^{\frac{1+\xi}{4}}\,\big)}{\vartheta_{4}\big(\,\frac{\theta}{2i\pi}+\frac{1}{4}\,,\,p^{\frac{1+\xi}{4}}\,\big)}\,.
p_{L}=\frac{m}{2R}+nR,\;p_{R}=\frac{m}{2R}-nR,\;H=\frac{m^{2}}{4R^{2}}+n^{2}R^{2},\;S=\frac{p_{L}^{2}}{2}-\frac{p_{R}^{2}}{2}=mn.
X=\left(\begin{array}{cccc}{0}&{0}&{L_{11}}&{L_{12}}\\{0}&{0}&{L_{21}}&{L_{22}}\\{L_{11}}&{L_{21}}&{-\,\frac{3}{2}\,e^{2}\,\phi_{1}^{2}-\,\frac{1}{2}\,e^{2}\,\phi_{2}^{2}}&{-e^{2}\,\phi_{1}\,\phi_{2}}\\{L_{12}}&{L_{22}}&{-e^{2}\,\phi_{1}\,\phi_{2}}&{-\,\frac{1}{2}\,e^{2}\,\phi_{1}^{2}-\,\frac{3}{2}\,e^{2}\,\phi_{2}^{2}}\\\end{array}\right)\,
\frac{1}{4}\sum_{i}\left[\left((A^{(i+1)}-A^{(i)})^{2}+4|W^{(i)}|^{2}\right)(|v_{+}|^{2}+|v_{-}|^{2})-4(W^{(i+1)+}W^{(i)-}v_{+}v_{-}^{*}+h.c.)\right].
W(\Sigma)-W({\tilde{\Sigma}}_{0})=iRn\int_{0}^{1}d\sigma\int_{0}^{2\pi}d{\tilde{\theta}}\int_{-\infty}^{\infty}d{\tilde{\rho}}{\frac{\zeta}{\lambda_{1}^{2}}}{\frac{({\,tilde\lambda}-1)^{2}}{{\tilde{\lambda}}}}\left({\frac{\partialg_{+}}{\partial\sigma}}{\frac{\partialg_{-}}{\partial\rho}}-{\frac{\partialg_{+}}{\partial\rho}}{\frac{\partialg_{-}}{\partial\sigma}}\right).
\tilde{G}_{\sigma}^{(n)}(\zeta)=(\sqrt{-1})^{-n}\sum_{k_{0},k_{1},\cdotsk_{2n}}t_{k_{1}}^{k_{0}}(u_{1}-u_{0})\otimes\cdots\otimest_{k_{2n}}^{k_{2n-1}}(u_{2n}-u_{0})\tilde{F}_{\sigma}^{(n)}(\zeta)^{k_{0}k_{1}\cdotsk_{2n}}.
K(t,t^{\prime})=K_{s}^{(o)}(t,t^{\prime})+\sum_{r=1}^{n}\int_{C_{r}}K_{s}^{(o)}(t,t^{\prime\prime})\frac{dt^{\prime\prime}}{2\pii}K(t^{\prime\prime},t^{\prime})\quad\mathrm{{for}}\;s=1,2...n.
\left(G_{4}\varrho_{m}a^{3(w+1)}\right)^{.}=G_{4}\varrho_{\delta\sigma}a^{3(w+1)}\left(2{\frac{\dot{c}}{c}}-{\frac{\dot{G}_{4}}{G_{4}}}\right)+{\frac{3kc\dot{c}a^{3w+1}}{4\pi}}.
\bar{\xi}^{k\bar{z}}(z)=\overline{{\xi^{kz}(z)}},~~~\xi^{kz}(z_{n}^{c})=\bar{\xi}^{k\bar{z}}(z_{n}^{c})=\delta_{kn},~~~\xi^{k\bar{z}}(z)=0,~~~\bar{\xi}^{kz}(z)=0
{\calD}_{\alpha}\xi^{A}=D_{\alpha}\xi^{A}+\frac{i}{2}\frac{{\epsilon_{\alpha}}^{\beta}}{\sqrt{-\gamma}}\partial_{\beta}\bar{X}^{\rho}E_{\mu}^{A}{{H_{\rho}}^{\mu}}_{\nu}E_{B}^{\nu}\xi^{B},
ds^{2}=-\left(1+{\frac{r^{2}}{\tilde{L}^{2}}}\right)dt^{2}+{\frac{dr^{2}}{1+{\frac{r^{2}}{\tilde{L}^{2}}}}}+r^{2}d\Omega_{p}^{2}+L^{2}d\Omega_{n}^{2},
T_{\mu\nu}=(-g)^{\frac{1}{4}}\left\{-\frac{1}{4}\gamma_{\mu\nu}\left[\gamma^{\rho\sigma}\left(g_{\rho\sigma}-g^{\kappa\tau}F_{\rho\kappa}F_{\tau\sigma}\right)-(p-3)\Lambda\right]+g_{\mu\nu}-g^{\rho\sigma}F_{\mu\rho}F_{\sigma\nu}\right\}.
=\frac1{16}{\calR}_{abcd}\bar{\Psi}^{a}(1+\gamma_{p})\Psi^{c}\bar{\Psi}^{b}(1+\gamma_{p})\Psi^{d}=\frac1{16}{\calR}_{abcd}\tilde{V}_{i}^{a}V_{j}^{c}\tilde{V}_{k}^{b}V_{l}^{d}\bar{X}^{i}(1+\gamma_{p})X^{j}\bar{X}^{k}(1+\gamma_{p})X^{l}~.
\begin{array}{c}{L=g_{\alpha\beta}(dX^{\alpha}\cdotdX^{\beta})+2g_{\alphaa}(dX^{\alpha}\cdotdX^{a})+g_{ab}(d\Phi^{a}\cdotd\Phi^{b})+}\\{+b_{\alpha\beta}dX^{\alpha}\wedgedX^{\beta}+2b_{\alphaa}dX^{\alpha}\wedged\Phi^{a}+b_{ab}d\Phi^{a}\wedged\Phi^{b}}\\\end{array}
\begin{array}{rl}{A_{SW}^{(i)}(z_{1},z_{2}):}&{{\cal\bar{H}}^{(i)}\longrightarrow{\calH}^{(i)},}\\{A_{NW}^{(i)}(z_{1},z_{2}):}&{{\calH}^{(i)}\longrightarrow{\cal\bar{H}}^{*(i)}.}\\\end{array}
\begin{array}{llcl}{}&{(1~0~0~0~0~1)}&{}\\{\mid\mathrm{gnd}>}&{(1~0~0~0~0~1)}&{=}&{\beta_{j}^{i+}}\\{\mid\mathrm{1st}>}&{(1~0~0~1~0~0)}&{=}&{\mathrm{SU(5)}}\\{}&{(0~0~0~0~0~2)}&{=}&{\mathrm{SU(2)}}\\{}&{(0~0~0~0~0~0)}&{=}&{\mathrm{U(1)}}\\{\mid\mathrm{2nd}>}&{(0~0~0~1~0~1)}&{=}&{\beta_{j}^{i-}.}\\\end{array}
\frac{1}{\sqrt{-g^{\prime}}}\partial_{\mu}\left(\sqrt{-g^{\prime}}g^{\prime\mu\alpha}g^{\prime\nu\beta}F_{\alpha\beta}\right)+\frac{e^{2}}{2}\varphi^{2}g^{\prime\mu\nu}A_{\mu}=-e\overline{{\Psi}}^{\prime}\gamma^{a}V_{a}^{\prime\mu}\Psi^{\prime}
\mathcal{L}=-\frac14F_{\mu\nu}^{2}-\fracN{4\pi}\varepsilon^{\mu\nu\rho}a_{\mu}\partial_{\nu}a_{\rho}+en_{e}A_{0}+i\psi^{\dagger}D_{0}\psi-\frac1{2m}\left|D_{k}\psi\right|^{2}+\psi^{\dagger}\mu\psi
n(L,\rho)=\frac{\sum_{K=1}^{\infty}KC\gamma^{K-1}(\Gamma(b(K-1)+2))^{-1}\rho^{b-2}(L-\tau\rho)^{b(K-1)+1}}{\sum_{K=0}^{\infty}\gamma^{K}(\Gamma(bK+1))^{-1}L^{bK+1}}
\left[\phi_{ij}(x^{-},x_{\perp}),\partial_{-}\phi_{kl}(y^{-},y_{\perp})\right]=\left\{\psi_{ij}(x^{-},x_{\perp}),\psi_{kl}(y^{-},y_{\perp})\right\}=\frac{1}{2}\delta(x^{-}-y^{-})\delta(x_{\perp}-y_{\perp})\delta_{il}\delta_{jk}.
{\mathcal{L}}(\xi,\xi)=2^{(D-1)/2}\pim^{D-1}\Gamma\left(\frac{D-1}{2}\right)\int_{0}^{\infty}dy\,\frac{\pi^{2}-3y^{2}}{(\pi^{2}+y^{2})^{3}}\frac{K_{(D-1)/2}(z)}{z^{(D-1)/2}},
\tilde{f}((q^{2}/\mu^{2},~p_{1}^{2}/\mu^{2},~p_{2}^{2}/\mu^{2})=1-\frac{\alpha}{2\pi}\int_{D}\frac{dk_{\perp}^{2}}{k_{\perp}^{2}}\frac{d\beta}{\beta}\tilde{f}((q^{2}/k_{\perp}^{2},~p_{1}^{2}/k_{\perp}^{2},~p_{2}^{2}/k_{\perp}^{2})~.
\begin{array}{l}{i)\qquad\chi^{(1)}=a^{(1)}+p^{F}\sum_{s=1}^{k+1}\frac{\eta_{s}^{F}}{|\vec{x}-\vec{x}_{s}|}\quad,\quad\chi^{(2)}=a^{(2)}+p^{H}\sum_{s=1}^{k+1}\frac{\eta_{s}^{H}}{|\vec{x}-\vec{x}_{s}|}}\\{ii)\qquad\chi^{(1)}=a^{(1)}+\sum_{s=1}^{k+1}\frac{p_{s}^{F}}{|\vec{x}-z_{s}|}\quad,\quad\chi^{(2)}=a^{(2)}+\sum_{s=1}^{k+1}\frac{p_{s}^{H}}{|\vec{x}-z_{s}|}}\\\end{array}
(\bar{C}_{\alpha},\bar{C}_{\beta})_{\Omega}=U_{\alpha\beta}^{\gamma}\bar{C}_{\gamma},\quad(\bar{C}_{\beta}^{\dagger},\bar{C}_{\alpha}^{\dagger})_{\Omega}=\bar{C}_{\gamma}^{\dagger}U_{\alpha\beta}^{\dagger\gamma},
{\calL}_{M}=\frac{\mu^{2}m^{2}}2A_{\mu}{}_{m}^{a}A_{\mu}{}_{-m}^{a}+\frac{\mu^{2}}2(m+1/2)^{2}A_{\mu}{}_{m+1/2}^{a^{\prime}}A_{\mu}{}_{m+1/2}^{a^{\prime}}.
\phi_{-}(\sigma)=L\int_{0}^{\sigma}\frac{dx}{r_{-}^{2}(x)}=\pm\sqrt{\frac{L^{2}}{m^{2}+L^{2}}}F[\psi_{-}(r_{-}(\sigma)),\;\sqrt{\frac{m^{2}}{m^{2}+L^{2}}}\;].
C_{\Omega}=2\pi^{1/4}\kappa^{-3/2}N^{1/2}\frac{\sqrt{m}r_{+}(2r_{+}^{2}+l^{2})(3r_{+}^{2}-l^{2})}{(r_{+}^{2}+l^{2})(2r_{+}^{2}-l^{2})},
F_{1}=\int_{-1}^{1}dyye^{1/{\tilde{g}}(y-1)}\left[1-\frac{1}{6}(y-1)-\frac{1}{6\tilde{g}}(y-1)^{2}\right]=\tilde{g}+\frac{1}{6}{\tilde{g}}^{2}+...
ds_{4}^{2}=(1+{\frac{4M}{r}})^{\frac{1}{2}}(dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\varphi^{2})-(1+{\frac{4M}{r}})^{-{\frac{1}{2}}}dt^{2}\,;\quade^{2\Phi}={\frac{e^{2\Phi_{0}}}{(1+{\frac{4M}{r}})^{\frac{3}{2}}}}\,,
f\starg=\frac{1}{(2\pi)^{2}}\sum_{m,n,P^{\prime},P^{\prime\prime}}\int_{-\pi}^{+\pi}d\theta^{\prime}d\theta^{\prime\prime}\delta_{P^{\prime},P_{\theta}+m}\delta_{P^{\prime\prime},P_{\theta}-n}e^{[-2iA(\tilde{\bfr},\tilde{\bfr}^{\prime},\tilde{\bfr}^{\prime\prime})]}f(\tilde{\bfr}^{\prime})g(\tilde{\bfr^{\prime\prime}})
l^{2}=\left(1+{\frac{c_{p}\,N}{(rM_{P})^{7-p}(R_{S}M_{P})^{2}(V_{T}M_{P}^{p})}}\right)R_{S}^{2}=R_{S}^{2}+{\frac{c_{p}\,N}{(rM_{P})^{7-p}M_{P}^{2}(V_{T}M_{P}^{p})}}
\frac{\partial}{\partialz_{i}}W_{\psi_{0}}(\psi_{1}\otimes\psi_{2}\otimes\psi_{3};z_{1},z_{2},z_{3})=-\bar{h}\sum_{j\neqi}\frac{t_{ij}}{z_{i}-z_{j}}W_{\psi_{0}}(\psi_{1}\otimes\psi_{2}\otimes\psi_{3};z_{1},z_{2},z_{3}),
\begin{array}{rl}{3H^{2}\=}&{{\displaystyle{\frac{2\pi\lambda\phi^{4}}{M_{\mathrm{p}}^{2}+8\pi|\xi|\phi^{2}}}\,}}\\{(1+6|\xi|)\3H\dot{\phi}\phi\=}&{{\displaystyle-\{\frac{\lambda\phi^{4}M_{\mathrm{p}}^{2}}{M_{\mathrm{p}}^{2}+8\pi|\xi|\phi^{2}}}\.}}\\\end{array}
Z_{v}^{X}\left(-\frac{1}{\tau}\right)=\vert\Gamma_{\calG}\vert^{-\frac{b_{2}(X)}{2}}\left(\frac{\tau}{i}\right)^{-\frac{\chi(X)}{2}}\cdot\sum_{u\inH^{2}(X,\Gamma_{\calG})}\zeta_{\calG}^{u\cdotv}Z_{u}^{X}(\tau),
\prod_{j=1}^{N}(1-w/w_{j})=\frac{F_{0}(w)}{F_{0}(x^{2r}w)},\qquad\prod_{j=1}^{N}(1-w_{j}/w)=\frac{G_{0}(1/x^{2r}w)}{G_{0}(1/w)}.
g_{s}l_{s}^{D-p-1}~k_{1}.k_{3}~k_{2}.k_{4}~\tilde{P}(\zeta)\intd^{D-p-1}p_{1}~\intd^{D-p-1}p_{2}~\frac{q^{2}e^{i({\vec{p}_{1}}\cdot{\vec{\epsilon}_{1}}+{\vec{p}_{2}}\cdot{\vec{\epsilon}_{2}})}}{(q^{2}+p_{1}^{2})(q^{2}+p_{2}^{2})}~\delta^{p+1}\left(\sum_{i}k_{i}\right)~.
I=\int_{\cal{M}}d^{4}\!x\sqrt{\tilde{g}}e^{\Phi}\left(\tilde{R}(\tilde{g})+\left(\nabla\Phi\right)^{2}-{\frac{1}{12}}H_{\mu\nu\rho}H^{\mu\nu\rho}-\left(\nablaT\right)^{2}+V(T)\right)\,
ds^{2}=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)d\tau^{2}+\left(\frac{4M}{r}-\frac{2Q^{2}}{r^{2}}\right)d\taudr+\left(1+\frac{2M}{r}-\frac{Q^{2}}{r^{2}}\right)dr^{2}+\phi^{2}d\Omega_{2}^{2}.
1_{l}=\left(\begin{matrix}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{1}\\\end{matrix}\right),~~1_{q}=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right).
Q^{\alpha}=g_{i\bar{j}}\dot{\bar{z}}^{\bar{j}}\chi^{i\alpha}=(P+K)_{i}\chi^{i\alpha}\qquad\qquad\bar{Q}_{\alpha}=g_{i\bar{j}}\dot{z}^{i}\bar{\xi}\bar{\chi}^{\bar{j}}=(\bar{P}+\bar{K})_{\bar{j}}\bar{\chi}_{\alpha}^{\bar{j}}
ds^{2}=X(\frac{d\eta^{2}}{1-\eta^{2}}-\frac{d\mu^{2}}{1-\mu^{2}})-(1-\mu^{2})^{1/2}(1-\eta^{2})^{1/2}[\chidy^{2}+\frac{1}{\chi}(dx-\lambdady)^{2}].
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline{{}~~A~~}&{~~B~~}&{~~C~~}&{~~D~~}&{~~E~~}&{~~F~~}&{~~G~~}&{~~H~~}&{~~I~~}&{~~J~~}\\\hline{-\frac{9}{4}\alpha}&{3\alpha}&{\frac{1}{4}\alpha}&{\frac{3}{4}\beta}&{\frac{1}{4}\beta}&{-\frac{9}{2}\alpha}&{\frac{3}{2}\beta}&{\frac{3}{2}\beta}&{12\alpha}&{\frac{15}{2}\alpha}\\\hline{}\\\end{array}\nonumber
\begin{array}{rcl}{\{\theta_{0}^{A},\theta_{0}^{B}\}}&{=}&{0\,,}\\{\{\theta_{0}^{A},\theta_{3}^{B}\}}&{=}&{-\delta^{AB}\,,}\\{\{\theta_{3}^{A},\theta_{3}^{B}\}}&{=}&{-c^{ABC}\hat{f}^{C}\,,}\\{\{\theta_{0}^{A},H_{initial}\}}&{=}&{0\,,}\\{\{\theta_{3}^{A},H_{initial}\}}&{=}&{-\partial^{i}f^{i\,A}\,.}\\\end{array}
L_{m}(\alpha)\equiv\left(\begin{array}{cc}{e^{-iV_{m}^{-}}}&{e^{\alpha}e^{iV_{m}^{+}}}\\{e^{\alpha}e^{-iV_{m}^{+}}}&{e^{iV_{m}^{-}}}\\\end{array}\right)\,,\quad\bar{L}_{m}(\alpha)\equiv\left(\begin{array}{cc}{e^{-iV_{m}^{+}}}&{e^{-\alpha}e^{iV_{m}^{-}}}\\{e^{-\alpha}e^{-iV_{m}^{-}}}&{e^{iV_{m}^{+}}}\\\end{array}\right)\,,
\left(\begin{array}{c}{q_{0}^{2}}\\{\bar{q}_{0}^{2}}\\\end{array}\right)=\frac{1}{\tilde{\gamma}\tilde{g}^{2}-\frac1{k_{T}}}\left(\begin{array}{c}{\frac12\tilde{\gamma}\tilde{g}^{2}k_{T}(m_{q_{1}}^{2}-m_{\bar{q}}^{2})+m_{\bar{q}}^{2}}\\{\frac12\tilde{\gamma}\tilde{g}^{2}k_{T}(m_{q_{1}}^{2}-m_{\bar{q}}^{2})-m_{q_{1}}^{2}}\\\end{array}\right)
(dG)_{11\bar{I}\bar{J}\bar{K}\bar{L}}=-4\sqrt{2}\pi\left(\frac{\kappa}{4\pi}\right)^{2/3}(J^{(1)}\delta(x^{11})+J^{(2)}\delta(x^{11}-\pi\rho)+\Sigma_{i=1}^{N}\hat{J}^{(i)}\delta(x^{11}-x_{i}))_{\bar{I}\bar{J}\bar{K}\bar{L}}.