harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e_{M\dot{M}}^{A\dot{A}}=e^{\frac{\rho(x)}{2}}\delta_{M}^{A}\delta_{\dot{M}}^{\dot{A}},\quad\omega_{A\dot{A},BC}=-\frac{1}{4}e^{-\frac{\rho(x)}{2}}(\epsilon_{BA}\partial_{\dot{A}C}\rho(x)+\epsilon_{CA}\partial_{\dot{A}B}\rho(x)),
ds_{10}^{2}=\tilde{f}_{++}^{-\alpha}dx_{//}^{2}+\tilde{f}_{+}^{2(p-8)/(7-p)}\tilde{f}_{++}^{\beta_{+}}(d\rho^{2}+\rho^{2}\tilde{f}_{+}^{2}d\Omega_{8-p}^{2})
\tilde{V}^{(\!1\!)}(\vec{\xi},\vec{\xi}^{\prime})\!=\!\delta(\xi_{3}\!-\!\xi_{3}^{\prime})e^{i(\xi_{1}\xi_{2}-\xi_{1}^{\prime}\xi_{2}^{\prime})}\!\int\!\!dx\,dy\,e^{-i\sqrt{{\mathcal{B}}}[(\xi_{2}-\xi_{2}^{\prime})x+(\xi_{1}-\xi_{1}^{\prime})y]}V^{(\!1\!)}(x,y,\xi_{3}).
\begin{array}{l}{\displaystyle\displaystyle\tilde{\calQ}_{\alpha}^{1}=(ip_{\alpha\beta}\tilde{W}^{\beta}-mW_{\alpha})[1+\mathrm{q}^{cl}(bP_{3}-\sqrt{1-b^{2}}\,P_{2}-P_{4})]}\\{\displaystyle\tilde{\calQ}_{\alpha}^{2}=(ip_{\alpha\beta}\tilde{V}^{\beta}-mV_{\alpha})[1+\mathrm{q}^{cl}(bP_{3}+\sqrt{1-b^{2}}\,P_{2}-P_{4})]\,.}\\\end{array}
\bigl\{T_{a},\,T_{b}^{(1)}\bigr\}_{(\phi,\pi)}+\bigl\{T_{a}^{(1)},\,T_{b}\bigr\}_{(\phi,\pi)}+\bigl\{T_{a}^{(1)},\,T_{b}^{(2)}\bigr\}_{(\eta)}+\bigl\{T_{a}^{(2)},\,T_{b}^{(1)}\bigr\}_{(\eta)}=0\,.
\prod_{a\inA\backslash\{1\}}\oint_{C_{a}}\frac{dw_{a}}{2\piiw_{a}}\left(\oint_{C_{1}}+2\pii\mathrm{Res}_{w_{1}=x^{-3}z_{1}}\right)\frac{dw_{1}}{2\piiw_{1}}\Psi_{b}^{(n)}(w|\zeta_{1}^{-1},\tilde{\zeta})Q_{b}^{(n)}(w|\zeta_{1}^{-1},\tilde{\zeta})\frac{xz_{1}^{-1}-w_{1}}{x^{-3}z_{1}-w_{1}}.
{\bf(p_{L}^{D})}_{i}={\calR}_{i}^{\,\,j}{\bf(p_{L})}_{j}\,\,\mathrm{with}\,\,{\calR}_{i}^{\,\,j}=-{\bfe^{a}}_{i}(e^{t}.e^{-1})_{a}^{\,\,b}{\bfe_{b}}^{j}\,.
G_{ab,cd}^{(k)}(x_{1},x_{2};x_{3},x_{4})=e^{ik\theta}\frac{1}{Z[0,0,0]}\left.\frac{\delta^{4}Z^{(k)}[\eta,\overline{{\eta}},J]}{\delta\overline{{\eta}}_{a}(x_{1})\delta\overline{{\eta}}_{b}(x_{2})\delta\eta_{c}(x_{3})\delta\eta_{d}(x_{4})}\right|_{\eta,\overline{{\eta}},J=0}\;.
1-\bar{S}_{F}\ast{\calV}=\left[1-S_{F}^{cov}\left(\gamma^{-}{A}_{-}+\gamma^{j}{A}_{j}\right)\right]\ast\left[1+\frac{e}{2}\gamma^{+}\frac{1}{i\partial_{-}}\ast{\calW}_{-1}[\widehat{a}]\left(\gamma^{-}{A}_{-}+\gamma^{k}{A}_{k}\right)\right],
\begin{array}{l}{\psi_{1\overline{{z}}}=q\varphi_{1}\quad\quad,}\\{\varphi_{1z}=-\overline{{q}}\psi_{1}\quad\,\,,}\\\end{array}\qquad\begin{array}{l}{\psi_{2\overline{{z}}}=\overline{{\widetilde{q}}}\varphi_{2}\quad\quad,}\\{\varphi_{2z}=-\widetilde{q}\psi_{2}\qquad.}\\\end{array}
<p_{2}|T_{\mu\nu}(x)|p_{1}>={\frac{e^{i(p_{2}-p_{1})\cdotx}}{\sqrt{4E_{2}E_{1}}}}\left[2P_{\mu}P_{\nu}F_{1}(q^{2})+(q_{\mu}q_{\nu}-g_{\mu\nu}q^{2})F_{2}(q^{2})\right]
\left\{\begin{array}{l}{\frac\partial{\partial\triangle_{j}}\left[\sum_{i=1}^{k}\triangle_{i}+\alpha\mathrm{H}_{k+1}^{\left(k\right)}\right]_{\triangle_{j}=\triangle_{j}^{\left(k+1\right)}}=0,\qquad\forall\;\;j\leqk}\\{\mathrm{H}_{k+1}^{\left(k\right)}=0}\\\end{array}\right\}
\epsilon_{\mu\nu\rho\sigma\tau}\int\delta^{(5)}(x-y)dy^{\alpha}\wedgedy^{\mu}\wedgedy^{\nu}\int\delta^{(5)}(x-\tilde{y})d\tilde{y}^{\rho}\wedged\tilde{y}^{\sigma}\wedged\tilde{y}^{\tau}\\.
{\calD}_{\alpha}=-\frac{1}{2}\epsilonm{\calL}_{\alpha}-{\calJ}_{\alpha}-\frac{1}{2}R{\calP}_{\alpha},\quad\nu=-5;\quad{\calD}_{\alpha}=-\frac{1}{2}\epsilonm{\calL}_{\alpha}-\frac{1}{2}{\calJ}_{\alpha}-\frac{1}{4}(1+R){\calP}_{\alpha},\quad\nu=-7.
\hat{a}_{BD}(\zeta)=\frac{e^{\pi\vert\zeta\vert/2}}{(e^{\pi\vert\zeta\vert}-e^{-\pi\vert\zeta\vert})^{1/2}}\hat{a}_{H}(\zeta)-\frac{e^{-\pi\vert\zeta\vert/2}}{(e^{\pi\vert\zeta\vert}-e^{-\pi\vert\zeta\vert})^{1/2}}\hat{a}_{H}^{\dagger}(\zeta)\,.
S_{bh}~=~{\frac{{\calA}_{h}}{4l_{p}^{2}}}~-~{\frac{3}{2}}~ln\left({\frac{{\calA}_{h}}{4l_{p}^{2}}}\right)~-~{\frac{1}{2}}~ln\left({\frac{\pi}{8(ln2)^{3}}}\right)~-~O({\calA}_{h}^{-1}).
\chi_{\mathcal{A}[\mathcal{C}_{2N}]}(\tau)=\frac{1}{2N}\Biggl[\eta(\tau)^{2}+\sum_{l=0}^{N-1}(-1)^{l}(2N-2l-1)\Lambda_{(2l+1)N,2N^{2}}(\tau)\Biggr]
\omega^{2}(\varrho)=\omega_{\mathrm{b}}^{2}(\varrho)+{\mathcal{O}}(\varrho^{4})\qquad\mathrm{with}\qquad\omega_{\mathrm{b}}^{2}(\varrho)\equiv\omega_{0}^{2}-\Omega\,\varrho^{2}
{\itS}_{gr}=-\frac{1}{{\itk}^{2}}\intd^{4}x\sqrt{-g}\Bigl(\tilde{R}(\Gamma)-2\Lambda+b_{1}Q_{\sigma\mu\nu}Q^{\sigma\mu\nu}+b_{2}Q_{\sigma\mu\nu}Q^{\nu\sigma\mu}+b_{3}Q_{\sigma}Q^{\sigma}\Bigr)
\Phi=\int_{-\infty}^{\infty}dk\sum_{M_{n},N_{n}=0}^{\infty}\sum_{A}\left(U_{k\{M,N\}A}a_{k\{M,N\}}^{A}+a_{k\{M,N\}}^{A\dagger}U_{k\{M,N\}A}^{*}\right)\,
G_{0}^{(2)}(\vec{r},z,z^{\prime})=\frac{1}{(2\pi)^{d-1}}\intd^{d-1}p\sum_{n}e^{i\vec{p}.\vec{r}}u_{n}(z)u_{n}^{*}(z^{\prime})\,G_{0,n}(\vec{p}),
<\prod_{k}\sigma_{n_{k}}>=\intd\Phi\int_{{\calM}_{g}}\prod_{i=1}^{3g-3}dm_{i}\int_{{{\bfC}^{g}/\Lambda}}\prod_{j=1}^{g}d\nu_{j}\intd\hat{m}d\hat{\nu}\,\prod_{i}\mathrm{e}^{q_{i}\tilde{\pi}(z_{i})}\,\mathrm{e}^{-S(m,\hat{m},\nu,\hat{\nu})}\prod_{k}\sigma_{n_{k}},
I_{i}[g,\varphi_{i}]=\int_{{\calM}_{\beta}}\varphi_{i}^{+}L_{i}\varphi_{i}\sqrt{g}d^{4}x~~~,~~~\bar{I}_{i}[\bar{g},\bar{\varphi}_{i}]=\int_{\bar{{\calM}}_{\beta}}\bar{\varphi}_{i}^{+}\bar{L}_{i}\bar{\varphi}_{i}\sqrt{\bar{g}}d^{4}x~~~,
f_{\hat{A}}(x,p)=\frac{1}{2\pi}\sum_{n}{}\intd\taue^{-in{x}/{R}-i\tau{Rp}/{\theta}}Tr_{C}\left[\hat{A}(\hat{x},\hat{p})e^{i(n{\hat{x}}/{R}+\tau{\hat{p}R}/{\theta})}\right].
(M)_{n_{1}n_{1}^{\prime}}=\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{2\pi}\Psi_{n_{1}n_{2}m}^{*}(\mu,\nu,\varphi;\delta_{1},\delta_{2})M\Psi_{n_{1}^{\prime}n_{2}^{\prime}m}(\mu,\nu,\varphi;\delta_{1},\delta_{2})\,\frac{\mu+\nu}{4}d\mud\nud\varphi.
L=-\frac{1}{8\pi}\frac{1}{\left(a^{2}+b^{2}\right)}\left(aH_{\kappa\lambda}-ib\frac{1}{2}\epsilon_{\kappa\lambda\rho\sigma}H^{\rho\sigma}\right)H^{\kappa\lambda}+\frac{1}{4\pi}\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}H^{\mu\nu}(\partial^{\sigma}-\partial^{\sigma}A^{\rho}),
\delta\left(\intd^{3}x\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}\right)=2\intd^{3}x\epsilon^{\mu\nu\rho}\deltaA_{\mu}\partial_{\nu}A_{\rho}+\intd^{3}x\epsilon^{\mu\nu\rho}\partial_{\nu}\left(A_{\mu}\deltaA_{\rho}\right)
{\calR}_{0}+{\calR}_{1}^{2}+\frac{1}{M}{\calR}_{0}{\calR}_{1}=\left(\begin{matrix}{S_{\mu\nu}}&{q_{\mu}}&{\frac{1}{M}q_{\mu}\tilde{\partial}}\\{-\tilde{\partial}q_{\nu}^{T}}&{\tilde{\partial}\tilde{\nabla}}&{0}\\{0}&{0}&{\tilde{\nabla}\tilde{\partial}}\\\end{matrix}\right)\.
\begin{array}{lll}{\alpha\beta=q^{-2}\beta\alpha\quad,\quad}&{\alpha\gamma=q^{2}\gamma\alpha\quad,\quad}&{{}[\alpha,\delta]=0\quad,}\\{{}[\beta,\gamma]=q^{-1}\lambda(\delta-\alpha)\alpha\quad,\quad}&{{}[\delta,\beta]=q^{-1}\lambda\alpha\beta\quad,\quad}&{{}[\gamma,\delta]=q^{-1}\lambda\gamma\alpha\quad,}\\\end{array}
S^{(N_{T}=8)}=S_{T}^{(N_{T}=8)}+\int_{E}d^{2}x\,\Bigr\{\mathrm{~\frac{1}{2}~}\partial^{\mu}A_{\mu}\partial^{\nu}A_{\nu}+\partial^{\mu}\bar{C}_{\alpha}\partial_{\mu}C^{\alpha}\Bigr\},
{\calH}_{2}=9\lambda_{1}^{2}\kappa_{2}^{2}(\phi^{4})^{2}(F_{ij}^{\alpha})^{2}+\frac{1}{2}(D_{i}\phi^{\alpha})^{2}+\frac{\lambda_{1}^{2}}{2}(D_{i}\phi^{a})^{2}+\frac{1}{2}\kappa_{2}^{2}(D_{[i}\phi^{a}D_{j]}\phi^{b})^{2}.
{\calL}=-\frac{T_{3}}{g}\sqrt{-\mathrm{det}(\eta_{\mu\nu}+(2\pi\alpha^{\prime}){\calF}_{\mu\nu})}+\frac{\chi_{0}T_{3}}{8}(2\pi\alpha^{\prime})^{2}\epsilon^{\mu\nu\lambda\sigma}{\calF}_{\mu\nu}{\calF}_{\lambda\sigma},
\dot{\Psi}^{R}=i[H^{R},\Psi^{R}]=\gamma_{0}(-i\gamma_{i}\partial_{i}-e\gamma_{i}V_{i}^{R}+m)\Psi^{R}+{\frac{e}{2}}\left\{\Psi^{R},{\frac{1}{\vec{\partial}^{2}-M^{2}}}J_{0}\right\}~,
Z^{(\mathrm{cov)}}(G,p,r_{l}^{(A)})=\sum_{\{h_{i}\}}\left[\Lambda(h)\right]^{2G-2+p}\prod_{A=1}^{p}\frac{\chi_{h}(r_{l}^{(A)})}{\prod_{l}l^{r_{l}^{(A)}}r_{l}^{(A)}!}~,
y_{1}=\rho_{0}^{2}\mathrm{exp}\left(1-{\frac{A}{B}}\right)\,,\,y_{\chi}=\sqrt{e}y_{1}\,,\,\chi=1+{\frac{B}{2\rho_{0}^{4}}}\mathrm{exp}\left({\frac{2A}{B}}-3\right)~.
\left\langle{W}\right\rangle_{0}={W}_{0}+\frac{1}{2}\,\frac{\partial^{2}{W}_{0}}{\partial\varphi_{1}^{2}}\,\left\langle\chi_{1}^{2}\right\rangle_{\,0}+\frac{1}{2}\,\frac{\partial^{2}{W}_{0}}{\partial\varphi_{2}^{2}}\,\left\langle\chi_{2}^{2}\right\rangle_{\,0}\,,
\left(-{\frac{1}{r}}{\frac{\partial}{\partialr}}r{\frac{\partial}{\partialr}}+{\frac{\left(m-\delta{\frac{r}{R}}\right)^{2}}{r^{2}}}-{\frac{g}{2}}\delta{\frac{1}{rR}}\right)\psi_{m}(r)=\epsilon\psi_{m}(r)\,.
U=\left(\begin{array}{cccc}{-\sqrt{\frac{r_{0}}{2M}}}&{\sqrt{\frac{2M}{r_{0}}}}&{0}&{0}\\{\sqrt{\frac{r_{0}}{2M}}}&{0}&{0}&{0}\\{0}&{0}&{0}&{\sqrt{\frac{r_{0}}{2M}}}\\{0}&{0}&{\sqrt{\frac{2M}{r_{0}}}}&{\sqrt{\frac{r_{0}}{2M}}}\\\end{array}\right)\,.
G2(u)=\frac{1}{y_{1}(bb-cc)}\left(-\bar{Y}z_{1}b\bar{g}-y_{1}\bar{e}\bar{e}-Yg_{1}\bar{e}c-\bar{X}z_{1}\bar{e}\bar{f}+Z\bar{x_{1}}c\bar{f}+2Y((\bar{e},c))\right).
V_{D}=V_{E}=V_{0}\intd^{2}z\left|\Phi_{X_{1}}(z)\right|^{2}\left|\Phi_{X_{2}}(z)\right|^{2}={\frac{V_{0}}{\sqrt{2\pi}}}{\frac{1}{L_{y}}}e^{-(X_{1}-X_{2})^{2}/2}.
\sum_{p_{1},\ldots,p_{i}\inT_{2}}\sum_{s_{1},t_{1},\ldots,s_{G},t_{G}\inS_{n}}N^{n(2-2G)-i}\left[\frac{1}{n!}\delta(p_{1}\cdotsp_{i}\Omega_{n}^{2-2G}P_{n}^{(N)}\prod_{j}s_{j}t_{j}s_{j}^{-1}t_{j}^{-1})\right].
W[\Phi_{0},a,M]=-{\frac{1}{64\pi^{2}}}\tau({\frac{4}{3}}\pia^{3})\Phi_{0}^{4}{\frac{1}{35}}Log{\frac{M^{2}}{\Phi_{0}^{2}}}(1+{\frac{70}{63}}{\frac{1}{\Phi_{0}^{2}a^{2}}})+...
\epsilon^{\mu\nu\rho\sigma}\partial_{\nu}R_{\rho\sigma}=\epsilon^{\mu\nu\rho\sigma}\partial_{\nu}(\vec{\Gamma}_{\rho}\times\vec{\Gamma}_{\sigma})\vec{T}=-\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\partial_{\nu}(\partial_{\rho}\Gamma_{\sigma}-\partial_{\sigma}\Gamma_{\rho})=0
\begin{array}{l}{<\chi_{+}\mid\hat{\Phi}(\zeta)\bar{\hat{\Phi}}(\zeta)\mid\chi_{+}>\equiv\Phi_{+}^{2}(\zeta)=\quad\Phi_{r}^{(\lambda,1,\alpha)}(\zeta)\Phi_{(\lambda,1,\alpha)}^{r}(\zeta),}\\{<\chi_{-}\mid\hat{\Phi}(\zeta)\bar{\hat{\Phi}}(\zeta)\mid\chi_{-}>\equiv\Phi_{-}^{2}(\zeta)=\Phi_{r}^{(\lambda,2,\alpha)}(\zeta)\Phi_{(\lambda,2,\alpha)}^{r}(\zeta).}\\\end{array}
\omega_{b}^{~a}={\nabla}u_{b}^{~\underline{b}}u_{\underline{b}}^{~a}\equiv(du_{b}^{~\underline{b}}+u_{b}^{~\underline{a}}\Omega_{\underline{a}}^{~\underline{b}})u_{\underline{b}}^{~a}\equiv\hat{\Omega}_{b}^{~a}.
U(t)=exp\left\{\int_{0}^{t}\,\frac{dt}{(4\pi)^{2}}\,\left[6\,f(t)-\frac{9}{2}\,g^{2}(t)-\frac{3}{2}\,g^{\prime2}(t)+\sum_{i}\,N_{i}\,h_{i}^{2}(t)\right]\right\}\,\,,
\hat{\zeta}_{a}|\zeta\rangle=\zeta_{a}|\zeta\rangle\quad\quad\hat{\eta}^{a}|\zeta\rangle=+\frac{\partial_{r}}{\partial\zeta_{a}}|\zeta\rangle\qquad\qquad\langle\zeta|\hat{\zeta}_{a}=\langle\zeta|\zeta_{a}\quad\quad\langle\zeta|\hat{\eta}^{a}=-\frac{\partial_{r}}{\partial\zeta_{a}}\langle\zeta|.
\begin{array}{l}{N=5,\quad\alpha=5,2,1,-1,}\\{\displaystyleT(f)=f\partial_{t}+\frac{1}{5}f^{\prime}(x\partial_{x}+2y\partial_{y})}\\{\displaystyle\qquad\qquad\qquad-\biggl(\frac{1}{5}wf^{\prime}+\frac{4}{25}y^{2}f^{\prime\prime}\biggr)\partial_{w},}\\{\displaystyleY(g)=g\partial_{y}-\frac{4}{5}g^{\prime}y\partial_{w},\quadX(h)=h\partial_{x},}\\{\displaystyleW(k)=k\partial_{w},}\\\end{array}
M_{ADM}^{(b)}(r_{h})=\frac{r_{h}}{2}+\frac{b^{2}r_{h}}{3}(r_{h}^{2}-\sqrt{r_{h}^{4}+a^{4}})+\frac{2q^{2}}{3}\int_{r_{h}}^{\infty}{\frac{ds}{\sqrt{a^{4}+s^{4}}}},
w_{n}=\sum_{m=0}^{n-2}w_{m}w_{n-2-m}-g^{2}w_{n+2}-Jw_{n-1}+f\sum_{p=1}^{\infty}\sum_{q=0}^{\infty}(-\lambdag)^{p+q}~{}_{p+q-1}C_{q}w_{n+p-2}w_{q},
j^{+}=\left(\begin{array}{ccc}{0}&{-1}&{0}\\{0}&{0}&{0}\\{-1}&{0}&{0}\\\end{array}\right)~~~j^{-}=\left(\begin{array}{ccc}{0}&{0}&{-1}\\{1}&{0}&{0}\\{0}&{0}&{0}\\\end{array}\right)
\delta\left(\begin{matrix}{x^{0}}\\{x^{1}}\\{{\bf1}}\\\end{matrix}\right)=\left(\begin{matrix}{a}&{b}&{u^{0}}\\{q^{-1}b^{\mathrm{T}}}&{a^{\mathrm{T}}}&{u^{1}}\\{0}&{0}&{{\bf1}}\\\end{matrix}\right)\otimes\left(\begin{matrix}{x^{0}}\\{x^{1}}\\{{\bf1}}\\\end{matrix}\right),
\begin{array}{ll}{z^{\prime}=z+\varepsilon\zeta\theta;}&{\theta^{\prime}=\theta+\varepsilon\zeta}\\{\bar{z}^{\prime}=\bar{z}-\bar{\varepsilon}\bar{\zeta}\bar{\theta};}&{\bar{\theta}^{\prime}=\bar{\theta}+\bar{\varepsilon}\bar{\zeta}.}\\\end{array}
A={\frac{\eta_{0}a_{0}+\cdots+\eta_{r}a_{r}}{\sqrt{\eta_{0}^{2}+\cdots+\eta_{r}^{2}}}},\qquad\widetilde{A}={\frac{\eta_{0}a_{0}^{\dagger}+\cdots+\eta_{r}a_{r}^{\dagger}}{\sqrt{\eta_{0}^{2}+\cdots+\eta_{r}^{2}}}}.
S_{\mathrm{NH}}^{(4)}=\frac{1}{8\piG_{N}}\intdTd\rhod\Omega_{2}\sqrt{-\gamma}\Biggl\{\gamma^{ab}\nabla_{a}\Phi\nabla_{b}\Phi+\frac{1}{2}R^{(2)}\Phi^{2}\Biggr\}.
{\calZ}_{2}^{(2)}=\int(J^{3})^{8}d^{8}\psi_{3}^{a}d^{8}\omega^{i-}dA_{3}^{+}dA_{3}^{-}{\frac{(A_{3}^{+})^{10}}{Pf(A_{1}^{i}A_{2}^{j}\gamma^{ij})}}Z^{\prime},
\{6j\}(\sigma_{F},R)\equiv\left\{\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{3}}\\{J_{1}+R}&{J_{2}+R}&{J_{3}+R}\\\end{array}\right\}\:\:\:\begin{array}{c}{\longrightarrow}\\{R\gg1}\\\end{array}\:\frac{(-1)^{\Phi}}{\sqrt{2R}}\:\left[\begin{array}{ccc}{j_{1}}&{j_{2}}&{j_{3}}\\{\mu_{1}}&{\mu_{2}}&{\mu_{3}}\\\end{array}\right],
F^{\prime}=\int\!dx~\left(J^{1~j}\Phi^{j}+{\frac{i}{2}}\bar{\chi}^{j}\gamma^{1}X^{j}-{\frac{i}{2}}(\bar{\chi}^{j}+\bar{X}^{j}\gamma^{p})\gamma^{1}\varepsilon^{jkl}\Phi^{k}(\chi^{l}-\gamma^{p}X^{l})\right).
\tilde{C}=-3g^{2}N(N^{2}-1)\delta^{ab}\int\frac{d^{4}k}{(2\pi)^{4}}d^{4}\theta\,\frac{1}{(k^{2}+m^{2})}\left\{V^{a}(p,\theta,{\overline{\theta}})V^{b}(-p,\theta,{\overline{\theta}})\right\}\,.
\psi(p_{0},\vec{p})=-\frac{1}{2\pii}\int_{w_{+}(\vec{p})}^{\infty}dq_{0}\frac{f(q_{0},\vec{p},W)}{p_{0}-q_{0}+io}+\frac{1}{2\pii}\int_{-\infty}^{w_{-}(\vec{p})}dq_{0}\frac{g(q_{0},\vec{p},W)}{p_{0}-q_{0}-io};
\left(-\frac{h^{2}}{8\pi^{2}M_{1}}\bigtriangledown_{1}^{2}-\frac{h^{2}}{8\pi^{2}M_{2}}\bigtriangledown_{2}^{2}+V_{n}(r)\right)\psi_{\mathrm{vib-rot}}=E_{t}\psi_{\mathrm{vib-rot}}~,
d\ell^{2}=-\varepsilon\frac{df^{2}}{\prod_{i\neqj,j+1}^{4}(z_{j}-z_{i})f(f-2\hat{\eta})}-\prod_{i\neqj,j+1}^{4}(z_{j}-z_{i})f(f-2\hat{\eta})d\psi^{2}
L_{inv}\equivL_{eff}+L_{WZ}=-{\frac{e^{2}}{4\pi}}\,\left[\,F_{\mu\nu}{\frac{1}{\Box}}\,F^{\mu\nu}-2\,{\bfa}\,A^{\mu}A_{\mu}\,-4\,{\bfa}\,\phi(x)\,\partial_{\mu}A^{\mu}+2\,{\bfa}\,\phi\,\Box\,\phi\,\right]
S_{III}\leq\left(\frac{2\pi-\sigma_{0}}{2\pi}\right)^{2\nu-1}\frac{2\pi-\sigma_{0}}{\pi}~\sum_{n\neq0}^{}\left(\frac{r}{|n|}\right)^{2\nu-1}\!|\phi_{n}(2\pi-\sigma_{0})|^{2}~S_{I}
\sum_{j=2}^{3}\phi_{j}^{a}f_{tt}^{j}=-\sum_{j=2}^{3}[2\phi_{jt}^{a}f_{t}^{j}+\sum_{k=2}^{3}(\phi_{jk}^{a}f_{t}^{k})f_{t}^{j}]-\phi_{tt}^{a}\quad\quad\quada=1,2,3
\begin{array}{rcl}{\hat{R}^{\hat{\mu}\hat{\nu}}}&{=}&{\hat{\Lambda}\hat{g}^{\hat{\mu}\hat{\nu}}-\hat{\chi}\left[g^{\rho\sigma}\delta_{\rho}{}^{\hat{\mu}}\delta_{\sigma}{}^{\hat{\nu}}-{\textstyle\frac{1}{\hat{d}-2}}\hat{g}^{\hat{\mu}\hat{\nu}}(\hat{g}^{\rho\sigma}\hat{g}_{\rho\sigma})\right]\sum_{n}T_{n}\delta(y-y_{n})\,,}\\{}&{}&{}\\{R^{\mu\nu}}&{=}&{\Lambdag^{\mu\nu}\,.}\\\end{array}
{\bfF}(\Box)\approx\frac{l^{2}\Box}{2}\left[\begin{array}{cr}{1}&{1/J_{2}[l\sqrt\Box/a]}\\{1/J_{2}[l\sqrt\Box/a]}&{-\frac2{l\sqrt{\Box}a}\frac{J_{1}[l\sqrt\Box/a]}{J_{2}[l\sqrt\Box/a]}\,}\\\end{array}\right].
F_{\mu\nu}\quad\Rightarrow\quad{\bfG}_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{(a)\,\ast}-\partial_{\nu}A_{\mu}^{(a)\,\ast}-i{\frac{e}{\hbar}}[A_{\mu}^{(b)},A_{\nu}^{(c)}]\quad,
\sqrt{-g}g^{\mu_{1}\nu_{1}}g^{\mu_{2}\nu_{2}}\cdotsg^{\mu_{d-p}\nu_{d-p}}{\tilde{F}}_{\nu_{1}\nu_{2}\ldots\nu_{d-p}}=\frac{1}{p!}\epsilon^{\mu_{1}\mu_{2}\ldots\mu_{d-p}\nu_{1}\nu_{2}\ldots\nu_{p}}F_{\nu_{1}\nu_{2}\ldots\nu_{p}},
\mathrm{Re}\Gamma_{\mathrm{b}}^{(2,1)}=1+gq_{3}^{-\varepsilon}\left(\frac{4}{3}-\frac{1}{3}\delta\right)\left(-\frac{3}{8\pi^{2}\varepsilon}+\mathrm{Q}+{\calO}(\varepsilon)\right)\,,
F_{8}^{{\mathcal{O}}_{1,1}^{0,1}|+++++---}=\frac{-iH^{1,1}(\sigma_{3}^{-})^{1/2}((\sigma_{1}^{-})^{2}\sigma_{4}^{+}+(\sigma_{3}^{-})^{2}+\sigma_{3}^{-}\sigma_{1}^{-}\sigma_{2}^{+})/(\sigma_{5}^{+})^{3/2}}{2^{7}\prod_{1\leqi<j\leq5}\mathrm{cosh}(\theta_{ij}/2)\prod_{6\leqi<j\leq8}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
\vartheta_{1}\bigl(\frac{w_{1}+\bar{w}_{2}}{2\piit}|\frac{i}{t}\bigr)=\vartheta_{1}\bigl(\frac{\tau_{12}-i\delta}{2\pit}-\frac{i}{2t}|\frac{i}{t}\bigr)=-ie^{\frac{\pi}{4t}+i\frac{\tau_{12}}{2t}}\vartheta_{4}\bigl(\frac{\tau_{12}-i\delta}{2\pit}|\frac{i}{t}\bigr)
R_{XYZ}{}^{W}=f_{iA}^{W}f_{Z}^{iB}{\calR}_{XYB}{}^{A}\,,\qquad\delta_{j}^{i}\,{\calR}_{XYB}{}^{A}=f_{W}^{iA}f_{jB}^{Z}\,R_{XYZ}{}^{W}\,,
E_{0}=-{\frac{1}{2}}\,\sum_{n=-\infty}^{\infty}\sqrt{\left({\frac{(2\,n+1)\,\pi}{2\,L}}\right)^{2}+m^{2}}=-\sum_{n=0}^{\infty}\sqrt{\left({\frac{(2\,n+1)\,\pi}{2\,L}}\right)^{2}+m^{2}}.
J^{N}=\intd^{2}{\bfx}[\epsilon_{ij}x_{i}({\frac{i}{2}}(\phi^{*}\partial_{j}\phi-\phi(\partial_{j}\phi)^{*})-{\frac{k}{2}}\epsilon_{mn}A_{m}\partial_{j}A_{n})+{\frac{k}{2}}A_{j}A_{j}]
\Phi_{Gn}^{\mathrm{out}}=\frac{e^{-i\omega_{n}^{\mathrm{out}}t}}{\sqrt{2\pi\omega_{n}^{\mathrm{out}}}}e^{ik_{n}^{\mathrm{out}}z}e^{im\phi}J_{\alpha|m|}(k_{n}^{\mathrm{out}}r)\,,
L_{fk}=\frac{i}{2}\overline{{\psi}}[\gamma^{a}V_{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}V_{a}^{\mu}]\psi
e^{\mu}=\frac{R^{2}}{\alpha^{2}}=\frac{\alpha^{2}f_{0}^{2}[x^{5}+(t-t_{0})/\alpha]^{2}}{R_{0}^{2}+\alpha^{2}f_{0}[x^{5}+(t-t_{0})/\alpha]^{2}},
\begin{array}{c}{u(\sigma^{+},\sigma^{-})=u_{0}+\frac{p^{+}}2\left[(\sigma^{+}+f(\sigma^{+}))+(\sigma^{-}-g(\sigma^{-}))\right]}\\{v(\sigma^{+},\sigma^{-})=v_{0}+\frac{p^{-}}2\left[(\sigma^{+}-f(\sigma^{+}))+(\sigma^{-}+g(\sigma^{-}))\right]}\\\end{array}
\widehat{S}=-\frac{1}{2}\gamma^{(0)}\otimes1,\quad\widehat{N}=\frac{1}{2}1\otimes\sigma_{3},\quad\widehat{T}{}^{i}=-\frac{1}{2}\gamma^{(i)}\otimes\sigma_{1},\quad\widehat{K}{}^{i}=-\frac{1}{2}\gamma^{(i)}\otimes\sigma_{2}.
({\gamma}^{\mu}i{\partial}_{\mu}-m_{1}){\psi}_{1}(x,t)-\frac{1}{2}e_{1}b(t){\gamma}^{1}{\psi}_{1}(x,t)+e_{1}\int_{0}^{\mathrm{L}}dyD(x,y|\mathrm{L})J^{0}(y,t){\gamma}^{0}{\psi}_{1}(x,t)=0,
\alpha^{2}=\left(1-\left({\frac{a}{r}}\right)^{3}\right)^{-1},\quad\beta^{2}=\frac19r^{2}\left(1-\left({\frac{a}{r}}\right)^{3}\right),\quad\gamma^{2}=\frac{\displaystyler^{2}}{12}
\varepsilon=-\frac{1}{2\pi}\int_{m}^{\infty}dz\frac{4\xiz^{2}-m^{2}}{\sqrt{z^{2}-m^{2}}}\frac{\tilde{B}_{a}z+\tilde{A}_{a}}{\tilde{B}_{a}z-\tilde{A}_{a}}e^{-2z(r-a)},\quadp=0,\quadD=1.
\begin{array}{rl}{\displaystyle(s_{\rho}\psi)^{A}\,(X_{1},...,X_{q{+}1})}&{:=\displaystyle\sum_{i{=}1}^{q{+}1}(-1)^{i+1}\rho(X_{i})_{.B}^{A}\,(\psi^{B}(X_{1},...,{\hat{X}}_{i},...,X_{q{+}1}))}\\{}&{\displaystyle+\sum_{{j,k=1\atopj<k}}^{q{+}1}(-1)^{j{+}k}\psi^{A}([X_{j},X_{k}],\X_{1},...,{\hat{X}}_{j},...,{\hat{X}}_{k},...,X_{q{+}1})\quad.}\\\end{array}
\frac{E^{i}}{c}=-\frac{\partial}{\partialx^{i}}\frac{\Phi}{c}-\frac{1}{c}\frac{\partial}{\partialt}A^{i}=F^{i0}\\,\\B^{i}=\epsilon^{ijk}\left[\frac{\partial}{\partialx^{j}}A^{k}-\frac{\partial}{\partialx^{k}}A^{j}\right]=-\epsilon^{ijk}F^{jk}\\,\\i,j,k=1,2,3,
Y_{m}(\mu,y)\approx\sqrt{\frac{\mu+\omega_{m}}{2\mu}}\frac{m}{2\omega_{m}}B_{m}=\frac{1}{2\sqrt{2\mu}}\frac{\sqrt{\mu+\sqrt{m^{2}+\mu^{2}}}}{\sqrt{m^{2}+\mu^{2}}}mB_{m},
S_{E}=-\frac{1}{16\piG_{d-1}}\int_{\Sigma}d^{d-1}x\sqrt{g}\left[R-me^{\frac{4\phi}{\sqrt{d-2}(d-3)}}-\frac{4}{d-3}(\partial\phi)^{2}-\frac{1}{4}e^{\frac{-4\sqrt{d-2}}{d-3}\phi}F^{2}\right],
\omega^{(p)}\equiv{\frac{1}{(p+1)!}}\;\omega_{\mu_{0}\;\cdots\;\mu_{p}}^{(p)}\;dY^{\mu_{0}}\wedge\cdots\wedgedY^{\mu_{p}}=\sqrt{-\widehatg}\;d\zeta^{0}\wedge\cdots\wedged\zeta^{p}
E[\zeta_{a},\bar{\zeta_{b}}]=\lambda^{2}\frac{2}{2j+1}\left[kj-2j^{2}\sum_{a,b,c,d=1}^{k}[h_{ab}(h^{-1})_{bc}h_{cd}(h^{-1})_{da}]\frac{(\bar{\zeta}_{d}\zeta_{a}+\bar{\zeta}_{b}\zeta_{c}-2\bar{\zeta}_{d}\zeta_{c})}{(1+\bar{\zeta}_{d}\zeta_{a})(1+\bar{\zeta}_{b}\zeta_{c})}\right]
Z_{N}^{inf}(x_{1},\cdots,x_{M})\equiv(x_{1}+x_{2}\cdots+x_{M})^{N}=\sum_{\lambda\atop{|\lambda|=N}}{\frac{N!}{\lambda_{1}!\cdots\lambda_{M}!}}~m_{\lambda}(x_{1},\cdots,x_{M})\,\,\,\,.
\begin{array}{l}{{\calD}_{+}{\calX}^{+}-{\calX}^{+}{\calD}_{+}=\mu_{+}~,~~~~~~\mu_{+}{\calX}^{+}=q^{2}{\calX}^{+}\mu_{+}~,}\\{\mu_{+}{\calD}_{+}=q^{-2}{\calD}_{+}\mu_{+}~,~~~~~~\mu_{+}\equiv1+(q^{2}-1){\calX}^{+}{\calD}_{+}~,}\\\end{array}
\pi_{i}=g_{ij}{\dot{x}}^{j}=p_{i}-\frac{i}{2}g_{ij,k}\psi^{j}\psi^{k}-\frac{i}{2}A_{i\alpha\beta}\lambda^{\alpha}\lambda^{\beta}+\frac{i}{2}c_{ijk}\psi^{j}\psi^{k}\quad.
S_{f}=T\intd^{2}x(i\;\partial_{-}\psi^{i}\psi^{i}+{\frac{1}{N}}\;b_{\alpha}^{i}b_{\alpha}^{i})\;;\;\;\;b_{\alpha}^{i}\equivD_{\alpha}\Psi^{i}|_{\theta=0}\;,
S=\int\frac{\mathrm{tr}\,\hat{F}_{\mu\nu}^{2}}{2g^{2}}\,d^{4}x=\int\frac{\mathrm{tr}\,\hat{F}_{\alpha\beta}\,\hat{F}^{\alpha\beta}}{2g^{2}}\,\sqrt{g}\,d^{4}q.
L_{l-\varepsilonN}={\frac{\nu-|l|+\beta-2\beta\eta_{l}}{-\nu-|l|+\beta-2\beta\eta_{l}}},\quad\eta_{l}={\frac{\Phi^{\prime}(A_{1},B_{1};\;\beta)}{\Phi(A_{1},B_{1};\;\beta)}}\,.
(\nabla_{\betar}v_{\alphaq}^{~\underline{\beta}})v_{\underline{\beta}}^{~\gammaq^{\prime}}={\frac{1}{4}}(\nabla_{\betar}u_{a}^{~\underline{a}})u_{\underline{a}}^{~i}(\gamma^{a})_{\alpha}^{~\gamma}(\gamma_{i})_{q}^{~q^{\prime}}\equiv{\frac{1}{4}}\hat{\Omega}_{\betar,a}^{~~~~i}(\gamma^{a})_{\alpha}^{~\gamma}(\gamma_{i})_{q}^{~q^{\prime}}.
0=m^{2}Q+\Box\,(4\sigma\frac{\partial^{2}Q}{\partial\sigma^{2}}+\sigma^{\prime}\,\frac{\partial^{2}Q}{\partial\tau^{2}}+4\tau\,\frac{\partial^{2}Q}{\partial\sigma\partial\tau}+2(d-1)\frac{\partialQ}{\partial\sigma}).
ds^{2}=-dt^{2}+e^{2\alpha(t)}\left[\left(\mathbf{\omega}^{1}\right)^{2}+e^{-2\sqrt3\beta_{s}(t)}\left(\mathbf{\omega}^{2}\right)^{2}+e^{2\sqrt3\beta_{s}(t)}\left(\mathbf{\omega}^{3}\right)^{2}\right],
\Omega^{2}=\frac{v^{(1)}v^{(2)}}{\tau_{2}^{(1)}\tau_{2}^{(2)}}\Bigl((\tau_{1}^{(1)}+\tau_{1}^{(2)})^{2}+(\tau_{2}^{(1)}-\tau_{2}^{(2)})^{2}\Bigr)-\Bigl(v^{(1)}\frac{\tau_{1}^{(1)}}{\tau_{2}^{(1)}}+v^{(2)}\frac{\tau_{1}^{(2)}}{\tau_{2}^{(2)}}\Bigr)^{2}.
W=\frac{1}{3}W_{0}\frac{H}{H_{0}}(\alpha\mu)^{2}f(\mu)\left(\delta_{\zeta,\zeta^{\prime}}\frac{1-\zeta}{2}S_{0}+\delta_{\zeta,-\zeta^{\prime}}\frac{\alpha}{4}\frac{1+\zeta}{2}S_{1}\right)\,.
T_{\mu\nu}=\frac12(F_{\mu\alpha}F_{\,\,\nu}^{\alpha}+F_{\nu\alpha}F_{\,\,\mu}^{\alpha})+\frac12(D_{\mu}\phi^{*}D_{\nu}\phi+D_{v}\phi^{*}D_{\mu}\phi)-\eta_{\mu\nu}{\calL}
S^{ab}=\tilde{S}^{ab}+\tilde{\tilde{S}}{}^{ab},\quad\tilde{S}^{ab}:=-\frac{i}{4}[\tilde{a}^{a},\tilde{a}^{b}],\quad\tilde{\tilde{S}}{}^{ab}:=-\frac{i}{4}[\tilde{\tilde{a}}{}^{a},\tilde{\tilde{a}}{}^{b}],
f(u)=\theta_{1}(2u)\,\frac{{[}\theta_{1}\theta_{4}{]}(u+\eta)}{{[}\theta_{2}\theta_{3}{]}(u)}\quad\mathrm{and}\quadC(u)={[}\theta_{2}\theta_{3}\theta_{4}{]}(0)\,{[}T_{2}^{(+)}T_{3}^{(+)}T_{2}^{(-)}T_{3}^{(-)}{]}(n|u).
\hat{z}\equiv\frac{m_{\mathrm{\scriptscriptstyleMT}}^{2}}{4\piT^{2}}\left(1+\frac{g_{\mathrm{\scriptscriptstyleMT}}^{2}}{\pi}\right)\left(\frac{2T}{m_{\mathrm{\scriptscriptstyleMT}}}\right)^{(1+g_{\mathrm{\scriptscriptstyleMT}}^{2}/\pi)^{-1}}.
\begin{array}{c}{lim}\\{\theta\rightarrow\theta_{0}^{+}}\\\end{array}\frac{\partial}{\partial\theta}ImI^{\prime}=\begin{array}{c}{lim}\\{\theta\rightarrow\theta_{0}^{-}}\\\end{array}\frac{\partial}{\partial\theta}ImI^{\prime}=\int_{0}^{1}dx\{-\frac{\varepsilon}{B^{2}+\varepsilon^{2}}\frac{\partialB}{\partial\theta}\}\Vert_{_{\theta_{0}}}
{\mathcalD}^{-1}={\mathcalD}_{0}^{-1}-{\mathcalD}_{0}^{-1}{\mathcalV}{\mathcalD}_{0}^{-1}+{\mathcalD}_{0}^{-1}{\mathcalV}{\mathcalD}_{0}^{-1}{\mathcalV}{\mathcalD}_{0}^{-1}-...
\begin{array}{l}{h_{3}^{A}=h_{2}^{B}\partial_{B}h_{1}^{A}-i\bar{\lambda}_{2i}\gamma^{A}\lambda_{1}^{i}-(1\leftrightarrow2)}\\{{}}\\{\lambda_{3}^{i\alpha}=h_{2}^{A}\partial_{A}\lambda_{1}^{i\alpha}+\lambda_{2}^{j\beta}D_{j\beta}\lambda_{1}^{i\alpha}-(1\leftrightarrow2)}\\\end{array}
ds_{bubble}^{2}=d\chi^{2}-dT^{2}+dW^{2}+\frac{1-A^{2}/r_{0}^{2p-2}}{(\sqrt{W^{2}-T^{2}}/r_{0})^{2p-2}-1}\frac{(WdW-TdT)^{2}}{W^{2}-T^{2}}+W^{2}d\Omega_{p-2}^{2}\.
P_{\hat{m}}=\frac{1}{16\piG_{10}}\oint_{\infty}d\Sigma^{\hat{n}}(\partial_{\hat{n}}h_{\hat{0}\hat{m}}-\partial_{\hat{0}}h_{\hat{n}\hat{m}}+\delta_{\hat{n}\hat{m}}\partial_{\hat{0}}h_{\hat{p}\hat{p}}-\delta_{\hat{n}\hat{m}}\partial_{\hat{p}}h_{\hat{0}\hat{p}})
\Lambda_{i}^{3r_{i}}=\mathrm{det}(a_{i}^{2}-m^{2})\phi(a_{i})^{r_{i}}\prod_{j\nei}(a_{j}^{2}-a_{i}^{2})^{r_{i}-2r_{j}}\Lambda_{N=2}^{4(N_{c}-1)-2N_{f}}
B=\left(\begin{matrix}{0}&{\beta}&{0}&{0}&{0}&{0}&{0}\\{\beta}&{0}&{-p_{1}}&{0}&{0}&{0}&{0}\\{0}&{p_{1}}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{-p_{2}}&{0}&{0}\\{0}&{0}&{0}&{p_{2}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{-p_{3}}\\{0}&{0}&{0}&{0}&{0}&{p_{3}}&{0}\\\end{matrix}\right)~,