harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Psi_{\mathrm{CT}}=\frac{1}{2e^{2}}\intd^{4}x\,\sqrt{g}\,\Bigr\{\xi_{-}^{\nu}(\nabla^{\mu}T_{\mu\nu}^{+}-iV^{\mu}T_{\mu\nu}^{+}-Y_{\nu}^{+})+(\nabla^{\mu}T_{\mu\nu}^{-}-iT_{\mu\nu}^{-}V^{\mu}-Y_{\nu}^{-})\xi_{+}^{\nu}\Bigr\},
\begin{array}{rcl}{fe}&{=}&{\langlee,S(f)\rangle\langle1,{\bar{t}}^{-1}\rangle1\cdot\bar{t}^{-1}}\\{}&{}&{+\langlet,S(1)\rangle\langle1,\bar{t}^{-1}\rangleef+\langlet,S(1)\rangle\langlee,f\ranglet\cdot1}\\{}&{=}&{-\langlee,f\rangle{\bar{t}}^{-1}+ef+\langlee,f\ranglet~.}\\\end{array}
\Phi(x,y)={\frac{1}{R}}\sum_{n_{1},n_{2}}\int{\frac{d^{3}k}{(2\pi)^{3}2\omega(k)_{n_{1},n_{2}}}}[a(k)_{n_{1},n_{2}}e^{-\imathk.x}e^{\imath{\frac{2\pi}{R}}(n_{1}y_{1}+n_{2}y_{2})}+h.c.]
\left.\;J^{(N)}\left(n;\nu_{1},\ldots,\nu_{N}\right)\right|_{\begin{array}{l}{{}_{\!\!\Sigma\nu_{i}=n}}\\{{}^{\!\!c_{jl}=\delta_{jl}}}\\\end{array}}=\mathrm{i}^{1-2\Sigma\nu_{i}}\;\pi^{n/2}\;\frac{\prod\Gamma\left({\textstyle{\frac{1}{2}}}\nu_{i}\right)}{2^{N-1}\prod\Gamma\left(\nu_{i}\right)}\;\frac{1}{\prodm_{i}^{\nu_{i}}}.
\langleV_{\alpha,f_{\alpha}}^{B}V_{\beta,f_{\beta}}^{F}V_{\gamma,f_{\gamma}}^{F}\rangle=\langleV^{B}V^{F}V^{F}\rangle\langle\sigma_{\alpha,f_{\alpha}}\sigma_{\beta,f_{\beta}}\sigma_{\gamma,f_{\gamma}}\rangle\;.
\Xi(\kappa)={\frac{1}{4\sqrt{\pi}}}\int_{0}^{\infty}{\frac{ds}{s^{3/2}}}\left(k_{H}(s)\Theta_{S}({\frac{s}{\kappa}})-\sqrt{\frac{\pi}{s}}(({\frac{1}{3}}-{\frac{s}{20}})\kappa+{\frac{s}{9}})\right)e^{-{s/4}}~~.
\Sigma_{1+2}=\intd^{3}x\frac{1}{2}\left(\begin{array}{cc}{A^{\mu}}&{\varphi}\\\end{array}\right)\left[\begin{array}{cc}{M_{\mu\nu}}&{T_{\mu}}\\{-T_{\nu}}&{-\square}\\\end{array}\right]\\left(\begin{array}{c}{A^{\nu}}\\{\varphi}\\\end{array}\right).
{\calH}=\frac{R}{g^{2}}\sum_{\nu=1,2,3,4,7}N^{\nu}\Gamma_{\nu}\;,\;R\equiv\sqrt{(X^{1})^{2}+(X^{2})^{2}+(X^{3})^{2}+(X^{4})^{2}+(X^{7})^{2}}\;,\;N^{\nu}\equiv\frac{X^{\nu}}{R}\;,
f^{-}(z)={\frac{1}{P^{+}}}\int_{0}^{z}\left({\frac{\alpha}{2}}{f^{\prime}}_{\mathrm{tr}}^{2}+{\frac{\beta}{2}}{h^{\prime}}^{2}-2\betah^{\prime\prime}\right)\,dz^{\prime}\;+\;c^{-}\;\;\;.
{\calL}_{eff}=\frac{R}{2k^{2}}+\left(\frac{1}{4g^{2}(\phi)}\right)_{ab}F_{\mu\nu}^{\alpha}F^{\mu\nu\;\;b}+i\frac{{\Theta(\phi)}}{32{\pi}^{2}}F_{\mu\nu}^{\alpha}{\tilde{F}}^{\mu\nu\;\;b}+\frac{1}{2}G_{ij}(\phi)D^{\mu}{\phi}^{i}D_{\mu}{\phi}^{j}+V(\phi).
\widetilde\Pi_{(\widetildep)}\Pi_{(p)}^{-1}\;=\;-(\mathrm{cos}\theta\;\Gamma^{6}+\mathrm{sin}\theta\;\Gamma^{8})(\mathrm{cos}\theta\;\Gamma^{7}+\mathrm{sin}\theta\;\Gamma^{9})\;\Gamma^{6}\Gamma^{7}\.
S=\frac1{g^{2}}\intd^{10}x\Bigl\{-\frac14F_{\mu\nu}^{(r)}F^{(r)\mu\nu}+\fraci2\bar{\psi}^{(r)}\Gamma^{\mu}\partial_{\mu}\psi^{(r)}-|D_{\mu}T|^{2}-V(T)\Bigr\},
\hat{V}-V=\frac{6^{\frac{1}{3}}}{4}q_{I}(\hat{G}^{IJ}-G^{IJ})q_{J}=-\frac{1}{8\cdot6^{\frac{1}{3}}}t^{2}\:(q_{I}G^{I2})^{2}\rightarrow0\,\qquadt^{2}\rightarrow0.
\hat{S}_{MCS}=\intd^{3}x[-\frac{1}{2}(\partial_{\alpha}(\hat{A}+\hat{a})_{\beta}-\partial_{\beta}(\hat{A}+\hat{a})_{\alpha})*\partial^{\alpha}(\hat{A}+\hat{a})^{\beta}+\frac{2\pi}{g^{2}}\epsilon^{\alpha\beta\gamma}(\hat{A}+\hat{a})_{\alpha}*\partial_{\beta}(\hat{A}+\hat{a})_{\gamma}].
\frac{G_{N}\,M}{(R_{s}^{\mathrm{rw}})^{d-2}}\sim\left(\frac{R_{\mathrm{BH}}(M)}{R_{s}^{\mathrm{rw}}}\right)^{d-2}\simg^{2}\left(\frac{M}{M_{s}}\right)^{\frac{4-d}{2}}
i\left<\bar{A}_{\mu}^{i}\bar{A}_{\nu}^{j}\tilde{A}_{\rho}^{a}\tilde{A}_{\sigma}^{b}\right>_{\mathrm{bare}}=ig^{2}f^{aic}f^{cjb}\left[2g_{\mu\nu}g_{\rho\sigma}-\left(1-\frac1\alpha\right)(g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\nu\rho})\right].
e^{2}<A^{\mu}(x)A^{\nu}(y)>\approxi[{\frac{a}{a^{2}\partial^{2}+4\alpha^{2}}}g^{\mu\nu}\delta(x-y)+{\frac{2\alpha}{\partial^{2}(a^{2}\partial^{2}+4\alpha^{2})}}\epsilon^{\mu\nu\lambda}\partial_{\lambda}\delta(x-y).
M=\left(\begin{matrix}{-\frac{G}{r}-\frac{mp}{2G+1}}&{2\frac{\sqrt{G(G+1)}}{2G+1}mp}\\{2\frac{\sqrt{G(G+1)}}{2G+1}mp}&{\frac{G+1}{r}+\frac{mp}{2G+1}}\\\end{matrix}\right)\quad\mathrm{and}\quad\bar{M}=\left(\begin{matrix}{\frac{G+2}{r}+\frac{mp}{2G+1}}&{2\frac{\sqrt{G(G+1)}}{2G+1}mp}\\{2\frac{\sqrt{G(G+1)}}{2G+1}mp}&{-\frac{G-1}{r}-\frac{mp}{2G+1}}\\\end{matrix}\right)
\int{\calL}_{sfield}=\frac14\intd^{2}x\intd^{2}\theta~\overline{{D}}{\bf\Phi}^{j}{\bfN}^{jk}D{\bf\Phi}^{k}=\frac14\intd^{2}x\intd^{2}\theta~\overline{{D}}{\bf\Phi}^{j}({\bfG}^{jk}-\gamma_{p}{\bfE}^{jk})D{\bf\Phi}^{k}~.
\frac{\ddot{a}}{a}(t)=-\frac{4\piG_{N}}{3}\left[3(P_{\star}(t)+P(t)e^{-4kb_{0}}+\frac{\hat{P}_{B}}{k}(t)(1-e^{-2kb}))+\rho_{\star}(t)+\rho(t)e^{-4kb}+\frac{\hat{\rho}_{B}}{k}(t)(1-e^{-2kb_{0}})\right].
{\frac{A}{8\piG}}{\frac{\beta}{T}}in^{3}\delta_{m+n}=J[\{{\hat{\xi}}_{m},{\hat{\xi}}_{n}\}_{\mathrm{\scriptsize~SD}}]+K[{\hat{\xi}}_{m},{\hat{\xi}}_{n}]=i(n-m)J[{\hat{\xi}}_{m+n}]+K[{\hat{\xi}}_{m},{\hat{\xi}}_{n}].
G^{W}(\sigma)=\int\!d^{2}x\e\\sigma\langleT_{\\mu}^{\mu}\rangle=\frac{1}{48\pi}\int\!d^{2}x\e\\sigma\left(\mathcal{R}\mp\frac{1}{2}\varepsilon^{ab}\nabla_{\mu}\omega_{ab}^{\\\mu}\right)
U=\left(\begin{matrix}{\frac{1}{\sqrt{2}}}&{-\frac{1}{\sqrt{2}}}&{0}&{0}\\{\frac{i}{\sqrt{2}}}&{\frac{i}{\sqrt{2}}}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right)
r(\tau)=r_{o}+\frac{4(6H^{2}r_{o}^{2}-1)e^{-\sqrt{6H^{2}r_{o}^{2}-1}\;(\tau-\tau_{o})}}{H(e^{-\sqrt{6H^{2}r_{o}^{2}-1}\;(\tau-\tau_{o})}-4H^{2}r_{o}^{2})^{2}-4H(6H^{2}r_{o}^{2}-1)}.
\begin{array}{lll}{}&{}&{0<(\phi-\sqrt{\phi^{2}-1})^{1/4}\leqx_{1}\leq(\phi+\sqrt{\phi^{2}-1})^{1/4}\;}\\{}&{\mathrm{~or~}}&{\phantom{0}-(\phi+\sqrt{\phi^{2}-1})^{1/4}\leqx_{1}\leq-(\phi-\sqrt{\phi^{2}-1})^{1/4}<0\;,}\\\end{array}
A_{n}=\Phi_{1}^{a_{1}}\cdots\Phi_{n}^{a_{n}}\langle0\mid\intd\sigma_{1}K^{a_{1}}(\sigma_{1})e^{ik_{1}\cdotX(\sigma_{1})}\cdots\intd\sigma_{n}K^{a_{n}}(\sigma_{n})e^{ik_{n}\cdotX(\sigma_{n})}\midB\rangle.
\begin{array}{l}{N_{\alpha,\beta}^{}\,\Bigl(\,\zeta(u-v)\,-\,\zeta(u+v)\,+\,2\!\zeta(v)\Bigr)\,A_{k}^{\alpha+\beta}(q,u)}\\{\mathrm{}-\,N_{\alpha,\beta}^{}\,\Bigl(\Phi(\alpha(q),u-v)\,A_{k}^{\alpha+\beta}(q,v)\,-\,\Phi(\alpha(q),u+v)\,A_{k}^{\alpha+\beta}(q,-v)\Bigr)}\\{\mathrm{}=~0~.}\\\end{array}
\Omega_{16}^{(4)}\equiv\left(\begin{array}{cccc}{\frac{1+\mathrm{ch}}{2}}&{\frac{1-\mathrm{ch}}{2}}&{0}&{\frac{\mathrm{sh}}{\sqrt{2}}}\\{}&{}&{}&{}\\{\frac{1-\mathrm{ch}}{2}}&{\frac{1+\mathrm{ch}}{2}}&{0}&{-\frac{\mathrm{sh}}{\sqrt{2}}}\\{}&{}&{}&{}\\{0}&{0}&{1}&{0}\\{}&{}&{}&{}\\{\frac{\mathrm{sh}}{\sqrt{2}}}&{-\frac{\mathrm{sh}}{\sqrt{2}}}&{0}&{\mathrm{ch}}\\\end{array}\right)\,.
G^{ij}({\bfx},{\bfx^{\prime}};\omega)=\sum_{lmn\lambda}{\frac{A_{lmn}^{(\lambda)i}({\bfx})A_{lmn}^{(\lambda)j*}({\bfx^{\prime}})}{\omega_{ln}^{(\lambda)2}-\omega^{2}-i\epsilon}}
\overline{J}_{\dot{\alpha}}^{0}=-{\frac{i}{2}}\overline{\sigma}^{0\dot{\beta}\beta}(\overline{D}_{\dot{\alpha}}R_{\beta\dot{\beta}}-2\varepsilon_{\dot{\alpha}\dot{\beta}}\overline{D}^{\dot{\gamma}}R_{\beta\dot{\gamma}})\bigr|_{\theta=\overline{\theta}=0}.
\sigma_{\mathrm{abs}}=A_{bh}\;{\frac{\omega}{2(T_{R}+T_{L})}}{\frac{e^{\frac{\omega}{T_{H}}}-1}{(e^{\frac{\omega}{2T_{R}}}-1)(e^{\frac{\omega}{2T_{L}}}-1)}}[1+O(\omega^{2}r_{1}^{2})].
G(\omega,\vec{k})=-\frac{\phi_{1}f_{2}\partial_{r}\phi_{2}^{(-)}-\phi_{2}^{(-)}f_{1}\partial_{r}\phi_{1}}{\phi_{1}f_{2}\partial_{r}\phi_{2}^{(+)}-\phi_{2}^{(+)}f_{1}\partial_{r}\phi_{1}}\.
W(\lambda_{i})=\frac{\eta}{\lambda_{i}}+\frac{\sqrt{a\lambda_{i}+\eta^{2}}}{\lambda_{i}}+\frac{a}{2N}\sum_{j}\frac{1}{\sqrt{a\lambda_{j}+\eta^{2}}}\,\frac{1}{\sqrt{a\lambda_{j}+\eta^{2}}+\sqrt{\vphantom{\lambda_{j}}a\lambda_{i}+\eta^{2}}}.
|g,n\rangle=\sum_{i}(\sum_{j}{{C_{g_{1}}}^{i_{1}...i_{n_{1}\!-\!1}}}_{j}{C_{g_{2}}}^{j\,i_{n_{1}\!+\!2}...i_{n_{1}\!+\!n_{2}}})|i_{1}\rangle\otimes...\otimes|i_{n_{1}+n_{2}}\rangle
(D_{\bar{i}}Q_{i})({\overline{Q}}^{j}{\overline{D}}^{\bar{j}})\epsilon_{jk}\epsilon_{{\bar{j}}{\bar{k}}}\left(C_{a_{1}a_{2}}\Sigma_{a_{1}}^{i{\bar{i}}}\Sigma_{a_{2}}^{k{\bar{k}}}+C_{a_{1}a_{2}}^{\prime}\Sigma_{a_{1}}^{i{\bar{k}}}\Sigma_{a_{2}}^{k{\bar{i}}}\right)~,
L_{+}^{C}=\left(\begin{array}{ll}{(qt^{-1})^{S_{3}}}&{\OmegaS^{-}}\\{0}&{(qt)^{-S_{3}}}\\\end{array}\right)\\\,L_{-}^{C}=\left(\begin{array}{ll}{(qt)^{-S_{3}}}&{0}\\{-\OmegaS^{+}}&{(qt^{-1})^{S_{3}}}\\\end{array}\right)\,
S_{\Psi}^{M}=\intd^{4}x(\pi_{m}\dot{A}^{m}+\dot{c}\bar{\calP}-{\calH}_{0}+A_{0}\partial_{i}\pi^{i}+\pi_{0}\partial^{i}A_{i}+i\bar{\calP}{\calP}-i\bar{c}\squarec)\,,
\mathrm{for}\;\Omega\ne\frac{1}{3}:\;\;\;\;e^{A}=e^{A_{0}}\left(t_{s}^{\prime}-\mathrm{sign}(m)t^{\prime}\right)^{n}\;,\;\;\;\;\chi=\chi_{0}\left(t_{s}^{\prime}-\mathrm{sign}(m)t^{\prime}\right)^{m}\;,
\frac{1}{2}\int_{p}p^{2}K^{-1}(p^{2}e^{2\tau})\phi_{p}^{(0)}\phi_{-p}^{(0)}+S_{I}+\tau\left\{\int_{p}S_{I;-p}^{\prime}K^{\prime}(p^{2})S_{I;p}^{\prime}-\int_{p}K^{\prime}(p^{2})S_{I;-p,p}^{\prime\prime}\right\},
{\varphi}_{k}(j)=\left.\left(\frac{\partial}{\partialx}\right)^{k}\left[e^{x}\theta\left(\left.\frac{j}{l}-x\sqrt{\frac{2}{\pil}}\right|\tau=\frac{\mathrmi}{l}\right)\right]\right|_{x=0}\,\,\,\,\,\,j=0,1,\ldots,l-1;\,\,k=0,1,\,\ldots,
M_{\alpha\beta}=\left(\begin{array}{ccccc}{-\frac{1+f^{2}}{1-f^{2}}}&{\frac{2f}{1-f^{2}}}&{}&{}&{}\\{\frac{2f}{1-f^{2}}}&{-\frac{1+f^{2}}{1-f^{2}}}&{}&{}&{}\\{}&{}&{1}&{}&{}\\{}&{}&{}&{\ddots}&{}\\{}&{}&{}&{}&{1}\\\end{array}\right)~~.
\langle\psi({\bfp}_{1})\psi^{+}({\bfp}_{2})\rangle=\langle\phi({\bfp}_{2})\phi^{+}({\bfp}_{1})\rangle=f_{1}\!\left(\frac{|{\bfp}_{1}|}{|{\bfp}_{2}|}\right)\frac{1+z_{1}\bar{z}_{2}}{\sqrt{|{\bfp}_{1}||{\bfp}_{2}|}}\,\delta_{z\bar{z}}(z_{1}-z_{2})
K=\int\prod_{i=1}^{N}du_{i}\prod_{i<j}\mathrm{exp}\Bigl[G_{Bij}k_{i}\cdotk_{j}+i\dot{G}_{Bij}(k_{i}\cdot\varepsilon_{j}-k_{j}\cdot\varepsilon_{i})+\ddot{G}_{Bij}\varepsilon_{i}\cdot\varepsilon_{j}\Bigr]\,\,\,\bigg\vert_{\mathrm{multi-linear}}
\Gamma_{\mathit{eff}}^{\prime}[\phi]=i\int\frac{d^{4}k}{(2\pi)^{4}}\mathrm{tr}\langlek|\g\\phi(\hat{x})\S(\hat{p})+\frac{1}{2}\g\,phi(\hat{x})\S(\hat{p})\g\\phi(\hat{x})\S(\hat{p})+...|k\rangle.
U_{\mathrm{eff}}(\phi)=U(\phi)-\frac{1}{12}\left[\left({\bf{p}}^{2}+m^{2}\right)^{3/2}\Big|_{0}^{\Lambda}-\left({\bf{p}}^{2}+m^{2}+\frac{\lambda}{2}\phi^{2}+\frac{\sigma}{4!}\phi^{4}\right)^{3/2}\Big|_{0}^{\Lambda}\right]\;.
{\calL(}X,P_{\mu})=-\frac{\eta^{2}}{2}(\nabla_{\mu}X\,\nabla^{\mu}X+X^{2}P_{\mu}P^{\mu})-\frac{1}{16\pie^{2}}F_{\mu\nu}F^{\mu\nu}-\xi\eta^{4}(X^{2}-1)^{2}
\left(\begin{array}{c}{\tilde{F}^{(n)}{}_{\mu\nu}}\\{F^{(n)}{}_{\mu\nu}}\\\end{array}\right)\longrightarrow\Lambda\left(\begin{array}{c}{\tilde{F}^{(n)}{}_{\mu\nu}}\\{F^{(n)}{}_{\mu\nu}}\\\end{array}\right)\,,
\begin{array}{l}{R(\theta-\theta^{\prime})K_{1}^{+}(\theta^{\prime})^{t_{1}}M_{1}^{-1}R(-\theta-\theta^{\prime}-2\eta)K_{1}^{+}(\theta)^{t_{1}}M_{2}=}\\{K_{1}^{+}(\theta)^{t_{1}}M_{2}R(-\theta-\theta^{\prime}-2\eta)M_{1}^{-1}K_{1}^{+}(\theta^{\prime})^{t_{1}}R(-\theta-\theta^{\prime})}\\\end{array}
\check{R}_{i}\;=\;\left(\begin{array}{cccc}{\sqrt{\nu-1}+\frac{q+q^{-1}}{2}}&{0}&{0}&{\frac{\eta-\eta^{-1}}{2}}\\{0}&{\sqrt{\nu-1}-\frac{q-q^{-1}}{2}}&{\frac{\eta+\eta^{-1}}{2}}&{0}\\{0}&{\frac{\eta+\eta^{-1}}{2}}&{\sqrt{\nu-1}+\frac{q+q^{-1}}{2}}&{0}\\{\frac{\eta-\eta^{-1}}{2}}&{0}&{0}&{\sqrt{\nu-1}-\frac{q+q^{-1}}{2}}\\\end{array}\right).
\left\langle{\calO}_{\alpha_{1}}\ldots{\calO}_{\alpha_{n}}\cdot\widetilde{\calO}_{\widetilde{\beta}_{1}}\ldots\widetilde{\calO}_{\widetilde{\beta}_{s}}\right\rangle=\#\left(\widetilde{L}_{\alpha_{1}}\cap\ldots\cap\widetilde{L}_{\alpha_{n}}\cap\widetilde{L}_{\widetilde{\beta}_{1}}\cap\ldots\cap\widetilde{L}_{\widetilde{\beta}_{s}}\right),
\mathrm{I_{\mu\nu}^{(1)}=(\dots)n_{\mu}\int\frac{d^{4}q}{(q^{2}+i\epsilon)}\int\frac{d^{3}\vec{k}}{\vec{k}^{2}(\vec{p}-\vec{k}-\vec{q})^{2}}\int\frac{dk_{0}~k_{0}\,k_{\nu}}{(k^{2}+i\epsilon)}~,\qquadk\cdotn=k_{0}\,,}
K_{uv}^{x}\,k_{\mathbf{I}}^{u}\,k_{\mathbf{J}}^{v}\,-\,{\frac{\lambda}{2}}\,\varepsilon^{xyz}\,{\calP}_{\mathbf{I}}^{y}\,{\calP}_{\mathbf{J}}^{z}\,=\,{\frac{1}{2}}\,f_{\phantom{\mathbf{K}}\mathbf{I}\mathbf{J}}^{\mathbf{K}}\,{\calP}_{\mathbf{K}}^{x}
\langlef|H|f\rangle=\langlef|\,\int_{-\infty}^{\infty}dz\,:T^{00}:\,|f\rangle=\frac12\,\int_{-\infty}^{\infty}dz\,\left\{\left\vert\frac{\partial\phi_{f}(x)}{\partialt}\right\vert^{2}+\left\vert\frac{\partial\phi_{f}(x)}{\partialz}\right\vert^{2}+m^{2}\left\vert\phi_{f}(x)\right\vert^{2}\right\}.
4\partial_{+}\partial_{-}\rho+2\partial_{+}\psi\partial_{-}\psi+\frac{1}{2}\partial_{+}\psi_{1}\partial_{-}\psi_{1}+2e^{-\psi_{1}-2\rho}\OmegaF_{2+-}^{2}+2e^{\psi_{1}-2\rho}\OmegaF_{+-}^{2}=0
d\Theta^{1\underline{\mu}}=v_{q}^{-\underline{\mu}}\pi_{q}^{1+}+v_{\dot{q}}^{+\underline{\mu}}\pi_{\dot{q}}^{1-}=e^{+2}\psi_{+2\dot{q}}^{1-}v_{\dot{q}}^{\underline{\mu}+}+e_{q}^{+}v_{q}^{\underline{\mu}-},
\mathrm{Magnetic~NS~5~near-horizon}:\quad\hat{x}^{\mu}=\frac{\hat{\ell}x^{\mu}}{\sqrt{\alpha^{\prime}}},\,hat{u}=\frac{\hat{\ell}r}{(\alpha^{\prime})^{3/2}},\\theta,\,hat{\ell}^{2}=g^{-1}\alpha^{\prime}\.
\mathrm{Re}\,\Pi^{\mathrm{(non\mathrm{-}planar)}}=\frac{2\pi^{\frac{n+1}{2}}}{(2\pi)^{n}}\left(|\bar{\theta}|\right)^{\frac{3-n}{2}}\int_{0}^{\infty}dz\,\frac{1}{z^{2}-p^{2}}\,z^{\frac{n-3}{2}}J_{\frac{n-3}{2}}(\frac{|\bar{\theta}|z}{2})
\frac{e}{4m}F_{\alpha\beta}X^{\alpha\beta}=\frac{e}{4m}F_{ij}X^{ij}=\frac{e}{4m}\epsilon_{ijk}B^{k}X^{ij}=\frac{e}{2m}B^{k}\left[\frac{1}{2}\epsilon_{ijk}X^{ij}\right]=\frac{e}{2m}B^{k}h(\lambda_{k})
\frac12{\frac{g_{2}^{2}v^{2}}{4}}(2W_{+}^{\mu}W_{\mu-}+W_{3}^{\mu}W_{\mu3})+{\frac{g_{2}g_{1}v^{2}}{4}}W_{3}^{\mu}W_{\mu4}+\frac12{\frac{g_{1}^{2}v^{2}}{4}}W_{4}^{\mu}W_{\mu4}
\begin{aligned}{\deltaY}&{{}=\sqrt{2}\xi\eta}\\{\delta\eta}&{{}=i\sqrt{2}\sigma^{m}\bar{\xi}\partial_{m}Y+\sqrt{2}\xiF}\\{\deltaH}&{{}=\frac{i}{\sqrt{2}}\left(\bar{\xi}\bar{\sigma}^{m}\partial_{m}\eta-\partial_{m}\bar{\eta}\bar{\sigma}^{m}\xi\right)}\\{\deltaC_{mnp}}&{{}=-\frac{\sqrt{2}}{16}\epsilon_{mnpq}\left(\bar{\xi}\bar{\sigma}^{q}\eta-\xi\sigma^{q}\bar{\eta}\right)}\\\end{aligned}
\sum_{r,s}e^{-i(r-s)\Omega_{G}t}\frac{\partial}{\partialt}\Bigl(b_{(n)}^{(r,s)}\Bigr)\hat{a}_{0}^{\daggerr}\hat{a}_{0}^{s}=i\sum_{r,s}e^{-i(r-s)\Omega_{G}t}b_{(n-1)}^{(r,s)}[\hat{a}_{0}^{\daggerr}\hat{a}_{0}^{s},\hat{H}^{\prime}].
\frac{\beta}{r}\sum_{\ell=1}^{\infty}\widetilde{f}\left(h\ell\right)=-\left(\frac{r}{\beta}\right)^{3}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\left[\frac{dI_{n}\left(\lambda\right)}{d\lambda}\right]_{\lambda=1}+H\left(m_{eff}\beta\right),
\mathrm{det}\,{\calO}=\left[\frac{b}{\alphar}\tilde{\lambda}^{2}+\frac{b^{2}}{4Q^{2}}\right]^{2}-\left[\frac{2Q_{M}b\alphar}{4Q^{2}}-4Q_{M}(1-\tilde{\lambda}^{2})\right]^{2}\,.
\begin{array}{cccc}{q_{2}=}&{q_{1}+p_{1}+p_{n}+p_{n-1}+}&{\ldots}&{+p_{3}}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{q_{j}=}&{q_{1}+p_{1}+p_{n}+p_{n-1}+}&{\ldots}&{+p_{j+1}}\\{\vdots}&{\vdots}&{}&{}\\{q_{n}=}&{q_{1}+p_{1}}&{}&{}\\\end{array}
1.4\Fermi\approx{\frac{e^{2}}{2m_{e}c^{2}}}=\left[{\frac{e^{2}}{\hbarc}}\right]{\frac{\hbar}{2m_{e}c}}=\left[{\frac{e^{2}}{\hbarc}}\right]^{2}{\frac{\hbar^{2}}{2m_{e}e^{2}}}\approx0.265\Angstrom
\Phi(x_{1}\otimesx_{2}\,,y_{1}\otimesy_{2})=(-1)^{\alpha(x_{2})\alpha(y_{1})}\:\Phi_{\varphi_{1}\psi_{1}}^{j_{1},\lambda_{1}}(x_{1},y_{1})\:\Phi_{\varphi_{2}\psi_{2}}^{j_{2},\lambda_{2}}(x_{2},y_{2})\,,
\dot{\Lambda}\left[1+\frac{\alpha^{\prime}e^{\phi}\phi^{\prime}}{2g^{2}r}\,(1-3e^{-\Lambda})\right]=\frac{r\dot{\phi}\phi^{\prime}}{2}+\frac{\alpha^{\prime}e^{\phi}}{g^{2}r}\,(1-e^{-\Lambda})\left(\dot{\phi}\phi^{\prime}+\dot{\phi}^{\prime}-\frac{\dot{\phi}\Gamma^{\prime}}{2}\right)
\psi_{R}(x^{+},x^{-},{\vec{x}}_{\perp})={\calP}_{x^{-}}e^{i\int_{0}^{x^{-}}dy^{-}\big[A_{-}(x^{+},y^{-},{\vec{x}}_{\perp})-i{\vec{\alpha}}_{\perp}\big({\vec{\partial}}_{\perp}+i{\vec{A}}_{\perp}(x^{+},y^{-},{\vec{x}}_{\perp})\big)\big]}\psi_{R}(x^{+},0,{\vec{x}}_{\perp})\,.
{\calL}_{Coul}^{free}=\frac{1}{2}A_{+}^{a}\partial_{\varrho}\partial^{\varrho}A_{-}^{a}+i\bar{\psi}_{-}^{\prime}\hat{\partial}\psi_{+}^{\prime}+i\bar{\psi}_{+}^{\prime}\hat{\partial}\psi_{-}^{\prime}
\Psi_{16}\rightarrowexp\left(i\frac{\alpha_{a}}{2}\Sigma^{a}(1-\Gamma_{11})\right)\Psi_{16},\quadC_{10}\Psi_{\overline{{16}}}\rightarrowexp\left(-i\frac{\alpha_{a}}{2}\left[\Sigma^{a}(1-\Gamma_{11})\right]^{*}\right)C_{10}\Psi_{\overline{{16}}}
\{f,\,g\}\equiv\sum_{k}\left({\frac{\partialf}{\partialq_{k}}}{\frac{\partialg}{\partialp_{k}}}-{\frac{\partialg}{\partialq_{k}}}{\frac{\partialf}{\partialp_{k}}}\right)=-\,{\frac{i}{\hbar}}\sum_{k}\left({\frac{\partialf}{\partiala_{k}}}{\frac{\partialg}{\partiala_{k}^{*}}}-{\frac{\partialg}{\partiala_{k}}}{\frac{\partialf}{\partiala_{k}^{*}}}\right).
\Gamma^{-2}(r)-\Gamma^{-2}(\bar{r})={\frac{2M\Omega^{2}}{c^{2}}}\left({\frac{1}{\bar{w}}}-{\frac{1}{w}}\right)={\frac{Mc^{2}}{2E^{2}}}\bar{w}\left(1-{\frac{\bar{w}}{w}}\right),
\tilde{\zeta}(s)=V\frac{\Gamma(s-3/2)}{\Gamma(s)}\left(\frac{m}{2\pi}\right)^{3/2}(-\mu)^{3/2-s}+V\left(\frac{m}{2\pi\beta}\right)^{3/2}\frac{\beta^{s}}{\Gamma(s)}Li_{5/2-s}(e^{\beta\mu})\;.
\frac{\left\langle\Psi\midH\mid\Psi\right\rangle}{\left\langle\Psi\mid\Psi\right\rangle}=\frac{\int\left[{\calD}g_{ij}^{\bot}\right]\intd_{{}}^{3}x\Psi_{1}^{\ast}\left\{g_{ij}^{\bot}\right\}{\calH}\Psi\left\{g_{kl}^{\bot}\right\}}{\int\left[{\calD}g_{ij}^{\bot}\right]\mid\Psi\left\{g_{ij}^{\bot}\right\}\mid^{2}}
\lambda(p^{\mu})\equiv\left(\begin{matrix}{\left(\zeta_{\lambda}\Theta_{[j]}\right)\phi_{L}^{*}(p^{\mu})}\\{\phi_{L}(p^{\mu})}\\\end{matrix}\right),\quad\rho(p^{\mu})\equiv\left(\begin{matrix}{\phi_{R}(p^{\mu})}\\{\left(\zeta_{\rho}\Theta_{[j]}\right)^{*}\phi_{R}^{*}(p^{\mu})}\\\end{matrix}\right)\quad.
e^{-A};\;\;\;\frac{\rho}{\chi},\;\frac{\rho}{\chi\Omega};\;\;\;\frac{\dot{\chi}}{\chi},\;\frac{\Omega\dot{\chi}^{2}}{\chi^{2}};\;\;\;\mathrm{and}\;\;\;\frac{\chi^{n}}{\Omega}\;\frac{d^{n}\Omega}{d\chi^{n}}\;\left(\frac{\dot{\chi}}{\chi}\right)^{n},\;\;\foralln\ge1
\bar{M}(\lambda)=\bar{L}_{N}(\lambda^{-1})\ldots\bar{L}_{1}(\lambda^{-1})\equiv\left(\begin{array}{cc}{\bar{A}(\lambda)}&{\bar{B}(\lambda)}\\{\bar{C}(\lambda)}&{\bar{D}(\lambda)}\\\end{array}\right)\,,\,\,N\in2{\mathbbN}\nonumber
v_{q^{\prime}m^{\prime}k^{\prime}}(t^{\prime},r^{\prime},\theta^{\prime},z^{\prime})=N_{1}\,e^{ik^{\prime}z^{\prime}+im^{\prime}\theta^{\prime}}e^{-i\omega^{\prime}t^{\prime}}J_{m^{\prime}}(q^{\prime}r^{\prime}),
\rho_{t}(\bar{\psi}_{f},\psi_{f}^{\prime})=\intD\bar{\psi}_{i}^{\prime}D\psi_{i}J(\bar{\psi}_{f},\psi_{f}^{\prime};t|\bar{\psi}_{i}^{\prime},\psi_{i};0)\rho_{0}(\bar{\psi}_{i}^{\prime},\psi_{i})
\partial_{t}\;(\mathrm{cos}\;\vartheta\;\phi^{1}-\mathrm{sin}\;\vartheta\;{\hat{\phi}}^{2})\Bigl\vert_{0}\;=\;\partial_{x}\;(\mathrm{sin}\;\vartheta\;\phi^{1}+\mathrm{cos}\;\vartheta\;{\hat{\phi}}^{2})\Bigl\vert_{0}\;=\;0\.
S_{EM}\rightarrowS_{EM[\partialM]}~=~\ointd\tau\intd^{2}r_{\perp}\left(\frac{1}{2}\Omega^{-}\nabla^{2}{\partial_{\tau}\Omega}^{+}-\frac{1}{2}\Omega^{+}\nabla^{2}{\partial_{\tau}\Omega}^{-}\right)~~,
\int_{0}^{T}\mathrm{dt}\,[P\dot{Q}-P^{2/3}+E_{0}]=E_{0}T-\int_{0}^{T}\mathrm{dt}\,\frac{1}{3}{P_{i}}^{2/3}=\frac{2}{3}E_{0}T-\frac{1}{6}m\lambda^{2}T^{3}\;.
\frac{1}{2\pi}L=\frac{1}{k}\left((\frac{J^{3}}{2})^{2}+J^{+}J^{-}\right)+\frac{1}{k}\eta^{\alpha\beta}F_{+\alpha}F_{-\beta}+\frac{d-1}{2C_{\rho}}\frac{B^{a}B^{a}}{2k}
S_{ext}=-\frac{1}{4}F_{\mu\nu}^{a}(1-\frac{D^{2}}{\Lambda^{2}})^{2}F_{a}^{\mu\nu}+A_{a\,\mu}^{*}D^{\mu}c^{a}+\frac{1}{2}c^{*a}f_{bc}^{a}c^{b}c^{c}-\frac{1}{2\alpha}b^{*\,a}b_{a}^{*}.
H_{\delta}={\frac{1}{2}}p^{2}+\lambdax^{2}-{\frac{1}{2}}m^{2}a^{2}+ga^{3}+\delta\left[{\frac{1}{2}}m^{2}(x+ia)^{2}-\lambdax^{2}+{\frac{1}{2}}m^{2}a^{2}-ga^{3}+ig(x+ia)^{3}\right].
I_{n}(\beta)=M\int\frac{d^{3}q}{(2\pi)^{3}}\frac{q^{2n}}{ME^{+}-q^{2}}=\frac{M}{2{\pi^{2}}}\int_{0}^{\beta}{dq}\frac{q^{2n+2}}{ME^{+}-q^{2}}.
\chi_{T}=\kappa^{-1}\pi^{1/2}Nmr_{+}^{-2}\frac{2r_{+}^{4}+5l^{2}r_{+}^{2}-l^{4}}{2r_{+}^{4}+3l^{2}r_{+}^{2}+l^{4}}\frac{2r_{+}^{2}+l^{2}}{2r_{+}^{2}-l^{2}}.
I_{\Psi}=-\int_{{\calM}_{b}}d^{5}x\sqrt{-g}\left[\frac{1}{2}g^{MN}\partial_{M}\Psi\partial_{N}\Psi+U(\Psi)\right]-\sum_{\sigma=\pm}\int_{\Sigma_{\sigma}}d^{4}y_{\sigma}\sqrt{-q_{\sigma}}V_{\sigma}(\Psi_{\sigma})
\left.\begin{array}{lcl}{T_{\;\;z}^{z}}&{=}&{V(\phi)-{{{{{\phi^{\prime}}}^{2}}}{{e^{-2\,a}}}}}\\{T_{\;\,i}^{i}}&{=}&{V(\phi)+{{{{{\phi^{\prime}}}^{2}}}{{e^{-2\,a}}}},}\\\end{array}\right.
\begin{array}{c}{d\theta^{\alpha},\qquaddx^{m}+i\theta\Gamma^{m}d\theta,}\\{d\psi_{\alpha}-\displaystyle\frac12x^{m}(d\theta\Gamma_{m})_{\alpha}+dx_{m}(\theta\Gamma^{m})_{\alpha}+\frac12(\theta\Gamma^{m}d\theta)(\theta\Gamma_{m})_{\alpha}}\\\end{array}
\delta((u_{\mathrm{B}}-u_{\mathrm{T}})\cdot(\rho_{\mathrm{B}}-\rho_{\mathrm{B}}^{\prime}))=\int\frac{d\alpha}{2\pi}e^{i\alpha(u_{\mathrm{B}}-u_{\mathrm{T}})\cdot(\rho_{\mathrm{B}}-\rho_{\mathrm{B}}^{\prime})}.
\beta_{2}^{(k)}=3\frac{\lambda_{+}^{k+1}-\lambda_{-}^{k+1}+2\lambda_{-}^{k}-2\lambda_{+}^{k}-\lambda_{-}^{k-1}+\lambda_{+}^{k-1}}{\lambda_{+}-\lambda_{-}}=3(d-2)\frac{\lambda_{+}^{k}-\lambda_{-}^{k}}{\lambda_{+}-\lambda_{-}}\,,
<A_{\mu}\left(x\right)\,A_{\nu}(y)>_{\xi=0}^{dual}\,\cong\,\left[P_{\mu\nu}\,+\,2\,i\,\bar{\theta}\,\epsilon_{\mu\nu\alpha}\,\partial^{\alpha}\right]\left(\frac{1}{\bar{\theta}^{2}}\,-\,1\right)\left(\frac{4}{\bar{\theta}^{2}}\right)\frac{e^{-\,\bar{\theta}\,R}}{4\,\pi\,R},
\left(\int_{\gamma_{\rho}^{+}}+\sum_{k}\int_{C_{\rho}(iy_{k})}\right)\frac{H_{\mu}^{(1)}(\lambdaz)}{H_{\nu}^{(1)}(z)}F(z)dz+\left(\int_{\gamma_{\rho}^{-}}+\sum_{k}\int_{C_{\rho}(-iy_{k})}\right)\frac{H_{\mu}^{(2)}(\lambdaz)}{H_{\nu}^{(2)}(z)}F(z)dz
\partial_{t}^{2}\chi_{t}^{i}=(\frac{1}{4}l^{n}{}_{m}^{2}+\taul\chi)\chi_{t}^{i}+(2-\chi\tau)(\partial_{t}\chi_{t})^{i}\,,\quad\partial_{t}^{2}\chi_{t}^{i}=\chi_{t}^{i}(\frac{1}{4}l^{n}{}_{m}^{2}+\taul\chi)+(\partial_{t}\chi_{t})^{i}(2-\chi\tau)\,,
Z=\frac{\sum_{i_{1}\cdotsi_{N_{f}},j_{1}\cdotsj_{N_{f}}}B^{[i_{1}i_{2}\cdotsi_{N_{c}}]}T_{j_{N_{c}+1}}^{i_{N_{c}+1}}\cdotsT_{j_{N_{f}}}^{i_{N_{f}}}{\bar{B}}_{[j_{1}j_{2}\cdotsj_{N_{c}}]}}{\mathrm{det}\left(T\right)}\bigg(\equiv\frac{BT^{N_{f}-N_{c}}{\bar{B}}}{\mathrm{det}\left(T\right)}\bigg),
Q_{r}{\rightarrow}Q_{r}^{+},\quad{Q_{ri}}{\rightarrow}Q_{ri}^{+}=\lbrack{Q_{r}^{+},J_{i}}\rbrack,\quad{G_{ri}}{\rightarrow}G_{ri}^{+}=\lbrack{Q_{r}^{+},N_{i}}\rbrack,
S_{M}=-M\intd\tau\left[1-\frac{\kappa}{2}s_{\mu\nu}{\dot{x}}^{\mu}{\dot{x}}^{\nu}-\frac{\kappa^{2}}{8}s_{\mu\nu}s_{\rho\sigma}{\dot{x}}^{\mu}{\dot{x}}^{\nu}{\dot{x}}^{\rho}{\dot{x}}^{\sigma}+{\calO}(s_{\mu\nu}^{3})\right],
I_{5}=\frac{1}{16\piG_{5}}\intd^{5}x\sqrt{-g^{(5)}}\left\{e^{-2\phi_{1}}\left[R^{(5)}+20e^{2\psi}+4(D\phi_{1})^{2}-5(D\psi)^{2}\right]-\frac{1}{2}e^{5\psi}(F_{1}^{(5)2}+F_{2}^{(5)2})\right\}
\begin{aligned}{\big[M_{AB}^{(8)},M_{CD}^{(8)}\big]}&{{}=\delta_{AC}M_{BD}^{(8)}\,,\quad}&{\big[M_{IJ}^{(5)},M_{KL}^{(5)}\big]}&{{}=\delta_{IK}M_{JL}^{(5)}\,,}\\{{}\big[M_{AB}^{(8)},Q\big]}&{{}=\frac{1}{8}\big(\Gamma_{AB}Q\big)\,,}&{{}\big[M_{IJ}^{(5)},Q\big]}&{{}=\frac{1}{8}\big(\Gamma_{IJ}Q\big)}\\\end{aligned}
S=\int_{M}d^{6}x\sqrt{G}G^{\lambda\rho}G^{\mu\sigma}G^{\nu\tau}\left(-\frac{3}{4}D_{[\lambda}B_{\mu\nu]}D_{\rho}B_{\sigma\tau}-\frac{\alpha}{2}D_{\lambda}B_{\rho\mu}D_{\nu}B_{\tau\sigma}\right)
\frac{\theta\,\Box\,\epsilon_{\mu\alpha\nu}\,\partial^{\alpha}\,W^{\mu}}{\left(1\,-\,4\,\alpha^{2}\,\Box\right)}\,\longrightarrow\,U_{\nu}\,=\,\theta\,\int\,d\lambda^{\mu}\,\epsilon_{\mu\alpha\nu}\,\partial^{\alpha}\,\Box\,(1\,-\,4\,\alpha^{2}\,\Box)^{-1}\delta^{3}\left(z-\lambda\right).
Z_{\;\;A_{1}}^{\alpha_{0}}=\left(Z_{\;\;\alpha_{1}}^{\alpha_{0}},Z_{\;\;\alpha_{1}}^{\alpha_{0}},Z_{\;\;\alpha_{1}}^{\alpha_{0}}\right),\;\varepsilon^{A_{1}}=\left(\begin{array}{c}{\varepsilon_{1}^{\alpha_{1}}}\\{\varepsilon_{2}^{\alpha_{1}}}\\{\varepsilon_{3}^{\alpha_{1}}}\\\end{array}\right),\;A_{1}=\left(\alpha_{1},\alpha_{1},\alpha_{1}\right).
{\calL}=\frac{\theta}{4}\epsilon^{\mu\nu\alpha}F_{\mu\nu}A_{\alpha}+\frac{i}{2}\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-\frac{i}{2}(\partial_{\mu}\bar{\psi})\gamma^{\mu}\psi-m\bar{\psi}\psi+e\bar{\psi}\gamma^{\mu}\psiA_{\mu},
\begin{array}{cc}{\hat{M}_{0}(z)Z_{0}=Z_{0}H(z)~~~~}&{~~~~H(z)=\left(\begin{array}{cc}{\tilde{\hat{\omega}}(z)-\frac{1}{2}\hat{\lambda}(z)}&{2\hat{\rho}^{j}(z)}\\{0}&{\hat{T}_{i}^{~j}(z)}\\\end{array}\right)}\\\end{array}
{\calL}_{\mathrm{inv}}={\frac{1}{2}}\mathrm{tr}\,F_{\mu\nu}F^{\mu\nu}-m\,\mathrm{tr}\,\epsilon^{\mu\nu\lambda}A_{\mu}(\partial_{\nu}A_{\lambda}+{\frac{2\,g}{3}}A_{\nu}A_{\lambda})