harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e_{2}=\frac1{4\pi}\epsilon_{abc}\left(d\hat{y}^{a}d\hat{y}^{b}\hat{y}^{c}-d(\Theta^{ab}\hat{y}^{c})\right)=\frac1{4\pi}\epsilon_{abc}(D\hat{y}^{a}\landD\hat{y}^{b}y^{c}-R^{ab}(\Theta)\hat{y}^{c})
ds^{2}=(H_{1}H_{2})^{\frac{3}{11}}[(H_{1}H_{2})^{-{\frac{8}{11}}}(-dy^{2}+dy_{1}^{2})+H_{1}^{-{\frac{8}{11}}}dy_{2}^{2}+H_{2}^{-{\frac{8}{11}}}dy_{3}^{2}+dx^{\alpha}dx^{\alpha}]
E(x)=\left(\begin{array}{cc}{I_{R}}&{2B(x)}\\{0}&{I_{L}}\\\end{array}\right)\;\;\;\;\;F_{R}(x)=\left(\begin{array}{c}{G_{R}(x)}\\{0}\\\end{array}\right)\;\;\;\;\;F_{L}(x)=\left(\begin{array}{cc}{0}&{G_{L}(x)}\\\end{array}\right).
E=\frac{C_{1}+C_{2}^{\infty}}{a}=\frac{C_{1}+C_{2}^{\infty}}{a_{phys}+\deltaa}=\frac{C_{1}}{a_{phys}}\frac{\left(1+C_{2}^{\infty}/C_{1}\right)}{\left(1+\deltaa/a_{phys}\right)}.
L\left(\widehat{A}_{\mu}(\overrightarrow{X},t),\dot{\overrightarrow{X}},\overrightarrow{X},\overrightarrow{P},\dot{\overrightarrow{P}})\right)=L_{DH}(A_{\mu}(\overrightarrow{\eta},t),\overrightarrow{\eta},\dot{\overrightarrow{\eta}},\overrightarrow{{\calP}},\dot{\overrightarrow{{\calP}}})).\,
f_{(\eta)}^{\vec{\alpha}\beta\gamma}=f^{\alpha[1]\betab[1]}\,\,f^{b[1]\alpha[2]b[2]}\,f^{b[2]\alpha[3]b[3]}\,\cdots\,\,f^{b[\eta-2]\alpha[\eta-1]b[\eta-1]}f^{b[\eta-1]\alpha[\eta]\gamma}
(G_{rs})_{0}=\left(\begin{matrix}{-\delta_{mn}}&{}&{(A_{n})_{0}\}\\{}&{}&{}\\{(A_{n})_{0}}&{}&{-e^{2\hat{\phi}_{0}}K_{0}}&{}\\\end{matrix}\right),\qquad(B_{rs})_{0}=\left(\begin{matrix}{0}&{}&{(A_{n})_{0}}\\{}&{}&{}\\{-(A_{n})_{0}}&{}&{0}&{}\\\end{matrix}\right).
\epsilon^{+}(z)=\frac{z}{2\pi}+\frac{i}{\pi}\sum_{n>0}\frac{e^{-in(z-i\varepsilon)}}{n},~~~~~\epsilon^{-}(z)=\frac{z}{2\pi}+\frac{i}{\pi}\sum_{n<0}\frac{e^{-in(z+i\varepsilon)}}{n}.
\frac1{r^{2}}\fracd{dr}\left(r^{2}\frac{dR_{q,l}}{dr}\right)-\frac{l(l+1)-q^{2}}{\alpha^{2}r^{2}}R_{q,l}-\frac{2M}{\alpha^{2}}\left(\fracKr-E\right)R_{q,l}=0\.
S_{*}=\frac{1}{\pix}\int\mathrm{str}\left\{-\frac{\kappa}{2}\tilde{J}^{*}D[2\tilde{J}/\kappa]c_{\mathrm{DS}}+\frac{1}{2}c_{\mathrm{DS}}^{*}\mathrm{ad}\left(c_{\mathrm{DS}}\right)c_{\mathrm{DS}}\right\},
\partial_{=}X^{\mu}-R_{\,\,\nu}^{\mu}\partial_{+\!\!\!+}X^{\nu}+2i(P_{\,\,\rho}^{\sigma}\nabla_{\sigma}R_{\,\,\nu}^{\mu}+P_{\,\,\gamma}^{\mu}g^{\gamma\delta}H_{\delta\sigma\rho}R_{\,\,\nu}^{\sigma})\psi_{+}^{\rho}\psi_{+}^{\nu}=0,
E^{(0)}\,(m_{0})\;-\;m_{0}\,\frac{\partialE^{(0)}\,(m_{0})}{\partialm_{0}}\;+\;\frac{1}{2}\,m_{0}^{2}\,\frac{{\partial}^{2}E^{(0)}\,(m_{0})}{\partialm_{0}^{2}}
S_{E}=-\frac{\beta}{8\piG_{d}}\int_{\Sigma}D_{i}J_{D}^{i}+\frac{1}{8\piG_{d}}\int_{M}d^{d}x\sqrt{\hat{g}}e^{-a\Phi}((p+1)g_{\tau\tau}^{-1}{\mathcalH}_{e}^{2}),
S=\frac{1}{2\lambda_{s}^{2}}\intdte^{\alpha+2\beta-\phi}\biggl\{\frac{1}{N}\biggl[-\dot{\phi}^{2}-2\dot{\beta}^{2}-4\dot{\alpha}\dot{\beta}+2\dot{\alpha}\dot{\phi}+4\dot{\beta}\dot{\phi}\biggr]+\frac{w\lambda_{s}^{2}}{4N^{3}}\biggl[8\dot{\phi}~\dot{\alpha}~\dot{\beta}^{2}-\dot{\phi}^{4}\biggr]\biggr\}
\frac{d}{d{\alpha}_{\pmp}}\longrightarrow\frac{\tilde{d}}{d{\alpha}_{\pmp}}\equiv\frac{d}{d{\alpha}_{\pmp}}\mp(1-\mathrm{N}^{2})\frac{e_{+}^{2}\mathrm{L}^{2}}{8{\pi}^{2}{\hbar}^{2}}\frac{1}{p}{\alpha}_{\mpp},
S=\int{\frac{d^{2}x}{2\pi}}\left[\psi_{R}^{\dagger}\partial\psi_{R}+\psi_{L}^{\dagger}\bar{\partial}\psi_{L}+\Lambda\left(\psi_{R}^{\dagger}\psi_{L}+\psi_{R}\partial\psi_{R}\psi_{L}^{\dagger}\bar{\partial}\psi_{L}^{\dagger}\right)\right]
\left\{{\frac{\partial^{2}}{\partialr^{2}}}+{\frac{5}{r}}{\frac{\partial}{\partialr}}-{\frac{l(l+4)}{r^{2}}}+\tilde{\omega}^{2}\left(1+{\frac{\tilde{R}_{B}^{4}}{r^{4}}}\right)\right\}\varphi_{B}^{l}=0,
\intd^{6}\xi\,\left(\sum_{a=0}^{5}\left|(\partial_{a}+iA_{a}-iA_{a}^{\prime})\chi\right|^{2}+\biggr(\frac{1}{4g_{\mathrm{YM},p}^{2}}+\frac{1}{4g_{\mathrm{YM},p^{\prime}}^{2}}\biggl)\sum_{I=1}^{3}(\chi^{\dagger}\tau^{I}\chi)^{2}\right)\,
{\calF}_{\bar{\mu}\bar{\nu}}^{i}=\partial_{\bar{\mu}}{\calA}_{\bar{\nu}}^{i}-\partial_{\bar{\nu}}{\calA}_{\bar{\mu}}^{i},\\\H_{\bar{\mu}\bar{\nu}\bar{\rho}}=(\partial_{\bar{\mu}}B_{\bar{\nu}\bar{\rho}}+2{\calA}_{\bar{\mu}}^{i}L_{ij}{\calF}_{\bar{\nu}\bar{\rho}}^{j})+\mathrm{cyc.\perms.}.
|\eta(q)|^{2}=\frac{1}{2}\sum_{s,t}\left[q^{\frac{3}{2}(\frac{s}{6}+t)^{2}}\bar{q}^{\frac{3}{2}(\frac{s}{6}-t)^{2}}-q^{\frac{3}{2}(\frac{s}{2}+\frac{t}{3})^{2}}\bar{q}^{\frac{3}{2}(\frac{s}{2}-\frac{t}{3})^{2}}\right].
\begin{array}{rcl}{F^{(m)}(\zeta_{1},\cdots,\zeta_{2N})}&{=\displaystyle\sum_{\iota_{1},\cdots,\iota_{2N}}}&{v_{\iota_{1}}\otimes\cdots\otimesv_{\iota_{2N}}F^{(m)}(\zeta_{1},\cdots,\zeta_{2N})_{\iota_{1},\cdots,\iota_{2N}}}\\\end{array}
M_{0}^{\mu\nu}\left(x\right)=x^{\mu}p^{\nu}-x^{\nu}p^{\mu}-\imath\sum_{n=1}^{\infty}\frac{1}{n}\left(\alpha_{-n}^{\mu}\alpha_{n}^{\nu}-\alpha_{-n}^{\nu}\alpha_{n}^{\mu}\right)\tag{5}
\Theta\left[\begin{array}{c}{r/k}\\{0}\\\end{array}\right]\Big(x\Big|k(\tau+1)\Big)=e^{i\pir^{2}/k}\,\Theta\left[\begin{array}{c}{r/k}\\{0}\\\end{array}\right]\Big(x\Big|k\tau\Big)\\\,\\\\mathrm{only\if}\k\\mathrm{is\even}\\\,
\sum_{\epsilon_{l}\leq0}u_{\epsilon_{l}l}\left(r\right)u_{\epsilon_{l}l}\left(r^{\prime}\right)+\int_{0+}^{\infty}\frac{\mathrm{d}k}{2\pi}\,u_{kl}\left(r\right)u_{kl}\left(r^{\prime}\right)=\delta\left(r-r^{\prime}\right)\,.
\left(1-{\frac{r_{0}}{r}}\right){\frac{r}{r_{0}}}{\frac{d}{dr}}\left(r^{2}{\frac{df_{\ell}}{dr}}\right)+\left(2\ell+\left(2\ell+1+i\;{\frac{k}{2\piT_{H}}}\right){\frac{r_{0}}{r}}\right){\frac{r^{2}}{r_{0}}}\;{\frac{df_{\ell}}{dr}}+\ell\left(-\ell+i\;{\frac{k}{2\piT_{H}}}\right)\;f_{\ell}=0
I_{\mathrm{gauge}}=\frac{k}{2\pi}\intd^{2}ztr[{\bfA}\overline{{\partial}}gg^{-1}+\overline{{{\bfA}}}g^{-1}\partialg+{\bfA}\overline{{{\bfA}}}+{\bfA}g\overline{{{\bfA}}}g^{-1}]
{\binom{1}{-1}}{\binom{1}{1}}{\binom{1}{-1}},\,\,\,\,\,\,\,\,\,\,\,{\binom{1}{1}}{\binom{1}{-1}}{\binom{1}{-1}},\,\,\,\,\,\,\,\,\,\,\,{\binom{1}{-1}}{\binom{1}{-1}}{\binom{1}{1}},
\varepsilon_{\mu\nu\lambda\rho}\partial_{\lambda}\partial_{\rho}\theta_{a}^{\mathrm{sing.}}(x)=2\pi\Sigma_{\mu\nu}^{a}(x)\equiv2\pi\int_{\Sigma_{a}}^{}d\sigma_{\mu\nu}\left(x^{(a)}(\xi)\right)\delta\left(x-x^{(a)}(\xi)\right).
\phi_{L}=(1-\gamma_{5})/2\left(\begin{array}{c}{\phi_{4}}\\{\phi_{5}}\\{\phi_{6}}\\{.}\\{.}\\\end{array}\right),\\\phi_{R}=(1+\gamma_{5})/2\left(\begin{array}{c}{\phi_{1}}\\{\phi_{2}}\\{\phi_{3}}\\{\phi_{4}}\\{.}\\\end{array}\right)
\left[\partial_{\mu}\partial^{\mu}-\partial_{y}^{2}+e^{-2\sigma}\left\{m^{2}+\left(\frac{3}{16}-\xi\right)R\right\}\right]\tilde{\varphi}+e^{-\frac{7}{2}\sigma}V^{\prime}(e^{\frac{3}{2}\sigma}\tilde{\varphi})=0.
m_{1}^{2}\;-\;\frac{1}{N-1}\sum_{I=2}^{N}\;m_{I}^{2}\;\;=\;\;\frac{e^{2}N}{\pi}\;\;\stackrel{!}{=}\;\;\frac{4N}{(f_{1}^{0})^{2}}\;P^{0}(0)\;.
S_{I,\psi}=-\frac{\sigma}{2}\int\frac{d\nu}{2\pi}\int_{-\bar{R}_{1}}^{\bar{R}_{2}}dr\tilde{\psi}_{I}^{*}\left[-\frac{\partial}{\partialr}\Sigma(r)\gamma^{-2}\frac{\partial}{\partialr}-(\nu^{2}-C)\Sigma(r)\right]\tilde{\psi}_{I}\,,
S=-\frac{1}{2\kappa_{5}^{2}}\int_{{\calM}_{5}}d^{5}x\sqrt{-g}\left[R+\frac{1}{2V^{2}}\partial_{\alpha}V\,\partial^{\alpha}V+\frac{1}{3V^{2}}\alpha^{2}\right]-\sum_{i=1}^{2}\frac{\sqrt{2}}{\kappa_{5}^{2}}\alpha_{i}\int_{{\calM}_{4}^{(i)}}d^{4}\xi_{(i)}\sqrt{-g}\,V^{-1}\,,
\begin{array}{l}{[3(3,\bar{3},1)+(3,1,\bar{3})]_{1}+}\\{+[3(1,1,1+8)+(\bar{3},1,3)]_{2}+}\\{+[3(1,3,\bar{3})+(1,1+8,1)]_{3}+}\\{+[(2(1,1,1+8)+(1,\bar{3},3)+(\bar{3},1,3)]_{4}+}\\{+[2(1,1,1+8)+2(1,\bar{3},3)]_{5}+}\\{+[2(\bar{3},1,3)+(1,1,1+8)+(1,\bar{3},3)]_{6}+}\\{+[4(1,3,\bar{3})]_{7}.}\\\end{array}
\frac{1}{2m_{0}}\;\frac{\partial}{\partial\alpha_{i}}\theta\left(\left.\left(\alpha^{T}\|c\|\alpha\right)\right|_{\alpha_{i}=0}+2m_{0}\alpha_{i}\sum_{l\neqi}^{}\frac{\alpha_{l}}{m_{l}}+\alpha_{i}^{2}\!-\!1\right).
\tilde{r}_{\pm}=\frac{(1+\alpha^{2})(M\pm\sqrt{M^{2}-\left(1-\alpha^{2}\right)Q^{2}})}{\left(1\pm\alpha^{2}\right)},\;\;\;a=\frac{2(1+\alpha^{2})J}{(1+\alpha^{2})\tilde{r}_{+}+(1-\alpha^{2}/3)\tilde{r}_{-}}.
\Gamma=\frac{1}{\sqrt{-g}f}\left[\begin{array}{cc}{i\gamma^{(4)}-\frac{ig}{4}\epsilon_{ijkl}\bar{\tilde{\calH}}^{i}\tilde{\calH}^{j}\gamma^{kl}}&{i\sqrt{-g}\bar{\tilde{\calH}}^{i}v^{j}\gamma_{ij}}\\{-i\sqrt{-g}\tilde{\calH}^{i}v^{j}\gamma_{ij}}&{-i\gamma^{(4)}+\frac{ig}{4}\epsilon_{ijkl}\tilde{\calH}^{i}\bar{\tilde{\calH}}^{j}\gamma^{kl}}\\\end{array}\right].
C_{v}=\frac{3k\langleN\rangle}{2}\left[1+\frac{\langleN\rangle}{2V}\lambda_{T}^{3}\delta(q)-2\frac{\langleN\rangle^{2}}{V^{2}}\lambda_{T}^{6}\left(4\delta^{2}(q)-\frac{\Gamma(q)}{12}\right)+...\right],
W^{(1)}=\frac{1}{2!}\int_{1\ldots6}V_{123}V_{456}(-2)\frac{\deltaW_{A}^{(0)}}{\deltaD_{36}^{-1}}\left(\frac{\delta^{2}W_{\psi}^{(0)}}{\deltaS_{12}^{-1}\deltaS_{45}^{-1}}+\frac{\deltaW_{\psi}^{(0)}}{\deltaS_{12}^{-1}}\frac{\deltaW_{\psi}^{(0)}}{\deltaS_{45}^{-1}}\right).
\left({\frac{\dot{a}}{a}}\right)^{2}={\frac{(C+E)^{2}g_{S}e^{2\phi}-|g_{00}|[g_{S}g(g^{2}h^{2}+b^{2})+\ell^{2}e^{2\phi}]}{4|g_{00}|g_{rr}g_{S}g(g^{2}h^{2}+b^{2})}}\left({\frac{g^{\prime}}{g}}\right)^{2}.
K_{a}=\sum_{b=1,b\neqa}^{N}\frac{1}{\sigma_{b}^{2}-\sigma_{a}^{2}}l_{ab}^{2}+p_{a}^{2}+(2-\sigma_{a}^{2})q_{a}^{2}-q_{a}^{2}\sum_{b=1}^{N}q_{b}^{2}
I_{g}=\frac{1}{4\piG}\intd^{2}x\epsilon^{\mu\nu}\Bigl(\eta_{a}D_{\mu}e_{\nu}^{a}+\eta_{2}\partial_{\mu}\omega_{\nu}+\eta_{3}(\partial_{\mu}a_{\nu}+\frac{1}{2}\epsilon_{ab}e_{\mu}^{a}e_{\nu}^{b})\Bigr)\,.
D_{1}\eta_{2}^{\alpha_{1}}=-\pi^{\alpha_{1}}+\frac12C_{\;\;\beta_{1}\gamma_{1}}^{\alpha_{1}}\eta_{1}^{\beta_{1}}\eta_{2}^{\gamma_{1}}+\frac12C_{\;\;\beta_{1}\gamma_{3}}^{\alpha_{1}}\left(\eta_{2}^{\beta_{1}}\eta_{1}^{\gamma_{3}}+\eta_{1}^{\beta_{1}}\eta_{2}^{\gamma_{3}}\right),
\dot{r}=p_{r}\\\,\\\\dot{p}_{r}=\frac{1}{r^{3}}p_{\theta}^{2}\,-\,\omega^{2}r\\\;\\\\dot{\theta}=\left(\frac{1}{r^{2}}+g^{2}\right)p_{\theta}\\\,\\\,dot{p}_{\theta}=0\\\,
\begin{array}{c}{[A_{i}^{~j},A_{k}^{~l}]=i(\delta_{k}^{~j}A_{i}^{~l}-\delta_{i}^{~l}A_{k}^{~j}+{\calE}^{jl}A_{ki}-\bar{\calE}_{ik}A^{jl})}\\{{}}\\{{[A_{i}^{~j},Q^{k}]=i({\calE}^{jk}\bar{Q}_{i}^{t}-\delta_{i}^{~k}Q^{j})}}\\{{}}\\{{[A_{i}^{~j},S^{k}]=i({\calE}^{jk}\bar{S}_{i}^{t}-\delta_{i}^{~k}S^{j})}}\\{{}}\\{\displaystyle{[A_{i}^{~j},P_{A}]=[A_{i}^{~j},K_{A}]=[A_{i}^{~j},M_{AB}]=0}}\\\end{array}
ds_{11}^{2}\to\left(\frac\rho{\rho_{0}}\right)^{2}(-dt^{2}+dx_{10}^{2})+\left(\frac{\rho_{0}}\rho\right)^{2}d\rho^{2}+\left(\frac{\rho_{0}}2\right)^{2}d\Omega_{2}^{2}+\cdots,
\partial_{t}\Gamma^{(A2)}=-\partial_{t}\delta_{k}(\partial_{\mu\nu})(\partial_{\rho}F_{\mu\rho})+\frac{1}{\alpha}\partial_{t}\zeta_{k}(\partial_{\mu}\partial_{\nu}A_{\nu})(\partial_{\mu}\partial_{\rho}A_{\rho})+\dots
\kappa=\frac{k}{m_{f}}\;\;;\;\;L^{2}=\frac{m_{i}^{2}}{m_{f}^{2}}=\frac{T_{i}^{2}-T_{c}^{2}}{T_{c}^{2}-T_{f}^{2}}\;\;;\;\;\tau=m_{f}t\;\;;\;\;\vec{x}=m_{f}\vec{r}
E_{V}=\frac{\hbarm}{2}\left\{(1-A_{V})+2\underset{\frac{(2-A_{V})\pi}{Lm}}{\overset{\Lambda+\frac{A_{V}\pi}{Lm}}{\int}}\frac{dz}{2\pi}Lm\sqrt{z^{2}+1}+\left(1+\sqrt{\Lambda^{2}+1}\right)\right\}+O(\frac{1}{L},\frac{1}{\Lambda}).
L_{n}(\lambda,t)=\left(\begin{array}{cc}{t^{Z_{n}-S_{n}^{3}}\;sh[\gamma(\lambda+iS_{n}^{3})]}&{iS_{n}^{-}\;sin\gamma}\\{iS_{n}^{+}sin\gamma}&{t^{-Z_{n}-S_{n}^{3}}\;sh[\gamma(\lambda-iS_{n}^{3})]}\\\end{array}\right)\;,
N(\chi)\simeq\int_{\chi_{\mathrm{end}}}^{\chi}M_{P}^{-2}\frac{V}{V^{\prime}}d\chi=\int_{\psi_{1\mathrm{end}}}^{\psi_{1}(\chi)}M_{P}^{-2}\frac{d\psi_{1}}{\frac{\lambda_{2}^{2}}{4\pi^{2}}\frac{1}{\psi_{1}}}=\frac{4\pi^{2}}{\lambda_{2}^{2}M_{P}^{2}}\left(\frac{\psi_{1}^{2}}{2}-\frac{\psi_{1\mathrm{end}}^{2}}{2}\right)=\frac{1}{2|\eta(\chi)|}-\frac{1}{2|\eta_{\mathrm{end}}|}\;,
\int_{\bar{\calM}_{g,s,n}}\prod_{i=1}^{s}[c_{1}({\calL}_{i})]^{d_{i}}=<\prod_{i=1}^{s}\sigma_{d_{i}}>_{n}=\int_{\pi_{n}^{-1}(p*)}\prod_{i=1}^{n}\omega_{i}^{d_{i}}.
\begin{array}{lcr}{(|W_{1}|^{2}+|Z_{2}|^{2})-(|Z_{1}|^{2}+|W_{2}|^{2})=R_{2}}\\{Z_{1}W_{1}+Z_{2}W_{2}=\overline{{Z}}_{1}\overline{{W}}_{1}+\overline{{Z}}_{2}\overline{{W}}_{2}=0,}\\\end{array}
S_{GF}=4g^{2}\int_{x}(-2i\alpha\in_{\mu\nu\lambda}b_{\mu}\partial_{\lambda}+\alpha^{2}(\in_{\mu\nu\lambda}\partial_{\nu}a_{\lambda})^{2}+i\alpha\in_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda})
v_{\bfk}=-\frac{-iH}{\sqrt{2}}\frac{\mathrm{e}^{-i\alphak\eta}}{\left(\alphak\right)^{3/2}}\left(1-\frac{1}{2}(\alpha^{2}-1)\,k^{2}\,\eta^{2}\right)\left(1+i\alphak\eta\right).
\left(\Gamma_{j}\right)_{ab}=\left(\begin{array}{cc}{0}&{\mathbb{E}_{j}}\\{-\mathbb{E}_{j}}&{0}\\\end{array}\right),\quad\quad\Lambda_{j}=\left(\begin{array}{cc}{1|\mathbb{E}_{j}}&{0}\\{0}&{1|\mathbb{E}_{j}}\\\end{array}\right).
A=\left(\begin{array}{ccc}{-2M-2\Sigma/\sqrt{3}}&{-2Q}&{2N}\\{}&{}&{}\\{2Q}&{4\Sigma/\sqrt{3}}&{2P}\\{}&{}&{}\\{2N}&{-2P}&{2M-2\Sigma/\sqrt{3}}\\\end{array}\right).
{W}_{\alpha}=\frac{1}{8}\;\bar{D}^{2}\,\left({e}^{-V}\!D_{\alpha}{e}^{V}\right)=i\left(\lambda_{\alpha}+i\theta_{\alpha}D-\theta^{\beta}\,G_{\alpha\beta}-i\theta^{2}{\calD}_{\alpha\dot{\alpha}}\bar{\lambda}^{\dot{\alpha}}\right)\,.
\ddot{\Phi}-\frac{2}{a}\,\dot{\Phi}_{,r}+\frac{1}{a^{2}}\,\Phi_{,rr}+H\dot{\Phi}=8\piG\,\dot{\varphi}\left(\delta\dot{\varphi}-\frac{1}{a}\,\delta\varphi_{,r}\right),
\intDz\mu[z]P_{\mu_{1}\nu_{1}}[z]\cdotsP_{\mu_{2n}\nu_{2n}}[z]=\frac{\left(a^{2}\right)^{2n}}{(2n-1)!!}\left[\hat{1}_{\mu_{1}\nu_{1},\mu_{2}\nu_{2}}\cdots\hat{1}_{\mu_{2n-1}\nu_{2n-1},\mu_{2n}\nu_{2n}}+{\,}\mathrm{permutations}{\,}\right].
D_{n}(\eta)D_{n^{\prime\prime}}(\eta)=\mathrm{e}^{-\eta^{2}/4}\sum_{m=0}^{\mathrm{min}(n,n^{\prime\prime})}\frac{n!n^{\prime\prime}!}{m!(n-m)!(n^{\prime\prime}-m)!}D_{n+n^{\prime\prime}-2m}(\eta),
\gamma_{\Psi}A^{*\mu}=0,\;\gamma_{\Psi}\phi^{*}=0,\;\gamma_{\Psi}C^{*}=-\partial_{\mu}A^{*\mu},\;\gamma_{\Psi}b^{*}=-\overline{{C}}^{*},\;\gamma_{\Psi}\overline{{C}}^{*}=0.
I(g,A)=\frac{2k}{\pi}\intd^{2}z(1-uv)[A-\frac{u\partialv-v\partialu}{4(1-uv)}][\overline{{A}}+\frac{u\overline{{\partial}}v-v\overline{{\partial}}u}{4(1-uv)}]-\frac{k}{4\pi}\intd^{2}z\frac{\partialu\overline{{\partial}}v-\partialv\overline{{\partial}}u}{1-uv}
{\begin{array}{l}{M5}\\{MW}\\{M2}\\\end{array}}\left[{\begin{array}{lllllllllll}{0}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{6}&{7}&{8}&{9}&{\natural}\\{0}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\natural}\\{0}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{5}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\natural}\\\end{array}}\right]\quad.
f^{(\pm)}=e^{\mpikx}+e^{\mpikx}\sum_{n=0}^{\infty}(\frac{-1}{\pi})^{n+1}\int_{1}^{\infty}d\alpha_{o}...\int_{1}^{\infty}d\alpha_{n}\frac{(\prod_{j=0}^{n}D(\nu;\alpha_{j})e^{-2\alpha_{j}x})}{[\prod_{j=0}^{n-1}(\alpha_{j}+\alpha_{j+1})][\alpha_{o}\pmik]}
\varepsilonb_{3}\left(\xi_{1}a_{1}-{\frac{5}{3}}\xi_{3}a_{3}\right)e^{\sigma|\alpha(2)}h_{\sigma|\alpha(3)}^{3}+\left(2\xi_{1}a_{1}^{2}-{\frac{4}{3}}\xi_{3}b_{3}^{2}\right)e^{\sigma|}{}_{\alpha}{}_{\sigma|\beta}^{1}-{\frac{1}{m}}e^{\mu|\alpha(2)}\partial_{\mu}\lambda_{\alpha(3)}^{3}=0,
S=-\sum_{I=1}^{K}\,\int\!d\tau_{I}\,(m_{0}^{I}\,\sqrt{v_{\mu}^{I}\,v_{I}^{\mu}}+\mathrm{tr}\,Z_{I}\lambda_{I}^{-1}{\dot{\lambda}}_{I})-\int\!d^{4}x\,\mathrm{tr}\,\biggl(j_{\mu}\,A^{\mu}+{\frac{1}{16\pi}}\,F_{\mu\nu}\,F^{\mu\nu}\biggr).
\rho(\eta)\sim\biggl(\frac{\eta_{1}}{\eta_{2}}\biggr)^{3}\int_{\calK}\mathrm{d}k_{\mathrm{phys}}\,k_{\mathrm{phys}}^{2}\omega_{\mathrm{phys}}(k)=\biggl(\frac{\eta_{1}}{\eta_{2}}\biggr)^{3}{\calO}(k_{\mathrm{c}}^{4}),
\intd^{2}\theta\,\theta^{2}\Bigr\{\frac{\deltaW_{m}}{\delta\bar{\Phi}_{A}(\theta)}+\Phi^{A}(\theta)\Bigr\}=0\qquad\Longleftrightarrow\qquad\frac{\deltaW_{m}}{\delta\eta_{A}}=\phi^{A},
\begin{array}{rcl}{i\partial_{t}\,M(t,t_{0})}&{=}&{e^{i(t-t_{0})\hat{H}_{0}}\,\left[\hat{H}-\hat{H}_{0}\right]\,e^{-i(t-t_{0})\hat{H}}}\\{}&{}&{}\\{}&{=}&{e^{i(t-t_{0})\hat{H}_{0}}\,\hat{H}_{\mathrm{int}}(t_{0})\,e^{-i(t-t_{0})\hat{H}_{0}}\,M(t,t_{0})}\\{}&{}&{}\\{}&{=}&{\hat{H}_{\mathrm{int}}^{(I)}(t)\,M(t,t_{0})\,}\\\end{array}
\hat{V}(\tau)=e^{-\frac{\gamma\hat{p}}{4\pi}\tau}\,\,e^{-\gamma\hat{q}}\,\,e^{-\frac{\gamma\hat{p}}{4\pi}\tau}+\omega^{2}\,\frac{e^{\frac{\gamma\hat{p}}{4\pi}\tau}}{\hat{p}}\,\,e^{\gamma\hat{q}}\,\,\frac{e^{\frac{\gamma\hat{p}}{4\pi}\tau}}{\hat{p}}.
{\frac{C_{1}(g,x,k)}{\sqrt{x(x+1)}}}\Big[v({\frac{k^{2}}{\mu^{2}}})x(1+x)+{\frac{g}{2}}(2x+1)+2gxB(g,x,k)\Big]+g(\phi_{c}(k)+A(g,x,k))=0,\,
S_{\mathrm{2-form}}=\intd^{4}x\,dz\sqrt{-g_{(5)}}\left[-\frac{1}{4}g^{MP}g^{NQ}B_{MN}^{\alpha}B_{PQ}^{\alpha}+\frac{1}{8g}\epsilon^{MNPQR}\epsilon_{\alpha\beta}B_{MN}^{\alpha}D_{P}B_{QR}^{\beta}\right],
B_{1}=(\int_{B^{3}}dva_{1}+\int_{S^{2}}dsc_{1})V^{1/2}=\int_{B^{3}}(\frac{\lambda}{2}\hat{\phi}^{2})^{3/2}dv+\frac{2}{3R}\int_{S^{2}}(\frac{\lambda}{2}\hat{\phi}^{2})^{1/2}ds
I_{eff}=\frac{k}{2\pi}\intd^{2}z[\overline{{\partial}}\varphi\partial\varphi+2\varphi+4\epsilone^{-2\varphi}\overline{{\partial}}x\partialx+4\epsilone^{-2\varphi}(x^{2}+4)(\overline{{\partial}}\varphi\partial\varphi+1]
\sum_{\pi(1,2,...n)}\mathrm{Tr}\Big[H^{(1)}H^{(l_{\pi(1)})}.....H^{(l_{\pi(n)})}\Big]\frac{1}{\tilde{x}_{0}-\tilde{x}_{\pi(1)}}\frac{1}{\tilde{x}_{\pi(1)}-\tilde{x}_{\pi(2)}}.....\frac{1}{\tilde{x}_{\pi(n)}-\tilde{x}_{0}}\;.
ds^{2}=-dt^{2}+a(t)^{2}~\left[{\frac{dr^{2}}{1-kr^{2}}}+r^{2}d\Omega^{2}\right]\;,\;d\Omega^{2}=d\theta^{2}+\mathrm{sin}^{2}\thetad\varphi^{2}\;,
{\calL}=\bar{\psi}_{i}(i{\not\!\partial}-e{\not\!\!A}-m)\psi_{i}-\chi_{\mu}^{*}\bar{\psi}_{i}^{c}\gamma^{\mu}\tau\psi_{i}-\chi_{\mu}\bar{\psi}_{i}\gamma^{\mu}\tau^{T}\psi_{i}^{c}+\frac{N}{g}\chi_{\mu}^{*}\chi^{\mu}~.
(\varepsilon_{5})^{2}=B^{2}\left[n^{2}+2\sigma\frac{(k\cdotn)^{2}}{\kappa^{2}}+\frac{k^{2}}{\kappa^{2}}\right]=B^{2}\frac{\sigma}{\kappa^{2}}\left[k^{2}+(k\cdotn)^{2}\right]=B^{2}\frac{\sigma}{\kappa^{2}}\left[k+n(k\cdotn)\right]^{2}
\Sigma^{2}=-2\bar{x}\frac{1}{N}\sum_{n=1}^{N}\frac{1}{(y_{n}^{2}+\bar{x}^{2})^{2}}\frac{d\bar{x}}{dt}=-\frac{1}{2\bar{x}}{\calL}_{2}^{\prime\prime}(\bar{x})\frac{d\bar{x}}{dt}\:,
\begin{array}{lcll}{(\widehat{I}_{i}=\partial\widehat{M}_{ti},\widehat{e}_{i})}&{=}&{([0,t],1)}&{\;\;\;i=1,3\;\;\;,}\\{(\widehat{I}_{i}=\partial\widehat{M}_{ti},\widehat{e}_{i})}&{=}&{([0,1],1)}&{\;\;\;i=2,4\;\;\;,}\\\end{array}
J_{mn}=\frac{1}{16\pi^{2}}\frac{1}{A^{4}}\eta_{nm}\Gamma(\frac{\epsilon}{2})\int\frac{dxdydz\delta(1-x-y-z)}{{(a_{2}x+a_{3}(y+z)-p^{2}{(x+y)}^{2})}^{\epsilon/2}}
F_{,\phi_{i}\phi_{j}}+F_{,\phi_{i}\dot{\phi}_{k}}{\frac{\partial\dot{\phi}_{k}}{\partial\phi_{j}}}~=~S_{,\rho\phi_{i}\phi_{j}}+S_{,\phi_{i}\phi_{j}\phi_{k}}\dot{\phi}_{k}+S_{,\phi_{i}\phi_{k}}{\frac{\partial\dot{\phi}_{k}}{\partial\phi_{j}}}
\begin{array}{ll}{W}&{=\varepsilon_{abc}Tr((X_{33}^{(a)}\otimesY^{(a)})(X_{33}^{(b)}\otimesY^{(b)})(X_{33}^{(c)}\otimesY^{(c)}))\varphi_{3}^{a}\varphi_{3}^{b}\varphi_{3}^{c}+...}\\{}&{=\varepsilon_{abc}Tr(X_{33}^{(a)}X_{33}^{(b)}X_{33}^{(c)})Tr(Z^{a}Z^{b}Z^{c})+\ldots,}\\\end{array}
g_{kk}=\left(\omega^{2}g^{E}+{\frac{k^{2}}{\epsilon\epsilon^{\prime}}}g^{H}+{\frac{1}{\epsilon}}{\frac{\partial}{\partialz}}{\frac{1}{\epsilon^{\prime}}}{\frac{\partial}{\partialz^{\prime}}}g^{H}\right)\bigg|_{z=z^{\prime}}.
\begin{array}{lll}{f_{0}}&{=}&{\tilde{f}_{0}^{-1}\;,}\\{f_{i}f_{0}^{-1/2}}&{=}&{\tilde{f}_{k-i}\tilde{f}_{0}^{-1/2}\;\;\;;i=1,\ldots,k-1\;,}\\\end{array}
\frac{\partialD_{1}^{\prime}}{\partialt^{\prime}}(t,t^{\prime};w,w^{\prime})=-i\lambda\int{\calK}(w^{\prime},w^{\prime\prime})e^{i[\omega(w^{\prime})-\omega(w^{\prime\prime})]t}D_{1}^{\prime}(t,t^{\prime};w,w^{\prime\prime})dw^{\prime\prime}
D\theta=d\theta+\frac{1}{4}\hat{\omega}^{\hat{r}\hat{s}}\Gamma_{\hat{r}\hat{s}}\theta+\frac{1}{2!3!4!}\hat{e}^{\hat{r}}(\Gamma_{\hat{r}}{}^{\hat{s}\hat{t}\hat{u}\hat{v}}-8\delta_{\hat{r}}^{[\hat{s}}\Gamma^{\hat{t}\hat{u}\hat{v}]})\thetaF_{\hat{s}\hat{t}\hat{u}\hat{v}}~.
\begin{array}{l}{\vec{a}\,^{\prime}=sin(2\pip(\sigma-t)/L+\alpha)\hat{x}+cos(2\pip(\sigma-t)/L+\alpha)\hat{z},}\\{\vec{b}\,^{\prime}=sin(2\piq(\sigma+t)/L+\beta)[cos\Psi\hat{x}+sin\Psi\hat{y}]+cos(2\piq(\sigma+t)/L+\beta)\hat{z}.}\\\end{array}
S^{(2,2)}=<\eta^{(+)}\in{\mathbbC}^{(3,2)}:\,(\eta_{i}^{(+)})^{2}+C_{\alpha\beta}\,\eta_{\alpha}^{(+)}\,\eta_{\beta}^{(+)}=\frac{1}{4}\,,(\eta_{i}^{(+)})^{\ddagger}=\eta_{i}^{(+)}\,,(\eta_{\alpha}^{(+)})^{\ddagger}=C_{\alpha\beta}\eta_{\beta}^{(+)}>
^{(4)}R_{\phantom{\alpha}\beta\gamma\delta}^{\alpha}=R_{\phantom{A}BCD}^{A}q_{A}^{\phantom{A}\alpha}q_{\phantom{B}\beta}^{B}q_{\phantom{C}\gamma}^{C}q_{\phantom{D}\delta}^{D}-K_{\phantom{\alpha}\gamma}^{\alpha}K_{\beta\delta}+K_{\phantom{\alpha}\delta}^{\alpha}K_{\beta\gamma},
\begin{array}{lll}{}&{c(\omega_{1},\omega_{2},\omega_{3},\omega_{4})+c(\omega_{2},\omega_{1},\omega_{3},\omega_{4})+c(\omega_{3},\omega_{4},\omega_{1},\omega_{2})+c(\omega_{4},\omega_{3},\omega_{1},\omega_{2})+}\\{}&{d(\omega_{1},\omega_{2},\omega_{3},\omega_{4})+d(\omega_{2},\omega_{1},\omega_{3},\omega_{4})+e(\omega_{1},\omega_{2},\omega_{3},\omega_{4})=0}\\\end{array}
S_{\beta_{1}\ldots\beta_{4}}^{N_{1}\ldotsN_{4}}:=\frac{\sum_{N}\C_{\beta_{1}\ldots\beta_{4}}^{N_{1}\ldotsN_{4}\,N}}{\sqrt{\sum_{N}\C_{\beta_{1}\ldots\beta_{4}}^{N_{1}\ldotsN_{4}\N}{C}_{\beta_{1}\ldots\beta_{4}}^{N_{1}\ldotsN_{4}\N}}}\\.
\langleg\rangle^{(i)}=\omega^{i}\displaystyle\frac{(q^{2};q^{2})_{\infty}^{4}}{(t^{2};t^{2})_{\infty}^{4}}\displaystyle\frac{(t^{2}\omega;t^{2})_{\infty}^{2}(t^{2}\omega^{-1};t^{4})_{\infty}^{2}}{(q^{2}\omega;q^{2})_{\infty}^{2}(q^{2}\omega^{-1};q^{4})_{\infty}^{2}}.
\frac{db_{11}(t)}{dt}=\frac{\beta_{b_{11}}(b_{11},b_{e})}{1-\gamma(b_{11},b_{e})};\qquad\frac{db_{e}(t)}{dt}=\frac{\beta_{b_{e}}(b_{11},b_{e})}{1-\gamma(b_{11},b_{e})}
|0\rangle,\,\,A_{\vec{k}}^{\dagger}|0\rangle,\,\,A_{\vec{k}}^{\dagger}A_{\vec{k}^{\prime}}^{\dagger}|0\rangle\,\,\mathrm{for}\,\,\vec{k}\not=\vec{k}^{\prime},\,\,(A_{\vec{k}}^{\dagger})^{2}|0\rangle,\,\,\cdots
F_{8}^{{\mathcal{O}}_{1,1}^{1,0}|+++-----}=\frac{-iH^{1,1}((\sigma_{1}^{+})^{2}\sigma_{4}^{-}+(\sigma_{3}^{+})^{2}+\sigma_{3}^{+}\sigma_{1}^{+}\sigma_{2}^{-})/(\sigma_{3}^{+})^{2}}{2^{7}\prod_{1\leqi<j\leq3}\mathrm{cosh}(\theta_{ij}/2)\prod_{4\leqi<j\leq8}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
\langleT^{\mu\nu}\rangle(z)=u\left(\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}\\{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{3}\\\end{array}\right)+g(z)\left(\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}\\{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{0}\\\end{array}\right),
||u_{\underline{{a}}}^{~\tilde{\underline{{a}}}}||\qquad\inSO(1,9)\qquad\Rightarrow\qquadu_{\underline{{a}}}^{~\tilde{\underline{{a}}}}\,eta^{\underline{{a}}\underline{{b}}}u_{\underline{{b}}}^{~\tilde{\underline{{b}}}}=\eta^{\tilde{\underline{{a}}}\tilde{\underline{{b}}}}\qquadand\qquad||v_{\underline{{\alpha}}}^{~\tilde{\underline{{\alpha}}}}||\inSpin(1,9)
A_{jk}(x+y)={\frac{\biggl|\begin{matrix}{{A_{jm}(x)/\psi_{jm}(x)}}&{{A_{jm}(y)/\psi_{jm}(y)}}\\{{A_{mk}(x)/\psi_{mk}(x)}}&{{A_{mk}(y)/\psi_{mk}(y)}}\\\end{matrix}\biggr|}{\biggl|\begin{matrix}{{1/\psi_{jm}(x)}}&{{1/\psi_{jm}(y)}}\\{{1/\psi_{mk}(x)}}&{{1/\psi_{mk}(y)}}\\\end{matrix}\biggr|}}
\left[\widehat{{\mathcalJ}}_{\ell/r}^{0}\left(t,\underline{{y}}\right)\,,~{\mathcalJ}^{0}\left(t,\underline{{x}}\right)\right]\;=\;\pm~i~\frac{q^{2}}{8\pi^{2}}~\left(\underline{{B}}\left(\underline{{y}},t\right)\,\cdot\,\underline{{\nabla}}_{\underline{{x}}}\right)~\delta\left(\underline{{x}}-\underline{{y}}\right)~,
d\Pi^{i}=-\frac12\Omega^{+2i}\Pi^{-2}-\frac12\Omega^{-2i}\Pi^{+2}-\Omega^{ij}\Pi^{j}-2i\pi_{q}^{1+}\gamma_{q\dot{q}}^{i}\pi_{\dot{q}}^{1-}+2i\pi_{q}^{2-}\gamma_{q\dot{q}}^{i}\pi_{\dot{q}}^{2+}.
{\frac{ig^{2}}{2L}}\oint_{\gamma}dx_{j}^{+}\oint_{\gamma}dx_{k}^{+}\,max\{t_{j},t_{k}\}=i\,g^{2}\oint_{\gamma}dx^{+}\,\oint_{\gamma}dy^{+}\,{\frac{|x^{+}+x^{-}-y^{+}-y^{-}|}{4L\sqrt2}}.
\partial_{5}\left\langle{\Phi}\right\rangle=-g^{2}\left[(\sum_{i}Q_{i}\phi_{i}^{\dagger}\phi_{i})\left(\delta(x^{5})-{\frac{1}{2\ell}}\right)+(\kappa+\sum_{j}Q_{j}\phi_{j}^{\dagger}\phi_{j})\left(\delta(x^{5}-\ell)-{\frac{1}{2\ell}}\right)\right]\.
M^{2}\left(h_{\alpha}{}^{\alpha}g^{\mu\nu}-h^{\mu\nu}\right)+\Theta^{\mu\nu}-\frac{e}{2}\left(\partial_{\sigma}T_{(\alpha\beta)}^{\mu}\epsilon^{\alpha\beta\sigma\nu}+\partial_{\sigma}T_{(\alpha\beta)}^{\nu}\epsilon^{\alpha\beta\sigma\mu}\right)=0\,.
{\calL}^{(1)}=\pi_{0}\dot{A}^{0}+\pi_{i}\dot{A}^{i}+\pi_{\theta}\dot{\theta}+B\dot{N}+\bar{\calP}\dot{\calC}+\bar{\calC}\dot{\calP}+\tilde{\Omega}_{2}\dot{\rho}-{\calH}^{(1)}
A_{-1}^{\mathrm{tr}}(s)={\frac{1}{2\sqrt{\pi}}}{\frac{\Gamma(s-{\frac{1}{2}})}{\Gamma(s+1)}}\Bigr[r_{+}^{2s}-r_{-}^{2s}\Bigr]\Bigr[\zeta_{R}(2s-3)-\zeta_{R}(2s-1)\Bigr],