harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S=\int{\frac{d^{2}x}{2\pi}}\left[\sum_{i=1}^{m}\psi_{L}^{i}\partial\psi_{L}^{i}+\psi_{R}^{i}\bar{\partial}\psi_{R}^{i}+\sum_{j=1}^{n}\beta_{L}^{j}\partial\gamma_{L}^{j}+\beta_{R}^{j}\bar{\partial}\gamma_{R}^{j}+g\left(\psi_{L}^{i}\psi_{R}^{i}+\beta_{L}^{j}\gamma_{R}^{j}-\gamma_{L}^{j}\beta_{R}^{j}\right)^{2}\right]
{\calZ}={\frac{y^{2}(q)\,\prod_{i<j}A_{ij}B_{ij}}{c^{2}(\mathrm{det}K)^{5}\,\prod_{i<j}(a_{i}-a_{j})}}={\frac{y^{2}(q)\,Q_{\delta}}{c^{2}(\mathrm{det}K)^{5}\,P(a)}},
J^{\bar{\imath}}\!_{j\dot{\alpha}}\!^{\beta}(\Phi,\Phi^{*})=J^{\bar{\imath}}{}_{j}(\Phi,\Phi^{*})\delta_{\dot{\alpha}\alpha}\epsilon^{\alpha\beta}\,\qquadJ^{\bar{\imath}}{}_{j}J^{j}{}_{\bar{k}}=-\delta^{\bar{\imath}}{}_{\bar{k}}
T:\Upsilon_{l,\,m}=-i\sigma_{y}\Upsilon_{l,\,m}^{*}=\left(\begin{array}{r}{-\beta_{l,\,m}^{*}}\\{\alpha_{l,\,m}^{*}}\\\end{array}\right)=(-1)^{l-m}\,\left(\begin{array}{c}{\alpha_{l,\,-m}}\\{\beta_{l,\,-m}}\\\end{array}\right).
S=\intd^{4}x\sqrt{\hat{g}_{4}}\Bigl\{\hat{R}_{4}+\frac{2\hat{\nabla}_{\mu}{\calA}\hat{\nabla}^{\mu}{\calA}^{\dagger}}{({\calA}-{\calA}^{\dagger})^{2}}+\frac{1}{8}Tr\bigl(l\hat{\nabla}m\bigr)^{2}\Bigr\}
\begin{array}{rcl}{S_{g}}&{=}&{{\displaystyle\int}d^{4}x\,e\left\{R(e,\omega)+6g^{2}+2e^{-1}\epsilon^{\mu\nu\rho\sigma}\bar{\psi}_{\mu}\gamma_{5}\gamma_{\nu}\left(\hat{\calD}_{\rho}+igA_{\rho}\sigma^{2}\right)\psi_{\sigma}-{\calF}^{2}\right.}\\{}&{}&{}\\{}&{}&{\left.+{\calJ}_{(m)}{}^{\mu\nu}({\calJ}_{(e)\mu\nu}+{\calJ}_{(m)\mu\nu})\right\}\,,}\\\end{array}
M_{pl}^{2}={\frac{M_{X}^{3}}{k}}e^{-2ky_{1}-2c}\left({\frac{1}{2}}e^{-2k(y_{2}-2y_{1})}-e^{-2k(y_{2}-y_{1})}+1-{\frac{1}{2}}e^{-2k(\pi\rho-y_{2})}\right)~.~\,
\hat{\mathrm{K}}_{(1)}\equiv-\frac{i\hbar}{2}\sum_{\stackrel{p\in\calZ}{p\neq0}}\frac{(-1)^{p}}{p}\sum_{\stackrel{q\in\calZ}{q\neq(0;-p)}}\mathrm{k}_{\mathrm{R}}(p,q)\cdot\rho_{\mathrm{tot}}(-q)\rho_{\mathrm{tot}}(p+q),
q_{a}(a,b,r)=-\frac{1}{2\pir^{n}S_{D}}\sum_{l=0}^{\infty}D_{l}\int_{m}^{\infty}\frac{z^{3}dz}{\sqrt{z^{2}-m^{2}}}\Omega_{a\nu}(az,bz)F_{\nu}^{(q)}\left[G_{\nu}^{(a)}(az,zr)\right],\quadq=\varepsilon,\,p,\,p_{\perp},
G_{\mu\nu}=4\,\left[\partial_{\mu}\psi\,\partial_{\nu}\psi-{\frac{1}{2}}(\nabla\psi)^{2}g_{\mu\nu}\right]+\kappa^{2}e^{-a\psi}\left[F_{\mu}^{\alpha}F_{\nu\alpha}-{\frac{1}{4}}F^{2}g_{\mu\nu}\right],
\frac{1}{2}\left[(D-2)\left(\left(\frac{f^{\prime}}{f}\right)^{\prime}+\left(\frac{f^{\prime}}{f}\right)^{2}\right)+\frac{(D-2)(D-5)}{4}\left(\frac{f^{\prime}}{f}\right)^{2}\right]=\alpha^{2}\frac{(D-2)^{2}+(D-2)}{8}\.
S_{CTP}[\varphi^{a}]=\intd^{4}x\\frac{1}{2}c_{ab}\{\partial_{\mu}\varphi^{a}\partial^{\mu}\varphi^{b}-m^{2}\varphi^{a}\varphi^{b}\}-\frac{\lambda}{4!}h_{abcd}\varphi^{a}\varphi^{b}\varphi^{c}\varphi^{d}
\partial_{\mu}J^{5\mu}=e{\frac{{\bfa}-1}{(2\pi)^{3}}}\epsilon^{\mu\nu\rho\sigma\tau\omega}F_{\mu\nu}\left[\,e^{2}\,F_{\rho\sigma}F_{\tau\omega}+g^{2}F_{\rho\sigma}^{5}F_{\tau\omega}^{5}\right]
ig\left[\delta^{ij}\delta^{kl}T_{JI}^{a}T_{LK}^{a}\gamma_{~\alpha}^{\mu~\beta}\gamma_{\mu\,\gamma}^{~~~\delta}~-~\delta^{il}\delta^{kj}T_{LI}^{a}T_{JK}^{a}\gamma_{~\alpha}^{\mu~\delta}\gamma_{\mu\,\gamma}^{~~~\beta}\right]
\intD{\calB}_{+}det\left(\begin{array}{cc}{0}&{i\partial_{t}+{\calB}_{+}}\\{\partial_{-}}&{0}\\\end{array}\right)=det\left(\begin{array}{cc}{0}&{i\partial_{t}}\\{i\partial_{-}}&{0}\\\end{array}\right)~\intDUe^{-5\dot{W}[U]},
[\]_{\calE}:=\sum_{n=1}^{\infty}(-1)^{n}[\]_{2n},\\[\]_{\calO}:=\sum_{n=0}^{\infty}(-1)^{n}[\]_{2n+1},\\b_{n}^{\dagger}=b_{-n},c_{n}^{\dagger}=c_{-n},\n\geq1.
\Omega^{MN}=\left(\begin{array}{cc}{\Omega^{AB}}&{-F^{KJ}\Omega^{AC}A_{K}^{a}(\partialQ^{a}/\partial\xi^{C})}\\{F^{KI}\Omega^{BD}A_{K}^{a}(\partialQ^{a}/\partial\xi^{D})}&{F^{IJ}}\\\end{array}\right)
A_{0}^{2}(g_{s}A_{\mu}^{a}\lambda^{a})(g_{s}A^{\mub}\lambda^{b})\,\rightarrowA_{0}^{2}[g_{n}^{2}(A_{\mu}^{\pm})^{2}+g_{z}^{2}(A_{\mu}^{z})^{2}]
(\theta^{-}\tilde{\gamma}^{ij}\theta^{-})q_{+10}^{k}\gamma_{\dot{B}A}^{ijk}\lambda_{A}=0,\\(\theta^{+}\gamma^{ij}\theta^{+})q_{-10}^{k}\gamma_{B\dot{A}}^{ijk}\lambda_{\dot{A}}=0.
e_{\mu}^{a}(\xi)=\delta_{\mu}^{a}+\xi^{\nu}\xi^{b}\int_{0}^{1}d\lambda~\lambda(1-\lambda)R_{b,\nu\mu}^{a}(\lambda\xi)+\xi^{\nu}\int_{0}^{1}d\lambda~\lambdaS_{\nu\mu}^{a}(\lambda\xi)
a_{i_{1}i_{2}\dotsi_{m}}\;=\;a_{i_{1}}-\frac{r_{i_{1}}^{2}\;(a_{i_{1}}-a_{i_{2}\dotsi_{m}})}{|a_{i_{1}}-a_{i_{2}\dotsi_{m}}|^{2}-r_{i_{2}\dotsi_{m}}^{2}},\;\;\;\;\;\;r_{i_{1}i_{2}\dotsi_{m}}\;=\;\frac{r_{i_{1}}^{2}\;r_{i_{2}\dotsi_{m}}}{|a_{i_{1}}-a_{i_{2}\dotsi_{m}}|^{2}-r_{i_{2}\dotsi_{m}}^{2}}
\delta^{2}(z-z^{\prime})=\alpha\sum_{n}\frac{e^{in(\phi-\phi^{\prime})}}{2\pi}\int_{o}^{\infty}\frac{r^{(\alpha-1)}}{\alpha}J_{\frac{|n|}{\alpha}}(\frac{k}{\alpha}r^{\alpha})kdk\frac{{r^{\prime}}^{(\alpha-1)}}{\alpha}J_{\frac{|n|}{\alpha}}(\frac{k}{\alpha}{r^{\prime}}^{\alpha})
{\calL}_{int}=\lambda_{1}\sum_{i=1}^{N}(\phi_{i}\phi_{i})^{2}+\eta_{1}\sum_{i=1}^{N}(\phi_{i}\phi_{i})^{3}+\lambda_{2}\sum_{i=1}^{N}(\phi_{i})^{4}+\eta_{2}\sum_{i=1}^{N}(\phi_{i})^{6}.
dE_{2\,non-disp.}\Bigl|_{V=const}=\frac{\varepsilon-1}{4\pi}\frac{\delta\varepsilon}{4\pi}\iint_{|{\bfr}_{C}-{\bfr}|\ge\lambda}dV({\bfr}_{C})dV({\bfr})\,\frac{-23}{4\pi|{\bfr}_{C}-{\bfr}|^{7}},
k_{i\mu\nu}=-{\frac{\mu}{4}}(f_{~\mu\lambda}^{Y}f_{~~\nu}^{i\lambda}+f_{~\nu\lambda}^{Y}f_{~~\mu}^{i\lambda})+{\frac{1}{2\mu}}(k_{4\mu}k_{i\nu}+k_{4\nu}k_{i\mu}).
P_{\alpha\rho}^{(\mu)}\left(p\right)P^{(\mu)\rho}{}_{\beta}\left(p\right)=P_{\alpha\beta}^{(\mu)}\left(p\right)\;\,,\;\,P_{\alpha\beta}^{(\mu)}\left(p\right)=P_{\beta\alpha}^{*(\mu)}\left(p\right)\;\,.
\langle\,:a_{n}^{i}\,(a_{m}^{j})^{\dagger}:\,\rangle_{\beta,\gamma}=\langle\,:(a_{m}^{j})^{\dagger}\,a_{n}^{i}:\,\rangle_{\beta,\gamma}=\frac{\delta^{ij}\,\delta_{nm}}{e^{\epsilon_{n}}-1}\,,
H=\sqrt{g_{tt}}\left[{\frac{P_{\phi}^{2}}{h(r)r^{2}}}+{\frac{P_{r}^{2}}{g_{rr}}}+{\frac{P_{\rho}^{2}}{g_{\rho\rho}}}+{\frac{(\rhoP_{\phi}-N\rho^{p})^{2}}{g_{\phi\phi}}}\right]^{\frac{1}{2}},
{\bar{\psi}}(x_{2})=(2\pi)^{-3/2}\intd^{3}p_{2}(m/E_{2})^{1/2}[b_{p_{2}r_{2}}^{*}{\bar{u}}^{r_{2}}(p_{2})e^{-ix_{2}.p_{2}}+d_{p_{2}r_{2}}{\bar{v}}^{r_{2}}(p_{2})\timese^{ix_{2}.p_{2}}];
\frac{2}{\alpha\beta}(\partial_{0}(\frac{\partial_{0}\beta}{\alpha})-\partial_{1}(\frac{\partial_{1}\alpha}{\beta}))+\frac{d}{d\Omega}(V(\Omega)\Omega)-\frac{1}{4}(1-\frac{\epsilon}{2})\Omega^{-\epsilon/2}F^{2}
\delta{\calP}_{i}^{\left(1\right)a}=-\epsilon_{0ijk}\pi^{jka},\;\delta{\calP}_{i}^{\left(2\right)a}=-\frac{1}{2}\epsilon_{0ijk}F^{jka},\;\delta{\calP}^{a}=-\partial^{i}{\calP}_{i}^{\left(2\right)a},
\begin{array}{lcl}{\partial{\bar{\partial}}\phi_{1}}&{=}&{Ae^{-\frac{1}{2\theta}(\phi_{1}+\frac{1}{2}\phi_{2})}}\\{\partial{\bar{\partial}}\phi_{2}}&{=}&{Be^{-\frac{1}{2\theta}(\phi_{1}+2\phi_{2})}}\\\end{array}
(c_{mn}^{pr}+c_{nm}^{pr})(a_{pt}^{i}a_{ry}^{j}+a_{py}^{i}a_{rt}^{j})+(c_{ty}^{qs}+c_{yt}^{qs})(a_{mq}^{i}a_{ns}^{j}+a_{nq}^{i}a_{ms}^{j}),
ds^{2}=\left(\prod_{i=1}^{i=n}H_{i}^{-2r_{i}}\right)Wdt^{2}-\left(\prod_{i=1}^{i=n}H_{i}^{-2r_{i}}\right)^{-\frac{1}{d-3}}\left[W^{-1}d\rho^{2}+\rho^{2}d\Omega_{(d-2)}^{2}\right]\,,
S_{P}[{\bar{x}};{\hat{g}}]={\frac{1}{4\pi\alpha^{\prime}}}\sum_{n=-\infty}^{\infty}{\frac{2n\pi}{\mathrm{sinh}(2n\pi/l)}}\left[(|x_{n}^{i}|^{2}+|x_{n}^{f}|^{2})\mathrm{Cosh}(2n\pi/l)-2\mathrm{Re}(x_{n}^{i}\cdotx_{n}^{f*})\right]\quad.
\Xi_{\mu\nu}^{\left(++\right)}\left(\varphi\right)=\frac{1}{2}\left(\mathbb{E}_{\mu}^{t}\left(\varphi\right)\mathbb{E}_{\nu}\left(\varphi\right)-\mathbb{E}_{\nu}^{t}\left(\varphi\right)\mathbb{E}_{\mu}\left(\varphi\right)\right)
B_{MN}=(-)^{M(N+\underline{{N}})}\partial_{N}z^{\underline{{N}}}\partial_{M}z^{\underline{{M}}}B_{\underline{{MN}}}\;,\;\;\;E_{\pm}^{\underline{{A}}}=E_{\pm}^{M}\partial_{M}z^{\underline{{N}}}E_{\underline{{N}}}^{\underline{{A}}}\;,
\tilde{\calA}\vphantom{\calA}_{0}^{z}(T)=\tilde{a}\vphantom{a}_{0}^{z}(T+\tilde{\Lambda})^{\frac{-2\alpha}{1+\alpha}}+\tilde{b}\vphantom{b}_{0}^{z}(T+\tilde{\Lambda})^{\frac{1-\alpha}{1+\alpha}}
S_{I}\simk^{2}~\frac{2^{2\nu}~\tau_{0}}{\Gamma^{2}(1-\nu)~(2-2\nu)~r^{2\nu-2}}~J_{\nu}^{2}(-r\tau_{0})\left(\frac{\tau_{f}}{\tau_{1}}\right)^{2-2\nu}
{\mathbbC}[W_{\chi^{-1}}]/I\simeq\oplus_{k\geq0}W_{\chi^{-k}}=\sum_{k\ge0}\,\left(\underbrace{\begin{array}{|c|c|c|c|c|}\hline{\phantom{\times}}&{\phantom{\times}}&{\phantom{\times}}&{\dots}&{\phantom{\times}}\\\hline{}\\\end{array}}_{3k}\,\otimes\,\underbrace{\begin{array}{|c|c|c|c|}\hline{\times}&{\times}&{\dots}&{\times}\\\hline{}\\\end{array}}_{2k}\right)\,.
\mathrm{STr}(F_{0}\tau_{3})^{m}(\tau_{a}X^{a})(\tau_{b}X^{b})=F_{0}^{m}X^{a}X^{b}\left[\frac{1}{m+1}\sum_{i=0}^{m}\mathrm{Tr}(\tau_{3}^{i}\tau_{a}\tau_{3}^{m-i}\tau_{b})\right]
\begin{array}{ccccc}{\delta_{i}}&{\Longleftrightarrow}&{(\mathbb{E}_{i}(\varphi))_{\mu\nu}}&{=}&{\delta_{0\mu}\delta_{i\nu}-\delta_{0\nu}\delta_{i\mu}-f_{i\mu\nu}^{\left(+\right)}(\varphi),}\\{1|\delta_{i}}&{\Longleftrightarrow}&{(1|\mathbb{E}_{i}(\varphi))_{\mu\nu}}&{=}&{\delta_{0\mu}\delta_{i\nu}-\delta_{0\nu}\delta_{i\mu}+f_{i\mu\nu}^{\left(-\right)}(\varphi),}\\\end{array}
\begin{array}{c}{\Gamma^{\|}=1\otimes\gamma^{\|}\otimes1}\\{\gamma^{\|}=\gamma^{\partial_{\varphi}}\gamma^{I\partial_{\varphi}}}\\{\gamma^{\partial_{\varphi}}={\frac{1}{\sqrt{|\dot{x}|^{2}+|\dot{y}|^{2}}}}(\dot{x}\gamma^{x}+\dot{y}\gamma^{y}+c.c.)}\\{\gamma^{I\partial_{\varphi}}={\frac{i}{\sqrt{|\dot{x}|^{2}+|\dot{y}|^{2}}}}(\dot{y}^{*}\gamma^{x}-\dot{x}^{*}\gamma^{y}-c.c.)}\\\end{array}
\partial_{\mu}(\Lambda^{a}\sigma^{\mu}\bar{\Lambda}^{a})=2C_{2}(G)\frac{1}{8\pi^{2}}\left(\frac{1}{4}F_{\mu\nu}^{a}\tilde{F}^{a\mu\nu}\right)-i(m_{\mathrm{R}}\Lambda^{a\alpha}\Lambda_{\alpha}^{a}-\mathrm{h.c})
S_{R}(P)=G_{R}\left(\frac{Q}{2}+iP\right)=\left(\frac{\pi\mu}{2}\gamma\left(\frac{1+b^{2}}{2}\right)b^{1-b^{2}}\right)^{-\frac{2iP}{b}}\frac{\Gamma\left(\frac{1}{2}+iPb\right)\Gamma\left(\frac{1}{2}+\frac{iP}{b}\right)}{\Gamma\left(\frac{1}{2}-iPb\right)\Gamma\left(\frac{1}{2}-\frac{iP}{b}\right)}.
{f^{(1)}}^{-1}=\left(\begin{array}{c|c}{f^{-1}-f^{-1}AC^{-1}\tilde{A}f^{-1}}&{-f^{-1}AC^{-1}}\\\hline{C^{-1}\tilde{A}f^{-1}}&{C^{-1}}\\\end{array}\right).
\begin{array}{c}{OR_{a}\circOR_{b}=(-1)^{O_{ab}}(OR_{a}\oplus_{2}OR_{b});}\\{OR_{a}\astOR_{b}=(-1)^{O_{ba}}(OR_{a}\oplus_{2}OR_{b});}\\{OC_{a}\diamondOC_{b}=(-1)^{O_{ab}}(OC_{a}\oplus_{2}OC_{b});}\\{OC_{a}\starOC_{b}=(-1)^{O_{ba}}(OC_{a}\oplus_{2}OC_{b}).}\\\end{array}
S=S_{0}+S_{I}=\frac{1}{4\pi\alpha^{\prime}}\int_{M}\!\!d^{2}z\sqrt{g}g^{ab}\partial_{a}x^{\mu}\partial_{b}x^{\nu}G_{\mu\nu}+\frac{1}{2\pi\alpha^{\prime}\mu}\int_{\partialM}edt\;H_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}
\bar{\rho}_{\mathrm{s}}(\zeta)\;=\;{\frac{8\beta}{\left[E(k)-(1-k^{2})K(k)\right]^{2}}}\left[k^{2}{\frac{\sqrt{1-\zeta^{2}}}{\sqrt{k^{2}-\zeta^{2}}}}K(k)-\sqrt{k^{2}-\zeta^{2}}\sqrt{1-\zeta^{2}}\Pi(\zeta^{2},k)\right]\;,
ds^{2}=H^{1/3}[H^{-1}(-d\tilde{t}^{2}+dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2}+dy_{4}^{2}+dy_{5}^{2})+d\tilde{y}_{11}^{2}+dx^{i}dx_{i}]
{\frac{\ddot{a}_{0}}{a_{0}}}=-{\frac{32\pi^{2}G_{5}^{2}}{9c^{6}}}(2\varrho^{2}c^{4}+\sigma\varrhoc^{2}+3\sigma\wp+3\varrho\wpc^{2})-{\frac{c^{2}a_{R\,0}^{\prime\prime}}{a_{0}}}+{\frac{32\pi^{2}G_{5}^{2}}{9c^{6}}}\sigma^{2},
|1\rangle,\,\,A_{\vec{k}}^{\dagger}|1\rangle,\,\,A_{\vec{k}}^{\dagger}A_{\vec{k}^{\prime}}^{\dagger}|1\rangle\,\,\mathrm{for}\,\,\vec{k}\not=\vec{k}^{\prime},\,\,(A_{\vec{k}}^{\dagger})^{2}|1\rangle,\,\,\cdots
\frac{1}{4\pi}\frac{e^{2}}{v_{eff}}(\Lambda/\Lambda_{0})=\frac{1}{4\pi}\frac{e^{2}}{v_{R}}\left(\frac{v_{0}^{2}}{v_{R}^{2}}+\frac{1}{8\pi}\frac{e^{2}}{v_{R}}\;log\frac{\Lambda}{\Lambda_{0}}\right)^{-1/2}
\begin{array}{l}{^{W_{k}=W\left[\left(F(u_{k},V_{k}(\sigma^{+}))+F(u_{k},V_{k}(\sigma^{-}))-F(u_{k},v_{k-1})\right),u_{k}\right]}}\\{_{\bar{W}_{k}=\bar{W}\left[\left(\bar{F}(U_{k}(\sigma^{+}),v_{k})+\bar{F}(U_{k}(\sigma^{-}),v_{k})-\bar{F}(u_{k-1},v_{k})\right),v_{k}\right].}}\\\end{array}
\deltag_{s}^{\mu_{1}\mu_{2}\cdots\mu_{s}}=\cdots+\left(2g_{2}^{(\mu_{1}\mu_{2}}\varepsilon_{s-1,s-2}^{\mu_{3}\cdots\mu_{s})}-\varepsilon_{s-1,s-1}^{\mu(\mu_{3}\cdots\mu_{s}}\partial_{\mu}g_{2}^{\mu_{1}\mu_{2})}+\partial^{(\mu_{1}}\varepsilon_{s-1,s-1}^{\mu_{2}\cdots\mu_{s})}\right).
S_{F}(x-x^{\prime})=\int\frac{d\omega}{2\pi}\frac{dk}{2\pi}\;\frac{ie^{-i\omega(t-t^{\prime})}}{\omega{}^{2}-k{}^{2}-m^{2}+i\epsilon}\left(\begin{array}{cc}{m}&{-i(\omega+k)}\\{i(\omega-k)}&{m}\\\end{array}\right)e^{ik(x-x^{\prime})}.
\mu_{i,i+j}\simeq\frac{\mu_{0}}{1+a\mu_{0}\left[\left|x_{i+j}-x_{i}\right|-\left|x_{j}\right|\right]}\approx\mu_{0}\left[1-a\mu_{0}\left(\left|x_{i+j}-x_{i}\right|-\left|x_{j}\right|\right)\right]
\frac{\varphi_{n+1}-2\varphi_{n}+\varphi_{n-1}}{T^{2}}+\left(\mu^{2}-\nabla^{2}\right)\frac{\left(\varphi_{n+1}+4\varphi_{n}+\varphi_{n-1}\right)}6={}_{{}_{\!\!\!\!\!\!\!c}}\;\,j_{n}.
\psi^{(+)}\stackrel{r\sim0}{\rightarrow}\left[c_{1}\;r^{\left({\frac{7}{4}}\right)}\left(1+{\calO}(r)\right)\,\,+\,c_{2}\;r^{\left(-{\frac{3}{4}}\right)}\left(1+{\calO}(r)\right)\,\right].
\frac{1}{A^{\alpha}B^{\beta}C^{\gamma}}=\frac{\Gamma(\alpha+\beta+\gamma)}{\Gamma(\alpha)\Gamma(\beta)\Gamma(\gamma)}\int_{0}^{1}dz\int_{0}^{z}dt\frac{t^{\gamma-1}(z-t)^{\beta-\alpha}}{[A+(B-A)z+(C-B)t]^{\alpha+\beta+\gamma}}\;.
{\delta(\bfn-\bfn^{\prime})}=\Sigma_{j,\mu,m}{\Omega_{j,j+{\frac{\mu}{2}},m}}^{(+)}{(\bfn)}{{\Omega_{j,j+{\frac{\mu}{2}},m}}^{(+)}{(\bfn^{\prime})}}\dagger+{\Omega_{j,j+{\frac{\mu}{2}},m}}^{(-)}{(\bfn)}{{\Omega_{j,j+{\frac{\mu}{2}},m}}^{(-)}{(\bfn^{\prime})}}\dagger
(k_{1}-k_{2})_{\mu}T_{\mu}^{AP}=\int\frac{d^{4}k}{(2\pi)^{4}}Tr\left\{\gamma_{5}\frac{1}{[\not{k}+\not{k}_{2}-m]}i(\not{k}_{1}-\not{k}_{2})\gamma_{5}\frac{1}{[\not{k}+\not{k}_{1}-m]}\right\},
L=\!\intd^{2}x\left(i\,\psi^{\dagger}\dot{\psi}\!+\!i\,\chi^{\dagger}\dot{\chi}\!-\!{\frac{1}{2}}|\nabla\psi|^{2}\!-\!{\frac{1}{2}}|\nabla\chi|^{2}\!-\!{\frac{v}{4}}[\psi^{\dagger}\!*\!\psi^{\dagger}\!+\!\chi^{\dagger}\!*\!\chi^{\dagger}]\!*\!\left[\psi\!*\!\psi\!+\!\chi\!*\!\chi\right]\right),
h_{1}^{2}v^{2}\rightarrowh_{2}^{2}v^{2}+(2h_{2}h_{3}+h_{3}^{2}\phi^{2})(v^{j}\phi^{j})^{2}\equivg_{1}v^{2}+g_{2}(v^{i}\phi^{i})^{2}~,
\psi_{\varphi\varphi\pi\varphi\ldots\pi_{N}}(\stackrel{\rightharpoonup}{x_{1}},\ldots,\stackrel{\rightharpoonup}{x_{N}})\rightarrow(-1)^{\#(\pi-\mathrm{indices})}\psi_{\varphi\varphi\pi\varphi\ldots\pi_{N}}(-\stackrel{\rightharpoonup}{x_{1}},\ldots,\stackrel{\rightharpoonup}{x_{N}})\\\.
\begin{array}{l}{\mu_{+}x^{+}=q^{2}x^{+}\mu_{+}~,~~~~~\mu_{+}\partial_{+}=q^{-2}\partial_{+}\mu_{+}~,}\\{\Lambdax^{k}=q^{2}x^{k}\Lambda~,~~~~~\Lambda\partial_{k}=q^{-2}\partial_{k}\Lambda~,~~~\mathrm{for}~k=0,\pm~.}\\\end{array}
\lambda=W(\mu)\sim\mu^{p}A^{(W,Q)}\equivN^{(W,Q)}(\mu){\frac{1}{W^{\prime}(\mu)}}{\frac{\partial}{\partial\mu}}[N^{(W,Q)}(\mu)]^{-1}=={\frac{1}{W^{\prime}(\mu)}}{\frac{\partial}{\partial\mu}}-{\frac{1}{2}}{\frac{W^{\prime\prime}(\mu)}{W^{\prime}(\mu)^{2}}}+Q(\mu)
U_{\mathrm{AP}}^{\mathrm{bf}}(\vec{x};C)=e^{-\mathrm{i}{\frac{\theta}{2}}\tau^{3}}U_{\mathrm{AP}}^{\mathrm{bf}}(D(e^{\mathrm{i}{\frac{\theta}{2}}\tau^{3}})\cdot\vec{x};e^{\mathrm{i}{\frac{\theta}{2}}\tau^{3}}Ce^{-\mathrm{i}{\frac{\theta}{2}}\tau^{3}})e^{\mathrm{i}{\frac{\theta}{2}}\tau^{3}}
\deltaG^{AB}\,\biggl(\partial_{A}\partial_{B}\varphi-\overline{{\Gamma}}_{AB}^{C}\,\partial_{C}\varphi\biggr)+\overline{{G}}^{AB}\,\biggl(\partial_{A}\partial_{B}\chi-\overline{{\Gamma}}_{AB}^{C}\partial_{C}\chi-\delta\Gamma_{AB}^{C}\partial\varphi\biggr)+\frac{\partial^{2}V}{\partial\varphi^{2}}\chi=0,
S=\intd^{2}\xi[p_{m}(V^{\mu}\partial_{\mu}x^{m}-\bar{\lambda}\gamma^{m}\lambda)+\phi\epsilon^{\mu\nu}\left(\partial_{\mu}A_{\nu}-V_{\mu}\Omega_{\nu}(V^{\lambda}\partial_{\lambda}x^{m})(\Omega^{\rho}\partial_{\rho}x_{m})\right)]
\Box\sum\pi_{\nu_{s-1}\nu_{s}}\sum_{n=0}^{[s/2]-1}\left[b_{n}^{\prime}\sum_{\mathrm{symm}^{\prime}}\pi_{\mu_{1}\mu_{2}}\cdots\pi_{\mu_{n-1}\mu_{n}}\pi_{\nu_{1}\nu_{2}}\cdots\pi_{\nu_{n-1}\nu_{n}}\pi_{\mu_{n+1}\nu_{n+1}}\cdots\pi_{\mu_{s-2}\nu_{s-2}}\right].
\Lambda=a+k_{\alpha\dot{\alpha}}x^{\alpha\dot{\alpha}}+\lambda^{ij}u_{i}^{+}u_{j}^{-}+4i(\theta^{+\alpha}\eta_{\alpha}^{i}+\bar{\eta}_{\dot{\alpha}}^{i}\bar{\theta}^{+\dot{\alpha}})u_{i}^{-}
\epsilon\left(\,g_{1}^{a_{11}}\,g_{2}^{a_{12}}\,g_{3}^{a_{13}},\:g_{1}^{a_{21}}\,g_{2}^{a_{22}}\,g_{3}^{a_{23}},\:g_{1}^{a_{31}}\,g_{2}^{a_{32}}\,g_{3}^{a_{33}}\,\right)\:=\:\epsilon(g_{1},g_{2},g_{3})
\int_{\Omega_{\rho}^{+}(\eta_{k})}\frac{H_{\mu}^{(1)}(\lambdaz)}{H_{\nu}^{(1)}(z)}F(z)dz+\int_{\Omega_{\rho}^{-}(\eta_{k})}\frac{H_{\mu}^{(2)}(\lambdaz)}{H_{\nu}^{(2)}(z)}F(z)dz=\left(2-\delta_{0\eta_{k}}\right)\pii\,{\mathrm{Res}}_{z=\eta_{k}}\frac{H_{\mu}^{(1)}(\lambdaz)}{H_{\nu}^{(1)}(z)}F(z),
b(m_{n})=J_{\alpha-\frac{1}{2}}(\frac{m_{n}}{k})\frac{N_{\alpha-\frac{1}{2}}(\frac{m_{n}e^{kr_{0}}}{k})}{N_{\alpha-\frac{1}{2}}(\frac{m_{n}}{k})}-J_{\alpha-\frac{1}{2}}(\frac{m_{n}e^{kr_{0}}}{k})=0,
{\calL}_{CS}={\textstyle\frac{1}{4}}m\varepsilon^{\alpha\beta\gamma}\Bigl(G_{\alpha\beta}^{a}A_{\gamma}^{a}-{\textstyle\frac{1}{3}}g\varepsilon_{abc}A_{\alpha}^{a}A_{\beta}^{b}A_{\gamma}^{c}\Bigr),
R_{MNPQ}=\left(-B_{\infty}+\left[\frac{H^{2}}{B_{\infty}r^{2}}\right]_{M,N,P,Q\in\{x^{\mu}\}}+\left[\frac{1}{r^{2}}\right]_{M,N,P,Q\in\{\theta^{a}\}}\right)(g_{MP}g_{NQ}-g_{MQ}g_{NP}),
\left\langle\,{\frac{\bar{U}_{a}U_{a}}{V}}\,\right\rangle^{1/3}\,\approx\,{\frac{\left\langle\,\left|\lambda_{a}^{\alpha}\lambda_{a\alpha}\right|\,\right\rangle^{2/3}}{g_{s}^{2/3}}}\,=\,{\frac{\left\langle\,\left|\lambda_{a}^{\alpha}\lambda_{a\alpha}\right|\,\right\rangle^{2/3}}{M_{s}^{2}g_{s}^{-{4/3}}}},
\begin{array}{l}{\Bigl[\kappa(\lambda)\otimes\kappa(\mu),r(a,\lambda,\mu)\Bigr]+\Bigl(I\otimes\kappa(\mu)\Bigr)r(a,\lambda,-\mu)\Bigl(\kappa(\lambda)\otimesI\Bigr)}\\{\phantom{\kappa(\lambda)\otimes\kappa(\mu)\,r(a,\lambda,\mu)-r(a,\lambda,\mu)\,}-\Bigl(\kappa(\lambda)\otimesI\Bigr)r(a,\lambda,-\mu)\Bigl(I\otimes\kappa(\mu)\Bigr)=0,}\\\end{array}
\begin{aligned}{}&{{}\alpha_{n}-\alpha_{-n+d}^{\prime}=a_{n},\qquad\qquad\qquad\qquad\beta_{n}+\beta_{-n+2d}^{\prime}=b_{n}}\\{}&{{}\mu^{n/d}\alpha_{n}-\mu^{(d-n)/d}\alpha_{-n+d}^{\prime}={\overline{c}}_{-n},\qquad\mu^{(n-d)/d}\beta_{n}+\mu^{(d-n)/d}\beta_{-n+2d}^{\prime}={\overline{d}}_{-n}.}\\\end{aligned}
g_{0}\left(\begin{array}{cc}{N_{1}}&{N_{2}}\\{M_{1}}&{M_{2}}\\\end{array}\right)=\frac{\mu^{-2\varepsilon}}{\sqrt{Z_{N_{1}}^{(L)}Z_{M_{1}}^{(R)}Z_{N_{2}}^{(R)}Z_{M_{2}}^{(L)}}}G\left(\begin{array}{cc}{N_{1}}&{N_{2}}\\{M_{1}}&{M_{2}}\\\end{array}\right).
S_{WZ}^{ND3}=\mu_{3}\int_{R^{3+1}}\mathrm{Tr}\left(C^{(4)}+i(2\pi\alpha^{\prime})i_{\Phi}i_{\Phi}B^{(6)}+\frac12C^{(2)}\wedgeB^{(2)}-(2\pi\alpha^{\prime})B^{(2)}\wedge{\tilde{\calF}}+\dots\right)\,,
A_{r}\({\bfV}):\quad\mathrm{Spec}(\bar{L})=g\left\{\begin{array}{ll}{0[2],\,\pm2,\,\pm4,\ldots,\,\pm(r-1)}&{r:\\mathrm{odd}}\\{0,\quad\,\pm1,\,\pm3,\ldots,\,\pm(r-1)}&{r:\\mathrm{even}}\\\end{array}\right\},
I=\intd^{5}x[\frac{1}{4}V^{MN}V_{MN}^{*}-\frac{1}{4}V^{MN*}[\partial_{M}A_{N}-\partial_{N}A_{M}]-\frac{1}{4}V^{MN}[\partial_{M}A_{N}^{*}-\partial_{N}A_{M}^{*}]].
H_{Cal}^{(2)}={\frac{1}{2}}\sum_{i=1}^{N}\left[-\partial_{i}^{2}+x_{i}^{2}\right]+\sum_{j<i}^{N}\fracg{(x_{i}-x_{j})^{2}}+\frac{\gamma}{\tau_{2}}+A\tau_{2}^{3}+B\tau_{2}^{2}+C\tau_{2},
{\frac{{\calM}^{2}}{\Lambda^{2}}}={\frac{{\calM}^{2}}{\mu^{2}}}{\frac{\mu^{2}}{\Lambda^{2}}},\qquad\mathrm{and}\qquad{\frac{m^{2}}{\Lambda^{2}}}={\frac{m^{2}}{\mu^{2}}}{\frac{\mu^{2}}{\Lambda^{2}}},
Y_{\psi}=\left(\begin{matrix}{1~}&{~0~}&{~0~}&{~0}\\{0~}&{~1~}&{~0~}&{~0}\\{0~}&{~0~}&{~-1/3~}&{~0}\\{0~}&{~0~}&{~0~}&{~-1/3}\\\end{matrix}\right);\quad\tau^{1}=~\left(\begin{matrix}{0~}&{~1~}&{~0~}&{~0}\\{1~}&{~0~}&{~0~}&{~0}\\{0~}&{~0~}&{~0~}&{~1}\\{0~}&{~0~}&{~1~}&{0}\\\end{matrix}\right)
\widehat{\Psi}(R(S),\langle\cdot\,,\cdot\rangle)=\otimes_{k=0}^{N}(\widehat{\Phi}_{k}^{*^{k+1}})^{(-1)^{k+1}}(w_{k}^{*^{k+1}})\,{\in}\,\otimes_{k=0}^{N}\Lambda^{max}H^{k}(R(S))^{*^{k+1}}\,.
G\,\bar{\psi}\gamma^{\mu}\psi\,\bar{\psi}\gamma_{\mu}\psi={\frac{\pi(\pi-g_{\mathrm{eff}}^{2})}{g_{\mathrm{eff}}^{2}}}\,\Bigl(\bigl(j^{0}\bigr)^{2}-{\frac{g_{\mathrm{eff}}^{2}}{\pi}}\,\bigl(j^{1}\bigr)^{2}\Bigr).
\begin{array}{rclrclrcl}{e_{t}}&{=}&{J_{1}+iJ_{2}}&{e_{u}}&{=}&{J_{6}+iJ_{7}}&{e_{v}}&{=}&{J_{4}+iJ_{5}}\\{f_{t}}&{=}&{J_{1}-iJ_{2}}&{f_{u}}&{=}&{J_{6}-iJ_{7}}&{f_{v}}&{=}&{J_{4}-iJ_{5}}\\{h_{t}}&{=}&{2J_{3}}&{h_{u}}&{=}&{(\sqrt{3}J_{8}-J_{3})}&{h_{v}}&{=}&{(\sqrt{3}J_{8}+J_{3}),}\\\end{array}
\psi(x)=\frac{Z}{\sqrt{2\pi}}\left(\begin{matrix}{\mathrm{\Large~:}e^{\mathrm{i}\,s\varphi_{\mathrm{R}}-\mathrm{i}\,\bar{s}\varphi_{\mathrm{L}}}\mathrm{\Large~:}}\\{\mathrm{\Large~:}e^{\mathrm{i}\,\bar{s}\varphi_{\mathrm{R}}-\mathrm{i}\,s\varphi_{\mathrm{L}}}\mathrm{\Large~:}}\\\end{matrix}\right),
\deltaT_{N}^{M}=\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{-\delta\rho}&{-(\rho+p)e^{\alpha_{0}}v_{,i}}\\{0}&{(\rho+p)e^{-\alpha_{0}}v_{,i}}&{\deltap\:\delta_{ij}}\\\end{array}\right)\:\delta(y).
q^{-{\frac{1}{16}}}\chi_{\frac{1}{16}}(q)=1+q+q^{2}+2\,q^{3}+2\,q^{4}+3\,q^{5}+4\,q^{6}+5\,q^{7}+6\,q^{8}+8\,q^{9}+10\,q^{10}+12\,q^{11}+15\,q^{12}+O(q^{13})\\
\begin{array}{c}{\dot{x}^{\mu}=\lambdae_{A}^{\mu}p^{A}\,\quad\frac{\displaystyleDp_{A}}{\displaystyled\tau}+eF_{AB}p^{B}=(\lambda/4)R_{ABCD}p^{B}S^{CD}\,}\\{\frac{\displaystyleDZ_{i}^{A}}{\displaystyled\tau}=0\,\quad\frac{\displaystyleD\psi_{a}}{\displaystyled\tau}=0}\\\end{array}
\bar{S}^{\prime}=\int_{0}^{t_{0}}\left[{\textstyle{\frac{1}{2}}}\epsilon_{\alpha\beta}\dot{\phi}_{\alpha}(t)\dot{\phi}_{\beta}(t)+\left(j_{\alpha}(t)+\tilde{L}_{\alpha}(t)\right)\phi_{\alpha}(t)\right]dt\,.
K(\theta)=\frac{s_{\frac{1}{2}}(\theta)s_{\frac{1+a}{2}}(\theta)s_{\frac{2-a}{2}}(\theta)}{s_{\frac{1-E}{2}}(\theta)s_{\frac{1+E}{2}}(\theta)s_{\frac{1-F}{2}}(\theta)s_{\frac{1+F}{2}}(\theta)}\,\,\,,
\prod_{\alpha=1}^{3}\left[{\frac{P^{\alpha}!Q^{\alpha}!R^{\alpha}!S^{\alpha}!U^{\alpha}!V^{\alpha}!(U^{\alpha}!+2I^{\alpha}!1)(2I^{\alpha}!+1)}{(V^{\alpha}!+2I^{\alpha}!+1)}}\right]^{1/2}\,.
{\calL}={\frac{1}{16\piG^{2}}}\intd^{4}x\sqrt{-det|g|}({\frac{{\tilde{\beta}}^{2}+u^{2}}{u^{2}}}r_{4}+{\frac{3}{2u^{2}}}\partial_{\mu}{\tilde{\beta}}\partial^{\mu}{\tilde{\beta}}+V({\tilde{\beta}}))
\begin{array}{lcl}{\pi^{ij}}&{=}&{\pi^{ijTT}+e^{-2\lambda}\sqrt{g}(\nabla^{i}Y^{j}+\nabla^{j}Y^{i}-g^{ij}\nabla_{k}Y^{k})+\frac{1}{2}\tau\sqrt{g}g^{ij}}\\{}&{=}&{e^{-2\lambda}p^{ij}+e^{-2\lambda}\sqrt{h}(\tilde{\nabla}^{i}Y^{j}+\tilde{\nabla}^{j}Y^{i}-h^{ij}\tilde{\nabla}_{k}Y^{k})+\frac{1}{2}\tau\sqrt{h}h^{ij},}\\\end{array}
\begin{array}{c}{ds^{2}=C(\eta)d\eta^{2}-a_{1}^{2}(\eta)dx^{2}-a_{2}^{2}(\eta)dy^{2}-a_{3}^{2}(\eta)dz^{2}}\\{C=(a_{1}a_{2}a_{3})^{2/3}}\\\end{array}
\psi^{N}(z)=(2\pi)^{-3/2}\int\left[v_{\A}^{\daggerN}({\bfp})a^{A}({\bfp})e^{-i(pz)}+v_{\B}^{N}({\bfp})a^{\daggerB}({\bfp})e^{i(pz)}\right]\frac{d{\bfp}}{2\omega({\bfp})}
S~=~\frac{1}{2e^{2}L}\intdt~\mathrm{Tr}\left(\dot{U}^{\dagger}\dot{U}\right)~=~\frac{1}{4e^{2}L}\intdt~\mathrm{Tr}\left(iT^{a}U^{\dagger}(t)\dot{U}(t)\right)^{2}
Q=s+\omega^{+}\delta^{+}+\omega^{-}\delta^{-}+\varepsilon^{+\mu}\delta_{\mu}^{+}+\varepsilon^{-\mu}\delta_{\mu}^{-}+v^{\mu}\partial_{\mu}-(\omega^{+}\varepsilon^{+\mu}+\omega^{-}\varepsilon^{-\mu})\frac\partial{\partialv^{\mu}}\,.
Z=C\,\left(\mu^{n_{c}-n_{f}}\Lambda_{1}^{3n_{c}-n_{f}}\right)^{1/(2n_{c}-n_{f})}\omega^{k},\qquad(k=1,2,\ldots2n_{c}-n_{f};\,\omega=e^{2\pii/(2n_{c}-n_{f})}),
V_{p}=-(1+\frac{2\mu}{z^{7-p}})^{\alpha+\beta_{+}}(1+\frac{\mu}{z^{7-p}})^{2(p-8)/(7-p)}+\frac{(8-p)(6-p)}{4z^{2}}-\mu^{2}\frac{(7-p)^{2}}{z^{2}(z^{7-p}+2\mu)^{2}}
\omega^{2}Z_{i(2)}\rightarrowZ_{i(2)},\\,omega^{2}\tilde{B}_{i(12)}^{2}\rightarrow\tilde{B}_{i(12)}^{2},\omega^{2}\tilde{\psi}_{i(12)}^{2}\rightarrow\tilde{\psi}_{i(12)}^{2}(i=1,2,3).
S_{curv}={\frac{1}{32\piG}}\intd^{11}x\sqrt{-g^{0}}R_{ab}^{0}\left(2h^{ac}h_{c}{}^{b}-h^{ab}h_{c}{}^{c}+{\frac{1}{4}}h_{c}{}^{c}h_{d}{}^{d}-{\frac{1}{2}}h^{ab}h_{ab}\right).
e^{-1}\mathcal{L}=R-\frac{1}{2}\sum_{i}(X_{i}^{-1}\partialX_{i})^{2}+\frac{4}{L^{2}}\sum_{i}X_{i}^{-1}-\frac{1}{4}\sum_{i}X_{i}^{-2}(F_{(2)}^{i})^{2}-\frac{1}{4}e^{-1}\epsilon^{\mu\nu\rho\sigma\lambda}F_{\mu\nu}^{1}F_{\rho\sigma}^{2}A_{\lambda}^{3}
S_{eff}^{A}[A^{\prime},c^{\prime},{\bar{c}}^{\prime},\epsilon]=S^{0}[A^{\prime}]-\intd^{4}x\biggl({\frac{1}{2\lambda}}(\eta\cdotA^{\prime\alpha})^{2}+{\bar{c}}^{\prime\alpha}\tilde{M}^{\alpha\beta}c^{\prime\beta}\biggr)+\epsilonO_{1}^{\prime}[A^{\prime},c^{\prime},{\bar{c}}^{\prime}]