harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\rho_{0}^{-1}\,=\,2\,-\,\lambda_{1}\,\frac{\lambda_{0}\,-\,\lambda_{1}}{\lambda_{0}\,+\,\lambda_{1}}\,{,}\quad\rho_{1}^{-1}\,=\,2\,+\,\lambda_{0}\,\frac{\lambda_{0}\,-\,\lambda_{1}}{\lambda_{0}\,+\,\lambda_{1}}\,{,}
f_{k}(x)={\frac{1}{\sqrt{4\pi\omega}}}\left\{\Theta(-x){\frac{1}{\sqrt{n}}}Te^{i(nkx-\omegat)}+\Theta(x)\left[e^{i(kx-\omegat)}-Re^{-i(kx+\omegat)}\right]\right\}.
\delta_{K}j_{T}^{0M_{1}\cdotsM_{p}}=\sum_{k=1}^{p}\partial_{\mu_{k}}\left[\frac1{p!}\epsilon^{0\mu_{1}\cdots\mu_{k}\cdots\mu_{p}}\cdot\phi_{,\mu_{1}}^{M_{1}}\cdots\delta_{K}\phi^{M_{k}}\cdots\phi_{,\mu_{p}}^{M_{p}}\right]\quad.
{}~~~~~\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{w(\omegau_{2}^{\prime}u_{3},\sigma+j_{2}+j_{3})w(u_{2}^{\prime\prime},\sigma)s(\sigma,j_{1})}{w(u_{2}^{\prime},\sigma+j_{2})w(u_{2}^{\prime\prime}u_{3},\sigma+i_{3})}\Bigg\}_{0}.
S=-\frac{1}{4\pi\alpha^{\prime}}\int\!{{d^{2}}\xi}[{R^{2}}{\partial_{\alpha}}{y}{\partial^{\alpha}}{y}+2kR{A_{\mu}}{\partial_{\alpha}}{y}{\partial^{\alpha}}{x^{\mu}}+(g_{\mu\nu}+{k^{2}}{A_{\mu}}{A_{\nu}}){\partial_{\alpha}}{x^{\mu}}{\partial^{\alpha}}{x^{\nu}}].
(ii)\quadN^{R}=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{a-1}&{a}&{0}\\{0}&{a}&{a-1}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right),\quadN^{NS}=\left(\begin{matrix}{a}&{0}&{0}&{a-1}\\{0}&{0}&{1}&{0}\\{0}&{1}&{0}&{0}\\{a-1}&{0}&{0}&{a}\\\end{matrix}\right),
\frac{dE}{dt}=\frac{2^{2-D}\pi^{-(D-5)/2}{\calG}\,(D-3)D}{\Gamma[(D-1)/2](D^{2}-1)(D-2)}\,{\biggl[}(D-1)\partial_{t}^{(D+2)/2}D_{ij}^{*}(t)\partial_{t}^{(D+2)/2}D_{ij}(t)-|\partial_{t}^{(D+2)/2}D_{ii}(t)|^{2}{\biggr]}\,.
\sigma^{2}=\left[\fraci{2m}\epsilon_{\mu\nu\alpha\beta}\frac14\left(\Gamma_{\mu}\Gamma_{\nu}-\Gamma_{\nu}\Gamma_{\mu}+\overline{{\Gamma}}_{\mu}\overline{{\Gamma}}_{\nu}-\overline{{\Gamma}}_{\mu}\overline{{\Gamma}}_{\nu}\right)\right]^{2}
\beta^{K}\=\-\frac{1}{2}+\left(\frac{\tau^{2}}{3}-\frac{h^{2}}{6}\right)+\left(\frac{2\tau^{2}h^{2}}{15}+\frac{\tau^{4}}{15}+\frac{h^{4}}{90}\right)+O(\epsilon^{5})\,
V^{\prime}=\frac{1}{8}(\chi^{\prime2}-(2\lambda)^{1/2}\sigma^{2})^{2}+\chi^{2}W_{\mu-}^{\prime}W_{\mu+}^{\prime}+\frac{1}{2}Z_{\mu}^{\prime2}\chi^{\prime2}+(t_{L}^{\prime+}t_{R}^{\prime}+t_{R}^{\prime+}t_{L}^{\prime})\chi^{\prime}
|h_{3}\rangle=({\bfP}^{\bar{z}})^{k}\sum_{n=0}^{k}\rho^{n}\frac{\Gamma(\frac{D}{2}+k-n-2)}{2^{n}n!\Gamma(\frac{D}{2}+k-2)}({\bfM}^{\bar{z}j}{\bfM}^{\bar{z}j})^{n}V_{0}|0\rangle_{1}|0\rangle_{2}|0\rangle_{3}\,,
\begin{array}{l}{D_{\mu}\psi_{i}^{a}=\partial_{\mu}\psi_{i}^{a}+\frac{1}{2}\omega_{\mu\nu\rho}\sigma^{\nu\rho}\psi_{i}^{a}+gf_{abc}A_{\mu}^{b}\psi_{i}^{c}}\\{D_{\mu}\phi^{a}=\partial_{\mu}\phi^{a}+gf_{abc}A_{\mu}^{b}\phi^{c}}\\{F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf_{abc}A_{\mu}^{b}A_{\nu}^{c}.}\\\end{array}
{\calL}^{int}=ie(\overline{\psi}_{1}\gamma^{\mu}\gamma^{5}\psi_{1}-\overline{\psi}_{2}\gamma^{\mu}\gamma^{5}\psi_{2})C_{\mu}+e(\overline{\psi}_{1}\gamma^{\mu}\psi_{1}+\overline{\psi}_{2}\gamma^{\mu}\psi_{2})B_{\mu}\,.
D(\Psi_{2}^{\prime\prime}+{\frac{Z_{2}Z_{6}}{K^{2}}}\omega^{2}\Psi_{2})+P_{2}^{2}\Psi_{2}^{\prime}+P_{3}^{2}{\calZ}_{6}^{\prime}+Q_{1}^{2}\Phi+Q_{2}^{2}\Psi_{2}+Q_{3}^{2}{\calZ}_{6}=0,
{\cal{F}}\left\{P^{\alpha}\right\}(x)=i2^{2\alpha+\frac{\nu}{2}}\frac{\Gamma\left(\alpha+\frac{\nu}{2}\right)}{\Gamma\left(-\alpha\right)}\left[\frac{1}{2}\left(Q+i0\right)^{-\alpha-\frac{\nu}{2}}-\frac{1}{2}\left(Q-i0\right)^{-\alpha-\frac{\nu}{2}}\right]
d\mu(\hat{\bboxp})=\frac{d^{3}\hat{p}}{2\hat{p}^{0}}={\frac{1}{m^{2}}}\,{\frac{d^{3}p}{2E({\bboxp})}},\,\,\,\,\,\,\hat{p}^{0}=\sqrt{1+\hat{\bboxp}^{2}}\,.
({1_{-}2_{-}})\equiv(g_{1})_{-}{}^{i}\epsilon_{ij}(g_{2})_{-}{}^{j}=e^{-i{(\phi_{1}+\phi_{2})/2}}\frac{\tau_{1}-\tau_{2}}{\sqrt{(1+|\tau_{1}|^{2})(1+|\tau_{2}|^{2})}}
\begin{array}{lll}{\delta_{\xi}q}&{=}&{\{q,\xi^{\alpha}C_{\alpha}\}}\\{\delta_{\xi}\lambda^{\alpha}}&{=}&{\dot{\xi}^{\alpha}-\xi^{\beta}\lambda^{\gamma}t_{\gamma\beta}^{\alpha}-\xi^{\beta}t_{\beta}^{\alpha}}\\\end{array}
ds_{5+1}^{2}=-\Big[1-({\frac{\mu}{r}})^{2}\Big]dt^{2}+\Big[1-({\frac{\mu}{r}})^{2}\Big]dx^{2}+\Big[1-({\frac{\mu}{r}})^{2}\Big]^{-2}dr^{2}+r^{2}d\Omega_{3}^{2},
\frac{(2-\alpha-\lambda_{1})J_{\alpha}(mz_{1})+mz_{1}J_{\alpha-1}(mz_{1})}{(2-\alpha-\lambda_{1})Y_{\alpha}(mz_{1})+mz_{1}Y_{\alpha-1}(mz_{1})}=\frac{(2-\alpha+\lambda_{2})J_{\alpha}(mz_{2})+mz_{2}J_{\alpha-1}(mz_{2})}{(2-\alpha+\lambda_{2})Y_{\alpha}(mz_{2})+mz_{2}Y_{\alpha-1}(mz_{2})}
{\calR}={\frac{p(p^{2}-1)}{16L_{p}^{2}}}\left[(p-3)\tilde{E}^{2}\tilde{P}_{\phi}^{-2{\frac{p-1}{p-3}}}\left({\frac{a}{L_{p}}}\right)^{2{\frac{p+1}{p-3}}}-(p-7)\left({\frac{L_{p}}{a}}\right)^{2}\right].
\omega_{j}=tr(\gamma^{-1}d\gamma)e^{2\pi\sqrt{-1}\frac{\sqrt{3}}{3}aq(j)\mathbf{\sigma}_{3}}(\gamma^{-1}d\gamma)e^{-2\pi\sqrt{-1}\frac{\sqrt{3}}{3}aq(j)\mathbf{\sigma}_{3}},
D(\frac{\vec{\phi}}x)|_{(\vec{z}_{i},t^{*})}=\left|\begin{array}{ccc}{-\sum_{j=2}^{3}\phi_{j}^{1}f_{1}^{j}}&{\phi_{2}^{1}}&{\phi_{3}^{1}}\\{-\sum_{j=2}^{3}\phi_{j}^{2}f_{1}^{j}}&{\phi_{2}^{2}}&{\phi_{3}^{2}}\\{\phi_{1}^{3}}&{\phi_{2}^{3}}&{\phi_{3}^{3}}\\\end{array}\right|_{(\vec{z}_{i},t^{*})}=0
|\varphi\tilde{\varphi}>=N_{f}^{2}exp\{-{\frac{1}{2}}\int_{x,y}[(\phi_{x}-\varphi_{y})f_{xy}(\phi_{y}-\varphi_{y})+(\tilde{\phi}_{x}-\varphi_{x})f_{xy}(\tilde{\phi}_{y}-\varphi_{y})]\}\;.
S_{6}^{\mathrm{\tiny{A}}}=\intd^{6}x\left[+\frac{1}{4g^{2}}\intd^{2}\theta\;W^{\alpha}W_{\alpha}+\mathrm{h.c.}+\intd^{4}\theta\frac{1}{g^{2}}\left((\sqrt{2}\bar{\partial}V-\bar{\phi})(\sqrt{2}\partialV-\phi)-\partialV\bar{\partial}V\right)\right]
*\vee\left(*{\bfd}\omega-\frac1c\frac{\partial}{\partialt}{\calI}_{*}\omega,\omega\right)-\vee\left({\calI}_{*}\delta\omega,{\calI}_{*}\omega\right)=\frac{4\pi}{c}*\vee\left({\calI}_{*}{\calJ},\omega\right)+
{}_{\frac{1}{2}}U_{a,c}^{\mathrm{alt}}(\theta)={}_{\frac{1}{2}}T_{a,b}^{\mathrm{alt}}(\theta,\alpha)\cdot{}_{\frac{1}{2}}K_{c,d}^{+}(\theta)\cdot{}_{\frac{1}{2}}T_{d,b}^{\mathrm{alt}^{-1}}(-\theta).
{\calH}_{\mu\nu\rho}=3\partial_{[\mu}B_{\nu\rho]}-3\bar{\lambda}\gamma_{a}\gamma_{a^{\prime}}\partial_{[\mu}\lambda\left(2\delta_{\nu}^{a}-\bar{\lambda}\gamma^{a}\partial_{\nu}\lambda\right)\partial_{\rho]}X^{a^{\prime}}\.
\left[b_{\eta^{\prime}}(p^{\prime\,\mu}),\,b_{\eta}^{\dagger}(p^{\mu})\right]_{\pm}\,=\,(2\,\pi)^{3}\,2\,p_{0}\,\delta^{3}({\bfp}\,-\,{\bfp}^{\prime})\,\delta_{\eta,\,-\,\eta^{\prime}}\quad.
|\vec{x},\vec{y}\rangle=\mathrm{exp}\left[\int_{0}^{\infty}\left(-\vec{\beta}^{+}\cdot\vec{\chi}^{+}+i\sqrt{2}\vec{x}\cdot\vec{\chi}^{+}+\sqrt{2}\vec{\beta}^{+}\cdot\vec{y}-i\vec{x}\cdot\vec{y}\right)\right]|0\rangle.
\intdxg^{1/2}\Re_{1}\delta\gamma(-\Box)\Re_{2}=\intdxg^{1/2}\int_{0}^{\infty}dm^{2}w(m^{2})\Bigl(\frac{1}{m^{2}-\Box}\Re_{1}\Bigr)\delta\Box\Bigl(\frac{1}{m^{2}-\Box}\Re_{2}\Bigr)\:.
\phi_{R}\rightarrowd_{1}\,e^{-a\mux}e^{aL\mu}\frac{\Gamma(2\mu)\Gamma(1-2\nu)}{\Gamma(1+\mu-\nu-\lambda)\Gamma(\mu-\nu+\lambda)}+d_{1}\,e^{a\mux}e^{-aL\mu}\frac{\Gamma(-2\mu)\Gamma(1-2\nu)}{\Gamma(1-\mu-\nu-\lambda)\Gamma(-\mu-\nu+\lambda)}
\Gamma^{(1,1)}(\Lambda,p_{i})=\frac{i\pi\alphaG}{8}\frac{p_{3}+p_{4}-|p_{3}-p_{4}|}{p_{3}+p_{4}+|p_{3}-p_{4}|}(\frac{\not\!p_{3}}{p_{3}}+\frac{\not\!p_{4}}{p_{4}})+\mathrm{permutation~in~}p_{i}\;.
\begin{aligned}{N_{2}^{a}}&{{}=n_{2}^{a}+m_{2}^{a},}\\{M_{2}^{a}}&{{}=n_{2}^{a}-m_{2}^{a},}\\{M_{3}^{a}}&{{}=m_{3}^{a}+bn_{3}^{a}.}\\\end{aligned}
\overline{{N}}_{-m,-n}^{rs}\approx-\frac{1}{2\pi}\frac{(-1)^{r(m+1)+s(n+1)}}{\alpha_{s}\omega_{rm}+\alpha_{r}\omega_{sn}}\sqrt{\frac{|\alpha_{r}\alpha_{s}|(\omega_{rm}-\mu\alpha_{r})(\omega_{sn}-\mu\alpha_{s})}{\omega_{rm}\omega_{sn}}}s_{rm}s_{sn}.
a_{j}({\mathsf{s}})=-\frac{\sqrt{\mathsfs}\sqrt{\Gamma_{e}({\mathsf{s}})\Gamma_{f}({\mathsfs})}}{{\mathsf{s}}-M_{Z}^{2}+i\sqrt{{\mathsf{s}}}\Gamma_{Z}({\mathsf{s}})}\approx\frac{R_{Z}}{{\mathsfs}-M_{Z}^{2}+i\frac{\mathsfs}{M_{Z}}\Gamma_{Z}}\,,
\alpha=\left(\begin{array}{cc}{-\frac{1}{4s}\rho_{1}}&{0}\\{0}&{0}\\\end{array}\right),\qquad\beta=\left(\begin{array}{cc}{-\frac{i}{4s}\left(\rho_{2}+i\rho_{3}\right)+A}&{(-2is)^{(k-3)/2}\tilde{p}}\\{(-2is)^{(k-3)/2}\tilde{q}}&{B}\\\end{array}\right),
T_{31}^{amn\;k}=t_{31}^{amnk}\left(\frac{\ell_{P}}{{\calL}}\right)^{2\left(1+\Upsilon\right)}\;{\calF}_{31},\qquad{\calF}_{31}=\left(\vartheta_{312}+\vartheta_{313}\left(\frac{\ell_{P}}{{\calL}}\right)^{(1+\Upsilon)}+....\right)
G_{4}(x_{1},x_{2},x_{3},x_{4})=[\prod_{i=1}^{4}\Gamma(\alpha_{i})]^{-1}(x_{12}^{2})^{\alpha_{4}-\frac{1}{2}D}(x_{14}^{2})^{\frac{1}{2}D-\alpha_{1}-\alpha_{4}}(x_{24}^{2})^{\frac{1}{2}D-\alpha_{2}-\alpha_{4}}(x_{34}^{2})^{-\alpha_{3}}F(u,v)
\frac{ds}{d\omega}=\frac{\gamma^{3}}{2\pi^{3}R_{0}^{3}}\;\frac{\omega^{2}+(\gammav\omega_{0})^{2}}{\omega^{2}}\;\frac{v^{3}w^{2}}{w^{2}+(2\gammav)^{2}}\;j_{\gamma}(w).
iG^{(0)}(x,y)=\int_{0}^{-Tm^{2}}\int_{\stackrel{x(0)=x}{x(T)=y}}{\calD}x(\tau)[m-{\frac{1}{2}}\gamma\cdot\dot{x}(T)]e^{-{\frac{1}{4}}\int_{o}^{T}d\tau\dot{x}^{2}(\tau)}.
{\beta_{h}}(g)=-{\frac{g^{3}}{16\pi^{2}}}\left(3N_{c}-N_{f}(1-\gamma)\right)\,,\qquad\gamma(g)=-{\frac{g^{2}}{8\pi^{2}}}{\frac{N_{c}^{2}-1}{N_{c}}}+O(g^{4}),
\begin{array}{lll}{L_{KI}^{(-)}L_{IJ}^{(+)}}&{-}&{(-1)^{P_{I}(P_{J}+P_{K})}L_{IJ}^{(+)}L_{KI}^{(-)}=}\\{}&{}&{(q-q^{-1})(-1)^{P_{I}P_{J}}L_{KJ}^{(+)}L_{II}^{(-)},(I<K<J)}\\\end{array}
\begin{array}{l}{R^{(1)}=R_{12}\quad\mathrm{or}\quadR_{21}^{-1}\quad(\kappa_{1}=q\quad\mathrm{or}\quadq^{-1})\quad,}\\{R^{(4)}=R_{21}\quad\mathrm{or}\quadR_{12}^{-1}\quad(\kappa_{2}=q^{-1}\quad\mathrm{or}\quadq)\quad.}\\\end{array}
<q|\hat{p}_{\alpha}|q^{\prime}>=\frac{-i\hbar}{g^{1/4}(q)}\frac{\partial}{\partialq^{\alpha}}\left(\frac{1}{g^{1/4}(q)}\delta^{(n)}(q-q^{\prime})\right)+\frac{1}{\sqrt{g(q)}}\,A_{\alpha}(q)\delta^{(n)}(q-q^{\prime})\\,
D^{+}(x_{1}-x_{2})=\int\frac{d^{3}k}{(2\pi)^{3}2\omega_{k}}\;\mathrm{e}^{-\mathrm{i}k^{+}(x_{1}-x_{2})}\;,\qquad\omega_{k}=\sqrt{\vec{k}^{2}+m^{2}}\;,
\left(\begin{array}{c}{\Psi_{1}}\\{\Psi_{2}}\\\end{array}\right)=\sqrt{\frac{1+x_{5}}{2}}\left(\begin{array}{c}{u_{1}}\\{u_{2}}\\\end{array}\right),\quad\left(\begin{array}{c}{\Psi_{3}}\\{\Psi_{4}}\\\end{array}\right)=\sqrt{\frac{1}{2(1+x_{5})}}(x_{4}-ix_{i}\sigma_{i})\left(\begin{array}{c}{u_{1}}\\{u_{2}}\\\end{array}\right),
u=\frac{1}{2\mu}\left(m\,v+3\,s\right)=\frac{1}{2\mu}\left(m\,v+\frac34\,\frac{\mu^{2}\Lambda_{1}^{3}}{v}\right)=\frac{\sqrt{m\Lambda_{1}^{3}}}{4}\left(\kappa+\frac{3}{\kappa}\right)\;.
\Gamma_{\mathrm{N,\{\nu_{j}\}}}=\frac{\hbar}{2}\left(-\frac{\lambda}{4}\right)^{\mathrm{N_{1}+N_{2}}}\left[\Phi\cdots\Phi\right]_{\star_{\mathrm{N_{1}}}}\left(-P\right)\widetilde{{\calK}}_{\mathrm{N}-d/2}\left(P\circP+1/\Lambda^{2}\right)\left[\Phi\cdots\Phi\right]_{\star_{\mathrm{N_{2}}}}(+P)~,
\frac{d^{2}\phi(r)}{dr^{2}}+\frac{1}{r^{2}\Lambda}\frac{d(r^{2}\Lambda)}{dr}{}~\frac{d\phi(r)}{dr}-\frac{1}{\Lambda^{2}}\left[\frac{l(l+1)}{r^{2}}-E^{2}\right]~\phi(r)~=~0~~.
\tilde{\calV}=\left(\begin{matrix}{\tilde{U}_{\alpha}^{T}}\\{\tilde{\bar{U}}{}^{\alpha\,T}}\\\end{matrix}\right)=\left(\begin{matrix}{\int_{A^{A}}\,\gamma_{\alpha}}&{\int_{B_{B}}\gamma_{\alpha}}\\{\int_{A^{A}}\,\bar{\gamma}^{\alpha}}&{\int_{B_{B}}\bar{\gamma}^{\alpha}}\\\end{matrix}\right)\,
L_{U(1)}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}(\partial_{\mu}\Phi^{*}-ieA_{\mu}\Phi^{*})(\partial_{\mu}\Phi+ieA_{\mu}\Phi)-\frac{e^{2}}{8}\left(|\Phi|^{2}-|\Phi_{0}|\right)^{2}
\delta{\calE}_{L}=\delta{\calE}^{2,0}-\delta{\calE}^{2,1}=\frac{K}{a_{0}^{6}}\left[\left(\frac{a_{0}}{\Lambda}\right)^{3}-\left(\frac{a_{0}}{\Lambda}\right)^{2}+\frac{1}{6}\left(\frac{a_{0}}{\Lambda}\right)\right]e^{-\Lambda/a_{0}}\,{.}
\Lambda=\left(\begin{array}{cccccc}{\ddots}&{}&{}&{}&{}&{}\\{}&{1}&{}&{}&{}&{}\\{}&{}&{\frac{a}{c}}&{}&{}&{}\\{}&{}&{}&{\frac{b}{d}}&{}&{}\\{}&{}&{}&{}&{1}&{}\\{}&{}&{}&{}&{}&{\ddots}\\\end{array}\right)
\begin{array}{ll}{\begin{array}{l}{Z[e,\pi,\lambda,\rho]=}\\\end{array}\int}&{[De_{i}^{a}][D\pi_{a}^{i}][D\lambda_{mn}][D\rho^{mn}]\J\\delta(\rho^{mn})\delta(\lambda_{mn}-\lambda_{mn}^{0})}\\{}&{\times\,\,det\,M_{\alpha\beta}\\delta(H_{\perp})\delta(H_{i})\delta(J_{ab})\exp\frac{i}{\hbar}S.}\\\end{array}
F_{\mathrm{R}}^{\mu\nu}:=\left(\begin{matrix}{0}&{-q^{-1}\psi_{\mathrm{R}}^{N}}\\{q\psi_{\mathrm{R}}^{M}}&{\varepsilon^{MN}{}_{C}\,\psi_{\mathrm{R}}^{C}}\\\end{matrix}\right).
q^{\frac{c}{24}}q^{-\frac{(s-1)(\nu-s)}{2(\nu-1)(\nu+1)}}b_{2p-s+1}^{s-1}=\sum_{m_{1},\ldots,m_{\nu-2}\geq0,\mathrm{restr.}}q^{mC^{-1}m-e_{s-1}C^{-1}m}\prod_{i=1}^{\nu-2}\frac{1}{(q)_{m_{i}}}
f=f_{1}f_{2}f_{K}=(1+{\frac{r_{1}^{2}}{r^{2}}})(1+{\frac{r_{5}^{2}}{r^{2}}})(1+{\frac{r_{K}^{2}}{r^{2}}}),~~~h=(1-{\frac{r_{0}^{2}}{r^{2}}}).
\sum_{\ell=0}^{\infty}\frac{(2q+2\ell)\Gamma(2q+\ell)}{\Gamma(\ell+1)}J_{q+\ell}(z)^{2}=\frac{\Gamma(2q+1)}{\Gamma(q+1)^{2}}\left(\frac{z}{2}\right)^{2q}
P_{2}^{1}=-2r^{2}K(-2r^{2}\hat{r}_{2}+6r^{2}\hat{r}_{6}+8r\hat{r}_{2}\hat{r}_{6}+3\hat{r}_{2}^{2}\hat{r}_{6}+\hat{r}_{2}\hat{r}_{6}^{2}),
\int{\calD}\psi{\calD}\overline{\psi}\quade^{-S_{f}}=\int{\calD}\psi{\calD}\overline{\psi}{\calD}A{\calD}[\Lambda]{\calD}\lambda\quade^{-S_{f}+F(A,\psi,\overline{\psi},\phi)+(A,d\lambda+i\astd\Lambda)}.
\alpha^{\prime}m^{2}=24\sum_{n>0}\frac{2n^{2}-H^{2}m^{2}\alpha^{2}}{\sqrt{n^{2}-H^{2}m^{2}\alpha^{2}}}+2N\frac{2-H^{2}m^{2}\alpha^{2}}{\sqrt{1-H^{2}m^{2}\alpha^{2}}}.
\Psi_{\alpha}^{\kappa}(\beta)=\Psi_{\beta_{n}\alpha_{n}}^{\kappa}\Psi_{\beta_{n-1}\alpha_{n-1}}^{\beta_{n}}\cdots\Psi_{\beta_{3}\alpha_{3}}^{\beta_{4}}\Psi_{\alpha_{1}\alpha_{2}}^{\beta_{3}}.
t_{b_{1}\dotsb_{n}}^{a_{1}\dotsa_{m}}=r_{\mu_{1}}^{a_{1}}\dotsr_{\mu_{m}}^{a_{m}}r_{b_{1}}^{-1\nu_{1}}\dotsr_{b_{n}}^{-1\nu_{n}}t_{\nu_{1}\dots\nu_{n}}^{\mu_{1}\dots\mu_{m}}.
K^{-}=-\frac{1}{2}x_{I}^{2}P^{-}+x^{-}D+\frac{1}{\partial^{+}}x^{I}\partial^{J}M^{IJ}-\frac{x^{i}}{2z\partial^{+}}[M^{zi},A]+\frac{1}{\partial^{+}}B\,,
\Phi=-\left(\phi_{0}+\frac{1}{2}u_{0}\right)\;,\;\overline{{\Phi}}=-\left(\phi_{0}-\frac{1}{2}u_{0}\right)\;,\;{\calV}=v_{0}+{\tilde{u}}_{0}\;,\;\overline{{\calV}}=-v_{0}+{\tilde{u}}_{0}\;.
\Omega_{+}=\left(\begin{matrix}{K^{1/2}}&{(q^{3/2}-q^{-1/2})X_{+}}\\{0}&{K^{-1/2}}\\\end{matrix}\right),\,\Omega_{-}=\left(\begin{matrix}{K^{-1/2}}&{0}\\{-(q^{1/2}-q^{-3/2})X_{-}}&{K^{1/2}}\\\end{matrix}\right)\.
\Phi^{\ell}(\tau,\bar{\tau},\phi)=\sum_{n=0}^{2\ell}a_{n}t_{0n}^{\ell}(\tau,\bar{\tau},\phi)=\frac{e^{-i\ell\phi}}{(1+|\tau|^{2})^{\ell}}\sum_{n=0}^{2\ell}b_{n}\tau^{n}\equive^{-i\ell\phi}\varphi(\tau,\bar{\tau})\,.
\Big(a_{\mu}^{(b)},T_{\mu\nu}a_{\nu}^{(b)}\Big)=\Big(\varepsilon_{\mu\nu}\partial_{\mu}a_{\nu}^{(b)},\frac{-1}{\triangle}\;\varepsilon_{\rho\sigma}\partial_{\rho}a_{\sigma}^{(b)}\Big)\;,
S^{\prime}={\frac{1}{16\piG_{10}}}\intd^{10}x\sqrt{-G^{dual}}e^{-{\frac{6}{7}}\phi}\left[{\calR}^{dual}+{\frac{16}{49}}\partial_{M}\phi\partial^{M}\phi-{\frac{1}{2\cdot8!}}F_{M_{1}\cdotsM_{8}}F^{M_{1}\cdotsM_{8}}\right].
G^{(7)M_{1}...M_{7}}(x)={\frac{T_{5}}{\sqrt{-g_{11}}}}\int_{{\calM}_{7}}d{\hat{x}}^{M_{1}}\wedge...\wedged{\hat{x}}^{M_{7}}\delta(x-{\hat{x}}(z)),\quadd{}^{*}G^{(7)}\equiv{~}^{*}J^{(6)},\quad(z~\epsilon~{\calM}_{7})
\left(\begin{matrix}{a_{D1}}\\{a_{D2}}\\{a_{1}}\\{a_{2}}\\\end{matrix}\right)\toM_{1}\left(\begin{matrix}{a_{D1}}\\{a_{D2}}\\{a_{1}}\\{a_{2}}\\\end{matrix}\right),\qquadM_{1}={\tilde{M}}_{1}^{4},\quad\,{\tilde{M}}_{1}=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{-1}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right)
\mathbb{L}_{AB}^{-1~\Lambda}d\mathbb{L}_{\Lambda}^{~AB}+g(\mathbb{L}^{-1})_{AB}^{~~\Lambda}(G^{IJ})_{\Lambda}^{~\Sigma}\mathbb{L}_{\Sigma}^{~CD}A_{IJ}=\hat{\mathcal{Q}}_{AB}^{~~~CD}+\hat{\mathcal{P}}_{AB}^{~~~CD}\,,
\Phi_{R}\left(p^{\mu}\right)=\Lambda_{R}\left(\dot{p}^{\mu}\top^{\mu}\right)\Phi_{R}\left(\dot{p}^{\mu}\right)=e^{i\vec{J}\cdot(\vec{\vartheta}-i\vec{\varphi})}~\Phi_{R}\left(\dot{p}^{\mu}\right),
G_{\mathrm{ir}}^{(n)}({\bf{r}}_{1},\ldots,{\bf{r}}_{n})=\left.\frac{\delta^{(n)}{\cal{G}}_{\mathrm{ir}}\{\varphi\}}{\delta\varphi({\bf{r}}_{1})\ldots\delta\varphi({\bf{r}}_{n})}\right|_{\varphi=0}\;.
\varepsilon^{i}\rightarrow{\varepsilon^{\primei}}={{W_{5}(p,q,r)}^{i}}_{j}\varepsilon^{j}=\varepsilon^{i}+\delta\varepsilon^{i}=\varepsilon^{i}+(pa_{1}+qa_{2}+ra_{3})\frac{p^{i}}{\omega}
Z[\eta,\overline{{\eta}}]=\int[dA]e^{-S_{g}[A]}\mathrm{det}[\not\!{\partial}-ie\not{\!\!A}]^{N}e^{\sum_{a}\intd^{2}xd^{2}y\:\overline{{\eta}}^{(a)}\!(x)\:G(x,y;A)\:\eta^{(a)}\!(y)}\;.
H=-\sum_{s=1}^{9}\,\vec{\nabla}_{s}^{2}+\sum_{s<t}\left(\vec{q}_{s}\times\vec{q}_{t}\right)^{2}+i\vec{q}_{s}\cdot\vec{\Theta}_{\hat{\alpha}}\times\vec{\Theta}_{\hat{\beta}}\,\gamma_{\hat{\alpha}\hat{\beta}}^{s}\,.
D_{ij,kl}^{ab,cd}={\frac{Q}{N\sum_{b=0}^{K-1}\tilde{B}(b)}}\sum_{\tilde{m}=0}^{P-1}\tilde{B}(i-j-Q\tilde{m})\mathrm{e}^{i{\frac{2\pi}{K}}(a-c)(i-j-Q\tilde{m})}\delta_{ab}\delta_{cd}\delta_{il}\delta_{jk}.
\begin{aligned}{S_{\mathrm{dilaton}}}&{{}=}&{-\frac{1}{2\kappa_{d}^{2}}\int\!\mathrm{d}^{d}x\,}&{{}e\,(\phi^{-1}\partial_{\mu}\phi)^{2}\,,}\\{S_{\mathrm{dilatino}}}&{{}=}&{-\frac{1}{\kappa_{d}^{2}}\int\!\mathrm{d}^{d}x\,}&{{}e\,\bar{\lambda}\Gamma^{\mu}D_{\mu}\lambda\,.}\\\end{aligned}
S_{\mathrm{cl}}=\mathrm{tr}\left(\hat{\bar{\sigma}}^{\dot{\alpha}}\bar{D}_{\dot{\alpha}}D_{\alpha}\hat{\sigma}^{\alpha}\right)+\mathrm{tr}\left({\hat{\bar{\sigma}}}^{\dot{\alpha}}\bar{D}_{\dot{\alpha}}\left(e^{-V}-1\right)D_{\alpha}{\hat{\sigma}}^{\alpha}\right)\,.
Solv_{r+1}\,\equiv\,Solv\left(E_{r+1(r+1)}/{\calH}_{r+1}\right)=Solv\left(SL(2,R)/O(2)\right)\,\oplus\,Solv\left(\frac{GL(r)}{SO(r)}\right)\,\oplus\,{\bar{\calW}}_{n_{r+1}}
(t^{-1})^{\prime}=t^{\mathrm{ij}}{\frac{\partial\pi^{\alpha}}{\partialp^{\mathrm{i}}}}{\frac{\partial\pi^{\beta}}{\partialp^{\mathrm{j}}}}\;{\frac{\partial}{\partial\pi^{\alpha}}}\otimes{\frac{\partial}{\partial\pi^{\beta}}}\;\inT_{\pi(\mathrm{p})}(\Sigma)\otimesT_{\pi(\mathrm{p})}(\Sigma)
\mathrm{e}^{-S_{\mathrm{ind}}\left(\Sigma\right)}={\frac{1}{Z\left(B_{\mu\nu}\right)}}\\int{\calD}B_{\mu\nu}\\mathrm{e}^{-S_{C}\left(B_{\mu\nu}\right)+i\int_{\Sigma}d\sigma_{\mu\nu}B_{\mu\nu}}
g^{2}\frac{1}{\hbar^{2}f(k^{2})k^{2}-m^{2}}\int\frac{d^{4}l}{(2\pi)^{4}}\tilde{\Omega}(l)\frac{1}{\hbar^{2}f[(k-l)^{2}](k-l)^{2}-m^{2}}\tilde{\Omega}(k-k_{i}-l)+\dots
\check{X}\check{Y}=\left(\begin{array}{ccc}{XY}&{-\mathrm{sgn}\,M_{1}|M_{1}|^{1/2}}&{-X\mathrm{sgn}\,N_{1}|N_{1}|^{1/2}}\\{|M_{2}|^{1/2}Y}&{X^{*}|_{{\calM}_{1}}}&{-|M_{2}|^{1/2}\mathrm{sgn}\,N_{1}|N_{1}|^{1/2}}\\{|N_{2}|^{1/2}}&{0}&{Y^{*}|_{{\calN}_{1}}}\\\end{array}\right),
L_{\phi}^{\mathrm{cutoff}}=-\frac{1}{4\pii}\intdz(z^{2}-\omega^{2})\frac{1}{z}\left[1-\frac{R_{p}}{R_{p}+\frac{\omegav_{1}\bar{\gamma}_{1}}{z^{2}+\omega^{2}v_{1}^{2}\bar{\gamma}_{1}^{2}}+\frac{\omegav_{2}\bar{\gamma}_{2}}{z^{2}+\omega^{2}v_{2}^{2}\bar{\gamma}_{2}^{2}}}\right]F(z)\,.
\begin{array}{rl}{}&{G(x_{1},x_{2},x_{3},x_{4})}\\{=}&{\displaystyle\intd^{4}x_{5}\intd^{4}x_{6}<0|T\left(\pi_{1}\pi_{2}\chi_{3}\chi_{4}(\pi\star\sigma\star\pi)_{5}(\chi\star\sigma\star\chi)_{6}\right)|0>,}\\\end{array}
(t_{n(r)j}t_{n(s)l})_{n(t)p}{}^{n(u)q}=\chi_{n(r)j}\chi_{n(s)l}\sum_{I}U_{n(r)j}{}^{I}U_{n(s)l}{}^{I}U_{n(t)p}{}^{I}(U^{\dagger})_{I}{}^{n(u)q}
{}\langlek_{1}k_{2}\cdotsk_{m}|a_{i}^{\dagger}a_{j}^{\dagger}|i_{1}i_{2}\cdotsi_{n}\rangle=\delta_{m,n+2}N_{iji_{1}i_{2}\cdotsi_{n}}^{k_{1}k_{2}\cdotsk_{m}}=\sum_{l=1}^{n}\delta_{i_{l}}^{k_{1}}N_{i_{1}\cdotsi_{l-1}i_{l+1}\cdotsi_{n}}^{k_{2}\cdots\cdots\cdotsk_{m}}
R_{abc}^{\bar{a}\bar{b}\bar{c}}R_{\bar{a}de}^{a\bar{d}\bar{e}}R_{\bar{b}\bar{d}f}^{bd\bar{f}}R_{\bar{c}\bar{e}\bar{f}}^{cef}-R_{cef}^{\bar{c}\bar{e}\bar{f}}R_{bd\bar{f}}^{\bar{b}\bar{d}f}R_{a\bar{d}\bar{e}}^{\bar{a}de}R_{\bar{a}\bar{b}\bar{c}}^{abc}
[{\calP}_{0},X_{0}]=i,\quad[{\calP}_{i},X_{0}]=\frac{i}\kappa\,{\calP}_{i},\quad[{\calP}_{i},X_{j}]=-i\,\delta_{ij}\,e^{2{\calP}_{0}/\kappa}+\frac{i}{\kappa^{2}}\left(\vec{{\calP}}\,{}^{2}\,\delta_{ij}-2{\calP}_{i}{\calP}_{j}\right),\quad[{\calP}_{0},X_{i}]=-\frac{2i}\kappa\,{\calP}_{i}
Z=\int{\calD}A^{\mu}{\calD}\pi_{\mu}{\calD}\phi{\calD}\pi_{\phi}{\calD}\theta{\calD}\pi_{\theta}\prod_{i,j=1}^{2}\delta(\widetilde{\Omega}_{i})\delta(\Gamma_{j})\mathrm{det}\mid\{\widetilde{\Omega}_{i},\Gamma_{j}\}\mide^{iS},
{\calL}_{I}=\frac{1}{2}\left\{\phantom{|}_{ab}^{\mu\nu}\right\}\omega_{\mu}{}^{ac}\omega_{\nu}{}^{bc}-\frac{1}{2}\left\{\phantom{|}_{abc}^{\mu\nu\alpha}\right\}\omega_{\mu}{}^{ab}\partial_{\nu}h_{\alpha}{}^{c}
\left\langleH\right\rangle_{\Omega}=\left\langle\Omega\right|\int{dx_{1}}\left({-\frac{1}{2}\Pi_{1}\left({1+\frac{{e^{2}}}{\pi}\frac{1}{{\partial^{2}}}}\right)^{-1}\Pi^{1}}\right)\left|\Omega\right\rangle.
{\bff}(x)=\left(\begin{array}{c}{u_{-}(x)}\\{u_{+}(x)}\\\end{array}\right),\;\;\hat{x}^{\mu}=x^{\mu}{\bfI}\;,\;\;\hat{\pi}_{\mu}=-\imath\partial_{\mu}{\bfI}\;,\;\;\hat{\psi}^{a}=\frac{\imath}{2}\gamma^{a},
{\calE}=\frac{1}{2}\midD_{0}\Phi\mid^{2}+\frac{1}{2}\midD_{i}\Phi\mid^{2}+\frac{1}{4}F_{ij}^{2}+\frac{1}{2}F_{0i}^{2}+\frac{\lambda}{8}(\mid\Phi\mid^{2}-1)^{2},
C_{\kappa}(\vec{x},\vec{y})\equiv\frac{\pi}{\kappa}\int_{-\infty}^{\infty}d\omega\Omega_{\kappa}(\omega)\sum_{mn}j_{mn}(\vec{x})j_{nm}(\vec{y})\delta(\omega+\kappa-\omega_{n})\delta(\omega-\omega_{m}),
{}^{(5)}g_{AB}=\left(\begin{array}{ccc}{-n^{2}}&{-a^{2}S_{i}}&{0\nonumber}\\{-a^{2}S_{j}}&{a^{2}\left[\delta_{ij}+F_{i|j}+F_{j|i}\right]}&{-a^{2}S_{yi}\nonumber}\\{0}&{-a^{2}S_{yi}}&{1\nonumber}\\\end{array}\right)\,,
\Lambda(h)=\left(\begin{array}{llll}{0}&{1}&{}&{0}\\{}&{\ddots}&{\ddots}&{}\\{}&{}&{\ddots}&{1}\\{h}&{}&{}&{0}\\\end{array}\right),\quad\Lambda(h)^{-1}=\left(\begin{array}{llll}{0}&{}&{}&{h^{-1}}\\{1}&{\ddots}&{}&{}\\{}&{\ddots}&{\ddots}&{}\\{0}&{}&{1}&{0}\\\end{array}\right).
S_{gr}=\intd^{2}z\sqrt{-\bar{g}}[{\frac{1}{2\psi}}\bar{g}^{\alpha\beta}\partial_{\alpha}\psi\partial_{\beta}\psi+2\bar{g}^{\alpha\beta}\partial_{\alpha}\psi\partial_{\beta}\sigma-\psi\bar{R}-{\frac{1}{4\kappa}}U(\psi)e^{2\sigma}],
\begin{array}{rcl}{g_{i,\kappa}}&{=}&{\frac{\kappa}{n}+\frac{i-1}{s}+\frac{\gamma}{s\,n}\,\,\,(i=1,2,\dots,s),}\\{g_{i,\kappa}}&{=}&{\frac{\kappa}{n}+\frac{n-i}{n-s}-\frac{\gamma}{n\,(n-s)}\,\,\,(i=1,2,\dots,n),}\\{h_{i,\kappa}}&{=}&{\frac{\kappa+i}{s}+\frac{\delta_{i}}{n}\,\,\,(i=1,2,\dots,n-1),}\\\end{array}
R_{\alpha\beta}^{\gamma\delta}=(\delta_{\alpha}^{\gamma}-\delta_{\alpha}^{1}\delta_{3}^{\gamma}h_{3}^{1}-\delta_{\alpha}^{2}\delta_{4}^{\gamma}h_{3}^{1})(\delta_{\beta}^{\delta}+\delta_{\beta}^{1}\delta_{2}^{\delta}g_{3}+\delta_{\beta}^{3}\delta_{4}^{\delta}g_{3})
T_{\rho\sigma\mu\nu}^{(4)abcd}(p_{1},p_{2};k_{1},k_{2})=C_{1}\gamma_{\rho\sigma\mu\nu}^{(1)}+C_{2}\gamma_{\rho\sigma\mu\nu}^{(2)}+C_{3}\gamma_{\rho\sigma\mu\nu}^{(3)}
\sigma^{2}=\left[\fraci{2m}\epsilon_{\mu\nu\alpha\beta}\frac14\left(\Gamma_{\mu}\Gamma_{\nu}-\Gamma_{\nu}\Gamma_{\mu}+\overline{{\Gamma}}_{\mu}\overline{{\Gamma}}_{\nu}-\overline{{\Gamma}}_{\mu}\overline{{\Gamma}}_{\nu}\right)\right]^{2}.
\mathrm{C}~~~:~~~\mathrm{\bf~Tr}_{{\calH}_{o}}\left[\frac{1}{2}(-1)^{F_{s}}e^{-2\pit\,\mathrm{H}_{o}}\right]-\mathrm{\bf~Tr}_{\tilde{\calH}_{o}}\left[\frac{1}{2}(-1)^{F_{s}}e^{-2\pit\,\mathrm{H}_{o}}\right],
E=\left(\begin{matrix}{\left(\begin{matrix}{E_{R}}&{0}&{0}\\{0}&{E_{L}}&{0}\\{0}&{0}&{E_{L}}\\\end{matrix}\right)\otimes1_{N}}&{0}\\{0}&{\left(\begin{matrix}{E_{R}}&{0}&{0}&{0}\\{0}&{E_{R}}&{0}&{0}\\{0}&{0}&{E_{L}}&{0}\\{0}&{0}&{0}&{E_{L}}\\\end{matrix}\right)\otimes{\bf1}_{3}\otimes1_{N}}\\\end{matrix}\right).
\hat{g}=\left({\frac{L}{u}}\right)^{2}\left(\begin{array}{lll}{}&{{\dot{u}}^{2}-u^{4}}&{\;\;\;\;{\dot{u}}{u^{\prime}}}\\{}&{\;\;\;{\dot{u}}{u^{\prime}}}&{{u^{\prime}}^{2}+u^{4}}\\\end{array}\right),
V(x_{2})=a+(\frac{e^{2}V_{1,1}}{2})\int_{B}^{x_{2}}dy\int_{A}^{y}dz|\phi_{0}(z)|^{2}+(\frac{e^{2}V_{0,0}}{2})\int_{B}^{x_{2}}dy\int_{A}^{y}dz|\phi_{1}(z)|^{2}.
\frac{1}{\sqrt{2}k}\left[1-\kappa\left(\frac{d}{dr}+\frac{\kappa+1}{r}\right)\right]R_{E\ell(\kappa)}(r)=\left(E+\frac{1}{2k^{2}}\right)^{\mathrm{~\frac{1}{2}~}}R_{E\ell(-\kappa)}(r).
[\hat{x}_{1}^{2},\hat{x}_{2}]=2ik\hat{x}_{1},\[\hat{x}_{1}^{3},\hat{x}_{2}]=3ik\hat{x}_{1}^{2},\,dots,\[\hat{x}_{1}^{n},\hat{x}_{2}]=nik\hat{x}_{1}^{n-1}\.
a_{D}={\frac{\sqrt2}{\pi}}\int_{1}^{{\calG}(a)+B}{\frac{dx\sqrt{x-{\calG}(a)-B}}{\sqrt{x^{2}-1}}},\qquada={\frac{\sqrt2}{\pi}}\int_{-1}^{1}{\frac{dx\sqrt{x-{\calG}(a)-B}}{\sqrt{x^{2}-1}}}.
\begin{array}{lcl}{\phi_{a}(b)}&{=}&{\inta^{-n/2}\bar{\psi}\left(\frac{x-b}{a}\right)\phi(x)d^{n}x,}\\{\phi(x)}&{=}&{C_{\psi}^{-1}\inta^{-n/2}\psi\left(\frac{x-b}{a}\right)\phi_{a}(b)\frac{dadb}{a^{n+1}},}\\\end{array}