harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
m{\frac{d^{2}x^{\beta}}{d\tau^{2}}}(\delta^{\alpha}{}_{\beta}+\mu^{2}\mathrm{D}_{\beta\mu}g^{\mu\alpha})+{\frac{dx^{\nu}}{d\tau}}{\frac{dx^{\lambda}}{d\tau}}\left(\Gamma_{\nu\lambda}{}^{\alpha}+\frac{1}{2}\mu^{2}g^{\alpha\gamma}(\partial_{\gamma}\mathrm{D}_{\nu\lambda}-\partial_{\nu}\mathrm{D}_{\gamma\lambda}-\partial_{\lambda}\mathrm{D}_{\gamma\nu})\right)=0,
V_{NBPS}(z)=2m^{2}\left(\begin{array}{cc}{\frac{2\alpha}{\lambda}-(1+\frac{2\alpha}{\lambda})sech^{2}(Mz)}&{0}\\{0}&{\frac{\alpha}{2\lambda}-\frac{3\alpha}{\lambda}sech^{2}(Mz)}\\\end{array}\right).
\left(\!\begin{array}{c}{\omega_{1,c}(t,y)}\\{\omega_{2,c}(-t,y)}\\\end{array}\!\right)\!=\!L^{-1}\!\left(\!\begin{array}{c}{v_{1,c}}\\{v_{2,c}}\\\end{array}\!\right)\!(t,y),\;\left(\!\begin{array}{c}{\omega_{2,(1,f)}(-t,y)}\\{\omega_{1,(1,f)}(t,y)}\\\end{array}\!\right)\!=\!L^{-1}\!\left(\!\begin{array}{c}{w_{1,(1,f)}}\\{w_{2,(1,f)}}\\\end{array}\!\right)\!(t,y),
A_{i}^{\mathrm{vp}}(x_{j})=-\frac{\Phi}{2\pi}\epsilon^{ij}\frac{x^{j}}{\rho^{2}}\Biggl[\frac{\alpha}{4\sqrt{\pi}}(m\rho)^{5}\,G_{13}^{30}\Biggl((m\rho)^{2}\Bigg|\begin{array}{rrr}{}&{0}&{}\\{-\frac{1}{2}}&{-\frac{1}{2}}&{-\frac{5}{2}}\\\end{array}\Biggr)\Biggr].
2[k_{\nu}k^{\mu^{\prime}}\delta_{\mu}^{\nu^{\prime}}+k_{\mu}k^{\mu^{\prime}}\delta_{\nu}^{\nu^{\prime}}+k_{\nu}k^{\nu^{\prime}}\delta_{\mu}^{\mu^{\prime}}+k_{\mu}k^{\nu^{\prime}}\delta_{\nu}^{\mu^{\prime}}]-\frac{8}{D}\delta_{\mu\nu}k^{\mu^{\prime}}k^{\nu^{\prime}}
\{g(\Lambda_{1}^{(0)}),g(\Lambda_{2}^{(0)})\}\begin{array}{}{t]{l}=\frack{2\pi}\int_{D}d\Lambda_{1}^{(0)}d\Lambda_{2}^{(0)}}\\{=\frack{2\pi}\int_{\partialD}\Lambda_{1}^{(0)}d\Lambda_{2}^{(0)}\}\\{=0\mathrm{~for~}\Lambda_{1}^{(0)},\Lambda_{2}^{(0)}\in{\calT}^{(0)}.}\\\end{array}
D_{\vec{m}\vec{n}}=D_{m_{1}n_{1}}^{\theta_{(1)}}(z_{1},\bar{z}_{1})D_{m_{2}n_{2}}^{\theta_{(2)}}(z_{2},\bar{z}_{2})\dotsD_{m_{p/2}n_{p/2}}^{\theta_{(p/2)}}(z_{p/2},\bar{z}_{p/2}),
\widehat{R}_{j,j+1}(\zeta)\widehat{R}_{j+1,j+2}(\xi\zeta)\widehat{R}_{j,j+1}(\xi)=\widehat{R}_{j+1,j+2}(\xi)\widehat{R}_{j,j+1}(\xi\zeta)\widehat{R}_{j+1,j+2}(\zeta).
{\calL}_{sfield}=\frac18\intd^{2}\theta~\mathrm{Tr}\overline{{D{\bfG}}}D{\bfG}=\frac12\left(Z^{2}+J_{\mu}\cdotJ^{\mu}+i\overline{{\chi}}\cdot\partial\!\!\!/\chi+i\varepsilon^{jkl}\overline{{\chi}}^{j}J\!\!\!\!/^{k}\chi^{l}+\frac14(\overline{{\chi}}\cdot\chi)^{2}\right).
\delta{\psi}=-i\theta\gamma_{5}\psi_{+}-i\theta\gamma_{5}{\frac{1}{i\partial^{+}}}\alpha_{\perp}\cdot\big(i\partial_{\perp}+gA_{\perp}\big)\psi_{+}-i\thetam_{F}\gamma_{5}\gamma^{0}{\frac{1}{i\partial^{+}}}\psi_{+}.
q_{k}=q_{0\left(k\right)}-\frac\nu{2W\left(q_{0\left(k\right)}\right)},\;\psi_{+\left(k\right)}=\lambda_{+}W^{k}\left(q_{0\left(k\right)}\right),\;\psi_{-\left(k\right)}=\lambda_{-}W^{-k}\left(q_{0\left(k\right)}\right),
\gamma^{\mu}=\left(\begin{array}{cc}{0}&{\sigma^{\mu}}\\{{\overline{\sigma}}^{\mu}}&{0}\\\end{array}\right)\,,\qquad\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\,.
b(\xi)={\frac{-{{\gamma}^{2}}}{2\,M\,\Lambda\,{{\xi}^{2}}}}+{\frac{{{\gamma}^{2}}\,{\sqrt{M\,\Lambda}}}{2\,{M^{2}}\,\Lambda\,\xi}}-{\frac{{{\gamma}^{2}}}{2\,M\,\Lambda\,{{\left({\frac{2\,{\sqrt{M\,\Lambda}}}{\Lambda}}+\xi\right)}^{2}}}}-{\frac{{{\gamma}^{2}}\,{\sqrt{M\,\Lambda}}}{2\,{M^{2}}\,\Lambda\,\left({\frac{2\,{\sqrt{M\,\Lambda}}}{\Lambda}}+\xi\right)}}
\varepsilon_{\psi}\sum_{l=h}^{\infty}(w-z)^{l-h}\langle\varphi,V(\psi,\zeta)\;V(S_{-l}\chi,z)\;\Omega\rangle-\sum_{m=-\infty}^{h-1}(w-\zeta)^{m-h}\langle\varphi,V(S_{-m}\psi,\zeta)V(\chi,z)\;\Omega\rangle;
{d^{\dagger}}^{(i)}(\eta^{n}\cdot\tau^{n}{\cal{G}})\;\;=\;\;\eta^{n}\cdot\tau^{n}\,{d^{\dagger}}^{(i)}{\cal{G}}+{\textsf{P}}_{N}(n)\;\eta^{n-1}\cdot\tau^{n-1}\,{d^{\dagger}}^{(i)}{\cal{G}}
J_{11}\equiv\psi_{1}^{\dagger}\psi_{1}=2\pik^{2}\Lambda^{2}\left[\frac{i}{e}<(\eta+\xi)_{i}\epsilon_{j}>\partial_{j}\tilde{E}_{i}-<\epsilon_{i}(\eta_{j}-\xi_{j})>\{A_{i},\tilde{E}_{j}\}\right]
S_{0}=\intd^{2}\xiV^{m}V^{n}\left(\partial_{m}x^{\underline{m}}-i\partial_{m}\bar{\theta}\Gamma^{\underline{m}}\theta\right)\left(\partial_{n}x_{\underline{m}}-i\partial_{n}\bar{\theta}\Gamma_{\underline{m}}\theta\right).
{\calL}=\sqrt{-g}\left\{e^{-2\phi}\biggl[R-4(\partial\phi)^{2}\biggr]+{\frac{(-)^{p+r+1}}{2(p+r+2)!}}F_{(p+r+2)}^{2}+{\frac{(-)^{p+s+1}}{2(p+s+2)!}}F_{(p+s+2)}^{2}\right\}\,,
\tilde{\calA}\vphantom{\calA}_{0}^{0}(T)=\tilde{a}\vphantom{a}_{0}^{0}(T+\tilde{\Lambda})^{\frac{-\alpha}{1+\alpha}}+\tilde{b}\vphantom{b}_{0}^{0}(T+\tilde{\Lambda})^{\frac{1}{1+\alpha}}
\begin{array}{l}{X_{J_{i}^{\prime}}\rfloor\Omega_{q}=-dJ_{i}^{\prime}~,}\\{[X_{J_{i}^{\prime}},X_{J_{j}^{\prime}}]=-X_{[J_{i}^{\prime},J_{j}^{\prime}]_{GPB}}~,}\\{\Omega_{q}(X_{J_{i}^{\prime}},X_{J_{j}^{\prime}})=[J_{i}^{\prime},J_{j}^{\prime}]_{GPB}~.}\\\end{array}
\intD{\calB}_{-}det\left(\begin{array}{cc}{0}&{i\partial_{+}}\\{i\partial_{t}+{\calB}_{-}}&{0}\\\end{array}\right)=det\left(\begin{array}{cc}{0}&{i\partial_{+}}\\{i\partial_{t}}&{0}\\\end{array}\right)\intDQ~e^{-(1+2N)\tilde{W}[Q]},
G=\frac{1}{2\cdot6^{2/3}}\left(\begin{array}{ccc}{T^{2}U^{2}}&{Uf(T,U)}&{Tf(U,T)}\\{Uf(T,U)}&{\\frac{1}{T^{2}}[1-2h(T,U)+f^{2}(T,U)]\}&{2h(T,U)\\frac{1-h(T,U)}{TU}}\\{Tf(U,T)}&{2h(T,U)\\frac{1-h(T,U)}{TU}}&{\\frac{1}{U^{2}}[1-2h(T,U)+f^{2}(U,T)]}\\\end{array}\right).
\langle0|J^{\mu}(x)|0\rangle\,=\,\frac{m}{|m|}\,\,\frac{e^{2}}{4\pi}\,\,^{*}F^{\mu}(x)\,=\,sign(m)\,\,\frac{e^{2}}{4\pi}\,\,^{*}F^{\mu}(x)
E_{\pm}^{2}=k^{2}+m^{2}+\frac{{\left(k^{2}+m^{2}\right)}^{2}\,{\bar{\theta}}^{2}}{2}\pm\frac{{\left(k^{2}+m^{2}\right)}^{\frac{3}{2}}\,{\bar{\theta}}\,{\sqrt{4+\left(k^{2}+m^{2}\right)\,{\bar{\theta}}^{2}}}}{2}\,.
sh_{2}+\nu=\partial^{i}\left(2\left(f_{abc}A_{i}^{a}A_{j}^{b}A_{k}^{c}+A_{\left[i\right.}^{a}F_{a\left.jk\right]}\right)\left(\partial^{\left[j\right.}A_{d}^{\left.k\right]}\right)\eta_{2}^{d}\right).
G_{\parallel}=\frac{V_{3}Nu^{4}}{2\pi^{2}\lambda^{2}}\left(\sqrt{1-\frac{2m}{u^{4}}-\frac{\lambda\Omega^{2}(u^{2}+l^{2})}{u^{4}}+\frac{2\sqrt{2m\lambda}l\Omega}{u^{4}}}-1-\frac{\sqrt{2m\lambda}l\Omega}{u^{4}}\right).
H_{B}(DD)=\sum_{j}\left(N_{B}(DD)-\frac{1}{12}\right)=\sum_{j}\left(\sum_{n=1}^{\infty}({\alpha_{-n}}^{-j}{\alpha_{n}}_{j}+{\alpha_{-n}}^{j}{\alpha_{n}}_{-j})-\frac{1}{12}\right).
S^{\mathrm{FB}}=-\left(\begin{array}{cccc}{1}&{1}&{e^{-2\pii\lambda}}&{e^{2\pii\lambda}}\\{1}&{1}&{e^{2\pii\lambda}}&{e^{-2\pii\lambda}}\\{e^{2\pii\lambda}}&{e^{-2\pii\lambda}}&{1}&{1}\\{e^{-2\pii\lambda}}&{e^{2\pii\lambda}}&{1}&{1}\\\end{array}\right).
\delta\Psi_{\hat{a}}=\partial_{\hat{a}}\epsilon+{\frac{1}{8}}H^{-1}(\partial_{\hat{a}}H)\epsilon+{\frac{1}{8}}WH^{{\frac{m-n}{4}}}\gamma^{\underline{{uxy}}}\gamma_{\underline{{\hat{a}}}}(i\sigma_{2})\epsilon=0\,.
\chi_{\alpha\beta}^{\phi}(t,t^{\prime};k)=\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}g_{\alpha\beta}^{\phi}(t,t^{\prime};k^{\prime})g_{\alpha\beta}^{\phi}(t,t^{\prime};k^{\prime}+k)\,,
E=\frac{U_{T}}{\pi\alpha}\left[\int_{1}^{\infty}\left(\frac{\sqrt{(y^{2}-{\alpha}^{2})(y^{4}+{\alpha}^{2}y^{2}-\theta{\alpha}^{4})}}{\sqrt{(y^{2}-1)(y^{4}+y^{2}-\theta{\alpha}^{6})}}-1\right)dy-1+\alpha\right].
H=\sum_{p,i}\frac{2\pi}{L}(p+1/2)\left(a_{i}^{\dagger}(p)a_{i}(p)-b_{i}^{\dagger}(p)b_{i}(p)\right)+\frac{g^{2}L}{16\pi^{2}}\sum_{ij,n\neq0}\frac{j_{ij}(n)j_{ji}(-n)}{n^{2}}\.
T^{+\bar{M}_{1}\bar{M}_{2}\bar{M}_{3}}=\frac1{3!}\varepsilon^{\bar{M}_{1}\bar{M}_{2}\bar{M}_{3}\bar{N}_{1}\bar{N}_{2}\bar{N}_{3}K_{1}\cdotsK_{8}}T_{\bar{N}_{1}\bar{N}_{2}\bar{N}_{3}}^{+}V_{\bar{K}_{1}}^{1}\cdotsV_{\bar{K}_{8}}^{8}.
t_{f}-t_{i}=\int_{r_{i}}^{r_{f}}\,du\,\frac{\pm1}{\sqrt{2\left[E_{r}-V(u)-\frac{L^{2}}{2u^{2}}\right]}}\\,\\L=\frac{\varphi_{f}-\varphi_{i}}{g^{2}(t_{f}-t_{i})+\int_{t_{i}}^{t_{f}}\,\frac{dt}{r^{2}(t)}}\\\,
\Omega_{\bar{1}}^{(1)}(x,y)=\intd^{2}y\Sigma^{(1)}(x,y)L_{i_{1}}^{(1)}(y)=-\kappa\epsilon_{ij}\partial^{i}\dot{A}^{j}-\kappa\partial_{1}\dot{\Phi}^{1}-{\sqrt\kappa}\partial_{2}\dot{\Phi}^{2}=0.
L=-(D^{\mu}\phi_{A})(\overline{{D_{\mu}\phi_{A}}})-(\tilde{D}^{\mu}\phi_{B})(\overline{{\tilde{D}_{\mu}\phi_{B}}})-\frac14F_{A}^{\mu\nu}F_{A\mu\nu}-\frac14F_{B}^{\mu\nu}F_{B\mu\nu}-V,
{\calV}_{\sigma}\left(\eta,\zeta,s\right)=1-\frac{1}{\eta}\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}\left(-\eta\right)^{n}\zeta^{m}\,\theta\left(s+\sigma-n\sigma-m\right)\,F_{nm}\left(s+\sigma-n\sigma-m,\sigma\right)\,,
+{\frac{1}{2}}({\hat{a}}_{(1)(2)}{\bar{\pi}}_{(1)}^{A}\pi_{(2)}^{A^{\prime}}+{\bar{\pi}}_{(1)}^{A}\pi_{(2)}^{A^{\prime}}{\hat{a}}_{(1)(2)})n_{AA^{\prime}}+{\frac{1}{2}}({\hat{a}}_{(2)(1)}{\bar{\pi}}_{(2)}^{A}\pi_{(1)}^{A^{\prime}}+{\bar{\pi}}_{(2)}^{A}\pi_{(1)}^{A^{\prime}}{\hat{a}}_{(2)(1)})n_{AA^{\prime}},
eq-10(\frac{1}{m^{2}}d\delta+\frac{1}{{\mu}^{2}}\deltad)F^{(q)}-F^{(q)}+G^{(q)}-\frac{1}{m^{2}}dJ^{(q-1)}-\frac{1}{{\mu}^{2}}\deltaK^{(q+1)}=0,
f(a)=\frac{a^{2}\left[k_{4}^{2}(\rho+p)+(k_{4}^{2}/\lambda)\rho^{2}+(k_{4}^{2}/\lambda)\rhop+(8{\calU}_{0}/\lambdak_{4}^{2})a^{-4}\right]-2C}{a^{5}\left[k_{4}^{2}(\rho+p)+(k_{4}^{2}/\lambda)\rho^{2}+(k_{4}^{2}/\lambda)\rhop+(8{\calU}_{0}/\lambdak_{4}^{2})a^{-4}\right]}.
{\calL}_{cs}=\varepsilon^{\mu\nu\kappa}\Gamma_{\kappa\lambda}^{\,\,\,\,\,\,\,\,\rho}(\partial_{\mu}\Gamma_{\rho\nu}^{\,\,\,\,\,\,\lambda}+\frac23\Gamma_{\mu\sigma}^{\,\,\,\,\,\,\,\lambda}\Gamma_{\nu\rho}^{\,\,\,\,\,\,\sigma})
R_{W}^{2}={\frac{\intd^{3}x<\!\psi|T_{00}|\psi\!>r^{2}}{\intd^{3}x<\!\psi|T_{00}|\psi\!>r^{2}}}=R_{2}^{2}+\Re\left({\frac{2C}{E_{2}}}{\frac{\alpha}{\beta}}\right).
S_{0}^{(D)}(k)=\frac{1-\lambda\,e^{i\pi\nu}\,\Xi(-2\nu)\,\left(E/\mu^{2}\right)^{\nu}}{1-\lambda\,e^{-i\pi\nu}\,\Xi(-2\nu)\,\left(E/\mu^{2}\right)^{\nu}}\;,
\Phi_{v}(\vec{x^{\prime}},t)\equiv\left(\frac{\omega}{2}\right)t+\left(\frac{m{\vec{v}}^{\2}}{2}\right)\tau(t)-m\vec{v}\cdot\vec{x^{\prime}}+\frac{m}{2}{\dot{a}}a(\vec{x^{\prime}}-\tau\vec{v})^{2}.
V(x)=4\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}))}(\sum_{j=0}^{n}\alpha_{j}).
\Gamma_{R}^{\phi^{-}}(p)=-i/\left\{(p^{2}+i\varepsilon)[K_{UD}(p)+H_{UD}(p)]+E_{UD}(p)-i(p^{2}-\bar{M}^{2})\sqrt{S_{UD}^{2}(p)-R_{UD}^{2}(p)}\right\}
p_{0\alpha}=p_{\alpha}-\frac{g}{2}(\tau^{+}A^{+}{}_{\alpha}+\tau^{-}A^{-}{}_{\alpha})+\frac{gg^{\prime}}{\sqrt{g^{2}+g^{2}}}QA_{\alpha}+\frac{g^{2}}{\sqrt{g^{2}+g^{2}}}Q^{\prime}Z_{\alpha}.
\dot{x}{}^{a}\rightarrow\dot{f}{}^{-1}\dot{x}{}^{a},\,\,\,\,\,\,\,\,\acute{x}^{a}\rightarrow\acute{\varphi}{}^{-1}\acute{x}{}^{a}-(\acute{\varphi}\dot{f}){}^{-1}\acute{f}\dot{x}{}^{a}.
{\calL}=-\frac12(\partial_{\mu}\Phi)^{2}+V(\Phi)=-\frac12(\partial_{\mu}\Phi)^{2}+\frac12m^{2}\Phi^{2}-\frac14bm^{2}g^{2}\Phi^{4}+\frac18cm^{2}g^{4}\Phi^{6}\,
I={\frac{1}{2\pii}}\oint_{|z|=1}\left[{\frac{2{\bfp}^{2}{\bfk}^{2}z-({\bfk}^{2}+{\bfp}^{2})|{\bfk}||{\bfp}|(z^{2}+1)/2}{{\bfk}^{2}{\bfp}^{2}(z-|{\bfp}|/|{\bfk}|+i\epsilon/(2|{\bfk}||{\bfp}|))^{2}(z-|{\bfk}|/|{\bfp}|-i\epsilon/(2|{\bfk}||{\bfp}|))^{2}}}\right].
2A\sum_{k=1}^{\infty}{\frac{1}{\lambda_{k}}}=\zeta(1)={\frac{\pi^{2}}{6}},\quad\qquad\quad4A^{2}\sum_{k=1}^{\infty}{\frac{1}{\lambda_{k}^{2}}}=\zeta(2)={\frac{\pi^{4}}{90}}.
X_{\phi}^{i}(x;y)=X^{i}(x)+\nabla_{r}X^{i}(x)\,y^{r}+\left(\frac12\,\nabla_{r}\nabla_{s}X^{i}(x)+\frac16\,R_{rjs}^{i}(x)X^{j}(x)\right)\,y^{r}y^{s}+\cdots,
\vec{q}\,^{(1)}=e^{-{\frac{(p_{0}^{(2)}+p_{0}^{(3)})}{\kappa}}}\vec{p}\,^{(1)}\quad\vec{q}\,^{(2)}=e^{-{p_{0}^{(3)}/\kappa}}\vec{p}\,^{(2)}\quad\vec{q}\,^{(3)}=\vec{p}^{(3)}
\zeta(\Omega^{s_{1}-1}\omega_{p_{1}}\Omega^{s_{2}-1}\omega_{p_{2}}\ldots\Omega^{s_{k}-1}\omega_{p_{k}}):=\zeta\left(\begin{array}{rlcl}{s_{1},}&{s_{2}}&{\ldots}&{s_{k}}\\{\lambda^{p_{1}},}&{\lambda^{p_{2}-p_{1}}}&{\ldots}&{\lambda^{p_{k}-p_{k-1}}}\\\end{array}\right)
\nu_{1}=\sum_{l_{1},\ldotsl_{n-1},s_{1},\ldotss_{n-1},s}q^{C_{s}}~~\vert\phi_{(l_{1},\ldotsl_{n-1},s_{1},\ldotss_{n-1},s)}^{(2)}\rangle\langle\phi_{(l_{1},\ldotsl_{n-1},s_{1},\ldotss_{n-1},s)}^{(1)}\vert
I=\frac{M}{2}\int_{0}^{s}d\tau\dot{x}_{i}^{2}+\frac{M}{2}\int_{0}^{s^{\prime}}d\tau^{\prime}\dot{y}_{j}^{2}+\frac{M}{2}(s+s^{\prime})+\frac{K_{R}}{2}\int_{0}^{s}d\tau\epsilon_{ij}x_{i}\dot{x}_{j}+\frac{K_{R}}{2}\int_{0}^{s^{\prime}}d\tau^{\prime}\epsilon_{ij}y_{i}\dot{y}_{j}.
{\calC}={\calC}^{-1}=\left(\begin{matrix}{-1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right)\quad,\quad{\calC}\alpha^{\mu}{\calC}^{-1}=\overline{{\alpha}}^{\mu\,\,\ast}\quad.
f_{\ell}(k,r)\equivw_{\ell}(kr)g_{\ell}(k,r)\quad\mathrm{and}\quad\phi_{\ell}(k,r)\equiv\frac{(-k)^{-\nu+\frac{1}{2}}}{2\nu}\frac{h_{\ell}(k,r)}{w_{\ell}(kr)}\,,
{\cal{S}}_{(r)}^{[p_{1},...,p_{N}]}\;\;=\;\;-\left(\prod_{i=1}^{N}\frac{1}{p_{i}!}\right)\intd^{D}x\;A^{{\mu}_{1}^{1}...{\mu}_{p_{1}}^{1}...{\mu}_{1}^{i}...{\mu}_{p_{i}}^{i}...{\mu}_{p_{N}}^{N}}\square\,A_{{\mu}_{1}^{1}...{\mu}_{p_{1}}^{1}...{\mu}_{1}^{i}...{\mu}_{p_{i}}^{i}...{\mu}_{p_{N}}^{N}}
{\calZ}(x)=1+\frac{\alpha}{4\pi}\frac{(4\pi)^{\epsilon}}{\Gamma(2-\epsilon)}\xi\int_{0}^{\infty}dy{\frac{y^{-\epsilon}}{x-y}}{\calZ}(y)\left[(1-\epsilon)\left(1-I_{1}^{D}\left({\frac{y}{x}}\right)\right)+{\frac{y}{y+x}}I_{1}^{D}\left({\frac{y}{x}}\right)\right].
C_{(\star,\bullet,\bullet^{\prime})}^{p}(1+|t|)^{{\hat{N}}p}\left(\prod_{1\leqi\leqp}(1+\rho_{i})^{2N}(1+|w_{i}|)^{2N}\right)|\underline{F}_{ij}|_{1\leqi,j\leqp}.\\\(\mathrm{A}.9)\notag
{\frac{\delta}{\deltaj_{a}^{B}}}\longrightarrow-\frac1\xi\tilde{I}_{aa^{\prime}}{\calF}_{a^{\prime}}\Bigl({\frac{\delta}{\deltaj_{b}^{\mu}}},{\frac{\delta}{\deltaJ}},{\frac{\delta}{\deltaJ^{\dagger}}},\hat{\Phi}\Bigr)
\begin{array}{lll}{(sxu\wedge[r])\,m\,[(\{s\}\wedgerzu)\,j\,(\{s\}\wedgerzd)]=}&{}&{}\\{\left[(sxu\wedge[r])\,m\,(\{s\}\wedgerzu)\right]\,j\,\left[(sxu\wedge[r])\,m\,(\{s\}\wedgerzd)\right]}&{=}&{}\\{(\{s\}\wedge[r]).}&{}&{}\\\end{array}
\partial_{\mu}\partial^{\mu}\xi^{5}(x)=\frac{1}{3}M^{-3}\bigg(1-\frac{4\alphak^{2}}{M^{2}}\bigg)^{-1}\bigg[\frac{S_{\mu}\,^{\mu}(x)}{2}+\frac{1}{k}\bigg(\frac{1-12\alphak^{2}/M^{2}}{1-4\alphak^{2}/M^{2}}\bigg)T_{55}(0)-\frac{1}{k}\int_{0}^{y_{m}}dy\,\partial^{\mu}T_{\mu5}\bigg]
ds^{2}=\alpha^{\prime}R^{2}\left[u^{2}(-\tilde{f}d\tilde{x}_{0}^{2}+d\tilde{x}_{1}^{2})+u^{2}\tilde{h}(d\tilde{x}_{2}^{2}+d\tilde{x}_{3}^{2})+\frac{du^{2}}{u^{2}\tilde{f}}+d\Omega_{5}^{2}\right],
G_{mn}(\Box_{2},\Box_{3})X_{2}X_{3}=-2\int_{-\infty}^{0}dq\,q^{m+n+1}\Bigl[\Bigl(\frac{d}{dq}\Bigr)^{m+1}{\calH}_{q}X_{2}\Bigr]\Bigl[\Bigl(\frac{d}{dq}\Bigr)^{n+1}{\calH}_{q}X_{3}\Bigr]\quad,
R\,=\,\left(\begin{array}{cccc}{Q^{-1}}&{0}&{0}&{0}\\{0}&{\lambda^{-1}}&{0}&{0}\\{0}&{\sigma}&{\lambda}&{0}\\{0}&{0}&{0}&{-Q}\\\end{array}\right)\,,\qquad\sigma\,=\,Q^{-1}-Q\,.
SS^{\star}=\frac{(1-\frac{1}{R^{2}})(1-\frac{1}{{R^{\star}}^{2}})}{(e^{i\pi\nu}-\frac{e^{-i\pi\nu}}{R^{2}})(e^{-i\pi\nu^{\star}}-\frac{e^{i\pi\nu^{\star}}}{{R^{\star}}^{2}})}
S_{P}=\frac{1}{4\pi}\int_{M}d^{2}\sigma\left[\frac{U^{2}}{R^{2}}(x^{\prime})^{2}+\frac{U^{2}}{R^{2}}(\dot{t})^{2}+\frac{R^{2}}{U^{2}}(U^{\prime})^{2}\right]
D_{n}=\left(\begin{array}{cc}{\hat{N}_{0n}^{12}(1,x)-\hat{N}_{0n}^{22}(1,x)\;\;\;}&{0}\\{\;\;\;\hat{N}_{0n}^{32}(1,x)-\hat{N}_{0n}^{22}(1,x)\;\;\;}&{0}\\{0}&{\;\;\;\hat{N}_{0n}^{23}(1,x)-\hat{N}_{0n}^{33}(1,x)\;\;\;}\\{0}&{\;\;\;\hat{N}_{0n}^{13}(1,x)-\hat{N}_{0n}^{33}(1,x)\;\;\;}\\\end{array}\right)
\frac{(W^{2}R^{3})^{\prime\prime}}{W^{2}R^{3}}=\frac{G^{2}}{R^{3}W^{4}}\frac{\partial}{\partialu}\left[\frac{G^{2}}{W^{2}}\frac{\partial}{\partialu}(W^{2}R^{3})\right]=\frac{3\dot{\sigma}}{R^{2}-1}+\mathrm{finite~terms}.
m_{3/2}=\frac{1}{\kappa}e^{\frac{\sqrt3}{2}K(S_{0},z_{0}^{a})}|g(z_{0}^{a})|^{\sqrt3}=\frac{1}{(2S_{R})^{\frac{\sqrt3}{2}}}\left(\frac{5}{2}\lambda\right)^{\sqrt3}\left(\frac{M_{GUT}}{M_{pl}}\right)^{3\sqrt3},
\hat{Z}_{\hat{L},\hat{L}^{\prime}}(\{\hat{q}_{N},\overline{{\hat{q}_{N}}}\})=\hat{Z}_{L,L^{\prime}}(\{q_{N},\overline{q}_{N}\})|Jac(\partialq_{N}/\partial\hat{q}_{N^{\prime}})|^{2}
R(\lambda,\mu)=\left(\begin{matrix}{1}&{0}&{0}&{d(\lambda-\mu)}\\{0}&{b(\lambda-\mu)}&{c(\lambda-\mu)}&{0}\\{0}&{c(\lambda-\mu)}&{b(\lambda-\mu)}&{0}\\{d(\lambda-\mu)}&{0}&{0}&{1}\\\end{matrix}\right)
\frac{c_{2}^{-\nu}}{c_{0}^{-\nu}}=\frac{1}{\displaystyleh^{-2}[4]-\frac{1}{\displaystyleh^{-2}[-\frac{4i\sqrt{5}}{3}(h/2)^{4}]-\frac{1}{\displaystyleh^{-2}[-12]\cdot\cdot\cdot}}}
\phi_{2}={\frac{2\phi_{0}\;(\eta_{ab}\;\zeta^{a}\;\zeta^{b})\;\left\{\eta_{ab}\;[x^{a}-x_{0}^{a}]\;\zeta^{b}\right\}}{(\eta_{ab}\;[x^{a}-x_{0}^{a}]\;[x^{b}-x_{0}^{b}]-\eta_{ab}\;\zeta^{a}\;\zeta^{b})^{2}+4(\eta_{ab}\;[x^{a}-x_{0}^{a}]\;\zeta^{b})^{2}}}.
\begin{array}{cl}{\displaystyle\prod_{a\inA^{\prime}}\oint_{C_{a}}\frac{dw_{a}}{2\pi\sqrt{-1}w_{a}}\Psi_{\sigma}^{(n)}(w|\zeta_{1},\cdots\zeta_{2n}){\calA}_{l}(v|u){\calB}_{\varepsilon,k,l}(v_{2n+1},\cdots,v_{2n+l}|u_{0},u_{2n})=0,}\\\end{array}
S_{\Psi}=\intd^{6}x\sqrt{^6g}\bar{\Psi}i\Gamma^{A}D_{A}\Psi\simeq\epsilon^{2}\int_{0}^{\infty}dz\phi^{7}(\phi^{\prime})^{3/2}\intd^{4}x\sqrt{-\eta}\bar{\psi}i\gamma^{\nu}\partial_{\nu}\psi.
d\mu_{B}\=\\prod_{k=1}^{4}{\frac{d\gamma^{[k]}}{\sqrt{2\pi}}}\\biggl(\mathrm{det}_{rs}\{\frac{2\pi}{e^{2}}}\intd^{3}x\Z_{m}^{a\[r]}\Z_{m}^{a\[s]}\biggr)^{1/2}\.
\int_{-\infty}^{\infty}d\tau{\calL}=\frac{16C_{4}}{15}\frac{v^{3}}{b^{6}}+\frac{256C_{6}}{315}\frac{v^{5}}{b^{10}}+\frac{2048C_{8}}{3003}\frac{v^{7}}{b^{14}}+O(v^{9}).
\delta^{(4)}(\dot{-}k_{3}\dot{+}(\dot{-}p)\dot{+}k_{3}\dot{+}p^{\prime})\sim\delta(p_{0}-p_{0}^{\prime})\,\,\delta^{(3)}(e^{\lambdap_{0}}\vec{k_{3}}-\vec{p}+\vec{k_{3}}+e^{\lambdak_{3,0}}\vec{p^{\prime}})~.
R_{ijkl}=q^{2}[g_{jl}\widetilde{M_{ik}}-g_{jk}\widetilde{M_{il}}+g_{ik}\widetilde{M_{jl}}-g_{il}\widetilde{M_{kj}}]+{\frac{q^{2}(c_{0}^{2}-1)}{c_{0}^{2}}}F_{ij}F_{ml}
H=J\sum_{<{\bfx},{\bfx}^{\prime}>}\left\{\lambdaS_{z}({\bfx})S_{z}({\bfx}^{\prime})+{\frac{1}{2}}\left[S^{+}({\bfx})S^{-}({\bfx}^{\prime})+S^{-}({\bfx}^{\prime})S^{+}({\bfx}^{\prime})\right]\right\}\\,
\epsilon^{a}t_{a}^{IJ}\phi_{0}^{J}=\phi_{0}^{I}(\epsilon^{a}t^{JK}\tilde{\phi}_{R}^{K})\approx\epsilon^{a}\frac{\partial\phi_{0}^{I}}{\partial\phi_{R}^{J}}t_{a}^{JK}\phi_{R}^{K}.
I_{eff}=\frac{1}{\pi}\intd^{2}z[\frac{k}{2}\\\frac{\partialw\overline{{\partial}}w}{w^{2}}+\frac{w^{2}}{w^{2}+4\lambda}(\partialf\overline{{\partial}}f-c^{2}\partialy\overline{{\partial}}y)+\frac{cw^{2}}{w^{2}+4\lambda}(\partialf\overline{{\partial}}y-\partialy\overline{{\partial}}f)]
Tr(K(s)-K_{0}(s))={\frac{\pi\tau\Phi_{0}^{4}a^{3}}{(4\pi\sigma)^{2}}}(\sigma^{{\frac{1}{4}}}\gamma(-{\frac{1}{4}},\sigma)-2\sigma^{{\frac{1}{2}}}\gamma(-{\frac{1}{2}},\sigma)+\sigma^{{\frac{3}{4}}}\gamma(-{\frac{3}{4}},\sigma)+{\frac{4}{3}})
S=\intd^{4}x\{\frac{1}{4}F_{\mu\nu}^{a}F^{a,\mu\nu}+\frac{1}{2}Tr\{(D_{\mu}\phi_{1})^{2}\}+\frac{1}{2}Tr\{(D_{\mu}\phi_{2})^{2}\}-\frac{\lambda}{4}(Tr\{\phi_{1}^{2}\}-Tr\{\phi_{2}^{2}\}-f^{2})^{2}\}\quad.
K_{i}=\sum_{n=1}^{N}Y_{J^{n}i,00}\;\;\;,\;\;\;A_{i\alpha\beta}=\sum_{n=1}^{N}N_{J^{n}i,\alpha\beta}\;\;\;,\;\;\;M_{i\alpha}=\sum_{n=1}^{N}Y_{J^{n}\alpha,0i}\;\;\;.
\frac{\partialV}{\partial\Lambda}=\frac{1}{\Lambda^{2}}\frac{\partialV}{\partial\psi_{\alphaa}^{\phantom{a}}}C^{\prime}(0)\frac{\partialV}{\partial\bar{\psi}_{a}^{\alpha}}-\tau\Lambda^{D-2}\frac{\partial^{2}V}{\partial\psi_{\alphaa}^{\phantom{a}}\partial\bar{\psi}_{a}^{\alpha}},
\begin{array}{rcl}{i\Sigma_{00}^{\mathrm{1PI}}(p)}&{=}&{\displaystylei\lambda^{2}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}V(k_{i},p)}\\{V(k_{i},p)}&{=}&{\displaystyle\sum_{x=a}^{h}V_{x},}\\\end{array}
c\frac{d}{dt}\left((1-\frac{1}{c^{2}}|\frac{d{\bfx}_{1}}{dt}|^{2})^{-1/2}\right)=\frac{1}{c}\sum_{i=1}^{3}\frac{dx_{1}^{i}}{dt}\frac{d}{dt}\left((1-\frac{1}{c^{2}}|\frac{d{\bfx}_{1}}{dt}|^{2})^{-1/2}\frac{dx_{1}^{i}}{dt}\right).
a_{1}^{\prime}(n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime}n_{1}n_{2}nk)=n_{1}(Q_{1}(n,k)+n_{1}-1)\delta_{n_{1},n_{1}^{\prime}+1}\delta_{n_{2}n_{2}^{\prime}}\delta_{nn^{\prime}}\delta_{kk^{\prime}}
\begin{array}{rcl}{U_{(I)}}&{=}&{\left(\begin{matrix}{1}&{1}\\{0}&{m_{1}}\\\end{matrix}\right);}\\{U_{(II)}}&{=}&{\left(\begin{matrix}{1}&{1}\\{{\frac{2m_{1}}{m_{2}+1}}}&{m_{1}{({\frac{1-m_{2}}{1+m_{2}}})}}\\\end{matrix}\right);}\\{U_{(III)}}&{=}&{\left(\begin{matrix}{1}&{1}\\{{\frac{2m_{1}}{m_{2}+1}}}&{m_{1}}\\\end{matrix}\right)}\\\end{array}
(2\pi)^{3}\phi({\hat{\xi}}{\hat{\eta}};P)=\sum_{123}D^{-1}({\hat{q}}_{12})\intd{{\hat{q}}_{12}}^{\prime\prime}K({\hat{q}}_{12},{{\hat{q}}_{12}}^{\prime\prime})\phi({\hat{\xi}}^{\prime\prime}{\hat{\eta}}^{\prime\prime};P)
d\mu_{0}=\frac{1}{256\pi^{2}}\,e^{-8\pi^{2}/{g^{2}}}\,(M_{\mathrm{PV}})^{6}\,\left(\frac{8\pi^{2}}{g^{2}}\right)^{2}\frac{\mathrm{d}^{3}\bar{\omega}}{8\pi^{2}}\,\mathrm{d}^{4}x_{0}\,\mathrm{d}^{2}\theta_{0}\,\mathrm{d}\rho^{2}\,\mathrm{d}^{2}\bar{\beta}\,.
\begin{array}{rcl}{{\bf{Q}}_{-{\frac{1}{k}}}^{-}\vert\Omega_{s}^{+}\rangle}&{=}&{0}\\{\{{\frac{({\bf{Q}}_{-{\frac{1}{k}}}^{+})^{r}}{[r]!}}\vert\Omega_{s}^{+}\rangle}&{;}&{o\ler\lek-1\}}\\\end{array}
\begin{array}{rcl}{G(z_{1},\cdots,z_{n})^{n\cdots\hat{i}\cdots1i}}&{=}&{(-\tau)^{1-i}H(z_{1},\cdots,z_{n}),}\\{G(z_{1},\cdots,z_{n})^{in\cdots\hat{i}\cdots1}}&{=}&{(-\tau)^{i-1}H(z_{1},\cdots,z_{n}).}\\\end{array}
ds^{2}\;=\;\frac{R^{2}}{x_{0}^{2}}\,(dx_{0}^{2}+|d{\bfx}|^{2}),\;\;\;\;R\equivl_{s}(2\pig_{s}N)^{1/4},\;\;\;\;{\bfx}\equiv(T,X_{1},X_{2},X_{3})
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},\;\;\;\;\;\;q(x)=\frac{1}{4\pi}\epsilon_{\mu\nu}F_{\mu\nu}=\frac{-i}{2\pi}\epsilon_{\mu\nu}\partial_{\mu}u_{A}^{*}\partial_{\nu}u_{A}.
g_{\mu\nu}\,=\,{\frac{1}{(x^{0})^{2}}}\delta_{\mu\nu}\,\,,\,\,\sqrt{g}\,=\,{\frac{1}{(x^{0})^{n+1}}}\,\,,\,\,g^{\mu\nu}\,=\,(x^{0})^{2}\delta^{\mu\nu}\,\,.
\begin{array}{l}{\Psi(u)=R({\bfB}_{f})\Psi_{f}(u_{f}),\quad\bar{\Psi}(u)=\bar{\Psi}_{f}(u_{f})\widetilde{R}^{+}({\bfB}_{f}),}\\{g(u)\nabla\Psi(u)=S(B_{f})R({\bfB}_{f})\gamma_{f}D\Psi_{f}(u_{f}),}\\{\left(\nabla\bar{\Psi}(u)\right)g(u)=S(B_{f})\left(D\bar{\Psi}_{f}(u_{f})\right)\gamma_{f}\widetilde{R}^{+}({\bfB}_{f}).}\\\end{array}
P^{-}={\frac{\mu^{2}q_{0}^{2}}{2}}+\mu^{2}\Sigma_{2}+{\frac{\lambdaq_{0}^{4}}{4!d}}+{\frac{\lambdaq_{0}^{2}\Sigma_{2}}{2!d}}+{\frac{\lambdaq_{0}\Sigma_{3}}{d}}+{\frac{\lambda\Sigma_{4}}{d}}\;.
\begin{array}{cccccccccc}{\alpha_{1}^{\pm}}&{=}&{(\pm2}&{\mp1}&{0}&{~~0}&{~~0}&{0}&{0}&{0),}\\{\alpha_{2}^{\pm}}&{=}&{(\mp1}&{\pm2}&{\mp1}&{~~0}&{~~0}&{0}&{0}&{0),}\\{}&{\vdots}\\{\alpha_{7}^{\pm}}&{=}&{(0}&{0}&{0}&{~~0}&{~~0}&{\mp1}&{\pm2}&{\mp1),}\\{\beta_{8}^{\pm}}&{=}&{(0}&{0}&{0}&{~~0}&{~~0}&{0}&{\mp1}&{0).}\\\end{array}
H_{p}=1+\sum_{i}{\frac{Q_{i}}{[|\vec{x}-\vec{x}_{0\,i}|^{2}+{\frac{4Q}{(p-1)^{2}}}|\vec{z}-\vec{z}_{0}|^{p-1}]^{\frac{p+1}{p-1}}}},\\\H_{p+4}={\frac{Q}{|\vec{z}-\vec{z}_{0}|^{3-p}}}.
\begin{array}{l}{\psi_{\tau}(x)^{cl}=exp\Big(\frac{i}{\hbar}W(\tau,x)\Big)\Phi\big(\xi(\tau,x)\big)exp\Bigg(-\int_{0}^{\tau}\frac{1}{2M}\BoxW(\tau-s,\xi(s))ds\Bigg)}\\\end{array}
f^{\stackrel{\underbrace{\mathrm{\scriptsize1\ldots~1}}}{m_{1}}\stackrel{\underbrace{\mathrm{\scriptsize~2\ldots~2}}}{m_{2}}\stackrel{\underbrace{\mathrm{\scriptsize~3\ldots~3}}}{m_{3}}}(u)\;,\qquad{m_{1}\geqm_{2}\geqm_{3}\geq0}
R=e^{-B(r)}\Omega-D_{1}A^{\prime\prime}-D_{2}B^{\prime\prime}-\frac{1}{4}(D_{1}A^{\prime}+D_{2}B^{\prime})^{2}-\frac{1}{4}(D_{1}{A^{\prime}}^{2}+D_{2}{B^{\prime}}^{2})~,
\left\langleA_{\alpha}^{n}(y)\left[\partial_{\mux}A_{\beta}^{m}(x)\right]\right\rangle=-\delta^{nm}\eta_{\alpha\beta}\partial_{\mux}\mathcalG(y,x)-i\epsilon^{3nm}\eta_{\alpha\beta}\partial_{\mux}\mathcalP^{*}(y,x).
\sum_{i=0}^{n}\frac{\partial^{2}\Phi}{\partialx^{i}\partialx^{i}}-\sum_{i,j=0}^{n}x^{i}x^{j}\frac{\partial^{2}\Phi}{\partialx^{i}\partialx^{j}}-2\sum_{i=0}^{n}x^{i}\frac{\partial\Phi}{\partialx^{i}}=0~.
\bar{S}=\intd^{p}x\sqrt{|\bar{g}|}\left[\frac{1}{2}{}^{(p)}\!\bar{R}-\frac{1}{2}G_{XY}\bar{g}^{\mu\nu}{\Phi^{X}}_{\!\!,\mu}{\Phi^{Y}}_{\!\!,\nu}-V(\Phi)\right],
\delta_{L}\psi(x)=-{\frac{i}{2}}\epsilon_{ab}\sigma^{ab}\psi,\quad\delta_{L}{e^{a}}_{\mu}(x)=\epsilon{^a}_{b}e{^b}_{\mu}+{\frac{\kappa}{4}}\varepsilon^{abcd}\bar{\psi}\gamma_{5}\gamma_{d}\psi(\partial_{\mu}\epsilon_{bc}).
\left[G_{a}^{(1)},G_{i}^{(1)}\right]=0,\;\left[G_{a}^{(1)},G_{i}^{(2)}\right]=0,\;\left[G_{a}^{(2)},G_{i}^{(1)}\right]=0,\;\left[G_{a}^{(2)},G_{i}^{(2)}\right]=0,
I_{n,k}\left(\xi\right)\;=\;\left(\left(\vec{\nabla}_{1}-i\vec{\nabla}_{2}\right)g_{n}\right)\left(\xi\right)\cdot\inth_{k}\left(\xi-\xi^{\prime}\right)\;\frac{\partial}{\partial\vec{\lambda}}\;\Phi\left(\xi^{\prime}\right)d^{6}\xi^{\prime}\,