harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{ll}{{[}J_{i},J_{j}]=i{\epsilon_{ij}}^{k}J_{k}}&{(i,j,k=1,2,3)}\\{{[}J_{i},S_{j}]=i{\epsilon_{ij}}^{k}S_{k}}&{}\\{{[}S_{i},S_{j}]=-i{\epsilon_{ij}}^{k}J_{k}(2\vec{J}^{2}-C_{2}-4)}&{}\\{{[}C_{2},J_{i}]=[C_{2},S_{i}]=0}&{\mathrm{with~}\vec{J}^{2}=J_{1}^{2}+J_{2}^{2}+J_{3}^{2}}\\\end{array}
\frac{1}{g^{2}}\left(\begin{array}{l}{\Gamma_{1}}\\{\Gamma_{2}}\\{\Gamma_{3}}\\\end{array}\right)=\left(\begin{array}{lll}{B_{11}}&{B_{12}}&{B_{13}}\\{B_{21}}&{B_{22}}&{B_{23}}\\{B_{31}}&{B_{32}}&{B_{33}}\\\end{array}\right)\left(\begin{array}{l}{\Gamma_{1}}\\{\Gamma_{2}}\\{\Gamma_{3}}\\\end{array}\right).
\begin{aligned}{[m]S_{\mathrm{D}0}}&{{}=-T_{\mathrm{D}0}\intd\tau+\eta^{2}T_{\mathrm{D}0}\intd\tau\left\{\frac{1}{2}\delta\dot{x}^{i}\delta\dot{x}^{j}\delta_{ij}+\frac{1}{2}\delta\dot{r}^{2}+\frac{1}{2}(\beta)^{2}\deltar^{2}\right\}}\\{}&{{}+{\calO}(\eta^{3})~.}\\\end{aligned}
\Pi_{h}^{ij}=-\left(\begin{matrix}{0}&{2h^{2}}&{2h^{1}}\\{-2h^{2}}&{0}&{((h^{1})^{2}+(h^{2})^{2}+1)-e^{2h^{0}}}\\{-2h^{1}}&{-((h^{1})^{2}+(h^{2})^{2}+1)+e^{2h^{0}}}&{0}\\\end{matrix}\right)
D_{\hat{\alpha}}\xi_{\hat{\beta}}=-{\frac{1}{4}}(\Gamma^{\mu\nu})_{\hat{\beta}\hat{\alpha}}\partial_{\mu}\xi_{\nu}+{\frac{1}{12}}C_{\hat{\beta}\hat{\alpha}}\partial_{\sigma}\xi^{\sigma}+{\frac{1}{12}}(\Gamma^{ij})_{\hat{\beta}\hat{\alpha}}D_{\sigma[ij]}\xi^{\sigma},
L_{gf}=\left(b^{\mu}\omega_{\mu\nu}F^{\nu}+\pi^{\mu}\zeta_{\mu\nu}f^{\nu}+d^{\mu}\varsigma_{\mu\nu}\rho^{\nu}-\frac{1}{2}b^{\mu}\omega_{\mu\nu}b^{\nu}-\frac{1}{2}\pi^{\mu}\zeta_{\mu\nu}\pi^{\nu}-\frac{1}{2}d^{\mu}\varsigma_{\mu\nu}d^{\nu}\right)\sqrt{-g}
\frac{\partial^{2}x^{k}}{\partialu_{i}\partialu_{j}}=\Gamma_{ij}^{i}\frac{\partialx^{k}}{\partialu_{i}}+\Gamma_{ji}^{j}\frac{\partialx^{k}}{\partialu_{j}},\quadi\nej;\qquad\quad\frac{\partial^{2}x^{k}}{\partialu_{i}^{2}}=\sum_{j=1}^{n}\Gamma_{ii}^{j}\frac{\partialx^{k}}{\partialu_{j}},
{\calF}_{*}:=\bigoplus_{s=0}^{M}{\calF}_{s},\quad{\calF}_{s}:=\left\{\oplus_{k_{1}<k_{2}<\cdots<k_{s}}{\bfC}[\gamma_{0}^{1},..,\gamma_{0}^{N}]c_{0}^{k_{1}}c_{0}^{k_{2}}\cdotsc_{0}^{k_{s}}|0\rangle\right\},
{\itL}_{gh}=\overline{c}^{\mu}\biggl(g_{\mu\nu}\nabla^{2}+R_{\mu\nu}-\frac{2\xi\varphi}{\alpha(\varphi)}(\nabla_{\nu}\varphi)\nabla_{\mu}-\frac{2\xi\varphi}{\alpha(\varphi)}(\nabla_{\mu}\nabla_{\nu}\varphi)\biggr)c^{\nu}
{\hat{x}}_{a}^{i}(\varepsilon)=e^{i{\hat{H}}\varepsilon}{\hat{x}}_{a}^{i}(0)e^{-i{\hat{H}}\varepsilon}\simeq{\hat{x}}_{a}^{i}(0)+{\frac{{\hat{\pi}}_{a}^{i}(0)}{m}}\varepsilon+{\frac{e}{2m^{2}}}\sum_{b=1}^{N}\epsilon^{ij}[{\hat{B}}({\hat{\bfx}}_{b})\delta_{ab}+{\frac{e}{\theta}}\delta({\hat{\bfx}}_{a}-{\hat{\bfx}}_{b})]{\hat{\pi}}_{b}^{j}(0)\varepsilon^{2}
\tilde{\delta}_{1}A_{\mu}=-\frac{\mathcal{E}_{L}^{2}}{\mathcal{E}_{C}^{2}}\partial_{\mu}C\zeta,\quad\tilde{\delta}_{1}a_{\mu}=-\frac{1-\mathcal{E}_{L}^{2}}{1-\mathcal{E}_{C}^{2}}\partial_{\mu}c\zeta.
V_{R}(\overline{\phi})=\frac{\lambda^{*}\Phi_{m}^{2}\overline{\phi}^{4}}{4!}+\frac{\lambda^{*\frac{3}{2}}\Phi_{m}^{4}}{8\pi\surd2}\overline{\phi}^{2}-\frac{\lambda^{*\frac{3}{2}}\Phi_{m}^{3}}{24\surd2\pi}\overline{\phi}^{3}
(N-j)\,\mathrm{Tr}\left[\bigl(\lambda_{+}^{(N+1)}\bigr)^{j}\bigl(\lambda_{-}^{(N+1)}\bigr)^{j}\right]=(N+j+1)\,\mathrm{Tr}\left[\bigl(\lambda_{+}^{(N)}\bigr)^{j}\bigl(\lambda_{-}^{(N)}\bigr)^{j}\right].
\begin{array}{c}{\displaystyle{\frac{1}{\mathrm{e}^{\lambda}+1}}=\displaystyle{\int_{C}}\displaystyle{\frac{\mathrm{d}z}{2\pii}}\zeta_{+}\left(z\right)\Gamma\left(z\right)\,\lambda^{-z}}\\{\left(\zeta_{+}(z)\equiv\sum_{k=1}^{\infty}\displaystyle{\frac{(-1)^{k+1}}{k^{z}}}=(1-2^{1-z})\zeta\left(z\right);\;C\mathrm{is\;the\;contour\;}\mathrm{Re}\;z>1-2\epsilon\right)}\\\end{array}
g^{-1}=-{\frac{1}{2}}{\frac{\kappa^{\frac{2}{3}}}{f}}\left(\begin{matrix}{1}&{\epsilon+\chi\psi}&{-2\sqrt2\psi}\\{\epsilon+\chi\psi}&{f^{2}+(\epsilon+\chi\psi)^{2}-f\chi^{2}\kappa^{-2}}&{2\sqrt2[f\chi\kappa^{-2}-\psi(\epsilon+\chi\psi)]}\\{-2\sqrt2\psi}&{2\sqrt2[f\chi\kappa^{-2}-\psi(\epsilon+\chi\psi)]}&{-8(f\kappa^{-2}-\psi^{2})}\\\end{matrix}\right),
\mu\equiv\frac{\sqrt3h}{\xi},\quad\chi\equiv\frac{\psi}{\xi},\quad\nu\equiv\frac{N}{\xi^{2}},\quad\zeta\equiv\frac{\pm3K}{\xi^{2}},\quad\lambda\equiv\frac{\pm\Lambda_{\mathrm{M}}}{\xi^{2}},\quad\frac{d}{dT}\equiv\xi\frac{d}{d\tau},
H^{2}=-\nabla_{u}^{2}+R^{2}m^{2}-R^{2}\frac{\partial^{2}}{\partials^{2}}-iR^{2}\vec{\Sigma}\cdot\vec{\omega}\frac{\partial}{\partials}-\frac{R^{2}}{2}\frac{\partial}{\partials}\left(\vec{\Sigma}\cdot\vec{\omega}\right)+{\calO}({\delta}^{3})\.
\begin{array}{lllllccl}{}&{s}&{t}&{u}&{v}&{x}&{y}&{w}\\{\lambda\}&{1}&{1}&{n}&{0}&{2(n+2)}&{3(n+2)}&{0}\\{\mu\}&{0}&{0}&{1}&{1}&{4}&{6}&{0}\\{\nu\}&{0}&{0}&{0}&{0}&{2}&{3}&{1}\\\end{array}
\hat{\tilde{\mathrm{P}}}_{\mathrm{R}}\rightarrow\hat{\tilde{\mathrm{P}}}_{\mathrm{R}}+\beta_{\mathrm{P,0}}+\sum_{p>0}(\beta_{\mathrm{P,+}}(p)\rho(p)+\beta_{\mathrm{P,-}}(p)\rho(-p))
a_{j}=\frac{1}{2}\left(\begin{array}{c}{f_{j}}\\{0}\\\end{array}\right)+\frac{1}{2}\left(\begin{array}{c}{0}\\{f_{j}^{\ast}}\\\end{array}\right),\\b_{j}=-\frac{i}{2}\left(\begin{array}{c}{f_{j}}\\{0}\\\end{array}\right)+\frac{i}{2}\left(\begin{array}{c}{0}\\{f_{j}^{\ast}}\\\end{array}\right)
S={\frac{1}{4}}\intTr(B+F)\wedge(B+F)={\frac{1}{4}}\intd^{4}x\;\varepsilon^{\mu\nu\sigma\rho}(F_{\mu\nu}^{a}+B_{\mu\nu}^{a})(F_{\sigma\rho}^{a}+B_{\sigma\rho}^{a}).
e^{1}=\frac{\ell^{2}~e^{2\rho}}{2~r_{ex}}~du,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,e^{2}=r_{ex}~dv-\frac{\ell^{2}~e^{2\rho}}{2~r_{ex}}~du,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,e^{3}=\ell~d\rho,
{\calG}(x-x^{\prime},0)_{\mu\nu,\gamma\delta}\,=\,\langleh_{\mu\nu}(x,0)\,h_{\gamma\delta}(x^{\prime},0)\rangle=\intd^{N}m|\sigma_{m}(0)|^{2}\langle\epsilon_{\mu\nu}^{m}(x)\epsilon_{\gamma\delta}^{m}(x^{\prime})\rangle\,.
\frac{4}{\varepsilon}\frac{\delta^{I\overline{{J}}}}{1+b_{I}^{2}}\left\{{\calF}\left(1-\nu_{I};\frac{e^{\tau_{2}}}{e^{\tau_{1}}}\right){}-{\calF}\left(\nu_{I};\frac{e^{\tau_{1}}}{e^{\tau_{2}}}\right)\right\}=\frac{4}{\varepsilon}\frac{\delta^{I\overline{{J}}}}{1+b_{I}^{2}}\pib_{I}~.
F_{w,1}^{w/2}={\frac{(-1)^{pw}}{(w-1)!}}\prod_{l=1}^{w-1}(l+wp)\Biggl(\biggl(\int_{{\overline{M}}_{1,1}}\lambda_{1}-w\psi_{1}\biggr)p(p+1)+\int_{{\overline{M}}_{1,1}}\lambda_{1}\Biggr),
K_{ab}^{}~=\;-\,\frac{1}{2}\left(E_{bb}\,+\,E_{\theta(b)\,\theta(b)}\right)+\,\frac{1}{2n}\,\mathbf{1}_{2n}^{}~~,~~M_{ab}^{}~=\;-\,\frac{1}{2}\left(E_{bb}\,-\,E_{\theta(b)\,\theta(b)}\right)
\Phi(t)=\sum_{r\ge1}\int_{0}^{t}\cdots\int_{0}^{t}dt_{1}\cdotsdt_{r}{\frac{(-1)^{\Theta_{r}}}{r^{2}}}{\binom{r-1}{\Theta_{r}}}^{-1}\{\psi(t_{r})\cdots\psi(t_{1})\}\,
\eta\rfloor\omega={\frac{1}{p!}}\eta_{i}\omega_{j_{1}\ldotsj_{p}}g^{ij_{1}}dx^{j_{2}}\cdotsdx^{j_{p}}-{\frac{1}{p!}}\eta_{i}\omega_{j_{1}\ldotsj_{p}}g^{ij_{2}}dx^{j_{1}}dx^{j_{3}}\cdotsdx^{j_{p}}+\cdots.
\intd\Omega_{(3)}\Psi_{j_{1}j_{2}m_{1}m_{2}}^{\dagger(l)}\Psi_{j_{1}^{\prime}j_{2}^{\prime}m_{1}^{\prime}m_{2}^{\prime}}^{(l^{\prime})}=\delta_{ll^{\prime}}\delta_{j_{1}j_{1}^{\prime}}\delta_{j_{2}j_{2}^{\prime}}\delta_{m_{1}m_{1}^{\prime}}\delta_{m_{2}m_{2}^{\prime}}.\nonumber
+2i\Theta\intd^{4}xd^{4}y\left(\frac{N-1}{N}\tilde{\Sigma}_{\mu\nu}(x)-s_{i}^{(\alpha)}\tilde{\Sigma}_{\mu\nu}^{i}(x)\right)j_{\mu}(y)\partial_{\nu}^{x}D_{m}(x-y)\Biggr],
\{E_{i\bar{j}},E_{k\bar{l}}\}=u^{2}v^{2}[1,i](1,\hat{\bar{j}})[1,k](1,\hat{\bar{l}})\left({\frac{G_{ik}E_{\bar{j}\bar{l}}}{(\hat{\bar{j}},\hat{\bar{l}})}}-{\frac{G_{\bar{j}\bar{l}}E_{ik}}{[i,k]}}\right).
D_{0}^{-1}(\phi;k)=k^{2}+m^{2}+{\frac{\lambda_{\Phi}}{2}}\phi^{2}+\,{\frac{\lambda_{\Phi}}{2}}\,\mathrm{~\sum~}\!\!\!\!\!\!\!\int_{p}^{(0)}D_{0}(\phi;p)~.
\gamma_{i}^{\beta}({-\triangle})\equiv\gamma_{i}({\beta}\sqrt{-\triangle})=\int_{0}^{\infty}{\frac{{\mathrm{d}}s}{s}}\left[\theta_{3}\Big(0,{\mathrm{e}}^{-\frac{\beta^{2}}{4s}}\Big)-1\right]f_{i}(-s\triangle),
{\calL}_{gf}=\sqrt{g}\left[-\frac{1}{\alpha}(-D_{\mu}W^{\mu+}+i\alphaM_{W}\Psi^{+})(-D_{\mu}W^{\mu-}-i\alphaM_{W}\Psi^{-})-\frac{1}{2\alpha}(-D_{\mu}Z^{\mu}+\alphaM_{Z}\Psi)^{2},\right]
\quad\int\frac{d^{4}k}{(2\pi)^{4}}\Bigg(\frac{1}{k^{2}}-\sum_{i}c_{i}\,\frac{1}{k^{2}-M_{i}^{2}}\Bigg)-\int\frac{d^{4}k}{(2\pi)^{4}}\Bigg(\frac{1}{(k+p)^{2}}-\sum_{i}c_{i}\,\frac{1}{(k+p)^{2}-M_{i}^{2}}\Bigg)=0,\quad
\displaystyle{\frac{m_{\pi}^{2}vj_{\pi}(q)}{m_{\pi}^{2}-q^{2}}(=\partial^{\mu}J_{\mu}(q))=\frac{q^{2}vj_{\pi}(q)}{m_{\pi}^{2}-q^{2}}+\partial^{\mu}\widehatJ_{\mu}(q).}
F(r)=-\frac{w}{\kappa}\left[B_{\frac{r^{\kappa}}{r^{\kappa}+1}}\left(\frac{-1}{p+1},1-\frac{2}{\kappa}\right)+2B_{\frac{r^{\kappa}}{r^{\kappa}+1}}\left(\frac{p}{p+1},-\frac{2}{\kappa}\right)\right]\!.
Z[J_{1},J_{2}]=e^{W[J_{1},J_{2}]}=\int\!D\Phi\,e^{-S[\Phi]+\int\!J_{1}(x)\Phi(x)\,d^{n}x+\frac{1}{2}\int\!J_{2}(x)\Phi^{2}(x)\,d^{n}x}
V_{eff}=\intdx\Biggl(\frac{1}{2}\pi_{c}^{2}+\frac{m^{2}}{2}\phi_{c}^{2}\Biggr)+\frac{\hbar}{2}\sum_{n=1}^{\infty}\Biggl(m^{2}+\Bigl(\frac{n\pi}{L}\Bigr)^{2}\Biggr)^{1/2}.
\rho_{m_{0}\rightarrow0+0+0}^{5}\!=\frac{1}{64\pi^{4}}\int_{0}\!\!\int_{0}\!\!\int_{0}\!\!d\mu_{1}d\mu_{2}d\mu_{3}\frac{\mu_{1}\mu_{2}\mu_{3}\theta(\lambda_{E})}{\lambda_{E}(\mu_{1}^{2},\mu_{2}^{2},\mu_{3}^{2})}\left(1\!-\frac{\mu_{1}+\mu_{2}+\mu_{3}}{m_{0}}\right).
X=q^{-1/2}X_{11},\;\;\;Y=q^{-1/2}X_{12},\;\;\;Z=\frac{q^{-1}X_{21}-qX_{22}}{\sqrt{q+q^{-1}}},\;\;\;T=\frac{X_{21}+X_{22}}{q\sqrt{q+q^{-1}}}.
\big(\left<f\right|S^{(3)}\left|i\right>\big)_{a}=\frac{-ie^{3}}{2}\bar{u}^{(r_{2})}(\vec{p}_{2})\int{\mathrmd}^{4}k[(I)_{a}+(II)_{a}]a_{\mu}(q)u^{(r_{1})}(p_{1})
X\toq^{m{\rho}^{\vee}}X{\bar{q}}^{m{\rho}^{\vee}}\,,\,\,\,{\rho}^{\vee}=\sum_{i}{\Lambda}_{i}^{\vee}\,\,\,(\,\Rightarrow\,\,E_{i}\toq^{m}E_{i}\,,\,\,F_{i}\to{\bar{q}}^{m}F_{i}\,)\,.
\hat{G}_{MN}=\left(\begin{matrix}{e^{a\varphi}g_{{\mu}{\nu}}+G_{{m}{n}}A_{{\mu}}^{(1)\,m}A_{{\nu}}^{(1)\,n}}&{A_{{\mu}}^{(1)\,m}G_{{m}{n}}}\\{A_{{\nu}}^{(1)\,n}G_{{m}{n}}}&{G_{{m}{n}}}\\\end{matrix}\right),
\frac{\mathrm{cn}(\psi)\pm\mathrm{i}k^{\prime}\mathrm{sn}(\psi)}{1+k\mathrm{sn}(\psi)}\;\;,\;\;\frac{\mathrm{cn}(\psi)\pm\mathrm{i}k^{\prime}\mathrm{sn}(\psi)}{1-k\mathrm{sn}(\psi)}
\frac{1}{q^{2}}\left\{-i\delta^{ab}\left[g_{\mu\nu}-{\frac{1}{q\cdotu}}(q_{\mu}u_{\nu}+q_{\nu}u_{\mu})+{\frac{q_{\mu}q_{\nu}}{(q\cdotu)^{2}}}\left({\frac{\alpha}{n_{0}^{2}}}q^{2}+1\right)\right]\right\}\,,
\left(\begin{array}{ccc}{1}&{-\bar{t}_{a}R}&{-\bar{t}_{a}R}\\{-\bar{t}_{b}R}&{1}&{-\bar{t}_{b}R}\\{-\bar{t}_{c}R}&{-\bar{t}_{c}R}&{1}\\\end{array}\right)\left(\begin{array}{c}{l_{aa}}\\{l_{ba}}\\{l_{ca}}\\\end{array}\right)=\left(\begin{array}{c}{0}\\{t_{ba}(=t_{ab})}\\{t_{ca}(=t_{ac})}\\\end{array}\right)
{\calL}=-\sum_{j=1}^{N}\left(\partial^{\nu}\Psi_{j}^{\ast}\partial_{\nu}\Psi_{j}+m_{0}^{2}\Psi_{j}^{\ast}\Psi_{j}-g\Phi^{p}\left(\Psi_{j}^{\ast}\Psi_{j}\right)^{q}\right)-\frac{1}{2}\left(\partial^{\nu}\Phi\partial_{\nu}\Phi+\mu_{0}^{2}\Phi^{2}\right)\,,
\Theta_{\alpha}^{+}=\theta_{\alpha}^{i}u_{i}^{+}+\eta^{+}\tilde{\lambda}_{\alpha}+{\bar{\eta}}^{+}{\bar{\tilde{\lambda}}}_{\alpha}-i\eta^{+}{\bar{\eta}}^{+}{\dot{\theta}}_{\alpha}^{i}u_{i}^{-},
E_{R}={\frac{(\epsilon-1)^{2}}{4\pia}}\sum_{l=1}^{\infty}\nu^{2}\int_{0}^{\infty}dz\left[F_{l}(\nuz)-{\frac{t^{4}}{4}}+{\frac{t^{10}}{8\nu^{2}}}(1+8z^{2}-5z^{4}+z^{6})\right],
\partial_{a}\rho_{n+1}^{\bar{c}}=\epsilon_{a}^{\bar{a}}\bar{\partial}_{\bar{a}}\rho_{n}^{\bar{c}}+\bar{\partial}_{\bar{a}}\big(\rho_{n}^{\bar{b}}\epsilon^{\bar{a}\bar{c}}\bar{\partial}_{\bar{b}}\partial_{a}\phi\big)
\{P_{(z,w)}(x^{-}),P_{(z^{\prime},w^{\prime})}(x^{-^{\prime}})\}_{PB}=-g_{st}^{2}(zw^{\prime}-z^{\prime}w)P_{(z+z^{\prime}.w+w^{\prime})}\delta(x^{-}-x^{-^{\prime}}),
{\calH}^{MWY}={\frac{1}{2}}\sum_{k=0}^{\infty}\left[{\frac{(-)^{k}}{g^{00}\sqrt{-g}}}(\pi_{k}^{2}+\phi_{k}^{2})-{\frac{g^{01}}{g^{00}}}\pi_{k}\phi_{k}^{\prime}\right]
{d^{\dagger}}(\eta^{n}\cdot\tau^{n}{\cal{G}})\;\;=\;\;\eta^{n}\cdot\tau^{n}\,{d^{\dagger}}{\cal{G}}+{\textsf{P}}_{s}(n)\;\eta^{n-1}\cdot\tau^{n-1}\,{d^{\dagger}}{\cal{G}}
K^{(2)}={\frac{l^{3}}{8a^{4}}}\left(R_{\\beta}^{\alpha}R_{\\alpha}^{\beta}-{\frac{2}{9}}R^{2}\right)-{\frac{l^{3}}{12a^{2}}}\left(R_{\\beta}^{\alpha}R_{\\alpha}^{\beta}-{\frac{1}{6}}R^{2}\right)\.
\varepsilon_{0}^{\beta\rightarrow0}={\verte\vert}^{2}H^{2}\left(\frac{1}{8\pi}-\frac{\beta^{2}m^{2}}{64\pi}\right)+\frac{7m^{2}}{4\pi\beta^{2}}-\frac{2}{3\pi\beta^{4}}+{\calO}(m^{4})+{\calO}(\beta^{2}m^{6})+{\calO}(H^{2}(\betam)^{4}).
U_{n+1}=\left(\begin{array}{ll}{e^{v_{n+1}}U_{n}}&{e^{v_{n+1}}U_{n}\sigma_{n}}\\{e^{v_{n+1}}{\sigma_{n}}^{T}U_{n}}&{e^{-nv_{n+1}}+e^{v_{n+1}}{\sigma_{n}}^{T}U_{n}\sigma_{n}}\\\end{array}\right)\,.
A({\bf{p}},\omega)=\int\mathrm{~}{d{\bf{x}}}\mathrm{~}\int_{-\infty}^{+\infty}\mathrm{~}dt\mathrm{~}e^{-i{\bf{p.x}}+i\mathrm{~}\omega\mathrm{~}t}\{exp[\frac{1}{4N_{0}}\sum_{{\bf{q}}}f_{{\bf{q}}}(-{\bf{x}},-t)](\rho+u({\bf{x}},t))-exp[\frac{1}{4N_{0}}\sum_{{\bf{q}}}f_{{\bf{q}}}({\bf{x}},t)]\rho\}
\begin{array}{ll}{\tilde{\omega}^{\prime}=}&{\left[\,(r_{x}+q_{xx})\,dp_{x}+(r_{x}-p_{xx})\,dq_{x}-(1+p_{xx}+q_{xx})\,dr\,\right]\wedge(dp+dq)}\\{}&{+\frac{1}{2}dq\wedgedq_{x}}\\\end{array}
\{F,G\}=-\intd^{2}\theta\,\Bigr\{\theta^{2}\frac{\partial^{2}\deltaF}{\partial\theta^{2}\delta\Phi^{A}(\theta)}\,\frac{\deltaG}{\delta\bar{\Phi}_{B}(\theta)}\gamma_{B}^{A}+(-1)^{\epsilon(F)\epsilon(G)}(F\leftrightarrowG)\Bigr\}.
E=[P_{11}^{2}+[2L_{1}L_{2}T^{(2)}+V_{2}]^{2}]^{1/2}\approxP_{11}+\frac{2L_{1}L_{2}T^{(2)}V_{2}}{P_{11}}+{\frac{2(L_{1}L_{2}T^{(2)})^{2}}{P_{11}}}
\int_{{\bfp}}\frac{p_{\mu}}{\left(p^{2}+2p\cdotP-M^{2}\right)^{\alpha}}=\frac{-i\pi^{n/2}P_{\mu}}{(2\pi)^{d}\Gamma(\alpha)\left(-P^{2}-M^{2}\right)^{\alpha-n/2}}\Gamma\left(\alpha-\frac{n}{2}\right)\;\;,
(C_{i}\cdotC_{j})=\left(\begin{array}{rrrrrr}{-2,}&{1,}&{0,}&{\cdots}&{0,}&{1}\\{1,}&{-2,}&{1,}&{\ddots}&{\ddots}&{0}\\{0,}&{1,}&{\ddots}&{\ddots}&{\ddots}&{\vdots}\\{\vdots}&{\ddots}&{\ddots}&{\ddots}&{1,}&{0}\\{0,}&{\ddots}&{\ddots}&{1,}&{-2,}&{1}\\{1,}&{0,}&{\cdots}&{0,}&{1,}&{-2}\\\end{array}\right).
\hat{\chi}_{r,s}^{(p,p^{\prime})}(q)=\hat{\chi}_{p-r,p^{\prime}-s}^{(p,p^{\prime})}(q)=\frac{1}{(q)_{\infty}}\sum_{j=-\infty}^{\infty}\left(q^{j(jpp^{\prime}+rp^{\prime}-sp)}-q^{(jp+r)(jp^{\prime}+s)}\right).
\rho\left(1+\frac{\rho}{\lambda}\right)=\frac{6K}{k_{4}^{2}}\frac1{a^{2}}-\frac{12{\calU}_{0}}{\lambdak_{4}^{4}}\frac1{a^{4}}-\rho+\frac{2(3H_{0}^{2}-\Lambda)}{k_{4}^{2}},
X^{+}={\frac{*(2{\calD}X^{i}\wedgef^{+i})}{*(f^{+i}\wedgef^{-i})}}={\frac{1}{\calR}}({\calD}_{-}X^{i}f_{+}^{~+i}-{\calD}_{+}X^{i}f_{-}^{~+i})
{\calC}_{{\bfq}_{1}{\bfq}_{2};{\bfp}_{1}{\bfp}_{2}}=O(1)\;\;\;\mathrm{for}\;\rule{0.14mm}{8.5mm}\,\sum_{j=1}^{2}(p_{j}^{0}-q_{j}^{0})\,\rule{0.14mm}{8.5mm}\lesssim1/L^{0}\;\mathrm{and}\;\rule{0.14mm}{8.5mm}\,\sum_{j=1}^{2}({\bfp}_{j}-{\bfq}_{j})\,\rule{0.14mm}{8.5mm}\lesssim1/L.
T_{\kappa}^{+}:=\delta_{1\kappa}\,\,,\,\,T_{\kappa}^{-}:=\delta_{2\kappa}\,\,,\,\,V_{\kappa}^{-}:=\delta_{3\kappa}\,\,,\,\,V_{\kappa}^{+}:=\delta_{4\kappa}
\left(\frac{sinh\frac{\gamma}{2}(u_{l}+2j\;i)}{sinh\frac{\gamma}{2}(u_{l}-2j\;i)}\right)^{L}=\prod_{k=1,k\neql}^{M}\frac{sinh\frac{\gamma}{2}(u_{l}-u_{k}+2i)}{sinh\frac{\gamma}{2}(u_{l}-u_{k}-2i)}
P_{a}(a,b,a)=\frac{-1}{8\pi^{2}a^{3}}\sum_{l=0}^{\infty}(2l+1)\int_{0}^{\infty}dz\frac{K_{\nu}^{(b)}(bz)/K_{\nu}^{(a)}(az)}{K_{\nu}^{(a)}(az)I_{\nu}^{(b)}(bz)-K_{\nu}^{(b)}(bz)I_{\nu}^{(a)}(az)},
\tilde{\Phi}_{\lambdam}^{\muV^{(l)}}(z)=g_{\lambda}^{\muV^{(l)}}(z)\int_{0}^{s_{1}\infty}d_{p}t_{1}J^{S}(t_{1})\cdots\int_{0}^{s_{l-m}\infty}d_{p}t_{l-m}\,J^{S}(t_{l-m})\phi_{m}^{(l)}(z),
\intdx\left(\,{\frac{1}{2}}\,\tilde{g}^{ij}{\frac{\delta^{2}\Psi}{\delta\phi^{i}\delta\phi^{j}}}\,+\,{\frac{1}{2}}\,g_{ij}\partial_{x}\phi^{i}\partial_{x}\phi^{j}\Psi\,\right)\,=\,i{\frac{\partial}{\partialt}}\Psi\,.
\pi_{,i}=\frac{3}{2}\pi^{kl}\gamma_{kl,i}-\frac{3}{2}\tilde{\pi}^{kl}\tilde{\gamma}_{kl,i}+\frac{3}{2}\pi^{\phi}\phi_{,i}-\frac{3}{2}\tilde{\pi}^{\phi}\tilde{\phi}_{,i}~~~.
L_{+}^{B}=\left(\begin{array}{ll}{q^{S^{3}}t^{Z}}&{\OmegaS^{-}}\\{0}&{q^{S^{3}}t^{-Z}}\\\end{array}\right)\\\,\\\L_{-}^{B}=\left(\begin{array}{ll}{q^{-S^{3}}t^{Z}}&{0}\\{-\OmegaS^{+}}&{q^{S^{3}}t^{-Z}}\\\end{array}\right)\,
\alpha=\left(\begin{array}{ccc}{\omega_{k}}&{0}&{0}\\{0}&{\omega_{k}^{-1}}&{0}\\{0}&{0}&{1}\\\end{array}\right)~~~~~~~~\beta=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{\omega_{2k^{\prime}}}&{0}\\{0}&{0}&{\omega_{2k^{\prime}}^{-1}}\\\end{array}\right)~~~~~~~~\gamma=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{0}&{i}\\{0}&{i}&{0}\\\end{array}\right)
S=\intd^{11}x{\calE}\lbrace\frac{1}{4}R({\calE})-\frac{1}{48}{\calF}^{2}+\frac{2}{(12)^{4}}\epsilon^{x_{1}...x_{11}}{\calF}_{x_{1}...x_{4}}{\calF}_{x_{5}...x_{8}}{\calA}_{x_{9}...x_{11}}\rbrace
S^{ij}(\theta-\sigma_{ij})=\left(\begin{array}{ccc}{\eta_{ij}(-1)_{\theta}}&{\eta_{ij}^{2}(-2)_{\theta}}&{\eta_{ij}^{3}(-3)_{\theta}}\\{\eta_{ij}^{2}(-2)_{\theta}}&{-(-3)_{\theta}(-1)_{\theta}}&{\eta_{ij}^{2}(-2)_{\theta}}\\{\eta_{ij}^{3}(-3)_{\theta}}&{\eta_{ij}^{2}(-2)_{\theta}}&{\eta_{ij}(-1)_{\theta}}\\\end{array}\right),
S=\frac{1}{2}\intd\tau\intd^{2}\sigma\biggl[\biggl(\partial_{0}X^{I}-\lbrace\omega,X^{I}\rbrace\biggr)^{2}-\frac{1}{2}\lbraceX^{I},X^{J},\rbrace\lbraceX^{I},X^{J}\rbrace\biggr]~~.
H^{F}=\sum_{n=-\infty}^{\infty}\left[\sum_{a=1}^{2}\left(\omega_{n}-\mu\right)F_{n}^{(a)}+\sum_{a=3}^{6}\omega_{n}F_{n}^{(a)}+\sum_{a=7}^{8}\left(\omega_{n}+\mu\right)F_{n}^{(a)}\right].
\left[-\left({\frac{d^{2}}{dr^{2}}}+{\frac{2}{r}}\:{\frac{d}{dr}}\right)+{\frac{K^{2}+1}{r^{2}}}\right]\left({\frac{r\phi_{1}}{K}}\right)=\alpha^{2}\left({\frac{r\phi_{1}}{K}}\right)
\mathrm{Tr}K^{(j)}(s)=\mathrm{Tr}\left[e^{-s\bigtriangleup^{(j)}}\right]={\frac{1}{(4\pis)^{d/2}}}\left(A_{0}^{(j)}+sA_{1}^{(j)}+s^{2}A_{2}^{(j)}+....\right)~~~
+{\frac{1}{8}}\Theta\Theta(\gamma^{a})_{\alpha\gamma}\psi_{a}^{\gamma}\omega_{\underline{{n}}}^{kp}(\sigma_{kp})^{\alpha\mu}+{\frac{1}{4}}\Theta\Theta\phi\psi_{\underline{{n}}}^{\mu}-{\frac{1}{8}}\Theta\Theta\omega_{\underline{{n}}}^{kp}Tr(\sigma_{kp}\gamma^{a})\psi_{a}^{\mu}
\Omega_{-2}^{+2i}=\frac14\nabla_{-p}\gamma_{p\dot{q}}^{i}\psi_{-2\dot{q}}^{2+},\quad\Omega_{+2}^{-2i}=-\frac14\nabla_{+p}\gamma_{p\dot{q}}^{i}\psi_{+2\dot{q}}^{1-}.
{\widetilde{{\bf\Gamma}}}_{jk}^{i}=\Gamma_{jk}^{i}+g^{in}\partial_{n}\theta_{\alpha}{\widetilde{{\bf\Gamma}}}_{jk}^{\alpha},\qquad{\widetilde{{\bf\Gamma}}}_{ij}^{\alpha}=-g^{\alpha\beta}\nabla_{i}\partial_{j}\theta_{\beta},
\begin{array}{rcl}{i\Sigma_{ij}(p)}&{=}&{\displaystylei\lambda^{2}\delta_{ij}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}[f_{0}(k_{i},p)+f_{12}(k_{i},p)K_{12}],}\\{i\Sigma_{00}(p)}&{=}&{\displaystylei\lambda^{2}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}[g_{0}(k_{i},p)+g_{1}(k_{i},p)K_{1}+g_{12}(k_{i},p)K_{12}],}\\\end{array}
\left.\begin{array}{llrl}{\gamma_{1,3,(\pm1,0,0)}}&{=}&{{\bf1}_{1}}&{}\\{\gamma_{1,3,(m,0,\pm1)}}&{=}&{-\alpha{\bf1}_{1}}&{\,\,\,\,\,(m=0,\,\pm1)}\\{\gamma_{1,{\tilde{7}}_{3},(\pm1)}}&{=}&{-\alpha{\bf1}_{1}}&{\,}\\{\gamma_{1,7_{4},(\pm1)}}&{=}&{\alpha{\bf1}_{1}}&{}\\\end{array}\right.
{\calA}(x)={\frac{1}{8\pi^{2}}}\left[R^{ab}{\mathrm{\tiny~\wedge~}}R_{ab}+2(T^{a}{\mathrm{\tiny~\wedge~}}T_{a}-R_{ab}{\mathrm{\tiny~\wedge~}}e^{a}{\mathrm{\tiny~\wedge~}}e^{b})\right].
A=e^{a}P_{a}+\sigma^{(0)\alpha}S_{\alpha}^{(0)}+\frac{1}{2}\omega^{ab}J_{ab}+\sigma_{a}^{(1)\alpha}S_{\alpha}^{(1)a}+\sigma_{ab}^{(2)\alpha}S_{\alpha}^{(2)ab}+\cdots\quad,
s=\left(\begin{array}{ccccc|c}{a_{1}}&{a_{2}}&{a_{3}}&{\cdots}&{a_{N-1}}&{b}\\{1}&{a_{1}}&{a_{2}}&{\cdots}&{a_{N-2}}&{0}\\{0}&{1}&{a_{1}}&{\ddots}&{\vdots}&{\vdots}\\{\vdots}&{\ddots}&{\ddots}&{\ddots}&{a_{2}}&{\vdots}\\{0}&{\cdots}&{0}&{1}&{a_{1}}&{0}\\\hline{0}&{\cdots}&{\cdots}&{0}&{c}&{d}\\\end{array}\right)\.
G_{d}^{(h)}=g_{d}\left(\rho^{2}+(z-z^{\prime})^{2}\right)+g_{d}\left(\rho^{2}+(z+z^{\prime})^{2}\right)-2h~e^{h(z+z^{\prime})}\int_{(z+z^{\prime})}^{+\infty}e^{-hs}g_{d}\left(\rho^{2}+s^{2}\right)ds~
(\lambda-3)v(\phi^{2})+\phi^{2}v^{\prime}(\phi^{2})=\frac{(N-1)v^{\prime}(\phi^{2})}{\left(2+2V^{\prime}(\phi^{2})\right)^{3/2}}+\frac{v^{\prime}(\phi^{2})+2\phi^{2}v^{\prime\prime}(\phi^{2})}{\left(2+2V^{\prime}(\phi^{2})+4\phi^{2}V^{\prime\prime}(\phi^{2})\right)^{3/2}}
\langle0|Q|0\rangle=\left(\begin{array}{c}{Q_{0}I_{N_{f}}}\\{0}\\\end{array}\right)\qquad\qquad\qquad\langle0|\bar{Q}|0\rangle=\left(\begin{array}{cc}{\bar{Q}_{0}I_{N_{f}}}&{0}\\\end{array}\right)
S_{WZ}^{ND1}=\mu_{1}\int_{R^{1+1}}\mathrm{Tr}\left(C^{(2)}+i(2\pi\alpha^{\prime})i_{\Phi}i_{\Phi}C^{(4)}-\frac12(2\pi\alpha^{\prime})^{2}(i_{\Phi}i_{\Phi})^{2}C^{(6)}+\dots\right)\,.
\left[\delta_{1},\delta_{2}\right]\chi_{\alpha}=\left(\varepsilon_{2}^{\prime}\sigma_{\nu}\bar{\varepsilon}_{1}^{\prime}\,-\varepsilon_{1}^{\prime}\sigma_{\nu}\bar{\varepsilon}_{2}^{\prime}\right)\partial^{\nu}\tilde{\partial}_{+}\chi\,_{\alpha}\,\,\,.
V(A_{k})=\frac{1}{2g^{2}}\,\left[\left(A_{1}^{b}A_{1}^{b}\right)\,\left(A_{2}^{c}A_{2}^{c}\right)-\left(A_{1}^{b}A_{2}^{b}\right)\,\left(A_{1}^{c}A_{2}^{c}\right)\right]\,.
\{\psi_{R}(x^{-}),\psi_{L}^{\dagger}(y^{-})\}={\frac{m}{4\pi}}\sum_{n}{\frac{1}{n+{\frac{1}{2}}-q}}e^{-i{\frac{\pi}{L}}(n+{\frac{1}{2}})(x^{-}-y^{-})}.
\langlep|j_{\mu}|p+k\rangleA^{\mu}(k)=\bar{\Psi}_{in}(p)\Biggl(\gamma_{\mu}F_{1}(k^{2})+\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]k^{\nu}F_{2}(k^{2})\Biggr)\Psi_{in}(p+k)A^{\mu}(k)
V_{1}(\sigma)=\alpha_{1}\frac{\sigma^{5}}{M_{\mathrm{pl}}}+\alpha_{2}\frac{\sigma^{\ast}\sigma^{4}}{M_{\mathrm{pl}}}+\alpha_{3}\frac{{\sigma^{\ast}}^{2}\sigma^{3}}{M_{\mathrm{pl}}}+\mathrm{{h.c.},}
T_{+}(x,\lambda)\prod(x-z)^{\kappa}=\left[-\frac{\lambda}{\kappa}T_{+}(z,\frac{\kappa^{2}}{\lambda})+\frac{1+\frac{\lambda}{\kappa}}{2}\sum_{i,\alpha=1}^{N,M}\frac{\kappa(\kappa-1)}{(x_{i}-z_{\alpha})^{2}}\right]\prod(x-z)^{\kappa},
Y_{1}\hat{R}^{-1}dK_{1}^{\dagger}\hat{R}=\hat{R}dK_{1}^{\dagger}\hat{R}^{-1}Y_{1}\quad\Longleftrightarrow\quad\hat{R}dK_{1}\hat{R}^{-1}\tilde{Y}_{1}=\tilde{Y}_{1}\hat{R}^{-1}dK_{1}\hat{R}\;.
M={\calM}+i\textstyle{\frac{1}{2}}\Omega\,\left(\begin{array}{cc}{0}&{0}\\{0}&{\delta_{~b}^{a}}\\\end{array}\right)+i\textstyle{\frac{1}{2}}\psi\,\left(\begin{array}{cc}{1}&{0}\\{0}&{\delta_{~b}^{a}}\\\end{array}\right)\,,
H=\intd^{2}{\bfr}\left[\frac{1}{2}\left(D_{i}Q^{1}\!\!\cdot\!\!D_{i}Q^{1}+D_{i}Q^{2}\!\!\cdot\!\!D_{i}Q^{2}-D_{i}Q^{3}\!\!\cdot\!\!D_{i}Q^{3}\right)+V(Q^{3})+A^{0}\tilde{G}\right].
\tilde{\calH}=\frac{1}{2}(\vec{E}^{a}\vec{E}^{a}+\vec{B}^{a}\vec{B}^{a})+\frac{1}{2}p^{\dagger}p+\frac{2}{(g\chi)^{2}}G_{\mathrm{rad}}^{a}G_{\mathrm{rad}}^{a}+\frac{1}{2}(\vec{\nabla}\chi)^{2}+\frac{g^{2}}{8}\chi^{2}\vec{A}^{a}\vec{A}^{a}+V(\chi^{2}/2)\.
a_{j}^{BW}({\mathsfs})=\frac{R_{Z}}{{\mathsfs}-{\mathsfs}_{R}}=\frac{R_{Z}}{{\mathsfs}-\bar{M}_{Z}^{2}+i\bar{M}_{Z}\bar{\Gamma}_{Z}}=\frac{R_{Z}}{{\mathsfs}-(M_{R}-i\frac{\Gamma_{R}}{2})^{2}}\,,\quadm_{0}^{2}\leq{\mathsfs}<\infty\,.
\hat{T}_{\mu}=\left(\frac{\hbar}{2}\right)^{(D-2)/2}\hat{p}_{\mu}\hat{T},\quad\hat{T}=\hat{\kappa}\frac{\hat{p}^{i}\Sigma^{i}}{\hat{\omega}}-\alpha,\quad\hat{p}_{0}=-\hat{\kappa}\hat{\omega},\quad\alpha=\left(\frac{\hbar}{2}\right)^{-(D-2)/2}\tilde{\alpha}.
\tilde{h}=\frac{1}{1+(au)^{7-p}},\\\a^{7-p}=\tilde{b}^{2}/R^{7-p},\\\R^{7-p}=\frac{1}{2}(2\pi)^{6-p}\pi^{-(7-p)/2}\Gamma[(7-p)/2]\hat{g}_{s}N.
D\;=\;\left(\begin{array}{cccc}{0}&{\eta_{12}U^{12}}&{0}&{-\eta_{14}U^{14}}\\{-\eta_{12}{U^{12}}^{\dagger}}&{0}&{\eta_{23}U^{23}}&{0}\\{0}&{-\eta_{23}{U^{23}}^{\dagger}}&{0}&{\eta_{43}{U^{43}}^{\dagger}}\\{\eta_{14}{U^{14}}^{\dagger}}&{0}&{-\eta_{43}{U^{43}}^{\dagger}}&{0}\\\end{array}\right).
U(\prod_{i=1}^{g}\left(g_{a_{i}}^{~}g_{b_{i}}^{~}g_{a_{i}}^{-1}g_{b_{i}}^{-1}\right))=e^{i\Theta_{s}^{\pm}}~,~V(\prod_{i=1}^{g}\left(g_{a_{i}}^{~}g_{b_{i}}^{~}g_{a_{i}}^{-1}g_{b_{i}}^{-1}\right))=e^{i\Theta_{j}}~,
\frac{\ddot{A}+\dot{A}^{2}}{\dot{A}^{2}e^{3A}}\left[\frac{\dot{A}^{2}e^{3A}}{\ddot{A}+\dot{A}^{2}}\left(\frac{F}{\dot{A}e^{A}}\right)^{\cdot}\right]^{\cdot}-\eta^{\mu\nu}k_{\mu}k_{\nu}\left(\frac{F}{\dot{A}e^{A}}\right)=0,
\int_{0}^{T}d\tau_{1}\,G(\tau_{1},\tau_{2})\,\frac{1}{2}\frac{\partial^{2}}{\partial\tau_{1}^{2}}x(\tau_{1})=x(\tau_{2})-\frac{1}{T}\int_{0}^{T}d\tau_{1}\,x(\tau_{1})\;,
\dot{P}=\left(\begin{array}{cc}{\dot{p}_{1}}&{\dot{p}_{2}}\\{\dot{p}_{4}}&{\dot{p}_{3}}\\\end{array}\right)=\{p_{1},P\}=i\gamma\left(\begin{array}{cc}{0}&{-p_{1}p_{2}}\\{p_{1}p_{4}}&{0}\\\end{array}\right)\quad.