harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\gamma_{i}(\beta\sqrt{-\triangle})=\int_{0}^{\infty}\!\!{\mathrm{d}}t\,\,g_{i}(t)\left[\frac{2\pi}{\beta{\sqrt{-\triangle}}\,t}\,\frac{1}{{\mathrmth}(2\pit/(\beta{\sqrt{-\triangle}}))}-\frac{1}{t^{2}}\right],
e^{\mu(t,x^{5})}=\frac{1}{(k+f)}\frac{[R_{0}R_{0}^{\prime}+(f^{\prime}/2)^{2}(t-\hat{t_{0}})^{2}-f\hat{t_{0}}^{\prime}(t-\hat{t_{0}})]^{2}}{[R_{0}^{2}+f(t-\hat{t_{0}})^{2}]},
\frac{d}{d\rho}\left(\frac{\chi^{2}e^{2\gamma}}{\sqrt{-\tilde{G}}V}\frac{dz}{d\rho}\right)={\frac{1}{2\sqrt{-\tilde{G}}}}\frac{\partial}{\partialz}\left(\frac{\chi^{2}e^{2\gamma}}{V}\right)\left[1+\left(\frac{dz}{d\rho}\right)^{2}\right],
S=\int\d^{3}x\left(\pi_{0}^{(0)}\dot{A}^{(0)0}+\sum_{n=0}^{\infty}\pi_{i}^{(n)}A^{(n)i}-\lambda^{0}\omega_{0}+A^{(0)0}\omega_{3}-\sum_{n=1}^{\infty}\lambda^{(n)i}\tilde{\omega}_{i}^{(n)}\right)
\langleT_{\mu\nu}(x)\,T_{\rho\sigma}(0)\rangle=c_{n}{\frac{(n/2)!(n/2-1)!}{(2\pi)^{n}(n+1)!}}{\prod}_{\mu\nu,\rho\sigma}^{(2)}\Box^{n/2-2}\left({\frac{1}{|x|^{n}}}\right),
|1\rangle,\,\,J_{+}(\vec{k})|1\rangle,\,\,J_{+}(\vec{k})J_{+}(\vec{k}^{\prime})|1\rangle\,\,\mathrm{for}\,\,\vec{k}\not=\vec{k}^{\prime},\,\,(J_{+}(\vec{k}))^{2}|1\rangle,\,\,\cdots
S_{GN}=\intd^{D}x\left[\frac{i}2\overline{\psi}\not\!\partial\psi-\frac{M}2\overline{\psi}\psi-\frac12\sigma\star(\overline{\psi}\star\psi)-\frac{N}{4g}\sigma^{2}-\frac{N}{2g}M\sigma\right].
Z(x_{T},z)=\sum_{i=1}^{n_{1}}{\frac{b^{(0)}}{(x_{T}^{2}+|z-z_{i}^{(1)}|^{2})^{2}}}+\sum_{i=1}^{n_{2}}{\frac{1-b^{(0)}}{(x_{T}^{2}+|z-z_{i}^{(2)}|^{2})^{2}}}+\mathrm{const}\intd^{2}w{\frac{|\partial\gamma(w)|^{2}}{(x_{T}^{2}+|z-w|^{2})^{2}}}.
\begin{aligned}{D_{8}=}&{{}\Big\{\sigma,\tau,g\,\Big|\,g\tau=\sigma\taug,~g^{2}=\tau^{2}=\sigma^{2}=1\Big\}}\\{Q_{8}=}&{{}\Big\{\sigma,\tau,g\,\Big|\,g\tau=\sigma\taug,~g^{2}=\tau^{2}=\sigma,\sigma^{2}=1\Big\}}\\\end{aligned}
{\lambda_{1}^{a}\bar{\lambda}_{2}^{b}=-{\frac{1}{16}}\bar{\lambda}_{2}\gamma_{\mu}\lambda_{1}\,\gamma_{ab}^{\mu}+{\frac{1}{96}}\bar{\lambda}_{2}\gamma_{\mu\nu\rho}\lambda_{1}\,\gamma_{ab}^{\mu\nu\rho}-{\frac{1}{3840}}\bar{\lambda}_{2}\gamma_{\rho_{1}\cdots\rho_{5}}\lambda_{1}\,\gamma_{ab}^{\rho_{1}\cdots\rho_{5}},}
{\calL}(\Phi,A_{\mu})=\left(\partial_{\mu}\Phi+iqA_{\mu}\Phi\right)\left(\partial^{\mu}\Phi^{*}-iqA^{\mu}\Phi^{*}\right)-m^{2}\Phi\Phi^{*}-{\frac{1}{4}}F^{\mu\nu}F_{\mu\nu},
S=\intd^{11}x\sqrt{-g_{11}}[{\frac{1}{4}}R-{\frac{1}{2^{.}4!}}\hat{F}_{M_{1}...M_{4}}^{(4)}\hat{F}^{(4)M_{1}...M_{4}}]+\int{\frac{1}{3}}A^{(3)}\wedgedA^{(3)}\wedgedA^{(3)}
\left.Q_{g_{\mu\nu}(y)}^{\alphax}\right\vert_{back}=-\frac{1}{2}\left(\frac{1}{\partial^{2}}\right)(\delta_{\mu\alpha}\partial_{\nu}+\delta_{\nu\alpha}\partial_{\mu}-\delta_{\mu\nu}\partial_{\alpha})\delta^{4}(x-y)\,
{\calA}_{0}\left(t,{\bfx}\right)=\frac{q}{{2\pi}}\frac{1}{{1+e^{2}a\left({1-12\alpha^{2}g^{2}\partial^{2}}\right)}}\left({K_{0}\left({\chi|{\bfx}-{\bfa}|}\right)-K_{0}\left({\chi|{\bfa}|}\right)}\right).
\begin{aligned}{e^{10\lambda}}&{{}=4}\\{e^{2\phi}}&{{}=\frac{1}{3}e^{-6\lambda}\approx0.14509}\\{m^{2}R^{2}}&{{}=2^{-4}3^{-6}\approx8.5734\times10^{-5}}\\\end{aligned}
E_{\tau}^{\mu}\equiv\left(\frac{\alpha}{\sqrt{e^{2\psi}\alpha^{2}-\beta^{2}r^{2}}},0,0,\frac{\beta}{\sqrt{e^{2\psi}\alpha^{2}-\beta^{2}r^{2}}}\right)\quad;\quadE_{\sigma}^{\mu}\equiv\left(0,\sqrt{1-\frac{b}{r}},0,0\right)
d_{4}^{N}=\frac{\bar{d}_{1}^{N}a_{2}^{N}}{\bar{a}_{1}^{N}}\frac{b_{2}^{N}\bar{c}_{1}^{N}-\bar{a}_{1}^{N}d_{2}^{N}}{c_{2}^{N}\bar{b}_{1}^{N}-a_{2}^{N}\bar{d}_{1}^{N}}.
K_{\mu}=J_{\mu}+\frac{n_{f}}{8\pi^{2}}\epsilon_{\mu\nu\alpha\beta}\{Tr[g^{2}A_{\nu}\partial_{\alpha}A_{\beta}-\frac{g}{3}A_{\nu}A_{\alpha}A_{\beta}]+\frac{1}{2}g^{2}Y_{\nu}\partial_{\alpha}Y_{\beta}\}
{\mit\Pi}={\calL}+c^{2}\Phi^{2}\left[s^{2}{\calK}+y^{2}{\calA}+{\frac{1}{2}}\big(\mu_{_\infty}^{2}-\mu^{2}\big)+{\frac{1}{2}}{\calK}\sigma^{2}\big(\Gamma_{_\infty}^{-2}-\Gamma^{-2}\big)\right]\.
{\calL}_{2}=e^{2}\left(-f^{abc}f^{ade}A_{\mu}^{b}A^{d\mu}A_{\nu}^{c}A^{e\nu}-\frac{\mu^{2}}{e^{2}v^{2}}\left|\Phi^{\prime}\right|^{4}+\frac{1}{2}\,A_{\mu}^{a}A^{b\mu}\Phi^{\prime\dagger}\{\lambda^{a},\lambda^{b}\}\Phi^{\prime}\right).
\delta_{\hat{\chi}}\hat{T}_{\hat{\mu}_{1}\cdots\hat{\mu}_{r}}\;=\;-\hat{\lambda}{}^{(n)}{}_{\hat{\mu}_{1}}\hat{k}_{(n)}{}^{\hat{\nu}}\hat{T}_{\hat{\nu}\hat{\mu}_{2}\cdots\hat{\mu}_{r}}-\cdots-\hat{\lambda}{}^{(n)}{}_{\hat{\mu}_{r}}\hat{k}_{(n)}{}^{\hat{\nu}}\hat{T}_{\hat{\mu}_{1}\cdots\hat{\mu}_{\scriptscriptstyler-1}\hat{\nu}}\,.
M_{\mathrm{D}2}^{\left(n\right)}=\sqrt{2}\frac{R_{2n-1}R_{2n}}{g_{s}\left(\alpha^{\prime}\right)^{\frac{3}{2}}}\left[1+\left(2\pi\alpha^{\prime}B_{n}\right)^{2}\right]^{\frac{1}{2}}.
\left<n;\,p_{i}^{\prime}{}^{+},{\vec{p}\,_{\perpi}^{\prime}},\lambda_{i}^{\prime}\right.\,\left\vertn;\,p_{i}^{~}{}^{\!\!+},{\vec{p}_{\perpi}^{~}},\lambda_{i}\right>=\prod_{i=1}^{n}16\pi^{3}p_{i}^{+}\delta(p_{i}^{\prime}{}^{+}-p_{i}^{~}{}^{\!\!+})\,delta^{(2)}({\vec{p}\,_{\perpi}^{\prime}}-{\vec{p}_{\perpi}^{~}})\,delta_{\lambda_{i}^{\prime}\lambda_{i}^{~}}\.
\begin{array}{ll}{\delta_{(\xi)}^{S}M^{\mu\nu}={\calL}_{\xi}(\sqrt{g}g^{\mu\nu}),}&{}\\{\delta_{(\xi)}^{S}L^{\mu\nu}={\calL}_{\xi}\lbrackM^{\mu\nu}-s(\sqrt{g}g^{\mu\nu})\rbrack.}\\\end{array}
\dot{x}^{\mu}=1/2[\acute{\psi}_{+}^{\mu}(\tau+\sigma)+\acute{\psi}_{-}^{\mu}(\tau-\sigma)];\;\;\;\;\dot{x}^{\mu}=1/2[\acute{\psi}_{+}^{\mu}(\tau+\sigma)-\acute{\psi}_{-}^{\mu}(\tau-\sigma)],
X_{nk\sigma}^{(1)l}=e^{i\sigma\omega_{lk}^{(0)}t}\int_{0}^{t}dt^{\prime}e^{-i\sigma\omega_{lk}^{(0)}t^{\prime}}\sum_{j,\sigma^{\prime}}V_{k\sigma,j\sigma^{\prime}}^{(1)}X_{nk\sigma^{\prime}}^{(0)l}.
H_{R}(DD)=\sum_{j}\left(N_{F}^{R}(DD)+\frac{1}{12}\right)=\sum_{j}\left(\sum_{n=1}^{\infty}n({d_{-n}}^{-j}{d_{n}}_{j}+{d_{-n}}^{j}{d_{n}}_{-j})+\frac{1}{12}\right)
\begin{aligned}{\lambda_{0}+F}&{{}=(3;1,1,1,1,1,1,1,1,0)}&{\lambda_{7}}&{{}=(3;1,1,1,1,1,1,1,0,0)}&{}\\{\lambda_{6}}&{{}=(3;1,1,1,1,1,1,0,0,0)}&{\lambda_{5}}&{{}=(3;1,1,1,1,1,0,0,0,0),}&{}\\\end{aligned}
\ddot{\vec{\phi}}=-g^{2}[(\vec{A_{1}^{2}}+\vec{A_{2}^{2}})\vec{\phi}-(\vec{A_{1}}\cdot\vec{\phi}\vec{A_{1}}+\vec{A_{2}}\cdot\vec{\phi}\vec{A_{2}}]-\frac{1}{2}\frac{\partialV}{\partial\vec{\phi}}.
\Psi=|k|^{3/2}\rhoe^{i\varphi},\,\,\,\,\,\,A_{0}=-{\frac{k^{2}}{2}}w-\partial_{t}\varphi,\,\,\,\,\,\,A_{x}=-k\,u+\partial_{x}\varphi,\,\,\,\,\,\,A_{y}=-k\,v+\partial_{y}\varphi,\,\,\,\,\,\,
K_{N}^{(N_{f})}(x,y,M)\=\\sqrt{w^{(N_{f})}(x,M)w^{(N_{f})}(y,M)}\,\sum_{n=0}^{N-1}\frac{(n+1)\,Z_{n}^{(N_{f})}(M)}{Z_{n+1}^{(N_{f})}(M)}p_{n}^{(N_{f})}(x,M)p_{n}^{(N_{f})}(y,M)\.
H=\frac{1}{2}\mathrm{p}^{2}+\frac{1}{2}m^{2}\mathrm{x}^{2}+\frac{1}{4}\lambda\mathrm{x}^{4}=\frac{1}{2}p^{2}+\frac{1}{2}m^{2}(x+\mathrm{x}_{0})^{2}+\frac{1}{4}\lambda(x+\mathrm{x}_{0})^{4}\;,
\begin{array}{rcl}{T_{L}(z_{1},z_{2})}&{=}&{\mathrm{Tr}_{0}\,K_{+}(z_{1}){\calT}(z_{1},z_{2})}\\{{\calT}(z_{1},z_{2})}&{=}&{\mathrm{Tr}_{0}\,{\calT}(-z_{1}-z_{2})^{-1}K_{-}(z_{1}){\calT}(z_{1}-z_{2}).}\\\end{array}
\int\intf=\int_{-\infty}^{\infty}d\lambda_{1}^{\prime}d\lambda_{2}^{\prime}d\lambda_{2}\tilde{f}(0,\lambda_{2})\delta(\lambda_{2}^{\prime}(1-w^{-1})-\lambda_{2}e^{-2i\lambda_{1}^{\prime}Q^{\prime}})+
<\hat{k}_{\theta}^{*}(M)(\zeta_{-}^{i}+\sum_{j=1}^{2}U_{ij}\zeta_{+}^{j})|\varphi>-<\zeta_{-}^{i}+\sum_{j=1}^{2}U_{ij}\zeta_{+}^{j}|\hat{k}_{\theta}^{*}(M)\varphi>=0,~i=1,2.
E_{0}+\sum_{\alpha\neq(0,0)}w_{\alpha}(u)E_{\alpha}(I_{\alpha})_{j}^{i}=L(u)_{j}^{i}=\sum_{k}\phi(u,z)_{k}^{i}\phi^{-1}(u,z)_{j}^{k}\partial_{u}-l\sum_{k}\partial_{u}\phi(u,z)_{k}^{i}\phi^{-1}(u,z)_{j}^{k},
\frac{1}{2}\left(\frac{\partial^{2}}{\partials_{pj}^{2}}+\frac{1}{s_{pj}}\frac{\partial}{\partials_{pj}}\right)\gamma(s_{pj},s_{qj}^{\prime},t)=\frac{\partial}{\partialt}\gamma(s_{pj},s_{qj}^{\prime},t)\,
\tau=\left(\begin{array}{ccc}{1}&{x_{2}}&{x_{3}}\\{0}&{1}&{x_{1}}\\{0}&{0}&{1}\\\end{array}\right)\left(\begin{array}{ccc}{y_{1}y_{2}}&{0}&{0}\\{0}&{y_{1}}&{0}\\{0}&{0}&{1}\\\end{array}\right)
I_{ab}=\left(\begin{matrix}{1}&{4}&{10}&{20}&{36}&{60}&{94}&{140}\\{0}&{1}&{4}&{10}&{20}&{36}&{60}&{94}\\{0}&{0}&{1}&{4}&{10}&{20}&{36}&{60}\\{0}&{0}&{0}&{1}&{4}&{10}&{20}&{36}\\{0}&{0}&{0}&{0}&{1}&{4}&{10}&{20}\\{0}&{0}&{0}&{0}&{0}&{1}&{4}&{10}\\{0}&{0}&{0}&{0}&{0}&{0}&{1}&{4}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{1}\\\end{matrix}\right)
M=-\frac{\gamma}{2}\intr^{2}{\mathcal{V}}d^{2}x=-\frac{\gamma}{4}\int_{{\bf{R}}_{0}^{2}}\partial_{i}q_{i}d^{2}x=\frac{\gamma}{4}\sum_{s=1}^{N}\int_{C^{s}}q_{i}n_{i}^{s}\,dl.
u\,=\,(\sqrt{\frac{\omega_{\eta}}{\omega_{b}}}-\sqrt{\frac{\omega_{b}}{\omega_{\eta}}})\sqrt{\omega_{b}^{2}-\omega_{t}^{2}}\;\;v\,=\,(\sqrt{\frac{\omega_{\eta}}{\omega_{b}}}+\sqrt{\frac{\omega_{b}}{\omega_{\eta}}})\sqrt{\omega_{b}^{2}-\omega_{t}^{2}}\;.
\begin{array}{l}{\Phi^{\frac{d-3}{2}}\partial^{\mu_{1}}\Phi\Phi^{s-1}{\bfV}_{i_{2}}^{\mu_{2}}\cdots{\bfV}_{i_{s}}^{\mu_{s}}-\mathrm{traces}_{(i)}=}\\{={\frac{2}{2s+d-3}}\nabla^{\mu_{1}}\left(\Phi^{s+{\frac{d-3}{2}}}{\bfV}_{i_{2}}^{\mu_{2}}\cdots{\bfV}_{i_{s}}^{\mu_{s}}-\mathrm{traces}_{(i)}\right)}\\\end{array}
d*G_{3}+d\tau\wedge\left[i\mathrm{e}^{\phi}*\,G_{3}+*\,H_{3}\right]-i{\tilde{F}}_{5}\wedgeG_{3}=-2i\kappa^{2}\left[\frac{\delta{\calL}_{b}}{\deltaB_{2}}-\tau\frac{\delta{\calL}_{b}}{\deltaC_{2}}\right]
\widetildeM_{\lambda}(j_{p},j_{q})=-i\,\langlej_{q},j_{k}|S^{(1)}|j_{p}\rangle=-e\int{d^{4}x}\bar{\psi}_{e}(j_{q},x)\;{A^{\ast}}_{\mu}^{\lambda}(\vec{k},x)\gamma^{\mu}\;\psi_{e}(j_{p},x)
E_{\mathrm{cl}}[\Phi]=\frac{v^{2}}{2}\intd^{3}r\left(\partial_{i}s\partial_{i}s+\partial_{i}\vec{p}\cdot\partial_{i}\vec{p}+\frac{\lambdav^{2}}{4}(s^{2}+\vec{p}\,^{2}-1)^{2}\right)\.
(\gamma^{\mu}{\calC})P_{\mu}=\left(\begin{array}{}{c]{cccc}i\partial_{+}}&{\ast}&{\ast}&{\ast}\\{\ast}&{i\partial_{+}}&{\ast}&{\ast}\\{\ast}&{\ast}&{i\partial_{-}}&{\ast}\\{\ast}&{\ast}&{\ast}&{i\partial_{-}}\\\end{array}\right)\,,
2^{3}6^{2}V^{2}=\left|\begin{array}{ccc}{2l_{1}^{2}}&{-l_{12}^{2}+l_{1}^{2}+l_{2}^{2}}&{-l_{13}^{2}+l_{1}^{2}+l_{3}^{2}}\\{-l_{12}^{2}+l_{1}^{2}+l_{2}^{2}}&{2l_{2}^{2}}&{-l_{23}^{2}+l_{2}^{2}+l_{3}^{2}}\\{-l_{13}^{2}+l_{1}^{2}+l_{3}^{2}}&{-l_{23}^{2}+l_{2}^{2}+l_{3}^{2}}&{2l_{3}^{2}}\\\end{array}\right|
d\omega^{k_{s},s}=-\frac{1}{2}c_{i_{\frac{s}{2}}j_{\frac{s}{2}}}^{k_{s}}\omega^{i_{\frac{s}{2}},\frac{s}{2}}\wedge\omega^{j_{\frac{s}{2}},\frac{s}{2}}-\sum_{p=0}^{\frac{s-2}{2}}c_{i_{p}j_{s-p}}^{k_{s}}\omega^{i_{p},p}\wedge\omega^{j_{s-p},s-p}\;;
\frac{1}{2}\tilde{g}_{ab}(\hat{\bfv}^{i}\cdot{\bfK}^{a})(\hat{\bfv}^{i}\cdot{\bfK}^{b})=\frac{2\pi^{2}}{3e^{2}}\hat{\bfv}^{i}\cdot\hat{\bfv}^{i}\frac{r}{1+2\mur}
\begin{array}{l}{(z_{0};z_{1},z_{2})={\frac{1}{h}}\left({\frac{1+x_{1}^{2}+h^{2}-\Phi_{\mu}^{2}}{2}}+i\Phi_{0}(x_{1},h);\right.}\\{\left.x_{1}+i\Phi_{1}(x_{1},h),\;\;i{\frac{1-x_{1}^{2}-h^{2}+\Phi_{\mu}^{2}}{2}}+\Phi_{2}(x_{1},h)\right)}\\\end{array}
\begin{array}{l}{\,\,\,\,\,[f_{1},f_{2}]_{*_{1}}(\xi)=d_{S}t_{1}(\xi|f_{1},f_{2})+}\\{+c_{12}\biggl(f_{1}(\xi)(1-{\frac{1}{2}}\overleftarrow\partial_{\alpha}\xi^{\alpha})\varepsilon^{[\beta]_{2}}[\partial_{\beta}]^{2}f_{2}(\xi)f_{1}(\xi)[\overleftarrow\partial_{\alpha}]^{2}\varepsilon^{[\alpha]_{2}}(1-{\frac{1}{2}}\xi^{\beta}\partial_{\beta})f_{2}(\xi)\biggr).}\\\end{array}
PHP|\lambda>=\left[\begin{array}{rr}{H_{11}}&{H_{12}}\\{H_{21}}&{H_{22}}\\\end{array}\right]\left(\begin{array}{r}{\alpha_{1}}\\{\alpha_{2}}\\\end{array}\right)=\lambda\left(\begin{array}{r}{\alpha_{1}}\\{\alpha_{2}}\\\end{array}\right),
\frac{G_{\mu\nu}}{\alpha^{\prime}}=\frac{1}{\alpha_{\mathrm{eff}}^{\prime}}H^{-1/2}\eta_{\mu\nu}\,\quadG_{OS}^{2}=gH^{\frac{3-p}{4}}\,\quad\Theta^{01}=\alpha_{\mathrm{eff}}^{\prime}\,
\hat{Q}^{a}(\xi_{\alpha})=i[T_{i0}^{a}+T_{ij}^{a}\xi_{\alpha}^{j}-T_{00}^{a}\xi_{\alpha}^{i}-T_{0j}^{a}\xi_{\alpha}^{j}\xi_{\alpha}^{i}]\frac{\partial}{\partial\xi_{\alpha}^{i}}+iJ_{\alpha}T_{00}^{a}+iJ_{\alpha}T_{0i}^{a}\xi_{\alpha}^{i}.
\delta^{m}(x,x^{\prime})=\frac{1}{2J_{0}}x^{m}\delta(x-x^{\prime})+\frac{1}{2{J_{0}}^{2}}(x^{m}+x^{m})\left(\begin{matrix}{\widetilde\phi^{0}(x)}\\{\widetilde\phi^{0}(x)^{\dagger}}\\\end{matrix}\right)\left(\begin{matrix}{\widetilde\phi^{0}(x^{\prime})^{\dagger}}&{\widetilde\phi^{0}(x^{\prime})}\\\end{matrix}\right)
D_{\mu\nu}=\frac{1}{p^{2}+M^{2}}(\epsilon_{\mu\nu\lambda}p_{\lambda}+\frac{p_{\mu}p_{\nu}}{M}+M\delta_{\mu\nu})\;,{}~~~~D_{\mu\nu}^{-1}=-\epsilon_{\mu\nu\lambda}p_{\lambda}+M\delta_{\mu\nu}\;.
dx^{i}dx^{j}\wedge{}^{*}(dx^{0}dx^{l}dx^{m})=x_{0}^{-(d-5)}\delta_{lm}^{ij}dx^{1}...\,dx^{d},\;\;\;\delta_{lm}^{ij}=\delta_{l}^{i}\delta_{m}^{j}-\delta_{m}^{i}\delta_{l}^{j},
f(u)=\frac{x^{-\frac{u^{2}}{2r}+\frac{r-1}{2r}u+\frac{1}{4}}}{C\sqrt{(x^{2r};x^{2r})_{\infty}}}\frac{(x^{4}z;x^{4},x^{2r})_{\infty}(x^{2r+2}z;x^{4},x^{2r})_{\infty}}{(x^{2}z;x^{4},x^{2r})_{\infty}(x^{2r}z;x^{4},x^{2r})_{\infty}}
\begin{aligned}{q_{i\,yy}}&{{}=-2(h-q_{i\,x})c_{x}+\biggl\{\frac{q_{i\,xx}^{2}-q_{i\,y}^{2}}{h-q_{i\,x}}+q_{i\,xxx}\biggr\}_{\!x}+{}}\\{{}}&{{}\{}+4(h-q_{i\,x})\sum_{j\nei}\bigl[(h-q_{j\,x})^{3}\wp^{\prime}(q_{i}-q_{j})-3(h-q_{j\,x})q_{j\,xx}\,\wp(q_{i}-q_{j})+q_{j\,xxx}\,\zeta(q_{i}-q_{j})\bigr]}\\\end{aligned}
m_{h}=\left(\begin{array}{cccc}{0}&{0}&{\xi_{1}}&{\xi_{4}}\\{0}&{0}&{\bar{\xi}_{4}}&{\bar{\xi}_{1}}\\{\xi_{1}}&{\bar{\xi}_{4}}&{\bar{\Phi}_{12}}&{\phi_{2}}\\{\xi_{4}}&{\bar{\xi}_{1}}&{\phi_{2}}&{\Phi_{12}}\\\end{array}\right),
\theta_{m}^{m}=4V-\left[4F\,\frac{\partialV}{\partialF}+3\phi\frac{\partialV}{\partial\phi}+4Y\frac{\partialV}{\partialY}+3\mathrm{Tr}\left[F_{T}\frac{\partialV}{\partialF_{T}}\right]+\mathrm{h.c.}\right]\,
\left.{\frac{\partial\Gamma_{n-1}}{\partial\kappa}}\right|_{\Phi,K}=\left.{\frac{\partialW_{n-1}}{\partial\kappa}}\right|_{J,K}=\left.{\frac{\partial_{r}<\chi_{n-1}>_{J}}{\partialK_{A}}}\right|_{J}J_{A}(-1)^{\varepsilon_{A}}.
\displaystyleZ_{\scriptscriptstyleCPI}^{\varphi^{4}}=\int{\cal{D}}\xi^{a}{\cal{D}}\lambda_{a}{\cal{D}}c^{a}{\cal{D}}\bar{c}_{a}\;e^{i\intd^{4}x\tilde{\cal{L}}^{\varphi^{4}}}
\frac{k^{\prime}}{k}=-\frac{\lambda\left\{2(\phi^{2})^{2}+1+2\phi^{2}\sqrt{(\phi^{2})^{2}+1}\right\}^{3/2}}{\phi^{2}\left\{2(\phi^{2})^{2}+1+2\phi^{2}\sqrt{(\phi^{2})^{2}+1}\right\}^{3/2}-1}
\int_{SO(4)}dg\D_{\alpha_{1}\beta_{1}}^{(N_{1})}(g)\ldotsD_{\alpha_{4}\beta_{4}}^{(N_{4})}(g)=\sum_{\Lambda}\\C_{\alpha_{1}\ldots\alpha_{4}}^{N_{1}\ldotsN_{4}\,\Lambda}\\{C}_{\beta_{1}\ldots\beta_{4}}^{N_{1}\ldotsN_{4}\,\Lambda}.
\begin{array}{llll}{X_{1}:}&{(b^{-1},a;D_{2})}&{X_{1}^{\prime}:}&{(c,a^{-1};-D_{3})}\\{X_{2}:}&{(a,b^{-1};-D_{1})}&{X_{2}^{\prime}:}&{(c^{-1},b;D_{3})}\\{X_{3}:}&{(a^{-1},c;D_{1})}&{X_{3}^{\prime}:}&{(b,c^{-1};-D_{2})\,}\\\end{array}
\left[a_{hij}(k),a_{h^{\prime}kl}^{\dagger}({\tilde{k}})\right]=\{b_{hij}(k),b_{h^{\prime}kl}^{\dagger}({\tilde{k}})\}=\{d_{hij}(k),d_{h^{\prime}kl}^{\dagger}({\tilde{k}})\}=\delta_{hh^{\prime}}\delta_{jl}\delta_{ik}\delta(k-{\tilde{k}}),
\begin{aligned}{{}}&{{}g(\beta;\sigma_{3},\sigma_{1})=}\\{}&{{}\frac{f(\sigma_{3},\sigma_{1})^{b^{-1}\beta/2}\Gamma_{b}(Q)\Gamma_{b}(Q-2\beta)\Gamma_{b}(2\sigma_{1})\Gamma_{b}(2Q-2\sigma_{3})}{\Gamma_{b}(2Q-\beta-\sigma_{1}-\sigma_{3})\Gamma_{b}(\sigma_{1}+\sigma_{3}-\beta)\Gamma_{b}(Q-\beta+\sigma_{1}-\sigma_{3})\Gamma_{b}(Q-\beta+\sigma_{3}-\sigma_{1})},}\\\end{aligned}
{\calL}_{k}\Sigma^{i}=\big(\iota_{k}\mathrm{d}+\mathrm{d}\iota_{k}\big)\,\Sigma^{i}=\big(k^{\mu}\partial_{\mu}\Sigma_{\nu}^{i}+\partial_{\nu}k^{\mu}\Sigma_{\mu}^{i}\big)\,\mathrm{d}x^{\nu}=0\,,
{\calZ}\left[J_{\mu\nu}\right]=1+\sum_{n=1}^{\infty}\frac{(ig)^{n}}{n!}\intd^{4}x_{1}\cdots\intd^{4}x_{n}J_{\mu_{1}\nu_{1}}(x_{1})\cdotsJ_{\mu_{n}\nu_{n}}(x_{n})\left<F_{\mu_{1}\nu_{1}}(x_{1})\cdotsF_{\mu_{n}\nu_{n}}(x_{n})\right>.
\left.\frac{1}{b}\frac{n^{\prime}}{n}\right|_{y_{i}-}^{y_{i}+}=\frac{2\rho_{i}+3p_{i}}{3M^{3}},\quad\left.\frac{1}{b}\frac{a^{\prime}}{a}\right|_{y_{i}-}^{y_{i}+}=-\frac{\rho_{i}}{3M^{3}}.
D_{+}^{a}\equiv\partial_{z}-\frac{i}{2}(A_{z}^{1}+\frac{1}{\sqrt{3}}A_{z}^{2}+\cdots+\sqrt{\frac{2}{a(a-1)}}A_{z}^{a-1}+\sqrt{\frac{2(a+1)}{a}}A_{z}^{a}),
\psi_{k}(x)\rightarrowC\left(e^{ikx}\frac{\Gamma(ik)\Gamma(2\lambda+2)}{\Gamma(\lambda+1+ik)\Gamma(\lambda+1)}+e^{-ikx}\frac{\Gamma(-ik)\Gamma(2\lambda+2)}{\Gamma(\lambda+1-ik)\Gamma(\lambda+1)}\right)
W(G_{j},N_{j})=\Pi_{j}\,\sum_{k=0}^{N_{j}}\left(\stackrel{\displaystyleN_{j}}{k}\right)\left(\stackrel{\displaystyleG_{j}+k-1}{k}\right)\left(\stackrel{\displaystyleG_{j}}{N_{j}-k}\right)p_{b}^{k}p_{f}^{N_{j}-k},
S_{\psi}^{(2)}=\sum_{n\leqp+1}\int_{n\mathrm{\tiny-brane}~\subseteq~\mathrm{\tiny~D-brane}}dy^{m_{1}}\cdotsdy^{m_{n}}~A_{m_{1},\cdots,m_{n}}(y_{1},\cdots,y_{p}),
\bigl<q|\hat{\eta}|\eta;t\bigr>=\Bigl[2i\hbarf(t)\tilde{\varphi}(t)\frac{\partial}{\partialq}+\Bigl(\frac{1}{f(t)}-2g(t)\tilde{\varphi}(t)\Bigr)q\Bigr]\bigl<q|\eta;t\bigr>=\eta\bigl<q|\eta;t\bigr>.
{\bfB}\equiv\left[\begin{array}{l}{B_{1}}\\{B_{2}}\\{B_{3}}\\\end{array}\right]=\left[\begin{array}{l}{y_{7}dy_{1}-y_{1}dy_{7}-y_{2}dy_{4}+y_{4}dy_{2}}\\{y_{1}dy_{4}-y_{4}dy_{1}-y_{2}dy_{7}+y_{7}dy_{2}}\\{y_{4}dy_{7}-y_{7}dy_{4}-y_{2}dy_{1}+y_{1}dy_{2}}\\\end{array}\right]\;\;.
\Lambda_{(4)}(a)=\frac{1}{a^{2}}\left(-\kappa_{(4)}^{2}\rho_{0}a^{-3\gamma+2}-\frac{\kappa_{(4)}^{2}\rho_{0}^{2}}{2\lambda}a^{-6\gamma+2}-\sigma_{0}^{2}a^{-4}-\frac{6{\calU}_{0}}{\lambda\kappa_{(4)}^{2}}a^{-2}+3k\right).
L=\mathrm{Tr}\left\{-\frac{1}{4}G^{\mu\nu}G_{\mu\nu}+\frac{1}{2}D_{\mu}S^{\dagger}D^{\mu}S\right\}+\frac{1}{2}D_{\mu}\phi^{\dagger}D^{\mu}\phi-V(S,\phi)\,.
m^{2}L^{2}={\frac{(p-1)^{2}}{(q-1)^{2}}}(k+1)(k+q-2)+(p-1)\left(1\pm\sqrt{1+2{\frac{p-1}{(q-1)^{3}}}(p+q-2)(k+1)(k+q-2)}\right)\,.
e^{-1}{\calL}=R+{\textstyle{\frac{15}{2}}}g^{2}-{\textstyle{\frac{1}{12}}}(S_{\hat{\mu}\hat{\nu}\hat{\rho}}^{I})^{2}+e^{-1}\left[{\textstyle{\frac{1}{2}}}g^{-1}S_{[3]}^{I}\wedgedS_{[3]}^{I}\right],
D_{x_{1}x_{2},y_{1}y_{2}}^{J_{1}J_{2},K_{1}K_{2};a}=\frac{1}{(x_{1}-y_{1})^{2a}(x_{1}-y_{2})^{2J_{1}-2a}(x_{2}-y_{1})^{2K_{1}-2a}(x_{2}-y_{2})^{2J_{2}-2K_{1}+2a}}
\tilde{T}=\left(\begin{matrix}{1}\\{z^{m_{1}}}\\{\vdots}\\{z^{m_{n-1}}}\\\end{matrix}\right)+\left(\begin{matrix}{1}\\{z^{m_{1}}}\\{\vdots}\\{z^{m_{n-1}}}\\\end{matrix}\right)\,\frac{1}{1+\sum_{i=1}^{n-1}\bar{z}^{m_{i}}z^{m_{i}}}\,\sum_{i=1}^{n-1}\bar{z}^{i}f_{i+1}-\left(\begin{matrix}{0}\\{f_{2}}\\{\vdots}\\{f_{n}}\\\end{matrix}\right).
\langle\partial_{\mu}j_{R}^{\mu}\rangle_{pert}=\frac{n}{16\pi^{2}}\varepsilon^{\mu\nu\alpha\beta}\mathrm{tr}F_{\mu\nu}F_{\alpha\beta}=-\frac{n}{32\pi^{2}}\mathrm{Im}\\mathrm{tr}\intd^{2}\theta_{1}d^{2}\theta_{2}\Phi^{2},
i\frac{\partial}{\partialt}\psi(\vec{x},t)\left|_{\vec{x}}\right.=i\left[\frac{\dot{a}(t)}{a(t)}\right]\\vec{r}\cdot\left(\frac{\partialf}{\partial\vec{r}}\left|_t\right.\right)+i\frac{\partialf}{\partialt}\left|_{\vec{r}}\right..
\Psi_{i}^{\dagger}\Psi_{j}+\Psi_{j}^{\dagger}\Psi_{i}=-\left(\Psi_{i}-\Psi_{j}\right)^{\dagger}\left(\Psi_{i}-\Psi_{j}\right)+\Psi_{i}^{\dagger}\Psi_{i}+\Psi_{j}^{\dagger}\Psi_{j}
U_{1}(\phi_{0},a_{1},...,a_{d})=\mu^{D}\sqrt{a_{1}\cdotsa_{d}}\sum_{s=1}^{\infty}f(D,d,s)g^{s}\phi_{0}^{2s}A_{d}^{c^{2}}(s-\frac{D-d}{2};a_{1},...,a_{d}),
T_{c}\,\frac{d}{dy}\left(\begin{matrix}{\psi_{n}}\\{\psi_{n-1}}\\\end{matrix}\right)=\left(\begin{matrix}{a_{11}(y)}&{a_{12}(y)}\\{a_{21}(y)}&{a_{22}(y)}\\\end{matrix}\right)\left(\begin{matrix}{\psi_{n}}\\{\psi_{n-1}}\\\end{matrix}\right),
k_{2}^{\nu}T_{\rho\sigma\mu\nu}^{abcd}(p_{1},p_{2};k_{1},k_{2})=-ig^{2}e^{\rho}(p_{1})e^{\sigma}(p_{2})[k_{1\mu}S_{\rho\sigma}^{abcd}+\widetilde{G}_{\rho\sigma\mu}^{abcd}]
H_{\tau}=\frac{1}{2}p^{2}(t)+V[x(t)]+\frac{\tau}{2}p(t)V^{\prime}[x(t)]+\frac{\tau^{2}}{12}\{p^{2}(t)V^{\prime\prime}[x(t)]+V^{\prime2}[x(t)]\}+O(\tau^{3}).
dS_{(\epsilon)}^{2}=\frac{-2}{H^{2}}e^{\alpha^{(\epsilon)}}d\sigma_{+}d\sigma_{-}\;\rightarrow\;\frac{-2}{H^{2}}e^{\hat{\alpha}^{(\epsilon)}}d\hat{\sigma}_{+}d\hat{\sigma}_{-}=d\hat{S}_{(\epsilon)}^{2},
M_{N}=Nm_{0}\,\frac{\sqrt{2}}{3}\left(\,\sqrt{1+\sqrt{1-\left(\frac{g^{2}N}{4\pim_{0}^{2}\alpha_{0}}\right)^{2}}}+\,\frac{4\pim_{0}^{2}\alpha_{0}}{g^{2}N}\,\sqrt{1-\sqrt{1-\left(\frac{g^{2}N}{4\pim_{0}^{2}\alpha_{0}}\right)^{2}}}\;\right)
W_{\mathrm{c.p.1}}^{2}=\frac{\alpha^{2}}{4}\,,\qquadW_{\mathrm{c.p.2}}^{2}=\frac{2^{4/3}}{9}\alpha^{2}\,,\qquad\frac{W_{\mathrm{c.p.1}}^{3}}{W_{\mathrm{c.p.2}}^{3}}=\frac{27}{32}\,,
G_{ABCD}=4!\partial_{[A}C_{BCD]}+{\frac{4!(1+\alpha)}{8}}{\frac{\kappa^{2}}{\sqrt{2}\lambda^{2}}}\epsilon(x^{11})\bigg[{\frac{1}{2}}\mathrm{tr}R_{[AB}R_{CD]}-\mathrm{tr}F_{[AB}F_{CD]}\bigg],
{\vertO_{2}\rangle}_{\nu\bar{\nu}}=\epsilon^{l_{1}\cdotsl_{k-2}}\,\epsilon^{t_{1}\cdotst_{k-2}}\,(J_{A_{1}B_{1}}\,\nu_{l_{1}}^{A_{1}}\,\nu_{t_{1}}^{B_{1}})\cdots(J_{A_{k-2}B_{k-2}}\,\nu_{l_{k-2}}^{A_{k-2}}\,\nu_{t_{k-2}}^{B_{k-2}})\,\vert\Phi\rangle
<{\calJ}_{\mu_{1}\cdots\mu_{s}}^{(s)}(x)~{\calJ}_{\nu_{1}\cdots\nu_{s}}^{(s)}(0)>={\prod}_{\mu_{1}\cdots\mu_{s},\nu_{1}\cdots\nu_{s}}^{(s)}\left({\frac{c_{s}(t)}{x^{4}}}\right)+\sum_{s^{\prime}=0}^{s-2}
\partial_{w}\left(\frac{3S^{\prime}}{S}-4\alpha\frac{{S^{\prime}}^{3}}{S^{3}}+12\alpha\frac{kS^{\prime}}{S^{3}}+12\alpha\frac{S^{\prime}\dot{S}^{2}}{N^{2}S^{3}}\right)+..=-\rho\delta(w)
S_{A,B}=\intd^{3}x\left\{-\frac{1}{4}F_{\mu\nu}(A)F^{\mu\nu}(A)+\frac{1}{2}m\epsilon^{\mu\nu\rho}\left(F_{\mu\nu}(A)B_{\rho}-\frac{1}{4}F_{\mu\nu}(B)B_{\rho}\right)\right\}
d\tilde{s}^{2}=-dt^{2}+\tilde{R}^{2}(\tilde{x}^{5})d\Omega_{k}^{2}+\tilde{\phi}(\tilde{x}^{5})[d\tilde{x}^{5}+\kappa\tilde{A}_{0}(\tilde{x}^{5})dt]^{2}
\hat{M}_{\mu\nu}{\bf\Psi}(x)=\left\{-i(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})-\frac{i}{4}\left(\begin{array}{cc}{[\gamma_{\mu},\gamma_{\nu}]}&{0}\\{0}&{[\gamma_{\mu},\gamma_{\nu}]}\\\end{array}\right)\right\}{\bf\Psi}(x)\;.
\left[{\frac{d^{2}}{dr^{2}}}+\left({\frac{h^{\prime}}{h}}+{\frac{3}{r}}\right){\frac{d}{dr}}\right]\phi+{\frac{\omega^{2}f}{h^{2}}}\phi-{\frac{l(l+2)}{hr^{2}}}\phi-{\frac{K^{2}f_{5}^{2}}{h}}\phi=0.
{\calL}_{HEC}^{(1)}(B,a)=\sum_{n=0}^{\infty}{\frac{(-1)^{n}}{4\pi^{2}}}\int_{s_{o}}^{\infty}{\frac{ds}{s^{3}}}\,e^{-ism^{2}+i(an)^{2}/s}\;[1+iseB\,L(iseB)]\;,
\begin{array}{lll}{\lbrack{\calH}_{\perp}(x),{\calH}_{\perp}(y)]}&{=}&{g^{ij}(x)\delta(x,y),_{i}{\calH}_{j}(y)-g^{ij}(y)\delta(y,x),_{i}{\calH}_{j}(x)}\\{\lbrack{\calH}_{i}(x),{\calH}_{j}(y)]}&{=}&{\delta(x,y),_{i}{\calH}_{j}(y)-\delta(x,y),_{j}{\calH}_{i}(y)}\\{\lbrack{\calH}_{\perp}(x),{\calH}_{i}(y)]}&{=}&{\delta(x,y),_{i}{\calH}_{\perp}(y)}\\\end{array},
\big[\Gamma_{GV\bar{G}}^{[r,0]}(p,k,-p^{\prime})\big]_{abc}^{\mu}=if_{abc}\big[p^{\mu}\tilde{F}_{0}^{[r,0]}(p,k,-p^{\prime})+k^{\mu}\tilde{F}_{1}^{[r,0]}(p,k,-p^{\prime})\big]
S^{E}=\int_{0}^{\beta}d\tau\left[\dot{Y}_{a}^{i}\dot{Y}_{a}^{i}+\theta_{a}^{T}\dot{\theta}_{a}+\frac{g^{2}}{4}\,Y_{a}^{i}Y_{b}^{j}Y_{f}^{i}Y_{g}^{j}F^{abc}F^{fgc}+g\,\theta_{b}^{T}\gamma^{j}Y_{c}^{i}\theta_{d}F^{cdb}\right].
{\calL}_{Weyl}^{m}=\partial_{+}{A}_{i}(\partial_{-}{A}_{i}-\partial_{i}{A}_{-})+\frac12\left(\partial_{+}{A}_{-}\right)^{2}-\frac14\left(\partial_{i}{A}_{j}-\partial_{j}{A}_{i}\right)^{2}-\frac{m^{2}}{2}A_{i}^{2}+{A}_{-}{j}^{-}+{A}_{i}j^{i}.
\stackrel{(J)}{\omega}_{1}=a_{J}\left(\left[\Phi^{\alpha}\right],\left[\mathcal{P}_{a_{0}}\right],\left[\mathcal{P}_{a_{1}}\right],\left[\mathcal{P}_{a_{2}}\right]\right)e^{J+1}\left(\eta^{(2)a},\eta_{a}^{ijk},C_{ijk}^{a}\right),
\left(\begin{array}{l}{c_{r}^{a}}\\{\overline{{c}}_{r}^{a}}\\{d_{r}^{a}}\\{\overline{{d}}_{r}^{a}}\\\end{array}\right)=\frac{1}{2}\left(\begin{array}{cccc}{i}&{1}&{i}&{1}\\{-i}&{1}&{-i}&{1}\\{-1}&{-i}&{1}&{i}\\{1}&{-i}&{1}&{i}\\\end{array}\right)\left(\begin{array}{l}{b_{r}^{a}}\\{\overline{{b}}_{r}^{a}}\\{b_{-r}^{a}}\\{\overline{{b}}_{-r}^{a}}\\\end{array}\right)
R_{n,m}(g^{\otimesn};f^{\otimesm})=\sum_{k=0}^{n}(-1)^{k}\frac{n!}{k!(n-k)!}\bar{T}_{k}(g^{\otimesk})\times_{\hbar}T_{n-k+m}(g^{\otimes(n-k)}\otimesf^{\otimesm}).
S=\int\left(-\frac{1}{4}\,R+\frac{1}{2}\,\partial_{\mu}\phi\,\partial^{\mu}\phi-\frac{1}{4}\,e^{2\phi}\,F_{\mu\nu}^{a}F^{a\mu\nu}+\frac{1}{8}\,e^{-2\phi}\right)\sqrt{-\bf{g}}\,d^{4}x,
<{\calJ}_{\mu\nu\rho}^{(3)}(x)~{\calJ}_{\alpha\beta\gamma}^{(3)}(0)>=\sum_{\mathrm{symm}}\left[b_{1}\pi_{\mu\nu}\pi_{\alpha\beta}\pi_{\rho\gamma}+b_{0}\pi_{\mu\alpha}\pi_{\nu\beta}\pi_{\rho\gamma}\right]\frac{1}{|x|^{4+2h}}.
H=\frac{1}{2}\sum_{j=1}^{N-1}\left\{\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y}+\frac{\lambda+\lambda^{-1}}{2}(\sigma_{j}^{z}+\sigma_{j+1}^{z})\right\}\pm\frac{1}{4}(\lambda-\lambda^{-1})(\sigma_{1}^{z}-\sigma_{N}^{z})
\begin{aligned}{P_{M}}&{{}=-\varepsilon_{ij}V_{+}^{i}\partial_{M}V_{+}^{j}}\\{G_{3}}&{{}=-\varepsilon_{ij}V_{+}^{i}F^{j}}\\{F_{5}}&{{}=dA+\frac{i}{16}\varepsilon_{ij}A^{i}\wedgeF^{j}}\\\end{aligned}