harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R_{\mu}^{0}=-\frac{1}{g^{2}}\,\lambda^{a}\sigma_{\mu}\bar{\lambda}^{a}+\frac{1}{3}\sum_{f}\left(\psi_{f}\sigma_{\mu}\bar{\psi}_{f}-2i\,\phi_{f}\stackrel{\leftrightarrow}D_{\mu}\bar{\phi}_{f}\right)\,.
\widetilde{n^{i}}\,{\frac{\partial}{\partialn^{i}}}T_{\k}^{j}=\Big({\frac{c_{_\mathrm{S}}^{2}}{c^{2}}}-{\frac{c^{2}}{c_{_\mathrm{S}}^{2}}}\Big)n^{j}\mu_{k}+\Big({\frac{c_{_\mathrm{S}}^{2}}{c^{2}}}-1\Big)(\rhoc^{2}+P)g_{\k}^{j}\.
Z_{0(m)}(\{k_{s},u_{s},v_{s}\},L)=\frac{\Theta^{5}[l_{1},l_{2}](0|\omega^{(r)})}{\Theta^{5}[\{0\},\{1/2\}](0|\omega^{(r)})}\prod_{(k)}\prod_{m=1}^{\infty}\frac{(1-k^{m-1/2})^{10}}{(1-k^{m})^{10}}
\frac{1}{2}\sum_{p>0}\{[\hat{v}_{\mathrm{H,+}}(p),\hat{\mathrm{G}}_{+}(p)]_{+}+[\hat{v}_{\mathrm{H,-}}(p),\hat{\mathrm{G}}_{-}(p)]_{+}\}=\nonumber\,\frac{1}{\mathrm{L}^{2}}{\hbar}^{2}\sum_{p>0}(\frac{d}{d{\alpha}_{p}}\frac{d}{d{\alpha}_{-p}}-\frac{1}{2}[\frac{\tilde{d}}{d{\alpha}_{p}},\frac{\tilde{d}}{d{\alpha}_{-p}}]_{+}),
{\calL}=-\frac{1}{4}F_{\mu\nu}^{a}F^{\mu\nu,a}-\frac{1}{2}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}-\frac{\lambda}{4}(\Phi^{a}\Phi^{a}-v^{2})^{2}\,\a=1,2,3
L_{4}^{F}=\frac{\kappa^{2}}{36}B_{\mu\nu\alpha\beta}^{F}B_{\mu_{1}\nu_{1}\alpha_{1}\beta_{1}}^{F}G^{\mu\mu_{1};\nu_{1}\nu}K^{\alpha\alpha_{1};\beta_{1}\beta}-\frac{\kappa^{2}}{12}C_{\mu\nu\rho;\alpha\beta}^{F}C_{~~~~~~\alpha_{1}\beta_{1}}^{F\mu\nu\rho}K^{\alpha\alpha_{1};\beta_{1}\beta}.
f^{i}([\varphi],g)=\frac{\delta^{L}F}{\delta\hat{\varphi}_{i}^{*}}\big|_{\phi^{*}=0},\;\;\;\;g^{\alpha}([\varphi],[C],g)=\frac{\delta^{L}F}{\delta\hat{C}_{\alpha}^{*}}\big|_{\phi^{*}=0}
{\calL}=\left\{\frac{1}{4g^{2}}\,\int\!\mathrm{d}^{2}\theta\,\mathrm{Tr}\,W^{2}+\mathrm{H.c.}\right\}+\frac{1}{4}\int\!\mathrm{d}^{2}\theta\mathrm{d}^{2}\bar{\theta}\,\bar{Q}^{f}e^{V}Q_{f}+\left\{\frac{m}{4}\int\!\mathrm{d}^{2}\theta\,Q_{\alpha}^{f}Q_{f}^{\alpha}+\mathrm{H.c.}\right\}\,.
\begin{array}{rcl}{u_{t}}&{=}&{\displaystyle{\frac{1}{2}u(-u_{x}+v_{x}+w_{x})}\,,}\\{v_{t}}&{=}&{\displaystyle{\frac{1}{2}v(+u_{x}-v_{x}+w_{x})}\,,}\\{w_{t}}&{=}&{\displaystyle{\frac{1}{2}w(+u_{x}+v_{x}-w_{x})}\,.}\\\end{array}
dS_{(\epsilon)}^{2}=\frac{1}{H^{2}}\eta_{\mu\nu}^{(\epsilon)}dq^{\nu}=-\frac{1}{2H^{2}}\eta_{\mu\nu}^{(\epsilon)}(\dot{q}^{\mu}\dot{q}^{\nu}-q^{\mu}q^{\nu})(-d\tau^{2}+d\sigma^{2}).
f_{N}(\zeta,k,\theta,\phi)=\sum_{i\neqj}^{N}\frac{(\theta_{i}-\theta_{j})\phi_{i}(\zeta_{i}\cdotk_{j})(2\alpha^{\prime})^{11/4}-1/2\phi_{i}\phi_{j}(\zeta_{i}\cdot\zeta_{j})(2\alpha^{\prime})^{9/2}}{x_{i}-x_{j}-\theta_{i}\theta_{j}}\,.
\widehat{D}\theta=d\theta+\frac{1}{4}\hat{\omega}^{\hat{r}\hat{s}}\Gamma_{\hat{r}\hat{s}}+\hat{e}^{\hat{r}}T_{\hat{r}}{}^{\hat{s}\hat{t}\hat{u}\hat{v}}\widehat{F}_{\hat{s}\hat{t}\hat{u}\hat{v}}\theta
\{(\mathcal{H}_{i})_{A_{N-1}},(\mathcal{H}_{j})_{A_{N-1}}\}=\{(\mathcal{H}_{i}^{\varepsilon})_{A_{N-1}},(\mathcal{H}_{j}^{\varepsilon^{^{\prime}}})_{A_{N-1}}\}=0,\\\\\\\varepsilon,\varepsilon^{^{\prime}}=\pm,\,\\i,j=1,\ldots,N.
+\left.\left.\left.\frac{n-1}{n!}D_{a_{1}}\ldotsD_{a_{n-2}}F_{a_{n-1}a_{n}}\xi^{a_{1}}\ldots\xi^{a_{n-1}}\partial_{\tau}\xi^{a_{n}}-\frac{1}{2}\frac{1}{n!}D_{a_{1}}\ldotsD_{a_{n}}F_{bc}\xi^{a_{1}}\ldots\xi^{a_{n}}\psi^{b}\psi^{c}\right)\right]\right),
\ddot{x}=-\Phi(x)\Phi^{\prime}(x)-\Phi^{\prime\prime}(x)\overline{{\psi}}\,\psi,\qquad\dot{\psi}=-\mathrm{i}\Phi^{\prime}(x)\psi,\qquad\dot{\overline{{\psi}}}=\mathrm{i}\Phi^{\prime}(x)\overline{{\psi}}.
{\calL}_{\mathrm{gauge}}\sim{\frac{(rM_{h})^{6}}{g_{h}^{2}}}\;\mathrm{tr}F^{2}\\\mathrm{and}\\\{\calL}_{\mathrm{grav}}\sim{\frac{r^{6}M_{h}^{8}}{g_{h}^{2}}}\;{\calR}\,
\Lambda^{3(N_{c}+1)-(N_{f}+N_{c}^{\prime})}\tilde{\Lambda}^{3(\tilde{N}_{c}+1)-(N_{f}+N_{c}^{\prime})}=(-1)^{N_{f}+N_{c}^{\prime}-N_{c}-1}\mu^{N_{f}+N_{c}^{\prime}},
\left[\mathrm{Tr}~e^{-H^{2}t}\right]_{\mathrm{div}}={\frac{\Gamma\left(1-\fracD2\right)}{(4\pi)^{(D-1)/2}}}{\frac{m^{D-4}}{2\kappat^{3/2}}}\int_{\Sigma}\left[\left(m^{2}-\left(\frac16-\xi\right)R\right)t-{\frac{\calP}{4\kappa^{2}}}\right].
\left({\frac{d\tilde{\lambda}}{d\lambda}}\right)^{-1}{\frac{d}{d\tilde{\lambda}}}\left({\frac{d\tilde{\lambda}}{d\lambda}}\right)={\frac{1}{2}}\partial_{y}\left({\calW}g_{\mu\nu}\right){\frac{dx^{\mu}}{d\tilde{\lambda}}}{\frac{dx^{\nu}}{d\tilde{\lambda}}}.
\frac{-2iM_{V}^{2}}{v}(\gammad-b-\frac{\gammad\epsilon}{2})\eta_{\mu\nu}+\frac{i\gammad\epsilon}{v}(p_{2}\cdotp_{3}\eta_{\mu\nu}-p_{2\nu}p_{3\mu})-\frac{i(2-\epsilon)\gammad}{\alphav}(p_{2\mu}p_{1\nu}+p_{3\nu}p_{1\mu})-\frac{i\gamma\epsilond}{\alphav}p_{2\mu}p_{3\nu}.
{\calM}_{ab}^{2}=2(T_{\hat{r}}{}^{\hat{s}\hat{t}\hat{u}\hat{v}}\widehat{F}_{\hat{s}\hat{t}\hat{u}\hat{v}}\theta)_{a}(\bar{\theta}\Gamma^{\hat{r}})_{b}-\frac{1}{4}(\Gamma^{\hat{r}\hat{s}}\theta)_{a}(\bar{\theta}S_{\hat{r}\hat{s}}{}^{\hat{t}\hat{u}\hat{v}\hat{w}}\widehat{F}_{\hat{t}\hat{u}\hat{v}\hat{w}})_{b}~.
B_{mw}B_{nw^{\prime}}^{\;\;\;\;\;\;\dagger}=\sum_{w^{\prime\prime}}(B_{mw^{\prime\prime}}f_{w^{\prime\prime},w,n,w^{\prime}}+f_{w^{\prime\prime},w^{\prime},m,w}^{*}B_{nw^{\prime\prime}}^{\;\;\;\;\;\;\dagger})+E_{mw;nw^{\prime}}(\phi)
W^{(1)}=\frac{b_{1}}{8}x_{1}^{8}+\frac{b_{2}}{8}x_{2}^{8}+\frac{b_{3}}{4}x_{3}^{4}+\frac{b_{4}}{4}x_{4}^{4}+\frac{b_{5}}{4}x_{5}^{4}-\psi_{0}\,x_{1}x_{2}x_{3}x_{4}x_{5}-\frac{1}{4}\psi_{s}\,(x_{1}x_{2})^{4}\.
\Phi={\frac{1}{2\pi}}\intd^{2}xB_{\mathrm{reg}}(\vec{x})={\frac{1}{2\pi}}\int_{0}^{2\pi}d\varphi\bigl[\vec{x}\times\vec{V}(\vec{x})\bigr]\biggm|_{r=0}^{r=\infty},
\begin{array}{cc}{\mathrm{v}_{\alpha\beta}=\textstyle{\frac{1}{2}}\epsilon_{\alpha\beta\gamma\delta}\tilde{\mathrm{v}}^{\gamma\delta}~~~~}&{~~~~\tilde{\mathrm{v}}^{\alpha\beta}=\textstyle{\frac{1}{2}}\epsilon^{\alpha\beta\gamma\delta}\mathrm{v}_{\gamma\delta}}\\\end{array}
G_{(1+2)}^{\mathrm{(TM)}}(\tau)=360\tau^{5}(e^{\Omega}-1)\int_{0}^{\infty}d\Omega^{\prime}\int_{0}^{\Omega^{\prime}}dK\,\frac{K\,\left[K^{2}-\Omega^{\prime}\left(\Omega+\Omega^{\prime}\right)\right]^{2}}{\sqrt{(\Omega^{2}-K^{2})\left[(\Omega+\Omega^{\prime})^{2}-K^{2}\right]}}\,\frac{e^{\Omega^{\prime}}}{\left(e^{\Omega^{\prime}}-1\right)\left(e^{\Omega+\Omega^{\prime}}-1\right)},
H_{nm}^{\{{\bfR}\}}(0,0)\approx\frac{(-1)^{\theta_{nm}}}{\sqrt{2\pi}}\Gamma\left(\frac{n+m+1}2\right)R_{11}^{n/2}R_{22}^{m/2}\left(\frac{|R_{12}|}{\sqrt{R_{11}R_{22}}}+1\right)^{\frac{n+m+1}2},
K^{\alpha\left\{\beta\sigma\right\}}\rightarrow\Omega_{\left(\mu\nu\xi\right)}^{\left\{\beta\sigma\right\}},\;\;\;\;\;\;K^{\alpha\left\{\beta\sigma\right\}}=\epsilon^{\alpha\mu\nu\xi}\Omega_{\left(\mu\nu\xi\right)}^{\left\{\beta\sigma\right\}}
H(Y_{1},Y_{2})=\sqrt{{\frac{1}{3}}\left[{\frac{Y_{2}^{2}}{2(1+2g_{c}Y_{1})}}+{\frac{\tilde{V}(Y_{1})}{(1+4g_{c}Y_{1})}}+{\frac{\rho_{b}}{M_{p}^{4}}}\right]}
S_{glue}=\int\,dx^{+}dx^{-}dx^{\perp}\Big[{\calE}^{a}F_{+-}^{a}+F_{-k}^{a}F_{+k}^{a}-\frac{1}{2}\big({\calE}^{a}{\calE}^{a}+F_{12}^{a}F_{12}^{a}\big)\Big],
\deltaV(R)=-\frac{2(D-2)\beta^{2}}{M^{4}}\int_{0}^{\infty}dy\,y^{3}\bar{\Omega}\frac{\left(e^{-Ry}-e^{-R\bar{\Omega}}\right)^{2}}{\left(1-e^{-2Ry}\right)\left(1-e^{-2R\bar{\Omega}}\right)}<0.
\tilde{s}_{q1}^{(2)}(\tilde{x},t)=-\frac{i}{12\mu}\bigl\{\bigl[\partial_{\tilde{x}}\psi^{*}(\tilde{x},t)\bigr]\tilde{x}\partial_{\tilde{x}}^{2}\psi(\tilde{x},t)-\bigl[\partial_{\tilde{x}}^{2}\psi^{*}(\tilde{x},t)\bigr]\tilde{x}\partial_{\tilde{x}}\psi(\tilde{x},t)\bigr\}.
H_{D}=\frac{1}{2}\hbar\omega\sum_{n=1}^{N}\left(1+\sigma_{3}^{(n)}\right)\stackrel{N\rightarrow\infty}{\longrightarrow}\hbar\omega{\calN}=\hbar\omegaa^{\dagger}a=H_{D}^{\mathrm{JC}}.
\tilde{{\bfE}}{}={}\frac{1}{2}\,\left({\bfu}\,\mathrm{e}^{\mathrm{i}\Theta}{}+{}{\bfu}^{*}\,\mathrm{e}^{-\mathrm{i}\Theta}\right)\quad,\qquad\tilde{{\bfB}}{}={}\frac{1}{2}\,\bar{\bfk}{}\times{}\left({\bfu}\,\mathrm{e}^{\mathrm{i}\Theta}{}+{}{\bfu}^{*}\,\mathrm{e}^{-\mathrm{i}\Theta}\right)\quad,
\omega_{\mu}^{\ab}=\frac{1}{2}e^{a\nu}e^{b\rho}\left[e_{\muc}(\partial_{\nu}e_{\rho}^{\c}-\partial_{\rho}e_{\nu}^{\c})-e_{\nuc}(\partial_{\rho}e_{\mu}^{\c}-\partial_{\mu}e_{\rho}^{\c})-e_{\rhoc}(\partial_{\mu}e_{\nu}^{\c}-\partial_{\nu}e_{\mu}^{\c})\right].
\frac{1}{\sqrt{-G}}\frac{\partial}{\partialu^{a}}(\sqrt{-G}\;G^{ab}\frac{\partial\Phi}{\partialu^{b}})+\frac{1}{h}\partial_{\xi}(h\;\partial_{\xi}\Phi)-2\lambda(\Phi^{2}-\frac{M^{2}}{4\lambda})\Phi=0.
a_{n}^{\mathrm{UV}}-a_{n}^{\mathrm{IR}}={\frac{1}{2^{{\frac{3n}{2}}-1}\,n\,\Gamma(n+1)}}\int\mathrm{d}^{n}x\,|x|^{n}\,\langle\Theta(x)\,\Theta(0)\rangle,
S=-\intd^{2}x\,\sqrt{g}\left[{\frac{1}{2}}g^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi+CR\Phi+V(\Phi)+{\frac{1}{4}}f(\Phi)g^{\mu\alpha}g^{\nu\beta}F_{\mu\nu}F_{\alpha\beta}\right].
\mathcal{E}_{m}:=(4\pi)^{-\bar{m}}\textstyle\frac1{2^{\bar{m}}\bar{m}!}\varepsilon^{i_{1}...i_{m};j_{1}...j_{m}}R_{i_{1}i_{2}j_{2}j_{1}}...R_{i_{m-1}i_{m}j_{m}j_{m-1}}.
E_{\mathrm{C}}\,=\,\frac{D-2}{2}\,\sum_{n=0}^{\infty}\,\omega_{n}\,=\,\frac{\pi(D-2)}{2\,R}\,\sum_{n=0}^{\infty}\,(n+1/2)\,=\,\frac{\pi(D-2)}{2\,R}\,\zeta(-1,1/2)\,=\,\,\frac{\pi(D-2)}{48\,R}\,{,}
\int_{C_{4}^{0}}\zeta_{0,n^{\prime}+1}(z_{1},z_{2},w;\tau)dw=\int_{C_{6}^{1}}\zeta_{0,n^{\prime}+1}(z_{1},z_{2},w;\tau)dw=\int_{C_{6}^{0}}\zeta_{0n^{\prime}}(z_{1},z_{2},w;\tau)dw
\sum_{n=0}^{N}\partial^{A,\alpha_{1}\cdots\alpha_{n}}{\calL}\cdot\partial_{\alpha_{1}\cdots\alpha_{n}}\left(N_{aA}-\partial_{\mu}\varphi_{A}\cdotX_{a}^{\mu}\right)+\partial_{\mu}\left({\calL}X_{a}^{\mu}\right)\equiv0.
A_{\mu\nu}(x)~\sim~\frac{B_{A}}{(x^{2})^{\beta_{A}}}\left[\eta_{\mu\nu}~+~\frac{2\beta_{A}(1-\rho)}{(2-2\beta_{A}+\rho)}\frac{x_{\mu}x_{\nu}}{x^{2}}~+~i\varthetah(\beta_{A})\frac{\epsilon_{\mu\nu\sigma}x^{\sigma}}{(x^{2})^{1/2}}\right]
u_{\alpha}=\frac\delta{\deltau_{\alpha}}=(x_{i}=\frac\delta{\deltax_{i}}=\frac\partial{\partialx_{i}}-N_{i}^{a}(x,y)\frac\partial{\partialy^{a}};y_{a}=\frac\delta{\deltay^{a}}=\frac\partial{\partialy^{a}})
r(\omega)=\frac{\omega}{\rho_{\gamma}}\frac{d\rho_{B}}{d\omega}\simeq\frac{\omega^{4}}{\rho_{\gamma}}|c_{-}(\omega)|^{2}\simeq\frac{\omega^{4}}{\rho_{\gamma}}\left(\frac{g_{re}}{g_{ex}}\right)^{2}
L^{(-)}=\left(\begin{array}{cc}{(-1)^{F}q^{-\left(\begin{array}{cc}{1}&{0}\\{0}&{0}\\\end{array}\right)}}&{0}\\{(q^{-1}-q)(-1)^{F}Y^{-}}&{(-1)^{F}q^{-\left(\begin{array}{cc}{0}&{0}\\{0}&{1}\\\end{array}\right)}}\\\end{array}\right)
{\calH}_{\mathrm{\tiny~CGNM}}=\sum_{i=1}^{N}\left\{\bar{\psi}^{i}\left[-i\gamma_{1}\partial_{1}\right]\psi^{i}\right\}-\frac{g^{2}}{2}\left\{\left[\sum_{i=1}^{N}\bar{\psi}^{i}\psi^{i}\right]^{2}-\xi\left[\sum_{i=1}^{N}\bar{\psi}^{i}\gamma_{5}\psi^{i}\right]^{2}\right\}\;\;,
w_{1}({y},{\bar{y}}\midx)=\sum_{n,m=0}^{\infty}\frac{\lambda^{1-|\frac{n-m}{2}|}}{2i\,n!m!}w_{1}^{\alpha_{1}\ldots\alpha_{n}}{}_{,}{}^{\dot{\beta}_{1}\ldots\dot{\beta}_{m}}(x){y}_{\alpha_{1}}\ldots{y}_{\alpha_{n}}{\bar{y}}_{\dot{\beta}_{1}}\ldots{\bar{y}}_{\dot{\beta}_{m}}\,
D_{\alpha}D_{\beta}\Omega-g_{\alpha\beta}D\cdotD\Omega+\frac{\gamma}{8}(g_{\alpha\beta}\frac{(D\Omega)^{2}}{\Omega}-2\frac{D_{\alpha}\OmegaD_{\beta}\Omega}{\Omega})+\frac{1}{2}g_{\alpha\beta}\OmegaV(\Omega)
V_{GEP}^{T}(\phi_{0})={\frac{F}{V}}=I_{1}+I_{1}^{\beta}-{\frac{\lambda}{8}}(I_{0}+I_{0}^{\beta})^{2}+{\frac{m^{2}}{2}}\phi_{0}^{2}+{\frac{\lambda}{4!}}\phi_{0}^{4}
\tilde{A}_{\mu\nu}^{-1}(k)~\sim~\frac{1}{\tilde{B}_{A}(1+\vartheta^{2})(k^{2})^{\beta_{A}-3/2}}\left[\eta_{\mu\nu}~-~\frac{k_{\mu}k_{\nu}}{k^{2}}{}~-~\vartheta\frac{\epsilon_{\mu\nu\sigma}k^{\sigma}}{(k^{2})^{1/2}}\right]
\delta\Omega^{R}={\frac{\kappa}{8\pi}}Tr\Bigl([\deltavv^{-1}]^{2}(0)\;{\calM}^{-1}\delta{\calM}\;+\;[\deltavv^{-1}](0)\;\bigl({\calM}^{-1}\delta{\calM}\bigr)^{2}\;+\;{\frac{1}{3}}\bigl(\delta{\calM}{\calM}^{-1}\bigr)^{3}\Bigr)\quad,
p_{1,1}=\{{\frac{-1}{2}\frac{1}{\midG\mid}\sum_{allg\inG}\sum_{h\inG\atop{I=odd}}\prod_{\tilde{\theta}_{i}^{g}=\tilde{\theta}_{i}^{h}=0}(1-{\frac{1}{q_{i}}})}\}-1
A_{z}:=\frac{1}{2}\left(A_{x}-iA_{y}\right)\,,\quadA_{\bar{z}}:=\frac{1}{2}\left(A_{x}+iA_{y}\right)=-\overline{{A_{z}}}\,,\quadA_{x}=2i\ImA_{z}\,,\quadA_{y}=2i\ReA_{z}\,,
S_{1}=2\,\frac{\sqrt{(|y|^{2}+|x_{0}|^{2})^{3}+(|y|^{2}+|x_{0}|^{2})^{2}I_{2}+(|y|^{2}+|x_{0}|^{2})(I_{2}^{2}-I_{4})/2+|I_{3}|^{2}}}{|y|^{2}+|x_{0}|^{2}}\,,
H_{\infty}=\sum_{n>0}n\bigg[{\frac{1}{\lambda}}\big(a_{n+1}^{\dagger}-(1-\delta_{n,1})a_{n-1}^{\dagger}-\delta_{n,1}\big)+a_{n}^{\dagger}+\sum_{p=1}^{n-1}a_{p}^{\dagger}a_{n-p}^{\dagger}\bigg]a_{n}+\mathrm{~n<0~terms}\
\left\{i\left[V_{a}^{\prime\mu}\gamma^{a}\left(\partial_{\mu}-ieA_{\mu}\right)+\gamma^{a}C_{ab}^{\primeb}+\frac{i}{4}\omega_{\mu}^{\primecd}\varepsilon_{abcd}\gamma^{5}\gamma^{b}V^{a\mu}\right]+\chi^{-1/2}U^{\prime}(\overline{{\Psi}}^{\prime}\Psi^{\prime})\right\}\Psi^{\prime}=0
\delta=\intd^{4}x\left((D_{\mu}\phi)_{\alphax}\frac{\delta}{\deltaA_{\mu\alphax}}-\frac{1}{2}f_{\alpha\beta\gamma}\phi_{\betax}\phi_{\gammax}\frac{\delta}{\delta\phi_{\alphax}}+iB_{\alphax}\frac{\delta}{\delta\psi_{\alphax}}\right)
\dot{r}^{2}+\tilde{V}(r)=0,\;\;\;\;\dot{t}=\frac{-E}{M-\frac{J^{2}}{4r^{2}}-\frac{r^{2}}{l^{2}}},\;\;\;\;\dot{f}=\frac{E}{\frac{l}{r^{2}}(M-\frac{J^{2}}{4r^{2}})-\frac{1}{l}}.
{\tilde{H}}_{coll}=\sum_{n>0}(1-\beta)na_{n}^{+}a_{n}+N\beta\sum_{n>0}a_{n}^{+}a_{n}+\sum_{n,n^{\prime}>0}(\betaa_{n}^{+}a_{n^{\prime}}^{+}a_{n+n^{\prime}}+a_{n+n^{\prime}}^{+}a_{n}a_{n^{\prime}}).
H_{2}^{-1}\Big(H_{6}\delta^{ij}\partial_{i}\partial_{j}b+2\delta^{ij}\partial_{i}H_{6}\partial_{j}b\Big)-\frac{\kappaT_{2}}{4\pi^{2}\alpha^{\prime}}(4N-M)\delta(x^{3})\cdots\delta(x^{5})=0\,,
\begin{array}{l}{e^{+2}\nabla\psi_{+2q}^{1+}+e^{-2}\nabla\psi_{-2q}^{1+}+e^{+}\nabla\chi_{q}^{1}+e^{-}\nablah_{-q}^{1+}+T^{+2}\psi_{+2q}^{1+}+T^{-2}\psi_{-2q}^{1+}+T^{+}\chi_{q}^{1}+T^{-}h_{-q}^{1+}}\\{=-\frac12\gamma_{q\dot{p}}^{i}\Omega^{+2i}\pi_{\dot{p}}^{1-},}\\\end{array}
\frac{\partial}{\partialt_{p}}\left(\begin{array}{cc}{f}\\{{\overline{f}}}\\\end{array}\right)=J_{1}\left(\begin{array}{cc}{{\delta}/{\deltaf}}\\{{\delta}/{\delta{\overline{f}}}}\\\end{array}\right)H_{p+1}=J_{2}\left(\begin{array}{cc}{{\delta}/{\deltaf}}\\{{\delta}/{\delta{\overline{f}}}}\\\end{array}\right)H_{p},
\frac{\partial^{n+1}f^{A}}{\partialz^{n+1}}=\sum_{\alpha=1}^{n}W_{\alpha}\frac{\partial^{\alpha}f^{A}}{\partialz^{\alpha}},\quad\frac{\partial^{n+1}\bar{f}^{A}}{\partial\bar{z}^{n+1}}=\sum_{\alpha=1}^{n}\bar{W}_{\alpha}\frac{\partial^{\alpha}\bar{f}^{A}}{\partial\bar{z}^{\alpha}}
\int_{S}\frac{d^{2}\tau}{\tau_{2}^{2}}F(\tau)=\int_{\bigcup\gamma_{cd}{\calF}}\frac{d^{2}\tau}{\tau_{2}^{2}}F(\tau)=\sum_{(c,d)=1}\int_{\gamma_{ab}{\calF}}\frac{d^{2}\tau}{\tau_{2}^{2}}F(\tau)\;.
B(x^{2})\sim\sum_{n=0}^{\infty}{\frac{\left(8\piGe^{2}H(x^{2})\right)^{n}}{n!\,(n+1)!}}={\frac{I_{1}(\,2\sqrt{8\piGe^{2}H(x^{2})\,}\,)}{\sqrt{8\piGe^{2}H(x^{2})\,}}}\quad\mathrm{for~|x^2|\ll~1/m^2~,}
V_{e}=\frac{e^{2y_{0}}\left[(16n^{2}+16n+3)e^{4ny}+32n^{2}+80n-6+(16n^{2}+16n+3)e^{-4ny}\right]}{4(e^{2ny}+e^{-2ny})^{2+\frac{1}{n}}}.
g^{1}=0,\quadg_{abc}^{2}=-g_{abc}^{5},\quadg^{3}=0,\quadg_{abc}^{4}=\frac{1}{2}f_{abc}^{1},\quadg_{abc}^{6}=-f_{abc}^{1},\quadg_{abc}^{7}=f_{abc}^{1}-g_{abc}^{5}.
z_{1}^{\mathrm{(LG)}}=a_{0}a_{5}^{-\frac13}a_{6}^{-\frac13}a_{7}^{-\frac16}a_{8}^{-\frac19}a_{9}^{-\frac1{18}},z_{2}^{\mathrm{(LG)}}=a_{1}a_{7}^{-\frac12}a_{9}^{-\frac12},\ldots
F_{\mu\nu}=\left(\begin{array}{cccc}{0}&{B_{3}}&{-B_{2}}&{E_{1}}\\{-B_{3}}&{0}&{B_{1}}&{E_{2}}\\{B_{2}}&{-B_{1}}&{0}&{E_{3}}\\{-E_{1}}&{-E_{2}}&{-E_{3}}&{0}\\\end{array}\right)_{\mu\nu}=i\left({\bbox{B\cdotM}}+{\bbox{E\cdotN}}\right)_{\mu\nu}\,
\mathbf{s}_{m}^{a}C^{\alpha_{1}bc}=-\epsilon^{ac}B^{\alpha_{1}b}-\epsilon^{ab}B^{\alpha_{1}c}+G_{\alpha_{0}\beta_{1}}^{\alpha_{1}}C^{\alpha_{0}a}C^{\beta_{1}bc}-\mathrm{~\frac{1}{2}~}H_{\alpha_{0}\beta_{0}\gamma_{0}}^{\alpha_{1}}C^{\alpha_{0}a}C^{\beta_{0}b}C^{\gamma_{0}c}.
z={\frac{1+{\frac{1}{2}}\bar{h}^{2}}{1-{\frac{1}{4}}h^{2}\bar{h}^{2}}},\qquad\bar{z}={\frac{1+{\frac{1}{2}}h^{2}}{1-{\frac{1}{4}}h^{2}\bar{h}^{2}}}.
\Theta_{[1/2]}\Xi_{[1/2]}=\Xi_{[1/2]}^{-1}\Theta_{[1/2]}=i\sigma_{1}sin\phi-i\sigma_{2}cos\phi=i\frac{\left[\bbox{\sigma}\times{\bfp}\right]_{3}}{\sqrt{p_{1}^{2}+p_{2}^{2}}}\quad,
\Gamma_{_C}^{\mu\nu}=g^{\mu\nu}\left(\square+\frac{a_{_C}e^{2}}{4\pi}\right)-\partial^{\mu}\partial^{\nu}-\frac{e^{2}}{4\pi}\left(\partial^{\mu}+\tilde{\partial}^{\mu}\right)\frac{1}{\square}\left(\partial^{\nu}+\tilde{\partial}^{\nu}\right)
t^{(Y_{2}-Y_{1})^{2}-1}\mathrm{\Large{\{}}-\eta_{\mu\nu}a_{-1}^{(1)\mu}a_{-1}^{(2)\nu}+c_{-1}^{(1)}b_{-1}^{(2)}-b_{-1}^{(1)}c_{-1}^{(2)}\mathrm{\Large{\}}}c_{1}^{(1)}|\downarrow\rangle^{(1)}c_{1}^{(2)}|\downarrow\rangle^{(2)}
E_{i}=p_{i}+\frac{m_{i}^{2}}{2p_{i}}+\left(p_{i}+\frac{m_{i}^{2}}{2p_{i}}\right)\left(\frac{1}{6\omega^{2}}\left(p_{i}^{2}+m_{i}^{2}\right)-\frac{1}{24}\right)(q-1)^{2},
\Pi_{\mathrm{T}}^{(2)}(0)=4e^{4}\int\frac{d^{3}q}{(2\pi)^{3}}\frac{1}{q^{2}+\kappa^{2}}\int\frac{d^{3}k}{(2\pi)^{3}}\,\frac{\partial}{\partialk_{i}}\left[\frac{k_{i}}{(k^{2}+m^{2})^{2}}-\frac{k_{i}(q^{2}+4m(\kappa-m))}{(k^{2}+m^{2})^{2}((k+q)^{2}+m^{2})}\right]=0
{\calL}_{\mathrm{K}_{\mu}}=\frac{-1}{8g^{2}}\left(\mathrm{Tr}\{{\calF}_{\ast}^{\mu\nu}{\calF}_{\mu\nu}\}-i\,\mathrm{Tr}\{{\calF}_{\ast}^{\mu\nu}{\calF}_{\mu\nu}\}\,i\right),
\{\eta^{MN}\}=\left(\begin{array}{cc}{g^{\alpha\beta}}&{-A_{\gamma}^{\mu}g^{\gamma\beta}}\\{-g^{\alpha\gamma}A_{\beta}^{\nu}}&{g^{\mu\nu}+g^{\gamma\delta}A_{\gamma}^{\mu}A_{\delta}^{\nu}}\\\end{array}\right)\,.
V[L,~f;~R_{1},R_{2},....~R_{n}]~=~{\frac{q^{\sum_{i=1}^{n}{a_{i}C_{R_{i}}}}~~{\prod_{i=1}^{n-1}S_{\Lambda_{R_{i}}\Lambda_{R_{i+1}}}}}{S_{00}~~{\prod_{i=2}^{n-1}~~S_{{\Lambda_{{R_{i}}}}0}}}},\nonumber
{\calJ}\overline{{A}}_{n}^{i}{\calJ}^{-1}=a_{n}^{i}\;\;\;;\;\;\;{\calJ}\overline{{\widetildeA}}_{n}^{\scriptscriptstyleL}{\calJ}^{-1}=L_{n}^{\scriptscriptstyleT}\;\;\;;\;\;\;{\calJ}\overline{{C}}_{n}{\calJ}^{-1}=c_{n}\;\;\;.
\left|\frac{\partial(a_{1}^{1},a_{1}^{2},a_{1}^{3},a_{2}^{1},a_{2}^{2},a_{2}^{3})}{\partial(\tilde{a}_{1}^{1},\tilde{a}_{1}^{2},\tilde{a}_{2}^{1},\tilde{a}_{2}^{2},\varphi^{1},\varphi^{2})}\right|_{\vec{\varphi}=0}=\:\tilde{a}_{1}^{1}\tilde{a}_{2}^{2}-\tilde{a}_{1}^{2}\tilde{a}_{2}^{1}\;.
A_{\mu\nu\eta\sigma\rho}^{(2)}(x)=\frac{e^{2}}{2}(A_{\mu}A_{\eta}\partial_{\sigma}F_{\nu\rho}^{0}-\partial_{\nu}A_{\rho}\partial_{\eta}A_{\mu}A_{\sigma}+A_{\mu}F_{\nu\eta}^{0}F_{\sigma\rho}^{0})\,\,\,,
\frac{1}{6}\Biggl|q^{-2}\frac{\partialq}{\partialT}V^{-1}\frac{\partialV}{\partial{T_{1}}}\Biggr|\;,\;\frac{1}{6}\Biggl|q^{-2}\frac{\partialq}{\partialT}V^{-1}\frac{\partialV}{\partial{T_{2}}}\Biggr|\ll1
U(r)=-\frac{1}{\pir^{2}}\int_{0}^{+\infty}\,\omega^{4}\alpha_{1}(i\omega)\,\alpha_{2}(i\omega)\,e^{-2\,\omegar}\Bigl[1+\frac{2}{\omegar}+\frac{5}{(\omegar)^{2}}+\frac{6}{(\omegar)^{3}}+\frac{3}{(\omegar)^{4}}\Bigr]d\omega,
\left.L_{\mathrm{soft}\,(2)}^{(2)}\right|_{T>0}=\left(\frac{2e^{2}}{3\pi^{2}}\right)A_{k}^{(1)}A_{j}^{(2)}T\!\sum_{m=-\infty}^{\infty}\int_{0}^{\infty}d\theta\,\theta^{2}\int_{0}^{1}dx\left(\frac{\theta^{2}\delta_{kj}-3p_{j}p_{k}x(1-x)}{[P^{2}x(1-x)+\theta^{2}+(k_{4}-xp_{4})^{2}]^{2}}\right)\.\nonumber\,
\iota_{\delta}(L(S))\circ\iota_{\delta}(L(X))=\left(\begin{matrix}{U_{0}(S,X)}&{U_{1}^{*}(S,X)}\\{U_{1}(S,X)}&{U_{0}^{*}(S,X)}\\\end{matrix}\right)\inUsp(n+m,n+m),
\Theta(t)=\left[\sum_{\delta\neq{\bf1}}\sum_{p=1}^{\infty}{\frac{l(\delta)/2}{\mathrm{Sinh}(pl(\delta)/2)}}e^{-p^{2}l^{2}/4t}\right]{\frac{e^{-M^{2}t}}{(\pit)^{1/2}}}\quad.
{\frac{Dp^{\mu}}{d\tilde{\lambda}}}\equiv{\frac{dp^{\mu}}{d\tilde{\lambda}}}+\Gamma_{\rho\sigma}^{\mu}{\frac{dx^{\rho}}{d\tilde{\lambda}}}p^{\sigma}=-\tilde{m}_{0}{\frac{{\calW}^{\prime}}{\calW}}{\frac{dy}{d\tilde{\lambda}}}{\frac{dx^{\mu}}{d\tilde{\lambda}}},
\langlea_{+}(w)c{\bar{c}}V_{n}^{0}(0)c{\bar{c}}V_{m}^{1}(1)c{\bar{c}}V_{r}^{\alpha}(\infty)\prod_{i=1}^{L}\intV_{n_{i}}^{0}\prod_{j=1}^{M}\intV_{m_{i}}^{1}{\frac{1}{R!}}(Q_{-})^{R}\rangle
\frac{{f^{\prime}}_{2}\left(x_{2}-\frac{2i\gamma_{2}\hbar}{\theta}x_{1}\right)}{f_{2}\left(x_{2}-\frac{2i\gamma_{2}\hbar}{\theta}x_{1}\right)}=\frac{\left(\frac{5}{4}-\frac{\gamma_{1}\gamma_{2}\hbar^{2}}{\theta^{2}}\right)x_{1}-i\left(\frac{3\theta}{8\gamma_{2}\hbar}+\frac{\gamma_{1}\hbar}{2\theta}\right)x_{2}}{\frac{2i\gamma_{1}\gamma_{2}\hbar^{2}}{\theta}-\frac{i\theta}{2}}
\hat{M}(u)=\frac{q^{-1}u^{2}-qu^{-2}}{q-q^{-1}}\left(\begin{array}{cc}{-uH_{1}+u^{-1}H_{2}}&{H_{3}}\\{\bar{H}_{3}}&{q^{-1}u^{-1}H_{1}-quH_{2}}\\\end{array}\right)
\begin{aligned}{C_{IJK}}&{{}=C_{IJK}^{(0)}+C_{IJK}^{(B)}}\\{G_{IJKL}}&{{}=G_{IJKL}^{(0)}+G_{IJKL}^{(B)}}\\{g_{IJ}}&{{}=g_{IJ}^{(0)}+g_{IJ}^{(B)}}\\\end{aligned}
\begin{array}{rcl}{\nu}&{=}&{\left(E_{\mathrm{e-vib}}^{\prime}-E_{\mathrm{e-vib}}^{\prime\prime}\right)/hc}\\{}&{=}&{(E_{0}^{\prime}-E_{0}^{\prime\prime})/hc+(G_{q^{\prime}}^{\prime}-G_{q^{\prime\prime}}^{\prime\prime})=\nu_{e}+\nu_{v}~,}\\\end{array}
{\frac{\partial{\calO}[\tilde{\phi}(\kappa),\kappa]}{\partial\kappa}}+\int\tilde{\delta}_{i\BRS}[\tilde{\phi}(\kappa),\kappa]\Theta^{\prime}[\tilde{\phi}(\kappa)]{\frac{\delta{\calO}[\tilde{\phi}(\kappa),\kappa]}{\delta\tilde{\phi}_{i}(\kappa)}}=0,
{}_{2}F_{1}(a\;,\;b\;;\;c\;;\;z)=1+{\frac{ab}{c}}\;z+{\frac{a(a+1)b(b+1)}{c(c+1)}}\;{\frac{z^{2}}{2!}}+{\frac{a(a+1)(a+2)b(b+1)(b+2)}{c(c+1)(c+2)}}\;{\frac{z^{3}}{3!}}+\cdots
M(1,2)\equiv\frac{2}{2k+Q_{g}}\sum_{I}\left(\begin{array}{cc}{0}&{T_{I}^{(2)}}\\{T_{I}^{(2)}}&{0}\\\end{array}\right)\otimes\left(\begin{array}{cc}{0}&{T_{I}^{(1)}}\\{T_{I}^{(1)}}&{0}\\\end{array}\right)
\sum_{i=1}^{N-1}{\frac{\displaystyle(h_{N}-y_{i}^{2})^{N-1-j}\,dy_{i}}{\displaystyle\prod_{j=1}^{N-1}(v_{j}-y_{i}^{2})}}=-\delta_{j,1}\,dx\,,\quad1\leqj\leqN-1.
\tilde{G}(z,w,x_{i})=\frac{\langle\partial_{z}X(z,\bar{z})\partial_{w}\bar{X}(w,\bar{w})\sigma_{+}(x_{1})\sigma_{-}(x_{2})\sigma_{+}(x_{3})\sigma_{-}(x_{4})\rangle}{\langle\sigma_{+}(x_{1})\sigma_{-}(x_{2})\sigma_{+}(x_{3})\sigma_{-}(x_{4})\rangle}
\frac{\partial^{3}{\calF}_{0}}{\partialt_{m}\partialt_{i}\partialt_{j}}=m\sum_{l=1}^{N-1}\frac{1}{l(N-l)}\frac{\partial^{2}{\calF}_{0}}{\partialt_{m-N}\partialt_{l}}\frac{\partial^{3}{\calF}_{0}}{\partialt_{N-l}\partialt_{i}\partialt_{j}},
U^{\mu}=-\Gamma_{\rho\sigma}^{\mu}(\dot{\eta}^{\rho}\dot{\eta}^{\sigma}-\eta^{\rho}\eta^{\sigma})-2\Gamma_{\rho\sigma,\lambda}^{\mu}\dot{q}^{\rho}\eta^{\lambda}\dot{\eta}^{\sigma}-\frac{1}{2}\Gamma_{\rho\sigma,\lambda\delta}^{\mu}\dot{q}^{\rho}\dot{q}^{\sigma}\eta^{\lambda}\eta^{\delta}.
H_{c}=\frac{1}{2}\,p^{i}p^{i}-\frac{1}{2}\,d^{i}d^{i}+d^{i}\,\frac{\partialv}{\partialq^{i}}+\bar{\psi}^{i}\psi^{j}\,\frac{\partial^{2}v}{\partialq^{i}\partialq^{j}}\,.
\Gamma_{\underline{\alpha}\underline{\beta}}^{i}=\left(\begin{array}{cc}{0}&{\delta_{\beta}^{\alpha}\gamma_{qq^{\prime}}^{i}}\\{\delta_{\alpha}^{\beta}(\gamma^{i})^{q^{\prime}q}}&{0}\\\end{array}\right)\,,\quadi=1,...,D-p-1\,,
^0\langle\,m\neqp\midp\,\rangle=\{\sum_{n}\!^{\prime}\,[1+Q(H_{0}-E_{p})^{-1}QVQ]_{mn}^{-1}\frac{V_{np}}{(E_{p}-E_{n}^{0})}\}^{0}\langle\,p\midp\,\rangle
\begin{array}{c}{k_{1}^{\mu}T_{\rho\sigma\mu\nu}^{(4)abcd}(p_{1},p_{2};k_{1},k_{2})=C_{1}\gamma_{\rho\sigma\nu}^{(1)}(k_{1})+C_{2}\gamma_{\rho\sigma\nu}^{(2)}(k_{1})}\\{+C_{3}\gamma_{\rho\sigma\nu}^{(3)}(k_{1})}\\\end{array}
W_{\nu}(Z,Z^{\prime})=w_{\nu}(\zeta)=\frac{e^{-i\pi\frac{d-1}2}}{(2\pi)^{\frac{d+1}2}}(\zeta^{2}-1)^{-\frac{d-1}4}Q_{\nu-\frac12}^{\frac{d-1}2}(\zeta).
\delta{\calL}=\intd\theta^{2}\left(\sum_{i,j=1}^{3}m_{ij}Tr(\Phi_{i}\Phi_{j})+h.c\right)=\sum_{i,j=1}^{3}\left(m_{ij}\lambda_{i}\lambda_{j}+m_{ik}m_{kj}^{*}\phi_{i}^{\dagger}\phi_{j}\right),
ds^{2}={\frac{1}{4q^{2}}}\left(q{\frac{d}{dq}}\right)^{2}K\,dq^{2}+\left(q{\frac{d}{dq}}\right)K\,\left[{\frac{2}{3}}d\hat{z_{i}}d\hat{\bar{z_{i}}}-{\frac{2}{9}}|\hat{z_{i}}d\hat{\bar{z_{i}}}|^{2}\right]+{\frac{4}{9}}\left(q^{2}{\frac{d^{2}}{dq^{2}}}\right)K\,|\hat{z_{i}}d\hat{\bar{z_{i}}}|^{2}.
\tilde{E}_{n\delta}^{(i)}=\sum_{\begin{array}{c}{k_{1}p_{1}+...+k_{m}p_{m}=n}\\{0<k_{1}<...<k_{m}}\\\end{array}}\frac{\left(q^{(\alpha_{i},\alpha_{i})}-q^{-(\alpha_{i},\alpha_{i})}\right)^{\sum_{i}p_{i}-1}}{p_{1}!\;...\;p_{m}!}(E_{k_{1}\delta}^{(i)})^{p_{1}}...(E_{k_{m}\delta}^{(i)})^{p_{m}}
{\Gamma}_{\mu}=-\frac{n_{\mu}}{{\partial}_{\bot}^{2}+n^{2}{\partial}_{+}^{2}}B-\frac{{\partial}_{\mu}}{{\partial}_{\bot}^{2}+n^{2}{\partial}_{+}^{2}}\left(C-n^{2}x^{-}B-\frac{n^{2}n_{+}}{{\partial}_{\bot}^{2}+n^{2}{\partial}_{+}^{2}}{\partial}_{+}B\right)
\tilde{R}_{~b\mu\nu}^{a}(\tilde{w})\equiv\partial_{\mu}\tilde{w}_{~b\nu}^{a}-\partial_{\nu}\tilde{w}_{~b\mu}^{a}+\tilde{w}_{~c\mu}^{a}\tilde{w}_{~b\nu}^{c}-\tilde{w}_{~c\nu}^{a}\tilde{w}_{~b\mu}^{c}