harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
<\tilde{\Sigma}(k_{1},\theta_{1})\tilde{\Sigma}(k_{2},\theta_{2})>=-\frac{1}{2N}I(k_{1}^{2})\frac{D^{2}-2m}{k_{1}^{2}-4m^{2}}\delta_{12}^{5}\simeq-\frac{16i}{N}\frac{D^{2}-2m}{\sqrt{-k_{1}^{2}+4m^{2}}}\delta^{3}(k_{1}-k_{2})\bar{\delta}_{12}.
\Gamma^{+}(\partial_{+}-\frac{1}{2}A\partial_{-})\psi_{I}+\Gamma^{-}\partial_{-}\rho_{I}+\Gamma^{J}\partial_{J}(\psi_{I}+\rho_{I})=J_{I}-\frac{1}{8}\Gamma_{I}\Gamma^{J}J_{J}~,
{\frac{d^{2}x^{\rho}}{d\lambda^{2}}}+\Gamma_{\mu\nu}^{\rho}{\frac{dx^{\mu}}{d\lambda}}{\frac{dx^{\nu}}{d\lambda}}+{\frac{{\calW}^{\prime}}{\calW}}{\frac{dx^{\rho}}{d\lambda}}{\frac{dy}{d\lambda}}=0,
ds_{BTZ}^{2}=-{\frac{(\rho^{2}-\rho_{+}^{2})(\rho^{2}-\rho_{-}^{2})}{\rho^{2}R^{2}}}dt^{2}+\rho^{2}(d\varphi-{\frac{J}{2\rho^{2}}}dt)^{2}+{\frac{\rho^{2}R^{2}}{(\rho^{2}-\rho_{+}^{2})(\rho^{2}-\rho_{-}^{2})}}d\rho^{2}.
\hat{\nu}(\hat{x})=e^{m\chi}\hat{n}(\hat{x})=\left(\frac{\alpha+\hat{x}}{\alpha-\hat{x}}\right)^{\frac{\zetam}{2\alpha(\alpha^{2}-1)}}\left(\frac{\hat{x}-1}{1+\hat{x}}\right)^{\frac{\zetam}{2(\alpha^{2}-1)}}\hat{n}(\hat{x})
\begin{array}{rcl}{Y^{k}}&{=}&{\alpha^{-\frac{k}{2}}S(A)w^{-k}Aw^{k}}\\{}&{=}&{\alpha^{-\frac{k}{2}}<(w^{-k})_{(1)},A>(w^{-k})_{(2)}w^{k},}\\\end{array}
\gamma_{b}^{a}\left(\partial\bar{A}^{b}-\theta_{c}^{b}\bar{\partial}A^{c}+\varphi_{cd}^{b}A^{c}\bar{A}^{d}+\mu_{ic}^{b}\bar{\partial}x^{i}A^{c}+\nu_{ic}^{b}\partialx^{i}\bar{A}^{c}+\xi_{ij}^{b}\partialx^{i}\bar{\partial}x^{j}+\rho_{i}^{b}\partial\bar{\partial}x^{i}\right)=0\,\,\,.
\mathcal{H}=\left(a_{i}^{\dagger}\,,\,b_{j}\right)\left(\begin{matrix}{\mathcal{E}_{il}}&{\mathcal{F}_{jl}^{\dagger}}\\{\mathcal{F}_{im}}&{-\mathcal{E}_{jm}^{T}}\\\end{matrix}\right)\left(\begin{matrix}{a_{l}}\\{b_{m}^{\dagger}}\\\end{matrix}\right).
X_{nk\pm}^{(1)l}(t)=\mpv_{k-,n-}^{-}E_{\pm\omega_{lk}^{(0)}+\Omega+\omega_{ln}^{(0)}}^{\mp\omega_{lk}^{(0)}}\mpv_{k-,n-}^{+}E_{\pm\omega_{lk}^{(0)}-\Omega+\omega_{ln}^{(0)}}^{\mp\omega_{lk}^{(0)}}
(x_{\alpha\dot{\alpha}})=\left(\begin{array}{cc}{z_{2}}&{z_{1}}\\{-\bar{z}_{1}}&{\bar{z}_{2}}\\\end{array}\right),\qquad(\bar{x}^{\dot{\alpha}\alpha})=\left(\begin{array}{cc}{\bar{z}_{2}}&{-z_{1}}\\{\bar{z}_{1}}&{z_{2}}\\\end{array}\right).
S_{eff}=\intd^{3}x\,\left\{\frac{|m|}{16\pi}\,\mathrm{tr}\,(\partial_{\mu}\hat{n})^{2}+i\frac{\mathrm{sgn}(m)}{4\pi}\,\epsilon^{\mu\nu\lambda}\,a_{\mu}\,\partial_{\nu}a_{\lambda}\right\},
ds^{2}=-\left(1+\frac{\tilde{r}^{2}}{\ell^{2}}\right)d\tilde{t}^{2}+\left(1+\frac{\tilde{r}^{2}}{\ell^{2}}\right)^{-1}d\tilde{r}^{2}+\tilde{r}^{2}d\Omega_{d-1}^{2}\,.
\Gamma_{\mathrm{gWZW}}[v,m]=\frac{1}{2}\Gamma_{\mathrm{LR}}[v=0,U=\mu]=\frac{1}{2}\left\langle-\frac{1}{3}(\mu^{-1}d\mu)^{3}\right\rangle=\frac{1}{2}\left\langleh_{\mathrm{WZW}}(\mu_{1},\mu_{2})d\mu^{2}\right\rangle\,,
Z(s)=\int{\calD}\psi{\calD}\bar{\psi}{\calD}b_{\mu}~F(\partialb)\delta(\varepsilon^{\mu\nu\rho}f_{\nu\rho}(b))~e^{-\intd^{3}x~\bar{\psi}(\partial\!\!\!/+i(s\!\!\!/+b\!\!\!/)+m)\psi}\;.
\Gamma_{\nu}=\varepsilon^{\mu,[\mu\nu]}+\varepsilon^{[\mu\nu],\mu}+\varepsilon^{\nu,0}+\varepsilon^{o,\nu}+\varepsilon^{\widetilde{\nu},\widetilde{0}}+\varepsilon^{\widetilde{0},\widetilde{\nu}}+\frac12e_{\mu\nu\rho\omega}\left(\varepsilon^{\widetilde{\mu},[\rho\omega]}+\varepsilon^{[\rho\omega],\widetilde{\mu}}\right),
\tau_{0}=\frac12\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\,,\quad\tau_{1}=\frac12\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)\,,\quad\tau_{2}=-\frac12\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,.
g_{\mu\nu}\equiv{\frac{\rho_{0}}{c_{0}}}\left[\begin{matrix}{-(c_{0}^{2}-v_{0}^{2})}&{\vdots}&{-v_{0}^{j}}\\{\cdots\cdots\cdots\cdots}&{\cdot}&{\cdots\cdots}\\{-v_{0}^{i}}&{\vdots}&{\delta_{ij}}\\\end{matrix}\right].
Q_{a,b}(\mu,\nu;N)=p_{a,b}(\mu,\nu;0;N)+\sum_{\lambda=1}^{\infty}(-1)^{\lambda}p_{a,b}(\mu,\nu;\lambda;N)+\sum_{\lambda=1}^{\infty}(-1)^{\lambda}m_{a,b}(\mu,\nu;\lambda;N).
\lbraceA,B\rbrace=\partial_{\xi}A\partial_{t}B-\partial_{t}A\partial_{\xi}B-(-1)^{p(A)}(\sum_{i=1}^{N}\partial_{\theta_{i}}A\partial_{\bar{\theta}_{i}}B+\partial_{\bar{\theta}_{i}}A\partial_{\theta_{i}}B).
\mathrm{Im}\Bigl[V_{\nu}^{*}\tilde{J}^{\nu}\Bigr]=\frac{1}{2}\partial^{\nu}\Biggl[(b_{1}-1)W^{\mu}V_{(\nu}V_{\mu)}^{*}+2W_{\nu}V_{\mu}^{*}V^{\mu}\Biggr].
D(\frac\phix)|_{(\vec{z}_{i},t^{*})}=\left|\begin{array}{ccc}{-\sum_{j=2}^{3}\phi_{j}^{1}f_{1}^{j}}&{\phi_{2}^{1}}&{\phi_{3}^{1}}\\{-\sum_{j=2}^{3}\phi_{j}^{2}f_{1}^{j}}&{\phi_{2}^{2}}&{\phi_{3}^{2}}\\{\phi_{1}^{3}}&{\phi_{2}^{3}}&{\phi_{3}^{3}}\\\end{array}\right|_{(\vec{z}_{i},t^{*})}=0
T_{\mu\theta}^{\Psi}=\frac{e^{\nu/2}|\chi(r)|^{2}}{2}\left(m-e^{-\lambda/2}\frac{\sqrt{\varphi}}{2}\left(\nu^{\prime}-\frac{\varphi^{\prime}}{\varphi}\right)\right)\overline{\psi}\Gamma_{\hat{\mu}}\psi
S=\intd^{4}x\sqrtg\left(-{\frac{M_{\mathrm{P}}^{2}}{16\pi}}{\calR}+{\frac{1}{2}}f_{0}^{2}\,(\partial_{\mu}\theta)^{2}\right)-{\frac{M_{\mathrm{P}}^{2}}{16\pi}}\int_{\partialV}d^{3}S(K-K_{0})\.
{\calV}\equiv\left(\begin{array}{c}{U_{\alpha}^{T}}\\{\bar{U}^{\alpha\,T}}\\\end{array}\right)\equiv\left(\begin{array}{cc}{e_{\alpha}^{A}}&{h_{A\alpha}}\\{g^{\alpha\bar{\beta}}\bar{e}_{\bar{\beta}}^{A}}&{g^{\alpha\bar{\beta}}\bar{h}_{A\bar{\beta}}}\\\end{array}\right)\,
{\bfP}={\bfp},\quad{\bfM}={\bfp}\times{\bfx}+\sum_{i=1}^{N}{\bfp}_{i}\times{\bfe}_{i}={\bfp}\times{\bfx}+\sum_{i=1}^{N}{{\bfp}^{\bot}}_{i}\times{\bfe}_{i}-\frac12\sum_{i,j=1}^{N}{\phi}_{ij}{\bfe}_{i}\times{\bfe}_{j}.
{\calJ}_{\mu\nu\rho}^{(3)}=C^{ij}\sum_{\mathrm{symm}}\left(3\partial_{\mu}\varphi^{i}\partial_{\nu}\partial_{\rho}\varphi^{j}+\delta_{\mu\nu}\partial_{\alpha}\partial_{\rho}\varphi^{i}\partial_{\alpha}\varphi^{j}-\frac{1}{3}\varphi^{i}\partial_{\mu}\partial_{\nu}\partial_{\rho}\varphi^{j}\right).
\hat{\calL}_{0}=-\intd^{\,3}u\,d^{\,3}v\sum_{\tau=\pm}\hat{\phi}^{(\tau)\dagger}({\bfx};x_{0})\hat{\tau}_{3}D_{\tau}(x;{\bfu},{\bfv})\hat{\phi}^{(\tau)}({\bfv};x_{0}),
\left(-\frac{1}{4}i\lambda\right)^{2}\int_{(\infty)}\frac{d^{4}p^{\mu}}{(2\pi)^{4}}\,\left(\frac{i}{p^{2}-m^{2}+i\epsilon}\right)\,\left(\frac{i}{(p+k)^{2}-m^{2}+i\epsilon}\right)\,
\Phi_{n+1}=\frac{-1}{1-\Phi_{n}^{2}}\left[\frac{d\Phi_{n}}{dr}-\frac{n}{r}\Phi_{n}\right]=\Phi_{n-1}-\frac{2}{1-\Phi_{n}^{2}}\frac{d\Phi_{n}}{dr},\;\;\;\;n\geq1.
{\calL}=\partial^{\mu}\varphi^{\ast}\partial_{\mu}\varphi+ieA^{\mu}\left(\varphi^{\ast}\partial_{\mu}\varphi-\varphi\partial_{\mu}\varphi^{\ast}\right)+e^{2}A^{\mu}A_{\mu}\varphi\varphi^{\ast}-m^{2}\varphi^{\ast}\varphi,
\Theta~\equiv~\frac{\mathrm{proper~length}}{\mathrm{proper~radius}}~=~\frac{\int_{t_{1}}^{t_{2}}{\sqrt{g_{\tau\tau}}~d\tau}}{\int_{r_{+}}^{r}\sqrt{g_{rr}}~dr}~=~(NN^{\prime})|_{r_{+}}(t_{2}-t_{1}),
\frac{(p+k_{3})_{\alpha}(p+k_{13})_{\beta}(p+k)_{\gamma}}{p^{4}(p+k)^{2}}-\frac{p_{\alpha}p_{\beta}p_{\gamma}p\left(k_{1}+2k_{3}\right)}{p^{6}(p+k)^{2}}+\ldots
S_{g}=-M_{Pl}^{2}\intR\sqrt{-g}d^{4}x=\eta_{ij}M_{Pl}^{2}\int\left(\Box\phi^{i}\Box\phi^{j}-\partial_{\alpha}\partial_{\beta}\phi^{i}\partial^{\alpha}\partial^{\beta}\phi^{j}\right)\sqrt{-g}d^{4}x~,
v_{\Lambda\Omega}=\frac{1}{2}\mathrm{{sgn}}(\sigma_{R_{ij}^{(\ell)}\Lambda}^{a})\,v_{\Lambda_{ij}\left(R_{ij}^{(\ell)}\Lambda\right)_{ij}}n_{\omega^{s}}(\Lambda_{i}-\ell)\bigl(n_{\omega^{s}}(\Lambda_{i}-\ell)-1\bigr)\,,
\int_{0}^{\infty}db\int_{0}^{\infty}dr\:\frac{b^{N-1}r^{N}}{\sqrt{b^{2}+r^{2}}^{\,N-1}}\,e^{-2b^{2}r^{2}}=\frac{\pi^{1/2}\,\Gamma(N/2-1)\Gamma(N/4+1/2)}{2^{3N/4+3/2}\Gamma(N/2-1/2)},
\Gamma_{\underline{l}\underline{n}}^{\underline{p}}(x)\equiv(\nabla_{\underline{l}}E_{\underline{b}}^{\underline{p}})E_{\underline{n}}^{\underline{b}}=\left(\partial_{\underline{l}}E_{\underline{b}}^{\underline{p}}-\Omega_{\underline{l}\underline{b}}^{~~\underline{c}}E_{\underline{c}}^{\underline{p}}\right)E_{\underline{n}}^{\underline{b}}.
\left(M_{b}^{\,\,\,\,a}\right)=\left(\begin{array}{cccc}{S\left(a\right)a}&{S\left(a\right)b}&{S\left(c\right)a}&{S\left(c\right)b}\\{S\left(a\right)c}&{S\left(a\right)d}&{S\left(c\right)c}&{S\left(c\right)d}\\{S\left(b\right)a}&{S\left(b\right)b}&{S\left(d\right)a}&{S\left(d\right)b}\\{S\left(b\right)c}&{S\left(b\right)d}&{S\left(d\right)c}&{S\left(d\right)d}\\\end{array}\right).
\Phi_{\ell}=\left\{\begin{array}{lll}{\tau_{0}=-9.06\times10^{-12},}&{\tau_{1}=0.344976,}&{\tau_{2}=6.77\times10^{-12},}\\{\tau_{3}=-0.0703883,}&{\tau_{4}=-2.79\times10^{-12},}&{\tau_{5}=0.00985764,}\\{e_{0}=0.0752118,}&{a_{0}=0.0348621,}&{f_{0}=0.00760330,}\\{j_{0}=-0.0146609,}&{|\mathrm{other~14~modes}|<10^{-12}}&{}\\\end{array}\right\}.
\begin{array}{l}{\mathrm{A}1)\qquad\qquadh_{1}=h_{2}=h_{3}=0\,,\nonumber}\\{\mathrm{A}2)\qquad\qquadh_{1},\,h_{2}\not=0,\\\h_{3}=0\,,\nonumber}\\{\mathrm{A}3)\qquad\qquadh_{1}=h_{2}=0,\\\h_{3}\not=0\,,\nonumber}\\{\mathrm{A}4)\qquad\qquadh_{1},\,h_{2},\,h_{3}\not=0\,.}\\\end{array}
F(x)G(x)=\left[\sqrt{x}\sum_{k=0}^{n_{c}}(-1)^{k}s_{2k}(m)x^{n_{c}-k}\right]^{2}+\left[\sum_{k=0}^{n_{c}}(-1)^{n_{f}-k}s_{2k+1}(m)x^{k}\right]^{2}.
{\frac{\tilde{z}}{2\beta}}-d+{\frac{\zeta}{2\beta}}\;=\;2\left[R_{n}(\zeta)+dQ_{n}(\zeta)\right]+(d-2)\sum_{k=0}^{n}{\frac{(-\zeta)^{k}}{k!}}\phi_{n}^{(k)}(-2\tilde{z})\;,
\Gamma_{a-c}^{i}(\vec{p},-\vec{p})=\varepsilon_{\beta}^{*}(-\vec{p},s)\Gamma_{a+b+c}^{i\alpha\beta}\varepsilon_{\alpha}(\vec{p},s)=\frac{3e^{3}}{8\pi\theta}\epsilon^{ij}q_{j}.
\gamma_{\beta_{L}}=\left(\eta_{\alpha_{L-1}}^{*}Z_{\;\;\beta_{L}}^{\alpha_{L-1}}+\alpha_{2}\Phi_{\alpha_{L-2}}^{*}\Phi_{\alpha_{0}}^{*}C_{\;\;\beta_{L}}^{\alpha_{L-2}\alpha_{0}}+\eta_{\alpha_{L+1}}^{*}A_{\beta_{L}}^{\;\;\alpha_{L+1}}\right),
\{q^{k},q^{l}\}^{\mathrm{red}}=0,\quad\{p^{k},q^{l}\}^{\mathrm{red}}=\delta^{kl},\quad\{p^{k},p^{l}\}^{\mathrm{red}}=-s\varepsilon^{klm}\frac{q^{m}}{r^{3}},
{\frac{1}{6}}(B^{\prime})^{2}-{\frac{1}{3}}B^{\prime\prime}-{\frac{1}{6}}A^{\prime}B^{\prime}\leqA^{\prime\prime}\leq-{\frac{1}{2}}(B^{\prime})^{2}+B^{\prime\prime}-{\frac{3}{2}}A^{\prime}B^{\prime}+2(A^{\prime})^{2}
<M>=2V\int_{}^{}\frac{e^{\beta(\mu-\frac{p^{2}}{2m})}(1+e^{\beta(\mu-\frac{q^{-2}p^{2}}{2m}})d^{3}p}{(2\pi\hbar)^{3}(1+2e^{-\beta(\frac{p^{2}}{2m}-\mu)}+e^{-\beta(\frac{p^{2}}{2m}(q^{-2}+1)-2\mu)})}.
Tr~(T_{x})=\frac{1}{4}\int_{\Theta}d\theta\int_{K_{1}}d\mu_{l}(k)~{\midk_{22}\mid}^{-2\sigma-1}~x(\underline{{\theta}}^{-1}~\underline{{k}}~\underline{{\theta}})~+~\frac{1}{4}\int_{\Theta}d\theta\int_{K_{2}}d\mu_{l}(k)~{\midk_{22}\mid}^{-2\sigma-1}~x(\underline{{\theta}}^{-1}~\underline{{k}}~\underline{{\theta}}).
4\partial_{+}\partial_{-}\rho-e^{2\rho}\Omega^{-4/3}+\partial_{+}f\partial_{-}f+\partial_{+}f_{1}\partial_{-}f_{1}+\partial_{+}f_{2}\partial_{-}f_{2}+\frac{1}{2}\partial_{+}f_{1}\partial_{-}f_{2}
{\tilde{\varphi}}^{\prime\prime}+{\frac{1}{f(r)}}\left({\frac{f(r)}{r}}+f^{\prime}(r)\right){\tilde{\varphi}}^{\prime}+{\frac{1}{f(r)}}\left[-{\frac{J\mu\omega}{r^{2}f(r)}}+{\frac{\omega^{2}}{f(r)}}+{\frac{(-M+{r^{2}/\ell^{2})}}{r^{2}f(r)}}\mu^{2}-m_{\varphi}^{2}\right]\tilde{\varphi}=0.
\frac{\delta\Sigma}{\deltac}+\varepsilon^{ab}\overline{{c}}^{a}\frac{\delta\Sigma}{\deltab^{b}}=-\partial^{2}\,\overline{{c}}-\partial^{\mu}A_{\mu}^{*}+\varepsilon^{ab}A_{\mu}^{a*}A^{b\mu}-\varepsilon^{ab}c^{a*}c^{b}+(\xi-\alpha)\varepsilon^{ab}b^{a}\overline{{c}}^{b}\;.
\mathrm{~Pole~Part~of}\intd^{d}k_{1}\,d^{d}k_{2}\,\,(A\otimesB)C=\frac{\calT}{2}\intd^{d}k_{1}\,d^{d}k_{2}(\mathrm{Tr}[A]\,\,\mathrm{Tr}[B]-\mathrm{Tr}[AB])C,
(\partial_{0}+\partial_{\phi})J_{a}(t,\phi)=0\\Rightarrow\J_{a}(t,\phi)=J_{a}(t-\phi)\\Rightarrow\J_{a}(\phi)=\frac{1}{2\piN}\sum_{m\in\cal{Z}}J_{am}e^{i\frac{m}{N}(t-\phi)}
\frac{1}{2Q\cdotk_{i}+k_{i}^{2}}=\frac{1}{2Q\cdotk_{i}}-\frac{k_{i}^{2}}{\left(2Q\cdotk_{i}\right)^{2}}+\frac{\left(k_{i}^{2}\right)^{2}}{\left(2Q\cdotk_{i}\right)^{3}}+\cdots.
\bar{\tau}_{1}\tau_{1}=\left(A-\sqrt{A^{2}-B^{3}}\right)^{1/3}+\left(A+\sqrt{A^{2}-B^{3}}\right)^{1/3}\,,\quad\bar{\tau}_{2}\tau_{2}=\frac{3B}{\bar{\tau}_{1}\tau_{1}}\,.
Z(t)=\int\prod_{i=1}^{3}\frac{dk_{i}}{(2\omega_{{\bfk}_{i}}}\frac{e^{-i(\omega_{{\bfk}_{1}}+\omega_{{\bfk}_{2}}+\omega_{{\bfk}_{3}})t}-1}{-i(\omega_{{\bfk}_{1}}+\omega_{{\bfk}_{2}}+\omega_{{\bfk}_{3}})}(2\pi)^{6}\delta({\bfk}_{1}+{\bfk}_{2}+{\bfk}_{3})\delta(0)
\kappa_{1}{}^{2}={\frac{\alpha^{2}\kappa_{2}{}^{4}(4\kappa_{2}\sqrt{M^{2}}S-4M^{2}+4m^{2}-\alpha^{2}\kappa_{2}{}^{4})}{(\sqrt{M^{2}}S-\alpha\kappa_{2}{}^{3})^{2}}}\,.
\Big|\!\Big|\!\Big|f^{(a,b)}\Big|\!\Big|\!\Big|_{b}\;:=\;\Bigg[\int\;\prod_{j=1}^{n_{b}}d^{D}z_{j}^{(b)}\;\bigg|f^{(a,b)}\bigg|^{N-1}\Bigg]^{\frac{1}{N-1}}\;,
g_{1}(r)=-\int_{r}^{\infty}dr^{\prime}\;\chi^{\mathrm{ANO}}(r^{\prime})K_{1}(\kappar^{\prime})\tilde{f}_{u}(r^{\prime}),\;\;g_{2}(r)=-\int_{r_{0}}^{r}dr^{\prime}\;\chi^{\mathrm{ANO}}(r^{\prime})I_{1}(\kappar^{\prime})\tilde{f}_{u}(r^{\prime}).
{\calL}_{eff}=-\frac14F^{a\mu\nu}F_{\mu\nu}^{a}+\frac12m^{2}A^{a\mu}A_{\mu}^{a}-\frac1{2\alpha}(\partial^{\mu}A_{\mu}^{a})^{2}+\bar{C}^{a}\partial^{\mu}(
Q_{p-1}\wedgeE^{b}\wedgeE^{a}={\frac{1}{(p+1)!}}E^{a_{0}}\wedgeE^{a_{1}}\wedge...\wedgeE^{a_{p}}\epsilon_{a_{0}a_{1}...a_{p}}e^{-{\frac{p-3}{2}}\phi}{\frac{\partial}{\partialF_{ab}}}(\sqrt{-det(\eta+F)})
S_{\phi^{2}}=\int_{x}\left(\phi_{\mu}^{\dagger}\a_{\mu\nu}^{-+}\\phi_{\nu}^{}+\phi_{\mu}^{}\a_{\mu\nu}^{+-}\\phi_{\nu}^{\dagger}+\phi_{\mu}^{}\a_{\mu\nu}^{++}\\phi_{\nu}^{}+\phi_{\mu}^{\dagger}\a_{\mu\nu}^{--}\\phi_{\nu}^{\dagger}\right),
{}_{3}F_{2}(\alpha_{1},\alpha_{2},\alpha_{3};\beta_{1},\beta_{2};z)=\sum_{n=0}^{\infty}\frac{(\alpha_{1})_{n}(\alpha_{2})_{n}(\alpha_{3})_{n}}{(\beta_{1})_{n}(\beta_{2})_{n}}\frac{z^{n}}{n!}
\lambda\Psi_{n}=L_{n}\Psi_{n},\;\;\;\lambda\tilde{\Psi}_{n+\frac{1}{3}}=\tilde{L}_{n+\frac{1}{3}}\tilde{\Psi}_{n+\frac{1}{3}},\;\;\;\lambda\hat{\Psi}_{n+\frac{2}{3}}=\hat{L}_{n+\frac{2}{3}}\hat{\Psi}_{n+\frac{2}{3}},
H_{-}\chi(r,\theta)=\left(\begin{array}{cc}{m}&{ie^{i\theta}\left(\partial_{r}+\frac{i}{r}\partial_{\theta}+\frac{eB}{2}r\right)}\\{ie^{-i\theta}\left(\partial_{r}-\frac{i}{r}\partial_{\theta}-\frac{eB}{2}r\right)}&{-m}\\\end{array}\right)\chi(r,\theta)=E\chi(r,\theta),
\mathrm{I}_{E}=\int\frac{d^{2\sigma}Q}{(2\pi)^{2\sigma}}\int\frac{d^{2\sigma}K}{(2\pi)^{2\sigma}}\,\frac{K_{\mu}}{(K^{2}+\vec{K}^{2})}\,\frac{Q_{\mu}}{(Q^{2}+\vec{Q}^{2})}~.
2\sum_{i=1}^{N}\delta\alpha_{i}\lambda_{i}=\sum_{i=1}^{N}\delta\alpha_{i}\lambda_{i}+\sum_{i=1}^{N}(\alpha_{i}-1)(\delta\lambda_{i}-\delta\lambda_{0})=\delta\left[\sum_{i=1}^{N}(\alpha_{i}-1)(\lambda_{i}-\lambda_{0})\right].
f^{\alpha_{1},\cdots,\alpha_{i}}=\sum_{(\beta_{1},\cdots\beta_{p},\gamma_{1},\cdots,\gamma_{q})}G_{\beta_{1},\cdots\beta_{p},\gamma_{1},\cdots,\gamma_{q}}^{\alpha_{1},\cdots,\alpha_{i}}(m_{1},\cdots,m_{p})n_{1}^{\beta_{1}}\dotsn_{p}^{\beta_{p}}l_{1}^{\gamma_{1}}\dotsl_{q}^{\gamma_{q}}
S_{gf}=-\frac{1}{64e^{2}}\intd^{4}x\,d^{4}\theta\,\Bigg(VD^{2}\bar{D}^{2}\Big(1+\frac{\partial^{2n}}{\Lambda^{2n}}\Big)V+V\bar{D}^{2}D^{2}\Big(1+\frac{\partial^{2n}}{\Lambda^{2n}}\Big)V\Bigg),
\left(\begin{array}{c}{\lambda_{1}^{ij}}\\{0}\\{0}\\{0}\\\end{array}\right)\;,\;\left(\begin{array}{c}{0}\\{\lambda_{2}^{ij}}\\{0}\\{0}\\\end{array}\right)\;,\;\left(\begin{array}{c}{0}\\{0}\\{\lambda_{3}^{ij}}\\{0}\\\end{array}\right)\;,
\Psi_{\tilde{p}_{i};N_{1}(k),\dots,N_{8}(k)}=exp(i\sum_{s=1}^{d}\tilde{p}^{s}q_{s})\prod_{k\in{\bfN}}(a_{1}^{\dagger}(k))^{N_{1}(k)}\cdot\dots\cdot(a_{8}^{\dagger}(k))^{N_{d-1}(k)}|0\rangle
S_{GF+FP}^{\prime}=-\intd^{4}xi\delta_{B}\left[\bar{C}^{a}\left\{D_{\mu}[a]A^{\mu}+{\frac{\alpha}{2}}B\right\}^{a}-i{\frac{\zeta}{2}}gf^{abi}\bar{C}^{a}\bar{C}^{b}C^{i}-i{\frac{\zeta}{4}}gf^{abc}C^{a}\bar{C}^{b}\bar{C}^{c}\right],
H=\intdx\left[\frac{1}{2}\left\{\pi_{1}^{2}+\pi_{\phi}^{2}+(\partial_{1}\phi)^{2}-e^{2}A_{1}^{2}\right\}+e\pi_{\phi}A_{1}+A_{0}(\partial_{1}\pi_{1}+e\partial_{1}\phi)+\lambda\pi_{0}\right]
{\frac{\partial}{\partialt}}c_{W}=-a{\frac{\partial}{\partiala}}c_{W}=2\intg_{\mu\nu}(x){\frac{\delta}{\deltag_{\mu\nu}(x)}}c=-{\frac{1}{V}}\int_{x}\int_{y}\sqrt{g(x)}\sqrt{g(y)}\langle\theta(x)\theta(y)\rangle
S_{E}\rightarrowS_{E[\partialM]}~=~\frac{1}{G}\int{\sqrtg}\left({\sqrth}R_{h}+\frac{1}{4}{\sqrth}h^{ij}\partial_{i}g_{\alpha\beta}\partial_{j}g_{\gamma\delta}\epsilon^{\alpha\gamma}\epsilon^{\beta\delta}\right)~.
\overline{{G}}_{\sigma}^{(n)}(\zeta^{\prime},x^{2}\zeta_{2n})^{\varepsilon^{\prime}\,\varepsilon_{2n}}=\overline{{G}}_{\sigma}^{(n)}(\zeta_{2n},\zeta^{\prime})^{\varepsilon_{2n}\,\varepsilon^{\prime}}\prod_{j=1}^{2n-1}\frac{\zeta_{2n}}{\zeta_{j}}.
D^{M}D_{M}\Phi=D^{M}D_{M}\phi(r)Y(y^{i}/r)=\frac{r^{2}f(r)}{R^{2}}[\partial^{2}/\partialr^{2}+(\frac{5}{r}+\frac{f^{\prime}}{f})\partial/\partialr)-\frac{12}{r^{2}f(r)}]\phi(r)Y(y^{i}/r)
Z(N_{0})=\sum_{\tiny\mathrm{triangulation}\:{\calT}}\frac{1}{\calS(T)}\int\prod_{i=0}^{N_{0}-1}\mathrm{d}X_{i}\mathrm{d}\bar{\theta}_{i}\mathrm{d}\theta_{i}e^{-S_{B}-S_{F}}\delta(X_{0})\delta(\bar{\theta}_{0})\delta(\theta_{0}),
S=\intd^{2}x\left\{{\frac12}{\partial}_{\mu}{\phi}{\partial}^{\mu}{\phi}+\frac{1}{2({\partial}_{\nu}a)({\partial}^{\nu}a)}\left[{\partial}^{\mu}a({\partial}_{\mu}{\phi}-{\epsilon}_{{\mu}{\sigma}}{\partial}^{\sigma}{\phi})\right]^{2}\right\},
\epsilon(r)={\frac{1}{2\sqrt{gr}}}e^{\frac{igt}{2}}e^{-{\frac{1}{2}}\gamma_{0}\gamma_{1}\gamma_{2}\theta}e^{-{\frac{1}{2}}\gamma_{2}\gamma_{3}\phi}e^{U+T}\Big(\sqrt{f-1}-i\gamma_{1}\sqrt{f+1}\big)(1-\gamma_{0})\epsilon_{0}
(\zeta^{2}-4\theta){\frac{d^{2}f_{n}}{d\zeta^{2}}}+2\zeta{\frac{df_{n}}{d\zeta}}+\left(\zeta^{2}-\lambda-4\theta-{\frac{4\theta\mu^{2}}{\zeta^{2}-4\theta}}\right)f_{n}=0.
\left(\begin{array}{cccccccc}{.}&{.}&{s_{1}}&{s_{2}}&{s_{3}}&{s_{4}}&{.}&{d_{1}}\\{.}&{.}&{s_{5}}&{s_{6}}&{s_{7}}&{s_{8}}&{d_{2}}&{.}\\{.}&{.}&{.}&{.}&{.}&{d_{3}}&{.}&{.}\\{.}&{.}&{.}&{.}&{d_{4}}&{.}&{.}&{.}\\{.}&{.}&{.}&{.}&{.}&{.}&{.}&{.}\\{.}&{.}&{.}&{.}&{.}&{.}&{.}&{.}\\{.}&{.}&{.}&{.}&{.}&{.}&{.}&{.}\\{.}&{.}&{.}&{.}&{.}&{.}&{.}&{.}\\\end{array}\right),
\left({\calX}{\bar{H}}-E_{i}\right)|i\,\rangle=\left[\Xi^{\dagger}{\calA}^{\dagger}H_{\mathrm{\scriptsize~I}}+\left({\calX}{\bar{H}}-E_{i}\right)\left(1-\Xi^{\dagger}{\calA}^{\dagger}\right)\right]|i\,\rangle\,.
\begin{array}{rcl}{}&{\displaystyle\vartheta_{1}(\tau,z)}&{=\sqrt{-1}\sum_{n\in{\bfZ}}(-1)^{n}q^{\frac{1}{2}(n-\frac{1}{2})^{2}}y^{n-\frac{1}{2}}}\\{}&{}&{\displaystyle=\sqrt{-1}q^{\frac{1}{8}}y^{-\frac{1}{2}}\prod_{n=1}^{\infty}(1-q^{n})(1-q^{n-1}y)(1-q^{n}y^{-1})\,,}\\\end{array}
\langle\phi^{2}(x)\rangle_{\beta,Ren.}=\langle\phi^{2}(x)\rangle_{\infty,Ren.}+\frac{1}{4\pi^{2}r^{2}}\sum_{n=1}^{\infty}\sum_{l=0}^{\infty}(2l+1)Q_{\nu_{l}-1/2}\left(1+\alpha^{2}\frac{n^{2}\beta^{2}}{2r^{2}}\right).
K_{0}^{(1)}=\sum_{n=1}^{\infty}I_{2n}=\intd^{4}\theta\frac{d^{4}k}{{(2\pi)}^{4}k^{2}}\sum_{n=1}^{\infty}\frac{1}{2n}{\Big(-\frac{g^{2}|\phi|^{2}}{K_{\Phi\bar{\Phi}}^{2}k^{2}}\Big)}^{n}
\bar{\Lambda}=8\pi\bar{G}_{N}[\frac{\sqrt{2\pi}\sigmaw-1}{\sqrt{2\pi}\sigma}V+\frac{6k^{2}}{k_{5}^{2}\bar{\sigma}^{4}}y(y-2\bar{y})+\frac{\bar{T}}{4\sqrt{2\pi}\sigma}].
\left.\mathrm{Tr}K^{(1/2)}(s)\right|_{{\calC}_{\beta}}={\frac{\beta}{2\pi}}\left.\mathrm{Tr}K^{(1/2)}(s)\right|_{R^{2}}-{\frac{\beta}{24\pi}}\left(\left({\frac{2\pi}{\beta}}\right)^{2}-1\right)~~~,
i^{3}\int_{klq}\frac{k^{\mu}k^{\nu}k^{\sigma}l^{\rho}q^{\lambda}q^{\psi}}{(k^{2}-m^{2})^{2}((k-l)^{2}-m^{2})(l^{2}-m^{2})(q^{2}-m^{2})^{2}}~.
\intd^{2}xe^{-i\beta\phi/\sqrt{4\pi}}\left(\Lambda\frac{\partial\tilde{g}}{\partial\Lambda}+2\tilde{g}(\Lambda)+\tilde{g}(\Lambda)(\frac{-i\beta}{\sqrt{4\pi}})^{2}+\pi\frac{\tilde{g}(\Lambda)^{2}}{2}\tilde{C}_{\Phi\Phi}^{\Phi}(\tilde{g}(\Lambda),\Lambda)(\frac{-i\beta}{\sqrt{4\pi}})^{2}\right)=0,\nonumber
N_{L}-1={\frac{1}{2}}(Q_{\mathrm{gr}}^{2}-Q_{\mathrm{vec}}^{2})={\frac{1}{2}}(P_{\mathrm{gr}}^{2}-P_{\mathrm{vec}}^{2})=(\vec{P}_{F}\cdot\vec{P}_{H})\,\qquadM^{2}={\frac{1}{2}}P_{\mathrm{gr}}^{2}\.
\frac{k}{2\pi}\partial_{+}gg^{-1}\equiva^{I}G^{I}\equiv\left(J^{3}\frac{\sigma^{3}}{2}+J^{+}\sigma^{+}+J^{-}\sigma^{-}\right)+\left(\frac{d-1}{2C_{\rho}}B^{a}T^{a}\right)+\left(F_{+\alpha}R^{+\alpha}+F_{-\alpha}R^{-\alpha}\right)
{\frac{1}{4\piG}}\int_{S^{2}\infty}*N=\overline{{\epsilon^{\prime}}}\left[P_{\mu}\gamma^{\mu}+{\frac{1}{2\sqrt{2}G}}e^{-\eta_{0}/2}(\overline{{V}}_{R}\overline{{L}}(Q-i\gamma^{5}P))^{\overline{{a}}}\Gamma^{\overline{{a}}}\right]\epsilon\.
_{\left(\Gamma_{j}\right)_{ab}=\left(\begin{array}{cc}{0}&{{\mathbbE}_{j}|{\mathbbE}_{j+1}}\\{-{\mathbbE}_{j}|{\mathbbE}_{j+1}}&{0}\\\end{array}\right)\;\\rightarrow\;\;\left(\Lambda_{j}\right)_{ab}=\left(\begin{array}{cc}{{\mathbbE}_{j+1}|{\mathbbE}_{j}}&{0}\\{0}&{{\mathbbE}_{j+1}|{\mathbbE}_{j}}\\\end{array}\right)}
(O^{-1})_{\mu\nu}^{\rho\sigma}={\frac{1}{2\Box}}\,\Bigl[\delta_{\mu\nu}^{\rho\sigma}-{\frac{4}{\Box}}\,\Bigl(1-{\frac{\beta}{2}}\Bigr)\,\delta_{[\mu}^{[\rho}\partial_{\nu]}\partial^{\sigma]}\Bigr]\,.
b_{\nu\sigma}=\frac{\kappa^{2}}{4}e_{a}^{\mu}e_{d}^{\rho}(\eta^{gf}\eta^{de}-\eta^{ge}\eta^{df})(\eta_{gc}\delta_{b}^{a}-\eta_{gb}\delta_{c}^{a})R_{\mu\nu}^{bc}R_{\rho\sigma}^{ef}+\mathrm{spin~terms}
{\frac{m_{i}^{(d)}}{m_{j}^{(d)}}}={\frac{\mu_{i}^{(d)}}{\mu_{j}^{(u)}}}\left[{\frac{\lambda_{j}^{(d)}\lambda_{j}^{(u)}}{\lambda_{i}^{(d)}\lambda_{i}^{(u)}}}\right]^{1/2}
S_{\mathrm{cl}}={\frac{1}{2\pi}}\intd\taud\sigma~\partial^{\alpha}X^{\mathrm{i}}\partial_{\alpha}X^{\mathrm{i}}={\frac{1}{2\pi}}\intd^{2}z(\partialZ^{i}\bar{\partial}\bar{Z}^{i}+\bar{\partial}Z^{i}\partial\bar{Z}^{i})
\delta\widetilde{Y}^{a}=\epsilon\left(x\right)\left[C\partial\widetilde{Y}^{a}-\frac{1}{2}\left(DC\right)D\widetilde{Y}^{a}+C\left(\partial\Psi\right)\widetilde{Y}^{a}-\frac{1}{2}\left(DC\right)\left(D\Psi\right)\widetilde{Y}^{a}\right],
e^{\tauL_{0}}=\left(\begin{matrix}{\strut1}&{\tau}&{\tau}\\{\strut\tau}&{1+{\frac{\tau^{2}}{2}}}&{{\frac{\tau^{2}}{2}}}\\{\strut-\tau}&{-{\frac{\tau^{2}}{2}}}&{1-{\frac{\tau^{2}}{2}}}\\\end{matrix}\right)\.
\Omega_{Q}(T_{\mathrm{BBN}})=\left[1+{\frac{1-\Omega_{Q}(T_{0})}{\Omega_{Q}(T_{0})}}\left({\frac{T_{\mathrm{eq}}}{T_{0}}}\right)^{3}\left({\frac{T_{\mathrm{BBN}}}{T_{\mathrm{eq}}}}\right)^{4}\left({\frac{T_{\mathrm{BBN}}}{T_{0}}}\right)^{-{\frac{k^{2}}{3}}}\right]^{-1},
\sigma^{2}(\sqrt{s}y,\sqrt{s}y^{\prime})=s\left((y-y^{\prime})^{2}-\fracs3R_{\mu\lambda\nu\rho}(p_{0})(y-y^{\prime})^{\mu}(y-y^{\prime})^{\nu}y^{\lambda}y^{\rho}+O(s^{3/2})\right)~~~,
{\bfT}_{l_{1}l_{2}}^{A}(x)=(-1)^{1+r_{A}}x^{r_{A}}e^{x}\frac{\Gamma(\lambda_{A}+1)}{\Gamma(\lambda_{A}+r_{A}+1)}L_{\lambda_{A}}^{r_{A}}(-x)L_{\lambda_{A}}^{r_{A}}(-x),
\left(\mu^{2}\frac{\partial}{\partial\mu^{2}}+\beta_{\rho}\frac{\partial}{\partial\rho}+\beta_{\xi}\frac{\partial}{\partial\xi}-\gamma_{\phi}\phi\frac{\delta}{\delta\phi}-\gamma_{\pi}\pi\frac{\delta}{\delta\pi}+\gamma_{\psi}\psi^{\mu\nu}\frac{\delta}{\delta\psi^{\mu\nu}}\right)V=0
\begin{array}{|c|}\hline{\vec{\Sigma}(\tau,\vec{\sigma}_{o})}\\{\vec{K}(\tau,\vec{\sigma}_{o})}\\\hline{}\\\end{array}\longrightarrow\begin{array}{|c|c|}\hline{\vec{\calQ}(\tau)}&{\Re^{\prime\,r}(\tau,\vec{\sigma}_{o})}\\{\vec{\calP}(\tau)}&{\wp^{\prime\,s}(\tau,\vec{\sigma}_{o})}\\\hline{}\\\end{array}.
F^{\mu\nu}=\left(\begin{array}{cccc}{0}&{-E^{1}}&{-E^{2}}&{-E^{3}}\\{E^{1}}&{0}&{-B_{3}}&{B_{2}}\\{E^{2}}&{B_{3}}&{0}&{-B_{1}}\\{E^{3}}&{-B_{2}}&{B_{1}}&{0}\\\end{array}\right)\;,
\tilde{C}(p)=-2\,\gamma\,g^{2}\delta_{a}^{b}\delta_{J}^{I}\int\frac{d^{4}k}{(2\pi)^{4}}d^{2}\thetad^{2}{\overline{\theta}}\frac{1}{k^{4}}\left\{\Phi_{I}^{a\dagger}(p,\theta,{\overline{\theta}}){\Phi}_{b}^{J}(-p,\theta,{\overline{\theta}})\right\}\,,
\mathrm{Dim~}{\calH}_{\mathrm{susy}}=\sum_{j=0}^{l_{1}}(2j+1)_{k}\left[\left(\begin{array}{c}{2l_{1}}\\{l_{1}-j}\\\end{array}\right)-\left(\begin{array}{c}{2l_{1}}\\{l_{1}-j-1}\\\end{array}\right)\right].
S=\frac{1}{g^{2}}\mathrm{Tr}\Bigl(\frac{1}{4}F_{jk}F^{jk}+\frac{1}{2}\sum_{\alpha}<\nabla_{j}\Psi^{\alpha},\nabla^{j}\Psi^{\alpha}>_{\calA}+V(<\Psi_{\alpha},\Psi_{\beta}>_{\calA})\Bigr)\,.
S\to{{\frac{aS+b}{cS+d}}},\\M\toM,\\g_{\mu\nu}\tog_{\mu\nu},\\{\calF}_{\mu\nu}^{i}\to(c\Psi+d){\calF}_{\mu\nu}^{i}+ce^{-2\varphi}(ML)_{ij}\tilde{\calF}_{\mu\nu}^{j},
\frac{d\lambda}{d\tau}\frac{d}{d\lambda}\left(\frac{d\lambda}{d\tau}\frac{dx^{\mu}}{d\lambda}\right)+\sum_{\alpha,\beta=0}^{3}\Gamma_{\alpha\beta}^{\mu}\frac{d\lambda}{d\tau}\frac{dx^{\alpha}}{d\lambda}\frac{d\lambda}{d\tau}\frac{dx^{\beta}}{d\lambda}=0
0=\int_{-{\infty}}^{\infty}dx^{-}\left[{\theta}_{-}^{\;\;+}\right]_{x^{+}=-{\infty}}^{x^{+}={\infty}}+\frac{1}{2}{\int}_{-{\infty}}^{\infty}dx^{+}\left[B(x)\frac{n_{-}}{m^{2}-n_{-}{\partial}_{+}^{\;\;2}}B(x)\right]_{x^{-}=-{\infty}}^{x^{-}={\infty}},