harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S=\int_{M_{D}}d^{D}x\left(\sum_{n\geq0}\left(\frac12\partial\phi_{n}\partial\phi_{n}^{*}-\frac12(m^{2}+M^{2}n^{2})\phi_{n}\phi_{n}^{*}\right)-\frac{\lambda}{3!}\sum_{n,n^{\prime}}\phi_{n}\phi_{n^{\prime}}\phi_{n+n^{\prime}}^{*}\right).
\mathbf{H}_{n}^{i}=\oint_{C_{0}}dz\,\gamma^{n}H^{i}(z)\quad,\quad\mathbf{E}_{n}^{\alpha}=\oint_{C_{0}}dz\,\gamma^{n}E^{\alpha}(z)\quad,\quadn\in\mathbf{Z}\quad,
\frac{\intdA_{\mu}e^{-\int_{0}^{1/T}F_{\mu\nu}^{2}/4}~e^{ie\int_{0}^{1/T}d\tau(\sum_{1}^{m}A_{0}(x_{j})-\sum_{1}^{n}A_{0}(y_{j}))}}{\intdA_{\mu}e^{-\int_{0}^{1/T}F_{\mu\nu}^{2}/4}}
L_{\omega}=\sum_{\omega^{\prime},\omega^{\prime\prime}=-N+1}^{N}\sqrt{|\omega^{\prime}\omega^{\prime\prime}|}C_{\omega^{\prime},\omega^{\prime\prime}}^{-\omega}\::\tilde{\alpha}_{\omega^{\prime}}\:\tilde{\alpha}_{\omega^{\prime\prime}}:+\sum_{\omega^{\prime}=-\infty}^{-N}\sqrt{|\omega^{\prime}(\omega+\omega^{\prime})|}:\tilde{\alpha}_{\omega^{\prime}}\:\tilde{\alpha}_{-\omega-\omega^{\prime}}:\,
{\calD}_{m_{1}m_{1}^{\prime}}^{j_{1}}(X){\calD}_{m_{4}m_{4}^{\prime}}^{j_{4}}(X)C_{\sigma\sigma^{\prime}}\left(\begin{array}{lll}{j_{1}}&{j_{2}}&{j}\\{m_{1}^{\prime}}&{m_{2}}&{\sigma^{\prime}}\\\end{array}\right)\left(\begin{array}{lll}{j_{3}}&{j_{4}}&{j}\\{m_{3}}&{m_{4}^{\prime}}&{\sigma}\\\end{array}\right)\frac{\mathrm{e}^{-m|x-y|}}{\left(\frac{2\pi}{m}|x-y|\right)^{3/2}},
\delta_{ij}=|\alpha_{i}|^{2}\delta_{ij}+\alpha_{i}\beta_{j}b_{i}b_{j}+\beta_{i}^{*}\alpha_{j}^{*}b_{i}^{\dagger}b_{j}^{\dagger}+\beta_{i}^{*}\beta_{j}b_{i}^{\dagger}b_{j}+q\alpha_{i}\alpha_{j}^{*}b_{j}^{\dagger}b_{i}.
<n_{+},n_{-}|z_{+},z_{-}>=\frac{1}{\sqrt{n_{+}!n_{-}!}}\,z_{+}^{n_{+}}\,z_{-}^{n_{-}}\,e^{-\frac{1}{2}\left[|z_{+}|^{2}+|z_{-}|^{2}\right]}\\\.
e^{\phi}={\frac{1}{N}}\left[(g_{f}^{2})^{\frac{1}{p-1}}\left({\frac{u}{\calR}}\right)\left({\frac{d_{p}^{1/(3-p)}}{c_{p}^{1/(p-1)}}}\right)\right]^{\frac{(p-1)(3-p)}{2(2-p)}}\,.
D\equiv{\mathrm{det}}\left(\begin{matrix}{-k^{2}}&{\omegak_{x}}&{\omegak_{y}}&{\omegak_{z}}\\{\omegak_{x}}&{k_{y}^{2}+k_{z}^{2}-\omega^{2}}&{-k_{x}k_{y}}&{-k_{x}k_{z}}\\{\omegak_{y}}&{-k_{x}k_{y}}&{k_{x}^{2}+k_{z}^{2}-\omega^{2}}&{-k_{y}k_{z}}\\{\omegak_{z}}&{-k_{x}k_{z}}&{-k_{y}k_{z}}&{k_{x}^{2}+k_{y}^{2}-\omega^{2}+{\frac{1}{\alpha}}}\\\end{matrix}\right)=0.
d_{\Lambda\Gamma\Sigma}=\frac{20}{27}\,F_{\Lambda}\,F_{\Gamma}\,F_{\Sigma}\,-\,\frac{2}{3}\,\mathcal{N}_{(\Lambda\Gamma}\,F_{\Sigma)}+\frac{8}{27}T_{ijk}\,g^{ip}g^{jq}g^{kr}\,h_{\Lambdap}\,h_{\Gammaq}\,h_{\Sigmar}
\langle\Psi|H|\Psi\rangle=\frac{{1}}{2}\frac{{g}}{N}\left[1+\frac{{\langle\phi^{2}\rangle}}{4}\right]^{2}\:\langle\pi^{2}\rangle+\frac{{1}}{2}\frac{{N}}{g}\left[1+\frac{{\langle\phi^{2}\rangle}}{4}\right]^{-2}\langle{\phi^{\prime}}^{2}\rangle,
\Theta\left(\begin{array}{c}{{\frac{n}{2p}}}\\{0}\\\end{array}\right)({\frac{z}{\tau}},-{\frac{2p}{\tau}})=\sqrt{\frac{-i\tau}{2p}}e^{\pii{\frac{z^{2}}{2p\tau}}}\sum_{\lambda=0}^{2p-1}e^{\frac{2\piin\lambda}{2p}}\Theta\left(\begin{array}{c}{{\frac{-\lambda}{2p}}}\\{0}\\\end{array}\right)(z,2p\tau)
\phi(x^{0}\,,\,\vec{x}\,)\,=\,c\,\intd^{n}x^{\prime}{\frac{(x^{0})^{n}}{\left((x^{0})^{2}\,+\,(\vec{x}\,-\,{\vec{x}}^{\prime}\,)^{2}\,\right)^{n}}}\phi_{0}({\vec{x}}^{\prime})\,\,.
A_{\mu}^{(3)}=\mp{\frac{2i}{g}}\,(H_{2}\,{\frac{v_{\mu}^{1}}{\rho_{1}}}+H_{3}\,{\frac{v_{\mu}^{2}}{\rho_{2}}})+\kappa\,(E_{21}^{\pm}\,R_{\mu}^{1}+E_{31}^{\pm}\,R_{\mu}^{2})\,\delta(R^{1}\cdotR^{2}).
\left[\kappa_{1}+\frac{\kappa_{2}}{2r^{2}}(d-3)(d-4)(1-N)\right]\left(\frac{\sigma^{\prime}}{\sigma}\right)=\frac{n_{d}}{8r}\left[\tau_{1}+\frac32\frac{\tau_{2}}{r^{2}}(d-3)(d-4)\left(\frac{w^{2}-1}{r}\right)^{2}\right]w^{2}\.
\frac{h}{r^{3}}\frac{d}{dr}(hr^{3}\frac{dR}{dr})+[\frac{(\omegar_{1}r_{K}r_{5})^{2}}{r^{6}}+\frac{\omega^{2}(r_{1}^{2}r_{5}^{2}+r_{1}^{2}r_{K}^{2}+r_{5}^{2}r_{K}^{2})}{r^{4}}]R=0.
Q=q_{0}+i\sigma_{k}q_{k}=\left(\begin{array}{cc}{q_{0}+iq_{3}}&{q_{2}+iq_{1}}\\{-q_{2}+iq_{1}}&{q_{0}-iq_{3}}\\\end{array}\right)\quad\mathrm{with}\quad|Q^{2}|:=q_{0}^{2}+\vec{q}^{2}=1.
{\mathrm{p.v.}}\int_{0}^{\infty}{F(x)\bar{J}_{\nu}(x)dx}=r_{1\nu}\left[F(z)\bar{J}_{\nu}(z)\right]+\frac{\pii}{2}\sum_{\eta_{k}=0,iy_{k}}\left(2-\delta_{0\eta_{k}}\right){\mathrm{Res}}_{z=\eta_{k}}F(z)\bar{H}_{\nu}^{(1)}(z).
{\tilde{A}_{n}}=\frac{1}{(n^{2}-1)}\,\left\{\frac{3}{2\omega_{0}}\,\sum_{i+j=n}{\tilde{A}}_{i}{\tilde{A}}_{j}+\frac{1}{2\omega_{0}^{2}}\,\sum_{i+j+k=n}{\tilde{A}}_{i}{\tilde{A}}_{j}{\tilde{A}}_{k}\right\}\,.
{\widetildeH}_{GU}=H_{c}+{\displaystyle\int}dx\left[\frac{\displaystyle\pi_{1}+(\alpha-1)\partial_{1}A_{1}}{\displaystyle{\alpha-1}}Q_{1}+\frac{1}{\displaystyle2e^{2}(\alpha-1)}(\partial_{1}Q_{1})^{2}+\frac{1}{\displaystyle2(\alpha-1)^{2}}Q_{1}^{2},\right]
\tilde{m}_{b}={\frac{\beta^{2}}{8\pi}}(1-{\frac{\tilde{m}_{0}^{2}}{\tilde{\mu}^{2}(\varphi_{0})}})[{\frac{1}{\sqrt{1-\tilde{m}_{b}^{2}}}}tan^{-1}\sqrt{{\frac{1+\tilde{m}_{b}^{2}}{1-\tilde{m}_{b}^{2}}}}-{\frac{\pi}{4}}]
S=\intd^{5}x\sqrt{g_{5}}e^{-\phi_{5}}\Bigl\{R_{5}+({\nabla}\phi_{5})^{2}-({\nabla}\sigma)^{2}-\frac{\zeta}{4}e^{2\sigma}{V}_{\mu\nu}^{2}-\frac{\zeta}{4}e^{-2\sigma}{H}_{\mu\nu}^{2}-\frac{1}{12}{H}_{5\mu\nu\lambda}^{2}\Bigr\}
S_{Q}^{(1)}=-\mu^{-\epsilon}\intd^{4}\thetad^{d}x\partial_{\alpha\dot{\alpha}}\sigma\partial^{\alpha\dot{\alpha}}\bar{\sigma}(\frac{18\xi_{1}^{2}{(4\pi)}^{4}}{Q^{4}\epsilon}+fin)\equivS_{Q_{div}}^{(1)}+S_{Q_{fin}}^{(1)}
\nabla^{2}{\bar{h}}_{\mu\nu}=-16\piG\left[T_{\mu\nu}+\langleT_{\mu\nu}\rangle_{\lambda=0}^{G}+\langleT_{\mu\nu}\rangle_{\lambda=0}^{M}+\langleT_{\mu\nu}\rangle_{a(\lambda)}+\langleT_{\mu\nu}\rangle_{b(\lambda)}\right],
\mathbf{E}(\mathbf{x})=\frac{1}{4\pi}\int\rho_{e}(\mathbf{x}^{\prime})\frac{\mathbf{r}}{r^{3}}\,d^{3}x^{\prime}\,+\frac{1}{4\pi}\int\mathbf{J}_{m}(\mathbf{x}^{\prime})\,{\bf\times}\,\frac{\mathbf{r}}{r^{3}}\,d^{3}x
{\hat{b}}_{\mu_{1}\dots\mu_{4}}^{(4)}=b_{\mu_{1}\dots\mu_{4}}^{(4)}\,,\qquad{\hat{b}}_{\mu_{1}\dots\mu_{3}6}^{(4)}=b_{\mu_{1}\dots\mu_{3}}^{(3)}\,,\qquad\mu=0,1,\dots,5\,.
\begin{array}{c}{\epsilon_{c}=k^{ab}M_{ab}+b^{a}P_{a}+f^{a}K_{a}+fD;}\\{M_{ab}=x_{a}p_{b}-x_{b}p_{a}\;;\;P_{a}=p_{a}\;;\;K_{a}=x^{2}p_{a}-2(xp)x_{a}\;;\;D=(xp).}\\\end{array}
H_{nm}^{\{{\bfR}\}}(0,0)\approx\frac{(-1)^{\frac{n+m}2}\epsilon^{-\frac{|n-m|+1}2}}{\sqrt{2\pi}}\Gamma\left(\frac{n+m+1}2\right)R_{11}^{n/2}R_{22}^{m/2}\left(r+\epsilon\right)^{\frac{n+m+1}2}.
\left(\begin{array}{c}{u_{t}}\\{v_{t}}\\\end{array}\right)={\calD}_{1}\left(\begin{array}{c}{{\frac{\deltaH_{1}^{(2)}}{\deltau}}}\\{{\frac{\deltaH_{1}^{(2)}}{\deltav}}}\\\end{array}\right)={\frac{1}{\gamma-2}}\,{\calD}_{2}\left(\begin{array}{c}{{\frac{\deltaH_{0}^{(2)}}{\deltau}}}\\{{\frac{\deltaH_{0}^{(2)}}{\deltav}}}\\\end{array}\right)
\hat{C}_{0\,0\,0}^{l_{3}l_{1}l_{2}}=\rho({l_{3},l_{1},l_{2}})\sum_{k}{\frac{(-1)^{k}}{k!}}{\frac{(l_{1}+k)!(l_{2}+l_{3}-k)!}{(l_{1}-k)!(l_{2}-l_{3}+k)!(l_{3}-k)!}},
\psi_{\pm}^{m}=u^{m(n)}\varphi_{\pm(n)}=\frac12u^{m[+2]}\varphi_{\pm}^{[-2]}+\frac12u^{m[-2]}\varphi_{\pm}^{[+2]}-u_{m}^{(i)}\varphi_{\pm}^{(i)}.
U(\tau)=\sum_{\ell=0}^{\infty}\tilde{U}^{(\ell)}(\tau)\:\nu^{\ell}=U^{(0)}(\tau)\:\left[{\calT}e^{-\frac{i\nu}{\hbar}\int_{0}^{\tau}dt\:\tilde{H}^{\prime}(t)}\right]\;.
D=(\frac{\omegar_{1}r_{5}r_{K}}{2r_{0}^{2}})^{2},C=(\frac{\omega^{2}(r_{1}^{2}r_{5}^{2}+r_{1}^{2}r_{K}^{2}+r_{5}^{2}r_{K}^{2})}{4r_{0}^{2}}).
{\frac{{\calW}_{na}\pi^{2\omega}e^{2i\pi\omega}}{g^{4}C_{F}C_{A}(2T)^{4-4\omega}(LT)^{2}}}={\frac{1}{2(\omega-1)^{2}}}+{\frac{1-\gamma}{(\omega-1)}}-1-2\gamma+\gamma^{2}+{\frac{\pi^{2}}{12}}+{\calO}(\omega-1)\,
\begin{array}{ccccc}{{\calK}}&{=}&{\left(\mathrm{i}\langle{\hat{\Omega}}\,\vert\,{\hat{\bar{\Omega}}}\rangle\right)}&{=}&{\left[\mathrm{i}\left({\bar{Y}}^{I}F_{I}-{\bar{F}}_{J}Y^{J}\right)\right]}\\\end{array}
I\rightarrow\frac{R}{2(4\pi)^{2}}\int_{1/\Lambda^{2}}^{\infty}dt\frac{1}{t^{2}}\left(e^{-t\rho^{2}}-1\right)=\frac{R}{2(4\pi)^{2}}\left[\rho^{2}\mathrm{Ei}\left(-\frac{\rho^{2}}{\Lambda^{2}}\right)+\Lambda^{2}(e^{-\rho^{2}/\Lambda^{2}}-1)\right],
{\bfM}({\calN})\equiv\left(\begin{matrix}{\mathrm{Im}\,{\calN}+(\mathrm{Re}\,{\calN})(\mathrm{Im}\,{\calN})^{-1}(\mathrm{Re}\,{\calN})}&{-(\mathrm{Re}\,{\calN})(\mathrm{Im}\,{\calN})^{-1}}\\{-(\mathrm{Im}\,{\calN})^{-1}(\mathrm{Re}\,{\calN})}&{(\mathrm{Im}\,{\calN})^{-1}}\\\end{matrix}\right),
\left.\mathrm{Tr}K^{(3/2)}\right|_{{\calC}_{\beta}\times\Sigma}=\left.\mathrm{Tr}K^{(3/2)}\right|_{{\calC}_{\beta}}\left.\mathrm{Tr}K^{(1/2)}\right|_{\Sigma}+\mathrm{Tr}\left.K^{(1/2)}\right|_{{\calC}_{\beta}}\left.\mathrm{Tr}K^{(3/2)}\right|_{\Sigma}~~~.
M\approx\left(\begin{array}{cccc}{0.67910}&{-2.82440}&{-0.08261}&{-0.26102}\\{-2.82440}&{82.15982}&{0.31499}&{1.03418}\\{-0.08261}&{0.31499}&{4.70145}&{0.02849}\\{-0.26102}&{1.03418}&{0.02849}&{-0.95150}\\\end{array}\right)
\begin{aligned}{\intd^{4}xd^{2}\theta\,W(\Phi)}&{{}=-\intd^{4}x\,\frac14D^{2}W(\Phi)\bigr|~,}\\{\intd^{4}xd^{2}\bar{\theta}\,\overline{W}(\bar{\Phi})}&{{}=-\intd^{4}x\,\frac14\bar{D}^{2}\overline{W}(\bar{\Phi})\bigr|~.}\\\end{aligned}
\begin{array}{l}{e^{2i\pin\alpha_{0}^{-}\alpha_{0}^{+}}|p^{+},p^{-},s_{0}>=|p^{+},p^{-},s_{0}>}\\{\alpha_{0}^{-}\alpha_{0}^{+}=p^{-}p^{+}=-r,\quadr=0,1,2,\cdots}\\\end{array}
\Lambda(\zeta;r)=\frac{1}{\zeta^{2}}\frac{\Theta_{q^{4}}(r\zeta^{2})\Theta_{q^{4}}(q^{2}r\zeta^{-2})}{\Theta_{q^{4}}(r\zeta^{-2})\Theta_{q^{4}}(q^{2}r\zeta^{2})}.
F(r)={\frac{(r-n)(35r^{5}+175nr^{4}+(300n^{2}+5l^{2})r^{3}+(25nl^{2}+100n^{3})r^{2}+(47n^{2}l^{2}-295n^{4})r-315n^{5}+35l^{2}n^{3})}{35(r+n)^{4}l^{2}}}
{\calL}=-\frac{1}{2}Q_{i}^{A}(x)\left[-(\nabla^{2})^{AB}\delta_{ij}+M^{2}(T)\delta^{AB}\delta_{ij}+\left(D_{i}D_{j}\right)^{AB}+2B_{ij}\varepsilon^{3AB}\right]Q_{j}^{B}(x)
X^{2}:=Y~,~~\Psi^{2}:=\chi~,~~\Psi^{0}:=\psi^{3}~,~~~\Psi^{\pm}(\equiv-\frac{1}{\sqrt{2}}(\Psi^{1}\pmi\Psi^{\rho})):=e^{\pmiH_{1}^{\prime}}~~.
[T_{\alpha},T_{\beta}]=\frac{i\hbar}{2}\left(U_{\alpha\beta}^{\gamma}T_{\gamma}+T_{\gamma}U_{\alpha\beta}^{\gamma}(-1)^{(\varepsilon_{\alpha}+\varepsilon_{\beta}+1)\varepsilon_{\gamma}}\right)+\left(\frac{i\hbar}{2}\right)^{2}[U_{\alpha\delta}^{\gamma},U_{\gamma\beta}^{\delta}](-1)^{\varepsilon_{\delta}(\varepsilon_{\beta}+1)}.
Z(q)=a\sum_{r-s\,even}\left(|\chi_{r,s}^{NS}(q)|^{2}+|\widetilde\chi_{r,s}^{NS}(q)|^{2}\right)\;+a\sum_{r-s\,odd}|\sqrt{2}\chi_{r,s}^{R}(q)|^{2}+b\,|\chi_{f}^{\widetildeR}|^{2}.
W_{\mathrm{eff}}=\intd^{4}x\left[-\frac{1}{4}F_{\mu\nu}^{2}-\frac{1}{2}m^{2}{\left(A_{\mu}-{\partial_{\mu}\phi^{2}}\right)}^{2}+i\bar{\psi}\gamma^{\mu}D_{\mu}^{+}\psi\right]+M_{1}+S_{\mathrm{gf}}
\kappa_{N\alpha}^{(+)}+e^{i\left(\phi_{(+)}-\phi_{(-)}\right)}J_{NM}\kappa_{M\alpha}^{(-)}=e^{i\left(\phi+2\phi_{(+)}\right)}\:\epsilon_{\alpha\beta}\left(\kappa_{N\beta}^{(+)}+e^{i\left(\phi_{(-)}-\phi_{(+)}\right)}J_{NM}\kappa_{M\beta}^{(-)}\right)
\chi^{i\alpha}\rightarrow\chi^{i\alpha}+\epsilong^{i\bar{j}}(\Omega_{2})_{\bar{j}\bar{k}}\bar{\chi}_{\alpha}^{\bar{k}}\qquad\qquad\bar{\chi}_{\alpha}^{\bar{i}}\rightarrow\bar{\chi}_{\alpha}^{\bar{i}}+\epsilong^{\bar{i}j}(\Omega_{2})_{jk}\chi^{k\alpha}
\left(\begin{matrix}{\Rel_{++++}+l_{++--}}&{\Iml_{++++}}&{2\Rel_{+++-}}\\{\Iml_{++++}}&{-\Rel_{++++}+l_{++--}}&{-2\Iml_{+++-}}\\{2\Rel_{+++-}}&{-2\Iml_{+++-}}&{-2l_{++--}}\\\end{matrix}\right)\,
S={\frac{\pi^{3}R^{5}}{2\kappa_{10}^{2}}}\left[\;\int_{0}^{1}\!du\,d^{4}x\,\sqrt{-g}\left({\calR}-2\Lambda\right)+2\int\!d^{4}x\,\sqrt{-h}K+a\int\!d^{4}x\,\sqrt{-h}\;\right]\,.
G(p_{1},p_{2},p_{3},p_{4})=-i{\frac{g^{2}}{4}}\sigma^{0}(p_{1})\sigma^{0}(p_{2})\sigma^{0}(p_{3})\sigma^{0}(p_{4})\Big[{\frac{1}{1+i{\frac{g^{2}}{4}}\int{\frac{d^{d}k}{(2\pi)^{d}}}\sigma^{0}(k)\sigma^{0}(k+p_{2}-p_{1})}}\Big]
\begin{array}{rcl}{e^{\frac{4}{3}\hat{\phi}}}&{=}&{-\hat{\hat{g}}_{yy}\,,}\\{}&{}&{}\\{\hat{g}_{\hat{\mu}\hat{\nu}}}&{=}&{e^{\frac{2}{3}\hat{\phi}}\left[\hat{\hat{g}}_{\hat{\mu}\hat{\nu}}-\hat{\hat{g}}_{\hat{\mu}y}\hat{\hat{g}}_{\hat{\nu}y}/\hat{\hat{g}}_{yy}\right]\,,}\\\end{array}
\big\{f,g\big\}=\Big(\frac{\partialf}{\partialx_{i}}\frac{\partialg}{\partialp_{i}}-\frac{\partialg}{\partialx_{i}}\frac{\partialf}{\partialp_{i}}\Big)+\theta\Big(\frac{\partialf}{\partialx_{1}}\frac{\partialg}{\partialx_{2}}-\frac{\partialg}{\partialx_{1}}\frac{\partialf}{\partialx_{2}}\Big).
Z={\frac{1}{4}}\sum_{r^{\prime}=1}^{2}\sum_{s^{\prime}=1}^{3}\sum_{r=1}^{h-2}\sum_{s=1}^{h-1}|\chi_{r^{\prime},s^{\prime}}^{(4)}(q)\;\chi_{r,s}^{(h)}(q)|^{2}.
\hat{\calL}_{\{1_{q}\}}^{\{0,0,0,a_{ll^{\prime}}\}-1}(z_{q})G_{N}^{\{r_{i}\}}(z_{i})=<(\frac{\partial}{\partialz_{q}}-r_{q}\cdotQ^{(1)}(z_{q}))\prod_{i}^{N}U^{r_{i}}(z_{i})>
e^{\lambda\varphi(\tau,\sigma)}=\left(\frac{1}{\sqrt{A^{\prime}(\xi_{+})}}\frac{B(\xi_{-})}{\sqrt{B^{\prime}(\xi_{-})}}-\frac{A(\xi_{+})}{\sqrt{A^{\prime}(\xi_{+})}}\frac{1}{\sqrt{B^{\prime}(\xi_{-})}}\right)^{-2\lambda}
\begin{aligned}{e^{10\lambda}}&{{}\approx2.4453}\\{e^{2\alpha}}&{{}\approx2.1367}\\{e^{2\phi}}&{{}\approx0.73162}\\{e^{2h}}&{{}\approx1.3430}\\{m^{2}R^{2}}&{{}\approx0.17815}\\\end{aligned}
I(p,\theta,\varepsilon)=i\int\frac{d^{4}k}{(\theta^{2}k_{0}^{2}-|\mathbf{k}|^{2}+i\varepsilon)[\theta^{2}(k+p)_{0}^{2}-|\mathbf{k}+\mathbf{p}|^{2}+i\varepsilon]}
\mathrm{exp}[M_{{\alpha}{\alpha}}(x-y)]=\frac{2}{m_{0}e^{\gamma}}\frac{i(-1)^{{\alpha}-1}}{x^{--}y^{-}+i{\varepsilon}(-1)^{{\alpha}-1}}e^{F_{\alpha}(|x^{--}y^{-}|)},
I_{ab}\{(N_{a}^{i},\overline{{N_{b}}}^{i-\frac{1-b_{2}}{2}})+(N_{a}^{i},\overline{{N_{b}}}^{i+\frac{1-b_{2}}{2}})+(\overline{{N_{a}}}^{i},N_{b}^{i-\frac{1+b_{2}}{2}})+(\overline{{N_{a}}}^{i},N_{b}^{i+\frac{1+b_{2}}{2}})\},
\int\delta(\theta-\theta^{i})d\theta=1\,\\int\theta\delta(\theta-\theta^{i})d\theta=\theta^{i}\,\\delta^{3}({\vec{\theta}}-{\vec{\theta}}^{i})=\delta(\theta_{a}-\theta_{a}^{i})\delta(\theta_{b}-\theta_{b}^{i})\delta(\theta_{c}-\theta_{c}^{i})
H=-\intd{\bfx}\int_{\Gamma_{+}+\Gamma_{-}}dk\int_{\Gamma_{+}^{\prime}+\Gamma_{-}^{\prime}}dk^{\prime}e^{ikx}e^{ik^{\prime}x}\frac{a(k)a(k^{\prime})}{f(-k^{2})f(-k^{2})}(k_{0}^{2}-k_{0}k_{0}^{\prime})\frac{f(-k^{2})-f(-k^{2})}{k^{2}-k^{2}}
H_{\lambda=q^{-1}}(q=e^{i\pi/N})=-\sum_{j=1}^{L}\sum_{n=1}^{N-1}\left(\frac{1}{sin(\frac{\pin}{N})}\left[\left(\Sigma_{j}^{+}\Sigma_{j+1}^{-}\right)^{n}+\left(\Sigma_{j}^{-}\Sigma_{j+1}^{+}\right)^{n}\right]+(1-\frac{2n}{N})e^{i\pin/N}Z_{j}^{n}\right)
{\frac{d\varphi}{dz}}={\frac{d\zeta}{d\varphi}}=A(\epsilon)[1-\varphi^{2}-{\frac{\epsilon}{2}}{\frac{1-{\frac{9}{4}}(\varphi-{\frac{\varphi^{3}}{3}})^{2}}{1-\varphi^{2}}}].
\Gamma_{div}^{lg}=\frac{1}{4\pi\epsilon}\intd^{d}x\,\sqrt{-g}\,\left\{\frac{24-n}{6}R-\left[8-n\Phi^{\prime}(\phi)^{2}\right]g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+2\Lambdae^{2\phi}\right\},
\intd\alpha\left|G_{\mathrm{B}}\Bigl(\alpha(u_{\mathrm{B}}-u_{\mathrm{T}})\Bigr)\right|^{2}=\intd\alpha\left|G_{\mathrm{B}}\Bigl(\alpha(u_{\mathrm{B}}^{0}-u_{\mathrm{T}}^{0}),0,0,\alpha(u_{\mathrm{B}}^{3}-u_{\mathrm{T}}^{3})\Bigr)\right|^{2}.
\int\left[{\frac{\delta^{2}G}{\delta\Theta^{2}}}\,-\,\left({\frac{\deltaG}{\delta\Theta}}\right)^{2}\,-\,\Theta\partial_{i}\partial_{i}\Theta\,-\,{\frac{1}{R}}\partial_{i}\partial_{i}R\right]dx\,=\,2E\,.
\frac{b^{2}}{K^{2}}dK^{2}+a^{2}\sigma_{1}^{2}+b^{2}\sigma_{2}^{2}+c^{2}\sigma_{3}^{2}+\left(m+\frac{1}{r}\right)[dr^{2}+r^{2}(\hat{\sigma}_{1}^{2}+\hat{\sigma}_{2}^{2})]+\frac{r}{1+mr}\hat{\sigma}_{3}^{2}
\begin{array}{lcl}{{F}_{2}}&{=}&{E\,}\\{{F}_{3}}&{=}&{\sqrt{B^{2}+{\bboxE}^{2}}\,}\\{{F}_{\pm}}&{=}&{\left\{\vert{\bbox{B+E}}\vert\pm\vert{\bbox{B-E}}\vert\right\}/2\.}\\\end{array}
y(1-y)\partial_{y}^{2}R(y)+\frac{1}{2}(2-5y)\partial_{y}R(y)+\left(\frac{\omega^{2}r_{c}^{2}}{4y}-\frac{\nu(\nu+1)}{4(1-y)}-\frac{m^{2}r_{c}^{2}}{4}\right)R(y)=0.
C^{\alpha\beta}(x,y)=\left(\begin{array}{cc}{\{\chi^{ij}(x),\chi^{mn}(y)\}}&{\{\chi^{ij}(x),\phi^{mn}(y)\}}\\{\{\phi^{ij}(x),\chi^{mn}(y)\}}&{\{\phi^{ij}(x),\phi^{mn}(y)\}}\\\end{array}\right),
\langle0|O_{i}^{V}O_{j}^{V}|0\rangle\sim\langle0|\Theta_{i}^{\partialV}\Theta_{j}^{\partialV}|0\rangle_{d-1}=\alpha_{ij}^{d}\int_{\partialV}\!\!d^{d-1}z_{1}\int_{\partialV}\!\!d^{d-1}z_{2}~\hat{n}_{1}\cdot\hat{n}_{2}\langle0|\mathrm{\large~\vartheta~}_{i}(\vec{z}_{1})\mathrm{\large~\vartheta~}_{j}(\vec{z}_{2})|0\rangle_{d-1}.
dS^{2}={\frac{1}{2f^{2}}}\left[df^{2}+(d\epsilon+\psi\,d\chi)^{2}\right]+{\frac{1}{2f}}\left(\kappa^{2}\,d\psi^{2}+{\frac{1}{\kappa^{2}}}\,d\chi^{2}\right)+{\frac{2}{3}}\,{\frac{d\kappa^{2}}{\kappa^{2}}}.
{\calL}=\frac{1}{2g}\left[\mathrm{Tr}\dot{X}_{mn}^{i}\dot{X}_{nm}^{i}+\frac{1}{2}\mathrm{Tr}\;(X_{mq}^{i}X_{qn}^{j}-X_{mq}^{j}X_{qn}^{i})(X_{nr}^{i}X_{rm}^{j}-X_{nr}^{j}X_{rm}^{i})\right]
B(\beta_{1},\ldots,\beta_{n})=\left(\begin{matrix}{0}&{\ldots}&{0}&{(-1)^{n+1}S_{n}}\\{1}&{\ddots}&{}&{(-1)^{n}S_{n-1}}\\{}&{\ddots}&{\ddots}&{\vdots}\\{0}&{\ldots}&{1}&{S_{1}}\\\end{matrix}\right).
\vec{x}_{23\,k}^{\prime}=\frac{\vec{x}_{21\,k}}{|\vec{x}_{21}|^{2}}-\frac{\vec{x}_{31\,k}}{|\vec{x}_{31}|^{2}}\,,\quad\vec{x}_{24\,k}^{\prime}=\frac{\vec{x}_{21\,k}}{|\vec{x}_{21}|^{2}}-\frac{\vec{x}_{41\,k}}{|\vec{x}_{41}|^{2}}\,.
I_{red4D}=\intd^{4}x[\frac{1}{4}F^{mn}F_{mn}-\frac{1}{2}F^{mn}[\partial_{m}A_{n}-\partial_{n}A_{m}]+\frac{1}{2}V^{m}V_{m}-V^{m}[\partial_{m}\phi-\muA_{m}]].
\widetilde{\calA}({\calC}_{\;\mathrm{O,P}})[S]=\int_{\mathrm{O}}^{\mathrm{P}}G_{A}(\mathrm{O},p(t^{\prime}),{\calC})F_{\mu\nu}(t^{\prime})T^{\mu}(t^{\prime})S^{\nu}(t^{\prime})G_{A}(\mathrm{O},p(t^{\prime}),{\calC}^{-1})dt^{\prime}
\begin{array}{ll}{D+\alpha\tilde{D}+\frac{e}{2}E_{I}|\sigma_{I}|^{2}+k=0,}\\{\tilde{D}+\alphaD+\frac{q}{2}Q_{i}|\phi_{i}|^{2}+\eta=0,}\\{\bar{G}_{i}=-\frac{\partialW}{\partial\bar{\phi}_{i}}=-m\phi_{i},}\\{\bar{F}_{I}=-\frac{\partialW}{\partial\bar{\sigma}_{I}}=-\tilde{m}\sigma_{I}.}\\\end{array}
\int_{0}^{T}d\sigma^{\prime}\left[\delta(\sigma-\sigma^{\prime})\frac{\partial^{2}}{\partial\sigma^{2}}-{\frac{1}{L}}B(\sigma^{\prime},\sigma)\right]z_{\mu}(\sigma^{\prime})=[\delta(\sigma-\tau_{2})-\delta(\sigma-\tau_{1})](\tau_{1}-\tau_{2})\frac{u_{\mu}}{L}
g^{-1}dg=\sum\omega_{n}^{L^{(\pm)}}L_{n}^{(\pm)}+\sum\omega_{r}^{G^{\pm(\pm)}}G_{r}^{\pm(\pm)}+\sum\omega_{n}^{U^{(\pm)}}U_{n}^{(\pm)}+\ldots={\tilde{g}}^{-1}d{\tilde{g}}\in{\widetilde{\calH}}\;,
[\rho({\bfx^{\prime\prime}},t),\rho({\bfx^{\prime\prime\prime}},t)]=4\pi^{2}\mu\Omega^{-1}i\delta({\bfx^{\prime\prime}},{\bfx^{\prime\prime\prime}})\epsilon_{lj}\frac{({\bfx}-{\bfx^{\prime\prime}})^{l}}{|{\bfx}-{\bfx^{\prime\prime}}|^{2}}\frac{({\bfx}-{\bfx^{\prime\prime\prime}})^{j}}{|{\bfx}-{\bfx^{\prime\prime\prime}}|^{2}}
\Big((\alpha|\alpha)-2\,\mathrm{ht}\,(\alpha)\Big)c_{\alpha}=\sum_{\stackrel{\beta^{\prime},\beta^{\prime\prime}>0}{\beta^{\prime}+\beta^{\prime\prime}=\alpha}}(\beta^{\prime}|\beta^{\prime\prime})c_{\beta^{\prime}}c_{\beta^{\prime\prime}}
\bar{\Phi}_{A}(\theta)=\bar{\phi}_{A}-\theta^{a}\phi_{Aa}^{*}-\theta^{2}\eta_{A},\qquad\frac{\delta}{\delta\bar{\Phi}_{A}(\theta)}=\theta^{2}\frac{\delta}{\delta\bar{\phi}_{A}}+\frac{\delta}{\delta\phi_{Aa}^{*}}\theta_{a}-\frac{\delta}{\delta\eta_{A}}.
{s_{\rho}}_{(2m_{s}-2)}=c^{i_{1}}\dotsc^{i_{2m_{s}-3}}\rho(X_{i_{1}})\dots\rho(X_{i_{2m_{s}-3}})-\frac{1}{(2m_{s}-2)!}{\Omega_{i_{1}\ldotsi_{2m_{s}-2}}^{(2m_{s}-1)}}^{\sigma}c^{i_{1}}c^{i_{2}}\ldotsc^{i_{2m_{s}-2}}\pi_{\sigma}\quad.
H_{\mathrm{spin\frac{1}{2}}}^{(1)}[N]=-\frac{i}{2\kappa^{2}}\intd^{3}xN(x)\left(\epsilon^{ijk}\epsilon^{abc}4\frac{\{A_{a}^{i}(x),V(x,\delta)\}\{A_{b}^{j}(x),V(x,\delta)\}}{\sqrt{q}(x)}\right)[\pi^{T}\tau_{k}{\calD}_{c}\xi-c.c.]
<A_{2}|\sum_{k_{1},k_{2}=-\infty}^{+\infty}\hat{U}(k_{1},k_{2})|A_{1}>=\sum_{k_{1},k_{2}=-\infty}^{+\infty}\,\frac{\sqrt{k}}{2\pi}\,e^{ikk_{1}A_{2}}\,e^{-ikk_{2}A_{1}}\,e^{-\frac{ik}{2\pi}A_{1}A_{2}}\\\.
M_{1}=\left(\begin{matrix}{0}\\{mG_{1}e_{R}}\\{nG_{2}u_{R}}\\{nG_{3}d_{R}}\\\end{matrix}\right)~,~~M_{2}=\left(\begin{matrix}{0}\\{mG_{4}\mu_{R}}\\{nG_{5}c_{R}}\\{nG_{6}s_{R}}\\\end{matrix}\right)\,.
\tilde{\Theta}=k^{2}(\Theta-\lambda[\varphi]\Theta_{0})+\left(\frac{2}{\pi}\right)^{1/2}\frac{k(\lambda[\varphi]k^{2\mho}-\varphi(0)p(k^{2})+\varphi^{\prime}(0)q(k^{2}))}{(p(k^{2})^{2}+k^{2}q(k^{2})^{2})^{1/2}}.
\Psi_{i,g_{1}}^{[L][M]}(x_{1})\Psi_{j,g_{2}}^{[M][N]}(x_{2})\=\\sum_{k}(x_{1}-x_{2})^{h_{i}+h_{j}-h_{k}}\Psi_{k,g_{12}}^{[L][N]}(x_{2})C_{(i,g_{1})(j,g_{2})(k,g_{12})}^{[L][M][N]}+\dots
{\calR}\left(\lambda,\mu\right)=\left(\begin{array}{cccc}{\frac{\lambda^{2}q-\mu^{2}q^{-1}}{\lambda^{2}-\mu^{2}}}&{0}&{0}&{0}\\{0}&{1}&{\frac{2\lambda\mu}{\lambda^{2}-\mu^{2}}\left(q-q^{-1}\right)}&{0}\\{0}&{\frac{2\lambda\mu}{\lambda^{2}-\mu^{2}}\left(q-q^{-1}\right)}&{1}&{0}\\{0}&{0}&{0}&{\frac{\lambda^{2}q-\mu^{2}q^{-1}}{\lambda^{2}-\mu^{2}}}\\\end{array}\right).
i\left<{}^{\ast}\!\bar{B}_{\mu\nu}^{i}\bar{A}_{\xi}^{a}\bar{A}_{\eta}^{b}\right>_{\mathrm{counter}}=-2\sigma_{\mathrm{R}}g_{\mathrm{R}}\delta_{\sigma}I_{\mu\nu,\rho\sigma}f^{iab},
Q_{\hat{\beta}}^{(1)}=-i\Theta_{\hat{\alpha}A}\gamma_{\hat{\beta}\hat{\alpha}}^{t}(e_{A}E_{t}\partial_{r}+\frac{1}{r}E_{t}M_{AB}e_{B}+\frac{1}{r}e_{A}M_{ts}E_{s})+\frac{1}{2}(\vec{y}_{s}\times\vec{y}_{t})\gamma_{\hat{\beta}\hat{\alpha}}^{st}\vec{\Theta}_{\hat{\alpha}}\quad,
E_{C}^{\mathrm{fixed}}(R\to\infty,T)=\frac{RT}{2}\sum_{n=-\infty}^{+\infty}\Omega_{n}=2\piRT^{2}\sum_{n=1}^{\infty}n=2\piRT^{2}\zeta(-1)=-\frac{\piRT^{2}}{6}\,{,}
{\calR}\,=\,\bigl(\sum_{n=0}^{l-1}\frac{q^{-\frac{n(n-1)}2}}{[n]!}e^{n}\otimesf^{n}\bigr)\bigl(\frac1l\sum_{i,j\in{\bfZ}/l}q^{-ij}k^{j}\otimes\alpha^{i}\bigr)
\langleB_{\mathrm{FF}}(y)|q^{L_{0}^{\mathrm{FF}}+{\tilde{L}}_{0}^{\mathrm{FF}}-{\frac{c_{\mathrm{FF}}}{12}}}|B_{\mathrm{FF}}(y)\rangle={\tilde{q}}^{-1/12}\prod_{n=1}^{\infty}\left(1-{\tilde{q}}^{2n}\right)^{-1},
W=-ic(\omega-\omega^{-1})\,{\frac{\eta_{1}+\eta_{2}^{1/N_{1}}+\eta_{3}^{1/N_{1}}+\eta_{1}^{-1}+\eta_{2}^{-1/N_{1}}+\eta_{3}^{-1/N_{1}}}{(1-2\xi)\left(\eta_{1}-\eta_{1}^{-1}\right)+(1+\xi)\left(\eta_{2}^{1/N_{1}}+\eta_{3}^{1/N_{1}}-\eta_{2}^{-1/N_{1}}-\eta_{3}^{-1/N_{1}}\right)}}\.
\begin{array}{c}{a^{A*}=a^{A}~~~~b^{A*}=b^{A}~~~~\omega_{AB}{}^{*}=\omega_{AB}~~~~\lambda^{*}=\lambda~~~~T^{\dagger}=-T}\\{{}}\\{\bar{\rho}_{i}=\rho^{i\dagger}\gamma^{0}~~~~~~~\bar{\varepsilon}_{i}=\varepsilon^{i\dagger}\tilde{\gamma}^{0}}\\\end{array}
\begin{array}{l}{m^{2}=\left(\mu+\frac{3}{2}\right)\left(\mu-\frac{5}{2}\right)=\mu^{2}-\mu-\frac{15}{4}}\\{m^{2}=\left(\mu+\frac{5}{2}\right)\left(\mu-\frac{3}{2}\right)=\mu^{2}+\mu-\frac{15}{4}.}\\\end{array}
\mathrm{d}s^{2}=\alpha^{\prime}\left[\left({\frac{u}{\calR}}\right)^{2}\mathrm{d}x_{d}^{2}+\left({\frac{\calR}{u}}\right)^{2}\mathrm{d}u^{2}+\mathrm{d}\Omega_{\tilde{d}+1}^{2}\right]\,.
S_{10}^{\prime}=g\int\!\!d^{3}x~f^{abc}\left(\varphi^{*a}\varphi^{b}c^{c}-\partial^{\mu}\varphi^{a}\varphi^{b}A_{\mu}^{c}+A^{*a}A^{b\mu}c^{c}-\frac12c^{*a}c^{b}c^{c}\right).
X=z\left(\frac{\partial^{2}}{\partialx^{2}}+\frac{\partial^{2}}{\partialy^{2}}\right)-x\frac{\partial^{2}}{\partialx\partialz}-y\frac{\partial^{2}}{\partialy\partialz}-\frac{\partial}{\partialz}-\alpha\frac{z}{r}+c_{1}\frac{r-z}{r(r+z)}-c_{2}\frac{r+z}{r(r-z)},
1+x^{2}\Big[{\frac{(\gamma_{1}^{\,1})^{2}}{2!}}s_{21}\Big]+x^{3}\Big[{\frac{(\gamma_{1}^{\,1})^{3}}{3!}}s_{31}+\gamma_{3}^{\,2}s_{32}\Big]+x^{4}\Big[{\frac{(\gamma_{1}^{\,1})^{4}}{4!}}s_{41}+{\frac{\gamma_{1}^{\,1}\gamma_{3}^{\,2}}{2}}s_{42}+\gamma_{4}^{\,3}s_{43}\Big],
\bar{S}\kappa_{e}\bar{R}=\kappa_{o}R\bar{R}=\kappa_{o}+(1+\bar{w}w)\kappa_{o}v\bar{v}\,,\quad\sum_{e>0}S_{eo}\kappa_{e}w_{e}=\kappa_{o}Rw_{e}=\kappa_{o}v(1+\bar{w}w)\,.
C_{2}=\frac{1}{2}\tilde{A}_{\mu}^{\nu}\tilde{A}_{\nu}^{\mu}=\frac{P(N+P)(N-1)}{2N},\;\tilde{A}_{\mu}^{\nu}=A_{\mu}^{\nu}-\frac{\delta_{\mu}^{\nu}}{N}\sum_{\lambda}A_{\lambda}^{\lambda}\;,
\begin{array}{c}{\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}D(k_{1}+k_{2})D_{m}(k_{2})\overline{{D_{m}(k_{1}+p)}}K_{12}.}\\\end{array}
\phi_{l}\left(x,y,v\right)=\sum_{n=0}^{p-1}\sum_{j=-\infty}^{\infty}e^{i{\vec{z}}\cdot{\hat{e}}\left(l+jq\right)}e^{i\alpha\left(l+jq+n/\kappa\right)}F_{n}\left(x,y\right)\delta\left(\kappav-2\pi\kappal-\beta-2\pi\left(jp+n\right)\right)
\int\frac{d(Reu_{j})d(Rev_{j})d(Imu_{j})d(Imv_{j})d\mu_{j}d\nu_{j}d\overline{\mu}_{j}d\overline{\nu}_{j}}{|u_{j}-v_{j}-\mu_{j}\nu_{j}|^{2}}[Z_{n-1}A_{1}^{m}+(u_{j}-v_{j})^{2}B]
G_{m}(x_{1},x_{2},x_{3},x_{4})=[\prod_{i=1}^{m}\Gamma(\alpha_{i})]^{-1}r_{12}^{\alpha_{4}-\frac{1}{2}D}r_{14}^{\frac{1}{2}D-\alpha_{1}-\alpha_{4}}r_{24}^{\frac{1}{2}D-\alpha_{2}-\alpha_{4}}r_{34}^{-\alpha_{3}}F(u,v)