harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\partialS_{m}(\zeta)}{\partial\zeta}=\hat{W}_{m}(\zeta)Y\quad\mathrm{with}\quad\hat{W}_{m}(\zeta)\equiv\mathrm{~\frac{1}{2}~}\epsilon_{ab}\mathbf{Q}_{m}^{b}(\zeta)\mathbf{Q}_{m}^{a}(\zeta)+m^{2},
g_{\Lambda\Pi}^{(4+n)}=\left[\begin{matrix}{\mathrm{e}^{-{\frac{1}{\alpha}}\varphi}g_{\lambda\pi}+\mathrm{e}^{\frac{2\varphi}{n\alpha}}\rho_{ij}A_{\lambda}^{i}A_{\pi}^{j}}&{\mathrm{e}^{\frac{2\varphi}{n\alpha}}\rho_{ij}A_{\lambda}^{i}}\\{\mathrm{e}^{\frac{2\varphi}{n\alpha}}\rho_{ij}A_{\pi}^{j}}&{\mathrm{e}^{\frac{2\varphi}{n\alpha}}\rho_{ij}}\\\end{matrix}\right],
\chi_{\left\langlep,\phi\right\rangle}(\tau)=\frac{1}{|\Omega_{p}|}\sum_{(x,y)\in\Omega_{p}^{\left\{2\right\}}}\overline{{\phi}}(x,y)\prod_{\xi\in\mathcal{O}(x,y)}\omega_{p_{\xi}}^{-\kappa_{\xi}/\lambda_{\xi}}\chi_{p_{\xi}}(\tau_{\xi})
S_{3}^{(0)}={\frac{1}{2}}\varepsilon_{ab}\gamma^{b}\gamma^{a}Y_{3}={\frac{1}{2}}\varepsilon_{ab}(S_{2}^{(0)},(S_{2}^{(0)},Y_{3})^{a})^{b},\quad\mathrm{ngh}(Y_{3})=-2,
{\calL}=\frac{1}{8\pi}\epsilon^{\mu\nu\lambda}(B_{\mu}^{a}\partial_{\nu}B_{\lambda}^{a}+\frac{1}{3}\epsilon^{abc}B_{\mu}^{a}B_{\nu}^{b}B_{\lambda}^{c})+\frac{1}{2g^{2}}B_{\mu}^{a}B^{\mua}
ds^{2}=fdu^{2}+\bar{f}d\bar{u}^{2}+\left(q^{2}|u|^{2q-2}\,e^{2\rho}+{\frac{f\bar{f}e^{-2\rho}}{q^{2}|u|^{2q-2}}}\right)dud\bar{u}+d\rho^{2}
F_{2}^{(3)}[1,2]=\frac{(2E_{1}+E_{2})(E_{1}+E_{2})(E_{1}-2E_{2}-E_{3})(E_{1}+E_{3})(2E_{1}-E_{2}+E_{3})(E_{2}+E_{3})}{(E_{1}-2E_{2})(2E_{1}-E_{2})(E_{1}-E_{3})(E_{2}-E_{3})(E_{1}-E_{4})(E_{2}-E_{4})}
\frac{d\rho}{dt}=\frac{1}{i\hbar}\left(\left(\begin{array}{cc}{0}&{1}\\{q}&{0}\\\end{array}\right)\rho\left(\begin{array}{cc}{q^{\frac{1}{2}}}&{0}\\{0}&{q^{-\frac{1}{2}}}\\\end{array}\right)-\left(\begin{array}{cc}{q^{\frac{1}{2}}}&{0}\\{0}&{q^{-\frac{1}{2}}}\\\end{array}\right)\rho\left(\begin{array}{cc}{0}&{q}\\{1}&{0}\\\end{array}\right)\right),
\begin{array}{ccc}{}&{}&{({\calK}^{\pm}\,(\psi^{0}))^{2}={\calK}^{\pm}\,(\psi^{0})\,,}\\{}&{}&{({\calK}^{\pm}\,(\psi^{0}))^{\ddagger}={\calK}^{\pm}\,(\psi^{0})\,,}\\{}&{}&{{\calK}^{+}\,(\psi^{0})\,{\calK}^{-}\,(\psi^{0})=0}\\\end{array}
\delta\rho=-{\frac{i}{2}}\bar{\partial}\bar{\varepsilon},\\\\\deltaw=-{\frac{i}{2}}e^{-2\rho}\bar{\partial}^{2}\bar{\varepsilon},\\\\\delta\bar{w}=i\varepsilon+{\frac{2iGL}{l}}e^{-4\rho}\bar{\partial}^{2}\bar{\varepsilon}.
f(E,p;{\tilde{L}}_{p})=-{\tilde{L}}_{p}^{-2}\left(e^{{\tilde{L}}_{p}E}+e^{-{\tilde{L}}_{p}E}-2\right)+\vec{p}^{2}e^{{\tilde{L}}_{p}E}+E^{2}-\vec{p}^{2}
Au(x)=\frac{1}{(2\pi)^{n}}\intd^{n}k\,\,e^{ikx}a(x,k)\widehat{u}(k)=\frac{1}{(2\pi)^{n}}\intd^{n}kd^{n}y\,\,e^{ik(x-y)}a(x,k){u}(y)\,\,,
\Gamma_{\nu\rho\kappa}^{\mu}=\nabla_{\nu}\Gamma_{\rho\kappa}^{\mu}=\partial_{\nu}\Gamma_{\rho\kappa}^{\mu}-\Gamma_{\nu\rho}^{\lambda}\Gamma_{\lambda\kappa}^{\mu}-\Gamma_{\nu\kappa}^{\lambda}\Gamma_{\rho\lambda}^{\mu}.
\begin{array}{rcl}{S}&{=}&{\displaystyle\intd^{8}zE^{-1}\left[\bar{Q}Q+\bar{Q}^{c}Q^{c}+G(\bar{Q}^{c}Q)(\bar{Q}Q^{c})+\bar{\xi}_{1}(\bar{Q}Q)+\bar{\xi}_{2}(\bar{Q}^{c}Q^{c})\right],}\\\end{array}
\cdot\left[(2n-1)(4n-1){\calG}_{n}+2\sum_{j=0}^{n-1}{\calG}_{n-j}\left({\calG}_{j+1}{\calG}_{0}c(j,n)-\sum_{k=0}^{j+1}{\calG}_{j+1-k}{\calG}_{k}d(j,k,n)\right)\right],\;n\ge0,
\dot{\zeta}=-\dot{\psi}-H\frac{(\delta\rho)^{.}}{\dot{\rho}}-3H^{2}(1+c_{s}^{2})\frac{\delta\rho}{\dot{\rho}}-\frac{\delta\rho}{\dot{\rho}^{2}}\left[4abr_{B}\dot{H}-2(abr_{B}H)^{.}\right],
\langle:\prod_{a=1}^{r}O^{I_{a}}(x_{1}^{(a)})::\prod_{b=1}^{r}O^{J_{b}}(x_{2}^{(b)}):\rangle=\sum_{s\inS_{r}}\prod_{a=1}^{r}\frac{\delta^{I_{a}J_{s(a)}}}{(x_{1}^{(a)}-x_{2}^{(s(a))})^{4}}+\mathcal{O}(\frac{1}{N^{2}})
\rho_{m_{0}\rightarrowm_{1}+\cdots+m_{N}}^{D}=\left(\prod_{i=1}^{N}\int\frac{d^{n}\!\vec{\mu_{i}}}{(2\pi)^{n}}\right)\intd^{n}\vec{k}\,\,\mathrm{e}^{i\vec{k}.\sum_{i}\vec{\mu_{i}}}\,\rho_{m_{0}\rightarrowm_{1\mu_{1}}+\cdots+m_{N\mu_{N}}}^{D-n}
\frac13(\lambda^{0^{\prime}})^{2}\varepsilon^{ij}\,\,\sum_{\pm}\,\left(\begin{array}{l}{2\bar{\varepsilon}_{\pm}\gamma^{m}\partial_{i}\theta_{\pm}\,\,\,\,\bar{\theta}_{\pm}\gamma_{m}\partial_{j}\theta_{\pm}}\\{-\bar{\varepsilon}_{\pm}\gamma^{m}\theta_{\pm}\,\,\,\,\partial_{i}\bar{\theta_{\pm}}\gamma_{m}\partial_{j}\theta_{\pm}}\\\end{array}\right),
\left[12uv{\calC}+{\frac{1}{3}}P(u,v,\Lambda){\calU}+D^{2}(1-a^{i}\partial_{i})\right]{\calF}_{l}=0=\left[12uv{\calC}+{\frac{1}{u}}P(u,v,\Lambda){\calV}+D^{2}(1-a^{i}\partial_{i})\right]{\calF}_{l},
\hat{F}(v_{1},\cdots,v_{k})=\frac{1}{k!}\sum_{\sigma\inS_{k}}\prod_{i=1}^{k}[v_{\sigma(i)}-2i+2]\prod_{i<j\atop\sigma(i)>\sigma(j)}\frac{[v_{\sigma(i)}-v_{\sigma(j)}-1]}{[v_{\sigma(i)}-v_{\sigma(j)}+1]}.
\Upsilon_{\mu\nu\alpha\beta}^{2}\equivg_{\{\mu\nu}g_{\alpha\beta\}}\int_{k}\frac{1}{k^{2}-m^{2}}-8\int_{k}\frac{k_{\mu}k_{\nu}k_{\alpha}k_{\beta}}{(k^{2}-m^{2})^{3}},
m\sim\nu\alpha^{\frac{1}{2\epsilon}}{\frac{\left[{\frac{3}{\pi}}(1+d\epsilon)\right]^{\frac{1}{2\epsilon}}}{(1+\gamma\epsilon^{2/3}-\epsilon)^{\frac{1}{\epsilon}}}}\sim\nu\left({\frac{\alpha}{\pi/3}}\right)^{\frac{1}{2\epsilon}}e^{1+{\frac{d}{2}}-\gamma\epsilon^{-1/3}}\;\;\;.
[\delta_{\zeta_{1}},\delta_{\zeta_{2}}]\psi=\Xi^{\mu}\partial_{\mu}\psi,\quad[\delta_{\zeta_{1}},\delta_{\zeta_{2}}]e{^a}_{\mu}=\Xi^{\rho}\partial_{\rho}e{^a}_{\mu}+e{^a}_{\rho}\partial_{\mu}\Xi^{\rho},
W=\lambda_{U}^{ij}Q_{i}{u^{c}}_{j}h_{2}+\lambda_{D}^{ij}Q_{i}{d^{c}}_{j}h_{1}n+\lambda_{E}^{ij}\ell_{i}{e^{c}}_{j}h_{1}+\lambda_{N}^{ij}L_{i}{\nu}_{j}^{c}h_{2}+\cdots
\lambda_{r}\Bigl(\frac{q^{2}}{\omega^{2}};\frac{m_{F}^{2}}{\omega^{2}}\Bigr)=\frac{\lambda}{1+\frac{N\lambda}{2\pi}\Bigl(f(q^{2}/m_{F}^{2})-f(\omega^{2}/m_{F}^{2})\Bigr)}\,.
{\frac{3\sigma^{\prime\prime}}{b_{0}^{2}}}\left(1-{\frac{4\alpha}{M^{2}b_{0}^{2}}}(\sigma^{\prime})^{2}\right)={\frac{\Lambda_{1}}{M^{3}b_{0}}}\delta(y)+{\frac{\Lambda_{2}}{M^{3}b_{0}}}\delta(y-\frac{1}{2})
I=(\frac{6}{\pi^{2}})^{4}[16x_{24}^{2}(\frac{1}{2s}-1)D_{4455}+\frac{64}{9}\frac{x_{24}^{2}}{x_{13}^{2}}\frac{1}{s}D_{3355}+\frac{16}{3}\frac{x_{24}^{2}}{x_{13}^{4}}\frac{1}{s}D_{2255}
\Gamma_{a}^{odd}=e^{2}M\epsilon_{\lambda\nu\sigma}\int_{0}^{\infty}ds\rho_{2}(s)\int\frac{d^{3}k}{(2\pi)^{3}}k_{\sigma}\frac{\bar{u}(p-q)[2(p-q)_{\nu}+k\!\!\!/\gamma_{\nu}]\gamma_{\mu}[2p_{\lambda}+\gamma_{\lambda}k\!\!\!/]u(p)}{k^{2}[k^{2}-s][(p-k)^{2}-m^{2}][(p-k-q)^{2}-m^{2}]}
\omega_{r}^{(2)}=48\rho^{3}dt_{0}\int_{S_{r}^{2}}{\frac{\rhodx_{0}^{i}+d\rho\,(x-x_{0})^{i}}{(t_{0}^{2}+\rho^{2}+({\bfx}-{\bfx}_{0})^{2})^{4}}}\varepsilon_{ijk}dx^{j}dx^{k}.
\frac{\partial\Gamma_{\Lambda}[\phi]}{\partial\Lambda}=-\frac{\Omega}{2}\mathrm{tr}\int_{0}^{\infty}dq\,\frac{q^{D-1}}{C(q,\Lambda)}\frac{\partialC(q,\Lambda)}{\partial\Lambda}\left\langle\left[\delta^{ab}+C.\frac{\delta^{2}\Gamma_{\Lambda}}{\delta\phi^{a}\delta\phi^{b}}\right]^{-1}(\mathbf{q},\mathbf{-q})\right\rangle
T=\sum_{a=1}^{N}\pi_{a}\dot{\lambda}_{a}-T=\frac{1}{2}\sum_{a=1}^{N}\frac{1}{g_{aa}}\pi_{a}^{2}=-2\sum_{a=1}^{N}\frac{A(\lambda_{a})}{\Lambda^{\prime}(\lambda_{a})}\pi_{a}^{2}
{\calZ}=\mathrm{Tr}(q^{H_{\overline{{\alphaV}}}^{L}}~{\overline{q}}^{H_{\overline{{\alphaV}}}^{R}}\xi(\alpha_{1})e^{2\pii\alphas}\eta({\overline{{\alphaV}}},{\calN}_{\overline{{\alphaV}}},{\vec{P}}_{\overline{{\alphaV}}}^{2}))~.
{\calU}(z_{1}){\calU}(z_{2})={\frac{64}{c}}\left({\frac{{\calG}{\calU}-\frac{8}{3}{\calG}{\calG}^{\prime}}{z_{1}-z_{2}}}\right)\quad\mathrm{~and~}\quad\bar{\calU}(z_{1})\bar{\calU}(z_{2})=-{\frac{64}{c}}\left({\frac{\bar{\calG}\bar{\calU}+\frac{8}{3}\bar{\calG}\bar{\calG}^{\prime}}{z_{1}-z_{2}}}\right)\;.
L_{S}=\frac{e^{3Ht}}{L^{3}}\left\{\frac{1}{2}\sum_{n}\left[{\dot{\phi}}({\bfk}_{n},t){\dot{\phi}}(-{\bfk}_{n},t)-(e^{-2Ht}{\bfk}_{n}^{2}+m^{2})\phi({\bfk}_{n},t)\phi(-{\bfk}_{n},t)\right]-V_{F}\right\},
W=-\,\frac{1}{16}\,n\:\Gamma\left(\frac{7-p}{2}\right)\,\left[2\sum_{i}\omega_{i}^{4}-\left(\sum_{i}\omega_{i}^{2}\right)^{2}\right]\prod_{i}\omega_{i}^{-1}\left(\frac{2}{b}\right)^{7-p}+O\left(\frac{1}{b^{9-p}}\right).
H_{n,1}^{\prime}=\frac{(-1)^{n}}{(n+1)!}\intd\sigmaJ_{L}^{a}(\sigma)(Ad_{\Phi})^{n}\left\{\frac{1}{2}\partial_{\mu}\partial_{\nu}\Phi^{a}(\tilde{X})\xi^{\mu}(\sigma)\xi^{\nu}(\sigma)\+\partial_{\mu}\Phi^{a}(\tilde{X})\xi^{\mu}(\sigma)\right\}.
({{\widehat\Lambda}^{\alphaA}})_{~v}^{u}={\frac{\rho}{\sqrt{x^{2}(x^{2}+\rho^{2})^{3}}}}\,\big(\mu^{Au}\,x_{~v}^{\alpha}+x^{\alphau}\bar{\mu}_{~v}^{A}\big)~~,
\langleT_{\sigma^{-},\sigma^{-}}\rangle=\left[\frac{\partialx^{-}}{\partial\sigma^{-}}\right]^{2}\langleT_{x^{-},x^{-}}\rangle=-\,\frac{N\lambda^{2}}{12\pi}\,\frac{1+\frac{4\lambda}{M}e^{-2\lambda\sigma^{-}}}{(1+\frac{2\lambda}{M}e^{-2\lambda\sigma^{-}})^{2}}\,.
E[\sigma]=-\sum_{\ell}N_{\ell}\int_{m}^{\infty}\frac{dt}{\pi}\,\sqrt{t^{2}-m^{2}}\int_{0}^{\infty}dr\,\left[\rho_{\ell}(it,r)\right]_{N}+\sum_{i=1}^{N}E_{\mathrm{FD}}^{(i)}+E_{\mathrm{CT}}\,,
D_{\alpha}^{\mathbfi}\left[-3i{\bar{\sigma}}^{0\dot{\alpha}\gamma}{\bar{D}}_{\dot{\alpha}{\mathbfi}}D_{\beta\gamma}T\right]=-24\left[\sigma_{\alpha}^{\mu0\,\,\gamma}\partial_{\mu}D_{\beta\gamma}T-\frac{1}{2}\partial^{0}D_{\alpha\beta}T\right].
\langleQ^{\#}\Psi_{n,k},Q^{\#}\Psi_{n,k}\rangle\;=\;\langle\left[Q^{\#},R_{n,k}\right]\Psi,\left[Q^{\#},R_{n,k}\right]\Psi\rangle\;\leq\;\left(\frac{40}{n}\right)^{2}\\langle\Psi,\Psi\rangle\,
R_{+}(q)=\left(\begin{matrix}{q^{1/2}}&{0}&{0}&{0}\\{0}&{q^{-1/2}}&{q^{1/2}-q^{-3/2}}&{0}\\{0}&{0}&{q^{-1/2}}&{0}\\{0}&{0}&{0}&{q^{1/2}}\\\end{matrix}\right)\.
e^{W_{0}}=e^{-\sum_{j=1}^{r}({\frac{a}{4}}q_{j}^{4}+{\frac{b}{2}}q_{j}^{2})}\prod_{j=1}^{r}(q_{j})^{g_{S}}\prod_{j<k}(q_{j}^{2}-q_{k}^{2})^{g_{M}}
{\bfP}\left(\hat{a}_{a_{1}}^{\dagger}\ldots\hat{a}_{a_{s}+1}^{\dagger}A_{\;\;\quad\quadb_{1}\ldotsb_{s}}^{a_{1}\ldotsa_{s+1}}\hat{a}^{b_{1}}\ldots\hat{a}^{b_{s}}\right)=\hat{a}_{a}^{\dagger}K^{a}
\sum_{1}^{\infty}\frac{1}{n!}\intdx_{1}\ldotsdx_{n}\tau(x,x_{1},\ldots,x_{n})\delta\phi(x_{1})\ldots\delta\phi(x_{n})+\frac{1}{\lambda}\delta\phi(x)+\frac{1}{2N}\bar{\psi}_{0}(x)*\psi_{0}(x)=0
(\mathrm{cof}\,A)^{ij}=\frac{1}{(p-1)!}\epsilon^{i{i_{2}}{i_{3}}\cdots{i_{p}}}\epsilon^{j{j_{2}}{j_{3}}\cdots{j_{p}}}A_{{i_{2}}{j_{2}}}A_{{i_{3}}{j_{3}}}\cdotsA_{{i_{p}}{j_{p}}}=\frac{\partial\mathrm{det}\,A}{\partialA_{ij}}.
{\frac{1}{2}}\stackrel{(-1,0)*}{q_{i(1)}}\stackrel{(-1,-1)}{{\bar{q}}_{j}}T_{\alpha_{1}}^{ij}\stackrel{(2,1)}{\pi^{\alpha_{1}}}+{\frac{1}{2}}\stackrel{(0,-1)*}{q_{i(2)}}\stackrel{(-1,-1)}{\bar{q}_{j}}T_{\alpha_{1}}^{ij}\stackrel{(1,2)}{\pi^{\alpha_{1}}}.
\frac{\partiale^{A(p_{s})}}{\partial{p_{s}}_{\mu}}=\sum_{n=0}^{\infty}\frac{\Big[A^{n}(p_{s}),\frac{\partialA(p_{s})}{\partial{p_{s}}_{\mu}}\Big]}{\big(n+1\big)!}e^{A(p_{s})},
{}_{P}F_{Q}\left(\begin{array}{c|}{a_{1},\cdots,a_{P}}\\{b_{1},\cdots,b_{Q}}\\\end{array}~z\right)=1+z\frac{a_{1}\ldotsa_{P}}{b_{i}\ldotsb_{Q}}~{}_{P+1}F_{Q+1}\left(\begin{array}{c|}{1,1+a_{1},\cdots,1+a_{P}}\\{2,1+b_{1},\cdots,1+b_{Q}}\\\end{array}~z\right)\;.
\begin{array}{rcl}{G_{b}^{(n)}(\zeta^{\prime\prime},\zeta_{2n-1},\zeta_{2n})|_{\zeta_{2n}=\epsilonx^{-1}\zeta_{2n-1}}}&{=}&{G_{b}^{(n-1)}(\zeta^{\prime\prime})\otimesu_{\epsilon}~~~~(\epsilon=\pm).}\\\end{array}
\tilde{H}_{T}=\intd^{3}x\left[\frac{1}{2}\tilde{\pi_{i}}^{2}+\frac{1}{4}\tilde{F_{ij}}^{2}+\frac{1}{2}m^{2}(\tilde{A^{0}}^{2}+\tilde{A^{i}}^{2})-\tilde{A_{0}}\tilde{\Theta_{2}}-\partial_{i}\tilde{A^{i}}\tilde{\Theta}_{1}\right].
\phi^{(i)}\left(x\right)=\sum_{n=1}^{N}\sum_{\left[k\right]}\int\widetilde{dp}^{\mu_{n}}\sqrt{\left|\lambda_{\left[k\right]}^{\mu_{n}}\left(p\right)\right|}\left[a_{\left[k\right]}^{\mu_{n}}\left(p\right)f_{\left[k\right]}^{\mu_{n}(i)}\left(p\right)e^{-ipx}+a_{\left[k\right]}^{+\mu_{n}}\left(p\right)f_{\left[k\right]}^{*\mu_{n}(i)}\left(p\right)e^{ipx}\right]\;\,.
\Pi^{\mu}(x)=\sum_{k}\sum_{r=0}^{3}\frac{i}{(2Vw_{k})^{1/2}}\left(k^{\mu}\epsilon_{r}^{0}({\bfk})-w_{k}\epsilon_{r}^{\mu}({\bfk})\right)\left(-a_{r}({\bfk})e^{-ikx}+a_{r}^{\dagger}({\bfk})e^{ikx}\right)
[a_{m},\tilde{a}_{n}]=[a_{m}^{\dagger},\tilde{a}_{n}^{\dagger}]=[\psi_{\dot{\alpha}},\tilde{\psi}_{\dot{\beta}}]_{+}=[\psi_{\dot{\alpha}}^{\dagger},\tilde{\psi}_{\dot{\beta}}^{\dagger}]_{+}=0
\left(\begin{matrix}{F^{(\pm)\Lambda}}\\{G_{\Sigma}^{(\pm)}}\\\end{matrix}\right)\,\rightarrow\,\left(\begin{matrix}{F^{(\pm)\prime\Lambda}}\\{G_{\Sigma}^{(\pm)\prime}}\\\end{matrix}\right)\,=\,\left(\begin{matrix}{A_{t}}&{B_{t}}\\{C_{t}}&{D_{t}}\\\end{matrix}\right)\left(\begin{matrix}{F^{(\pm)\Lambda}}\\{G_{\Sigma}^{(\pm)}}\\\end{matrix}\right)
\begin{array}{lllllll}{g_{rr}}&{=}&{\displaystyle\frac{l^{2}}{r^{2}}+O(r^{-4})\;}&{}&{g_{tt}}&{=}&{\displaystyle-\frac{r^{2}}{l^{2}}+O(1)\;}\\{g_{tr}}&{=}&{O(r^{-3})\;}&{}&{g_{\varphi\varphi}}&{=}&{r^{2}+O(1)\;}\\{g_{\varphir}}&{=}&{O(r^{-3})\;}&{}&{g_{t\varphi}}&{=}&{O(1)\;}\\\end{array}
\m^{2}(\beta)=m_{0}^{2}+\lambda_{0}F(D)\left(m(\beta)\right)^{D-2}\int_{m(\beta)\beta}^{\infty}d\tau\left((\frac{\tau}{m(\beta)\beta})^{2}-1\right)^{\frac{D-3}{2}}\frac{1}{e^{\tau}-1}.
\mathrm{~\frac{1}{6}~}\epsilon_{ABCD}^{\!~~~~~~~~abcd}F_{\!~~~~cd}^{CD}=F_{AB}^{\!~~~~ab}\qquad\Leftrightarrow\qquad\Omega_{ABCD}F_{\!~~~~ab}^{CD}=0,\qquad\epsilon_{AB}F_{\!~~~~ab}^{AB}=0.
\tilde{N}_{(\rho^{\prime},y^{\prime})\,(\rho,y)}{}^{(\rho^{\prime\prime},y^{\prime\prime})}=Q_{(\rho,y)}{}^{(r,x)}Q_{(\rho^{\prime},y^{\prime})}{}^{(r^{\prime},x^{\prime})}Q_{(\rho^{\prime},y^{\prime})}{}^{(r^{\prime\prime},x^{\prime\prime})}N_{r\,r^{\prime}}^{(A_{g-2})}{}^{r^{\prime\prime}}\,(\tilde{N}_{x})_{x^{\prime}}{}^{x^{\prime\prime}}
\tau_{+}=\left(\begin{array}{cc}{0}&{\sqrt2}\\{0}&{0}\\\end{array}\right)\quad\tau_{-}=\left(\begin{array}{cc}{0}&{0}\\{\sqrt2}&{0}\\\end{array}\right)\quad\tau_{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)
\hat{\nabla}_{\hat{\mu}}\hat{F}_{(\hat{d})}{}^{\hat{\mu}\hat{\rho}_{1}\cdots\hat{\rho}_{d}}+{\textstyle\frac{2\chi}{\sqrt{|\hat{g}|}}}\sum_{n}\mu_{n}{\displaystyle\int}d^{d}\xi_{n}\epsilon^{i_{1}\cdotsi_{d}}\partial_{i_{1}}\hat{X}^{\hat{\mu}_{1}}\cdots\partial_{i_{d}}\hat{X}^{\hat{\mu}_{d}}\delta^{\hat{d}}(\hat{x}-\hat{X}_{n})=0\,,
e^{2(x+{\frac{i}{2}}\overrightarrow{\partial}_{p})}\,K_{i\left(P-p\right)}(e^{x+X})\,K_{i\left(P+p\right)}(e^{x-X})=e^{2x}\,K_{1+i\left(P-p\right)}(e^{x+X})\,K_{-1+i\left(P+p\right)}(e^{x-X})\,\;.
\rho_{N}(p)=2\ImI_{N}(p)\simeq\frac{\pi^{(N+1)/2}}{(4\pi)^{(N-1)\ell}}\frac{\varpi^{\ell-3/2}}{\sigma^{N/2+\ell(N-1)}}\frac{(p^{2}/\sigma^{2}-1)^{\ell(N-1)-(N+1)/2}}{\Gamma((N-1)(\ell-1/2))}
T_{\nu}^{\mu}=\left[\begin{array}{ccc}{-(\rho+\delta\rho)}&{}&{a(\rho+P)(v+B)_{|j}}\\{}&{}&{}\\{-a^{-1}(\rho+P)v^{|i}}&{}&{(P+\deltaP)\delta_{j}^{i}+\delta\pi_{j}^{i}}\\\end{array}\right]\,,
\left[\begin{array}{ccc}{(1+q(\alpha,\omega))\left[z+\frac{(m-1)(m+2)}{2m}y\right]}&{z}&{\frac{(m-1)(m+2)}{2m}y}\\{2}&{q(\beta,\omega)+z\Pi_{1}+2mNz^{2}\Pi_{2}}&{\frac{(m-1)(m+2)}{2m}y(\Pi_{1}+2Ny\Pi_{2})}\\{2}&{z(\Pi_{1}+2Ny\Pi_{2})}&{\left[q(\gamma,\omega)+\frac{(m-2)(m+4)}{2m}Ny^{2}\Pi_{2}\!\!\right.}\\{}&{}&{\left.+\frac{(m-2)}{2m}y\Pi_{1}+2Nyz\Pi_{2}\right]}\\\end{array}\right]
\rho(y_{0})=\frac{k^{k}}{k!}\frac{(n-k)^{n-k}}{(n-k)!}\frac{n!}{n^{n}}\frac{x^{n}}{n!}\nonumber\,=\sqrt{\frac{n}{2\pik(n-k)}}\frac{x^{n}}{n!}(1+{\calO}(n^{-1})).
{}^{(4)}\!R(g+\langle\delta_{2}g\rangle)={}^{(4)}\!R(g)-2\kappa_{5}^{2}\langleT^{\mathrm{GW}}{}_{5}^{5}\rangle+\frac{\dot{B}^{2}}{a^{2}}\left(\frac{\langle\delta_{2}V\rangle}{V}-\frac{\langle\dot{\delta}_{2}V\rangle}{\dot{V}}\right),
D^{c}(x_{1},t)\equiv-\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}dq_{0}\int_{-\infty}^{\infty}dq_{1}\frac{1}{q_{0}^{2}-q_{1}^{2}+i\epsilon}e^{-\frac{i}{\hbar}cq_{0}t}e^{\frac{i}{\hbar}q_{1}x_{1}}
P_{\epsilon}^{(i)}(z_{\pm};r)=\epsilon\frac{(q^{4};q^{4})_{\infty}}{(q^{6};q^{4})_{\infty}}\oint_{C_{\epsilon}^{(i)}}\frac{w^{1-i}dw}{2\pi\sqrt{-1}}\frac{q^{2i}(1-q^{2})z_{-}^{(1+i)/2}z_{\epsilon}^{i/2}z_{+}^{i/2}Y^{(i)}(z_{\pm},w;r)}{(w-z_{-})(w-q^{2}z_{\epsilon})(w-q^{4}z_{+})}.
\left({\tilde{\rho}}_{ij}\right)^{(1)}=\left(\begin{array}{ccc}{f_{ij}}&{\delta_{ij}}&{{\frac{\partial\Omega^{j}}{\partialq_{i}}}}\\{-\delta_{ij}}&{0}&{{\frac{\partial\Omega^{j}}{\partialp_{i}}}}\\{-{\frac{\partial\Omega^{i}}{\partialq_{j}}}}&{-{\frac{\partial\Omega^{i}}{\partialp_{j}}}}&{0}\\\end{array}\right)
\langle\phi_{k}\:(\int_{\Sigma_{L}}\!Q^{+}{\bar{Q}}^{+}{\bar{\phi}}_{\bar{l}})\;|(\int_{\Sigma_{R}}\!Q^{-}{\bar{Q}}^{-}\phi_{i})\;\phi_{j}\rangle-\langle\phi_{k}\:(\int_{\Sigma_{L}}\!Q^{-}{\bar{Q}}^{-}\phi_{i})\;|(\int_{\Sigma_{R}}\!Q^{+}{\bar{Q}}^{+}{\bar{\phi}}_{\bar{l}})\;\phi_{j}\rangle
\begin{array}{rcl}{\pm\,\alpha_{kl}\!\!}&{=}&{\!\!\pm\,(e_{k}-e_{l})~\qquad~(1\leqk<l\leq5)}\\{\pm\,\beta_{kl}\!\!}&{=}&{\!\!\pm\,(e_{k}+e_{l})~\qquad~(1\leqk<l\leq5)}\\{\pm\,\gamma_{\pm,\dots,\pm}\!\!}&{=}&{\!\!\pm\,{\frac{1}{2}}\left(e_{8}-e_{7}-e_{6}+{\displaystyle{\sum_{k=1}^{5}}}\,(-1)^{s(k)}\,e_{k}\right)}\\\end{array}~,
\langle\hat{\rho}(x,0)\hat{\rho}(y,t)\rangle_{c}=-{\frac{2}{\pi^{2}\beta^{2}}}{\frac{[(x-y)^{2}-\pi^{2}\beta^{2}t^{2}]}{[(x-y)^{2}+\pi^{2}\beta^{2}t^{2}]^{2}}}.
\intd^{n}xJ_{A}(x)\langle\,N_{d^{A}}\Big[{\frac{\delta^{L}S}{\delta\phi_{A}^{\ast}(x)}}\Big]\,\rangle_{_J}\,=\,i\hbar\,\intd^{n}x\langle\,N_{n}[{\calA}(x)]\rangle_{_J}
\phi^{\alpha}(x)\phi^{\beta}(0)=\frac{C_{\phi}}{x^{2\eta}}\delta^{\alpha\beta}+\frac{g_{\phi\phi{\calO}}}{C_{\calO}}\frac{1}{(x^{2})^{\eta-\frac{1}{2}\eta_{o}}}{\calO}(0)\delta^{\alpha\beta}+\cdots,
\rho(w)=\frac{d{\calT}(w)}{dw}=\frac{d{\calT}_{1}(w)}{dw}-\frac{1}{2}\int_{-\frac{\kappa}{4}}^{\frac{\kappa}{4}}\frac{d{\calT}_{2}(w-\overline{{w}})}{dw}\rho(\overline{{w}})\,d\overline{{w}}.
\delta_{\chi}^{(1)}M_{A}^{\phantom{A}B}=\chi^{D}[Q_{D},M]_{A}^{\phantom{A}B}=\left(\begin{matrix}{-i(\chi_{A}(\Psi^{\top}C)^{B}-\Psi_{A}(\chi^{\top}C)^{B})}&{-m_{A}^{\phantom{A}D}\chi_{D}}\\{-i(\chi^{\top}C)^{D}m_{D}^{\phantom{C}B}}&{0}\\\end{matrix}\right)
\omega^{(1)}=\int_{S^{3}}\hat{Q}={\frac{3}{4}}{\frac{\rho^{3}[\rhodx_{0}^{4}-x_{0}^{4}d\rho]}{[\rho^{2}+(x_{0}^{4})^{2}]^{\frac{5}{2}}}},\quad\quad{\omega^{(1)}}^{\prime}=\int_{{S^{3}}^{\prime}}\hat{Q}={\frac{3}{4}}{\frac{\rho^{3}[\rhodx_{0}^{3}-(x_{0}^{3})d\rho]}{[\rho^{2}+(x_{0}^{3}-\bar{x})^{2}]^{\frac{5}{2}}}}.
\left(\begin{array}{c}{\mathrm{{\small~number~of~different}~}}\\{\mathrm{{\small~Weyl~particles}~}}\\\end{array}\right)^{2}=\left(\begin{array}{c}{\mathrm{{\small~number~of~}~}}\\{\mathrm{{\small~gauge~bosons}~}}\\\end{array}\right)
S_{NUT}=\frac{(4\pi)^{(n-2)/2}\beta}{16\pi^{3/2}\ell^{2}}\Gamma\left(\frac{3-n}{2}\right)\Gamma\left(\frac{n}{2}\right)N^{n-3}\left[(n-1)(n-2)N^{2}-(n-3)\ell^{2}\right]
\partial_{z}\partial_{\bar{z}}\phi_{cl}={\frac{1}{2\gamma}}e^{\gamma\phi_{cl}}-{\frac{2\pi}{\gamma}}\left[\sum_{k=1}^{m}(1-q_{k})\delta^{(2)}(z-z_{k})+\sum_{k=m+1}^{m+n}\delta^{(2)}(z-z_{k})\right].
{\calL}_{\ast}=\omega\left(\frac1{\alpha^{\prime}}V(X)+g^{ij}\partial_{i}X^{A}\partial_{j}X^{B}G_{AB}(X)+\omega^{ij}\partial_{i}X^{A}\partial_{j}X^{B}H_{AB}(X)+D(X)R+O(\alpha^{\prime})\right).
M_{N}^{c}(s;a_{1},\ldots,a_{N};\alpha_{1},\ldots,\alpha_{N})\equiv\sum_{n_{1},\ldots,n_{N}=1}^{\infty}(a_{1}n_{1}^{\alpha_{1}}+\cdots+a_{N}n_{N}^{\alpha_{N}}+c^{2})^{-s},
\hat{L}_{-1-\frac{n}{Q}}~,\quad\bar{\hat{L}}_{-1-\frac{n}{Q}}~,\quad\hat{K}_{-\frac{n}{Q}}^{-}~,\quad\bar{\hat{K}}_{-\frac{n}{Q}}^{-}~,\quad\hat{G}_{-\frac12-\frac{n}{Q}}^{-\pm}~,\quad\bar{\hat{G}}_{-\frac12-\frac{n}{Q}}^{-\pm}~.
S(x)\,=\,\frac{1}{2}\,\left(\sum_{n=-\infty}^{+\infty}\,-\,\int_{-\infty}^{+\infty}\!\!dn\right)\,\sqrt{n^{2}\,+x}\,=\,-\frac{\sqrt{x}}{\pi}\,\sum_{n=1}^{\infty}\,\frac{1}{n}\,K_{1}(2\pin\sqrt{x})\,{,}
\langle{\calO}\rangle_{A}=\langle{\calO}\rangle_{L}+\int_{0}^{1}d\kappa\intD\phi\sum_{i}\biggl(\tilde{\delta}_{1,i}[\phi]+\kappa\tilde{\delta}_{2,i}[\phi]\biggr)\Theta^{\prime}[\phi]{\frac{\delta{\calO}}{\delta\phi_{i}}}e^{iS_{\mathrm{eff}}^{M}},
\left(\begin{array}{cccccccc}{0}&{0}&{0}&{-x}&{0}&{0}&{-w_{1}}&{w_{0}}\\{0}&{0}&{0}&{q}&{0}&{0}&{2w_{2}}&{-w_{1}}\\{0}&{0}&{0}&{0}&{0}&{0}&{-6w_{3}}&{2w_{2}}\\{0}&{0}&{0}&{0}&{0}&{0}&{h}&{-u}\\{0}&{0}&{0}&{0}&{0}&{0}&{q}&{-x}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{y}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\\end{array}\right).
F_{\tiny\mathrm{div}}^{C}[g,\beta,D]=-\eta\,{\frac{\Gamma\left(1-\fracD2\right)}{(4\pi)^{D/2}}}\,\,{\frac{\pi^{2}m^{D-4}}{3\kappa\beta^{2}}}\,\,\int_{\Sigma}\left[f_{1}m^{2}-\left(p_{1}{\frac{4\pi^{2}}{\kappa^{2}\beta^{2}}}{\calP}+p_{2}R+p_{3}{\calQ})\right)\right]~~~.
H=\frac{1}{2}e^{-2A_{0}(y)}(\dot{\phi}^{2}+(\vec{\nabla}\phi)^{2})+\frac{1}{2b_{0}^{2}}(\phi^{\prime})^{2}e^{-4A_{0}(y)}+\frac{1}{2}m^{2}\phi^{2}e^{-4A_{0}(y)}.
\Pi_{\mathrm{gg}}^{\mu\nu}(p,p_{3})=\frac{g^{2}}{L}\sum_{q_{3}}\int\frac{d^{3}q}{(2\pi)^{3}}\frac{N_{0}^{\mu\nu}+N_{1}^{\mu\nu}+N_{2}^{\mu\nu}}{(q^{2}-q_{3}^{2}-M^{2}+i\epsilon)((q-p)^{2}-(q_{3}-p_{3})^{2}+i\epsilon)}
\bigl\{\Phi(\sigma_{1},\tau),{\frac{\partial}{\partial\tau}}\Phi(\sigma_{2},\tau)\bigr\}_{\mathrm{P.B.}}=4\pi\,\delta(\sigma_{1}-\sigma_{2}),\quad\bigl\{\Phi(\sigma_{1},\tau),\Phi(\sigma_{2},\tau)\bigr\}_{\mathrm{P.B.}}=0
S_{eff}[A_{\mu}]={\frac{1}{4g_{4}^{2}}}\intd^{D}xF_{\mu\nu}F^{\mu\nu}+{\frac{1}{2}}\int{\frac{d^{D}q}{(2\pi)^{D}}}A_{\mu}(q)\Pi^{\mu\nu}(q^{2})A_{\nu}(-q),
A_{2}^{-}\eta_{2-}^{(0)}=0\Rightarrow\eta_{2-}^{(0)}(x)\propto\frac{1}{{sinh\left(\sqrt{2}mx\right)\left(2+sech^{2}\left(\frac{m}{\sqrt2}x\right)\right)}}=\eta^{(0)}(x;0,\beta).
\begin{array}{l}{A_{\mu}^{\prime}=UA_{\mu}U^{-1}+\fracigU\partial_{\mu}U^{-1},}\\{Q_{\mu}^{\prime}=UQ_{\mu}U^{-1},\,\,\,J_{\mu}^{\prime}=UJ_{\mu}U^{-1},}\\{C_{\mu}^{\prime}=UCU^{-1},\,\,\,\overline{{\xi}}^{\prime}=U\overline{{\xi}}U^{-1},}\\{\overline{{C}}^{\prime}=U\overline{{C}}U^{-1},\,\,\,\xi^{\prime}=U\xiU^{-1}.}\\\end{array}
ds^{2}=-{\tilde{n}}^{2}({\tilde{x}}^{\mu},{\tilde{z}})\,d{\tilde{\tau}}^{2}+{\tilde{a}}^{2}({\tilde{x}}^{\mu},{\tilde{z}})\,d{\tilde{x}}_{i}^{2}+{\tilde{b}}^{2}({\tilde{x}}^{\mu},{\tilde{z}})\,d{\tilde{z}}^{2}.
{\bf\nu}_{i}=\left(\begin{array}{cccc}{1}&{1}&{0}&{0}\\{1}&{2}&{1}&{0}\\{1}&{1}&{2}&{1}\\{1}&{0}&{1}&{1}\\{1}&{0}&{-1}&{0}\\{1}&{2}&{1}&{2}\\\end{array}\right),\quad{\bf\nu}_{\alpha}^{\ast}=\left(\begin{array}{cccc}{1}&{0}&{0}&{0}\\{1}&{1}&{-1}&{0}\\{1}&{2}&{-2}&{1}\\{1}&{1}&{-1}&{1}\\\end{array}\right).
p_{1}={\frac{2(1+\epsilon)}{\epsilon^{2}+3}},\,p_{2}={\frac{2(1-\epsilon)}{\epsilon^{2}+3}},\,p_{3}={\frac{\epsilon^{2}-1}{\epsilon^{2}+3}},\,\mathrm{and}\sum{p_{i}}=\sum{{p_{i}}^{2}}=1.
\Omega_{H}=-\left.{\frac{1}{{\omega^{0}}_{\phi}}}\right|_{H}={\frac{a^{2}}{2J}}\left[r_{+}-M+{\frac{\left(M^{2}+\Sigma^{2}-P^{2}-Q^{2}\right)\left(M+\Sigma/\sqrt{3}\right)}{\left(M+\Sigma/\sqrt{3}\right)^{2}-Q^{2}}}\right]^{-1}.
Z_{0}=[\mathrm{det}(-\epsilon_{\mu\nu\lambda}p_{\lambda}+M\delta_{\mu\nu})]^{-1}[\mathrm{det}(-\epsilon_{\mu\nu\lambda}p_{\lambda}+\frac{1}{\rho}p_{\mu}p_{\nu})]^{-\frac{1}{2}}\mathrm{det}(p^{2})\;.
r_{j}(w)=w^{n(a)}\prod_{a}\frac{E(-x^{\frac{6sa}{g}}/w,x^{12s})E(-x^{\frac{6s(g-a)}{g}}/w,x^{12s})}{E(-x^{\frac{6sa}{g}}w,x^{12s})E(-x^{\frac{6s(g-a)}{g}}w,x^{12s})}.
\frac{1}{8}\bar{D}^{2}\left(Z_{i}{S_{i}}^{\dagger}e^{V}{S}_{i}\right)=\frac{1}{2}S_{i}\,\frac{\partial{\calW}}{\partialS_{i}}+\frac{T_{i}}{16\pi^{2}}\mathrm{Tr}\,W^{2}\;.
p(u,s)={\frac{1}{2}}\int_{0}^{u}{\frac{dt\sqrt{1-s^{2}(1-t^{2})}}{\sqrt{t}(1-t^{2})}}=-i\sqrt{u}\sqrt{s^{2}-1}F_{1}\left({\frac{1}{4}},1,-{\frac{1}{2}};{\frac{5}{4}};u^{2},{\frac{s^{2}u^{2}}{s^{2}-1}}\right)\,.
\Lambda_{\pm}(x_{0},r;q)=\frac{1-q}{2}\left\{\sum_{n=0}^{\infty}\frac{(\frac{\rho_{+}}{\rho_{-}};q)_{2n}}{(q;q)_{2n+1}}\biggl(i\frac{(1-q)\rho_{-}}{2}\biggr)^{2n}\pm(\rho_{+}\leftrightarrow\rho_{-})\right\},
\begin{array}{c}{\sigma_{,m}{\sigma,}^{m}=4\Box\,\sigma\,;\,\tau_{,m}{\tau,}^{m}=\Box\,\sigma^{\prime}\,;\,\sigma_{,m}{\tau,}^{m}=2\Box\,\tau}\\{{\sigma,^{m}}_{m}=2\Box\,(d-1)\,;\,{\tau,^{m}}_{m}=0\,;{\sigma^{\prime},_{m}}=0,}\\\end{array}
{\calG}(i)^{-1}{\calZ}(z){\calK}(z;i)=\left(\begin{array}{cc}{1}&{0}\\{ix_{-}^{\prime}{\cdot\tilde{\sigma}}^{t}}&{-2\theta^{\primea}{}^{t}}\\{2\bar{\theta}_{b}^{\primet}}&{\delta_{b}^{~a}}\\\end{array}\right)=
\int{\frac{d^{D}k}{i(2\pi)^{D}}}{\frac{k_{\mu}}{(m^{2}+2p\cdotk-k^{2})^{a}}}={\frac{\Gamma(\epsilon+a-2)}{(4\pi)^{2-\epsilon}\Gamma(a)}}p_{\mu}(m^{2}+p^{2})^{2-a-\epsilon},
\tilde{\mathcalJ}_{m}^{R}(q_{\perp})=\frac{\pi}{2}\int\frac{d^{2}x_{\perp}}{(2\pi)^{2}}e^{-iq_{\perp}x_{\perp}}(({\mathcalA}_{1}^{R})_{m}(x_{\perp})+({\mathcalA}_{2}^{R})_{m}(x_{\perp})+2({\mathcalA}_{3}^{R})_{m}(x_{\perp})-4({\mathcalA}_{4}^{R})_{m}(x_{\perp})),