harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S=\intd^{4}x\,\sqrt{-g}\,{\bar{\psi}}_{i}\left[i\,\delta_{ij}\,\left({\tilde{\gamma}^{\mu}}\,\partial_{\mu}+\frac{3}{2}\,\frac{\dot{a}}{a}\,{\tilde{\gamma}^{0}}\right)-M_{ij}\right]\psi_{j}\,.
J_{r}^{(n)}(t;\{q\};L)\approx\frac{\hat{S}_{r}^{(n)}(\{q\};L)+\hat{\Sigma}_{r}^{(n)}(\{q\};L)\vartheta}{z-z_{0}}\equiv\frac{\hat{J}_{r}^{(n)}(\{q\};L;\vartheta)}{z-z_{0}}\,.
+{\frac{1}{2m^{2}}}\Pi_{\theta}^{2}+{\frac{m^{2}}{2}}\partial_{i}\theta\partial_{i}\theta+m^{2}(\partial=_{i}A_{i})\theta-A_{0}(\partial_{i}\Pi_{i}+{\frac{\mu}{2}}\epsilon_{ij}\partial=_{i}A_{j}+{\frac{m^{2}}{2}}A_{0}),
(\lambda-3)v(\phi^{2})+v^{\prime}(\phi^{2})\phi^{2}=\frac{N-1}{(2+2V^{\prime}(\phi^{2}))^{3/2}}v^{\prime}(\phi^{2})+\frac{v^{\prime}(\phi^{2})+2\phi^{2}v^{\prime\prime}(\phi^{2})}{(2+2V^{\prime}(\phi^{2})+4\phi^{2}V^{\prime\prime}(\phi^{2}))^{3/2}}
X_{2}X_{3}=\frac{1}{\Box_{2}}J_{2}^{\;\alpha}J_{3}^{\;\beta}\quad\mathrm{or}\quadJ_{2}^{\;\alpha}\frac{1}{\Box_{3}}J_{3}^{\;\beta}\quad\mathrm{or}\quad\nabla_{\alpha}\nabla_{\beta}\Bigl(\frac{1}{\Box_{2}\Box_{3}}J_{2}^{\;\alpha}J_{3}^{\;\beta}\Bigr)\quad.
\left\langle\xi(x)\xi(x^{\prime})\right\rangle\equivN(x,x^{\prime})=\left(\frac18\right)\left[\left\langle\left\{\Phi^{2}(x),\Phi^{2}(x^{\prime})\right\}\right\rangle_{\left\{\phi\right\}}-2<\Phi^{2}>_{\left\{\phi\right\}}(x)<\Phi^{2}>_{\left\{\phi\right\}}(x^{\prime})\right]
E_{M}\,=\,E_{O}\,+\,\frac{l}{4}\left(\frac{a_{3}}{r}\right)^{2}\sum_{n=1}^{\infty}\frac{\left(\omega_{n2}^{2}\,-\,\omega_{n1}^{2}\right)}{\lambda_{n}^{2}}\left[\omega_{n2}^{4}A_{n2}^{2}\,-\,\omega_{n1}^{4}A_{n1}^{2}\right].
\frac{\partial^{n}V}{\partial\Phi^{n}}_{\left|\Phi=\Phi_{c}^{\pm}\right.}=\left[\frac{\partial^{n}W}{\partial\Phi^{n}}\left(\frac{27}{2}n\frac{\partial^{2}W}{\partial\Phi^{2}}-24W\right)+\ldots\right]_{\left|\Phi=\Phi_{c}^{\pm}\right.},
Z^{*}W_{\pi^{0}}(K)Z=e^{i90m}b_{1}z_{1}^{*}z_{1}+e^{i90m}b_{2}z_{2}^{*}z_{2}+\frac12e^{i90m}b_{3}[z_{1}^{*}(z_{1}+z_{2})+z_{2}^{*}(z_{1}+z_{2})]
K(q_{f}\midq_{i})=\int{\calD}q~{\calD}p\int_{0}^{\infty}\!\!dT~e^{\frac{i}{\hbar}\int_{0}^{1}d\tau\,[\,p_{\mu}\dot{q}^{\mu}-T(\Gamma^{\mu}(q)p_{\mu}-m)\,]}.
{J^{\prime}}_{pzm}^{i}=-\frac{e}{2L}{\Gamma^{\prime}}^{i}-\frac{e^{3}}{(2L)^{2}}{\calM^{\prime}}^{2}{\calG}_{(\perp)}[{\calM^{\prime}}^{2}]\ast{\Gamma^{\prime}}^{i}\;,
\begin{array}{lllll}{e_{2}}&{=}&{\mathrm{dim}E_{2}}&{=}&{n^{2}(n^{2}+1)/2}\\{e_{3}}&{=}&{\mathrm{dim}E_{3}}&{=}&{n^{2}(n^{2}+1)(n^{2}+2)/6}\\\end{array}
S_{\Gamma}=-\frac{1}{\kappa}\int\sqrt{-g}e^{\alpha\phi/M_{p}}(\zeta+b_{g})g^{\mu\nu}R_{\mu\nu}(\Gamma)=-\frac{1}{\kappa}\int\sqrt{-\tilde{g}}\tilde{g}^{\mu\nu}R_{\mu\nu}(\Gamma),
\intD{\calB}_{-}det\left(\begin{array}{cc}{0}&{i\partial_{+}}\\{\partial_{t}+{\calB}_{-}}&{0}\\\end{array}\right)=det\left(\begin{array}{cc}{0}&{i\partial_{+}}\\{i\partial_{t}}&{0}\\\end{array}\right)~\intDQe^{-5\tilde{W}[Q]},
\rho=-3\gamma\,{\bfB}\cdot{\bfgrad}\,({\bfE}\cdot{\bfB})\,\\\,\\\mathrm{and}\,\\\,\\\,{\bfj}=3\,\gamma\,\biggl[\,{\bfB}\partial_{0}({\bfE}\cdot{\bfB})-{\bfE}\times{\bfgrad}({\bfE}\cdot{\bfB})\biggr]
\begin{array}{c}{J^{+}=q^{-J^{0}+\frac{1}{2}}\a_{1}^{+}a_{2}}\\{J^{-}=q^{-J^{o}-\frac{1}{2}}a_{2}^{+}a_{1}}\\{2J^{0}=N_{1}-N_{2}}\\{\left[J^{+},J^{-}\right]=[2J^{0}]_{q}=[N_{1}-N_{2}]_{q}}\\{\left[N,J^{\pm}\right]=[N,J^{0}]=0.}\\\end{array}
{\calL}_{4}\equiv{\calL}_{5}\left(\varphi,\partial_{\mu}\varphi,\nabla_{\mu\nu}\varphi;B_{\rho},\tilde{F}^{(1)},F^{(2)},\tilde{C}^{(1)},\tilde{C}^{(2)};\tilde{D}^{(2)},\tilde{\tilde{G}}^{(2)},\tilde{G}^{(3)}\right),
\langle\phi(z,\bar{z})\phi(z^{\prime},\bar{z}^{\prime})\rangle={\frac{1}{4\pi}}\int_{0}^{\infty}{\frac{ds}{s}}e^{-{\frac{d^{2}}{4s}}-m^{2}s}={\frac{1}{2\pi}}K_{0}(d^{2}m^{2})~,
\delta{\calE}_{2}^{2,0}=\frac{K}{4a_{0}^{6}}\left\{\left[4\left(\frac{a_{0}}{\Lambda}\right)^{2}-10\left(\frac{a_{0}}{\Lambda}\right)\right]e^{-\Lambda/a_{0}}-11Ei\left(-\frac{\Lambda}{a_{0}}\right)\right\}\,{,}
\partial_{r^{*}}^{\,2}f_{l,m}\left(r\right)=\left(1-{\frac{2M}{r}}\right)\left(m^{2}+{\frac{2M}{r^{3}}}+\frac{l\left(l+1\right)}{r^{2}}\right)f_{l,m}\left(r\right)\;.
\Phi_{0}=\hat{\Phi}_{0}+\beta_{1}\xi_{0}^{y}+\dot{\hat{T}}_{0}\,\\Psi_{0}=\hat{\Psi}_{0}-\alpha_{1}\xi_{0}^{y}-\dot{\alpha}_{0}\hat{T}_{0}\,\N_{0}=\hat{N}_{0}+\xi_{1}^{y}+\beta_{1}\xi_{0}^{y}
\left\langle0\right|F_{n}^{12}(t)F_{-n}^{12}(t^{\prime})\left|0\right\rangle=-\frac{1}{2tt^{\prime}}\left\langle\left\{\frac{\partial}{\partialv_{n}}+i\frac{\partial}{\partialx_{n}}\right\}_{t}\left\{-\frac{\partial}{\partialv_{n}}+i\frac{\partial}{\partialx_{n}}\right\}_{t^{\prime}}\right\rangle+\frac{1}{t}\delta(t-t^{\prime}).
G_{m}^{\mu\nu\alpha\beta}(x-x^{\prime})=\int{\frac{dp^{4}}{(2\pi)^{4}}}{\frac{{\frac{1}{2}}(g^{\mu\alpha}g^{\nu\beta}+g^{\mu\beta}g^{\nu\alpha})-{\frac{1}{3}}g^{\mu\nu}g^{\alpha\beta}+{\calO}(p)}{p^{2}-m^{2}-i\epsilon}}e^{-ip(x-x^{\prime})}~,
\cdot\deltaY\!\bigg(\frac{\hbar}{i}\frac{\delta}{\delta{\calJ}_{i}},\,\psi^{\alpha},\phi_{Aa}^{*},\bar{\phi}_{A};\,(-1)^{\varepsilon_{i}}\frac{1}{i\hbar}{\calJ}_{i},\,\frac{\delta_{l}}{\delta\psi^{\alpha}},\,\frac{\delta_{l}}{\delta\phi_{Aa}^{*}},\,\frac{\delta_{l}}{\delta\bar{\phi}_{A}}\bigg){\calZ}\!({\calJ},\psi,\phi^{*},\bar{\phi}).
\Phi\equiv\left(\begin{array}{c}{\phi^{+}(x)}\\{\frac1{\sqrt2}(H(x)+i\chi(x))}\\\end{array}\right)\qquad\tilde{\Phi}\equivi\tau_{2}\Phi^{*}=\left(\begin{array}{c}{\frac1{\sqrt2}(H(x)-i\chi(x))}\\{-\phi^{-}(x)}\\\end{array}\right)
\frac{dx^{i}}{dt}=\{x^{i},Q\}_{1}=\{x^{i},H_{0}\}_{0}\equiv\xi_{H}^{i},\quad\frac{d\theta^{i}}{dt}=\{\theta^{i},Q\}_{1}=\frac{\partial\xi_{H}^{i}}{\partialx^{j}}\theta^{j}.
Z_{\;\;\nu_{1}\ldots\nu_{p-k-1}}^{\mu_{1}\ldots\mu_{p-k}}=\frac1{\left(p-k-1\right)!}\partial_{}^{\left[\mu_{1}\right.}\delta_{\;\;\nu_{1}}^{\mu_{2}}\ldots\delta_{\;\;\nu_{p-k-1}}^{\left.\mu_{p-k}\right]},\;k=0,\ldots,p-1.
{\calD}_{2}=\left(\begin{array}{cc}{-2D-2D^{-1}\Phi_{1}D^{-1}+D^{-1}(D^{2}\Phi_{0})D^{-1}}&{D^{3}-D(D\Phi_{0})+D^{-1}\Phi_{1}D}\\{-D^{3}-(D\Phi_{0})D-D\Phi_{1}D^{-1}}&{-D^{2}\Phi_{1}-\Phi_{1}D^{2}}\\\end{array}\right)
E^{d/2}=\left(\frac{d}{2}\right)^{d/2}\left(\frac{V_{p}V(S^{d-1})}{16\piG}\right)^{-(d-4)/2}h^{-(d-2)^{2}/2}\left(\frac{S}{4\pi}\right)^{d-2}\prod_{i}\left(1+\left(\frac{2\piJ_{i}}{S}\right)^{2}\right)
\begin{matrix}{\{X,Y\}=(-1)^{\varepsilon(X)\varepsilon(Y)+d_{X}d_{Y}}\{Y,X\},}\\{\{X,YZ\}=\{X,Y\}Z+(-1)^{\varepsilon(Y)\varepsilon(Z)+d_{Y}d_{Z}}\{X,Z\}Y.}\\\end{matrix}
V_{gen}^{(1)}=\frac{eH}{2\pi\beta}\sum_{l=-\infty}^{+\infty}\int_{-\infty}^{+\infty}\frac{dp_{3}}{2\pi}\sum_{n=0,\sigma=0,\pm1}^{\infty}log[\beta^{2}(\omega_{l}^{2}+\epsilon_{n,\sigma,p_{3}}^{2}+\Pi(T,H))],
\partial_{++}\partial_{--}\tilde{\phi}=-e^{\tilde{\phi}}(e^{\tilde{\phi}}+i\tilde{\psi}_{-}\tilde{\psi}_{+}),\qquad\partial_{++}\tilde{\psi}_{-}=-e^{\tilde{\phi}}\tilde{\psi}_{+},\qquad\partial_{--}\tilde{\psi}_{+}=e^{\tilde{\phi}}\tilde{\psi}_{-}.
ds^{2}\,=\,-dt^{2}\,\left(1-\frac{2m}{\rho}+\frac{q^{2}}{\rho^{2}}\right)+d\rho^{2}\,\left(1-\frac{2m}{\rho}+\frac{q^{2}}{\rho^{2}}\right)^{-1}+\rho^{2}\,d\Omega^{2}
K=\left[{\frac{dT}{dz}}\right]_{\bar{z}^{+}}-\left[{\frac{dT}{dz}}\right]_{\bar{z}^{-}}=\left[{\frac{dT}{ds}}\right]_{s(\bar{z})}\,\left[\left({\frac{ds_{2}}{dz}}\right)_{\bar{z}^{+}}-\left({\frac{ds_{1}}{dz}}\right)_{\bar{z}^{-}}\right].
\left[\fracd{dx}\,(1-x^{2})\,\fracd{dx}-\frac{m^{2}-\sigma_{z}\,m\,x+\frac14}{1-x^{2}}\right]\left(\begin{array}{c}{\alpha_{\lambdam}}\\{\beta_{\lambdam}}\\\end{array}\right)=-\left(\lambda^{2}-\frac14\right)\,\left(\begin{array}{c}{\alpha_{\lambdam}}\\{\beta_{\lambdam}}\\\end{array}\right).
\left(\Gamma_{45}\right)_{a}^{\,b}\;\varepsilon_{b}\,=\,-i\varepsilon_{a}\,,\,\left(\Gamma_{3}\right)_{a}^{\,b}\;\varepsilon_{b}\,=\,\pm\varepsilon_{a}\,\rightarrow\,\bar{g}\left(\frac{a}{\sqrt{2}}\,\pm\,b\right)\,=\,1,
\gamma^{0}=\left(\begin{matrix}{0}&{i}\\{-i}&{0}\\\end{matrix}\right)\quad\gamma^{1}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right)\quad\gamma^{*}=\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right)\quad.
\bar{f}_{\pm}=-\frac{\kappa_{5}^{2}e^{-2A_{\pm}-\alpha_{\pm}^{(0)}}}{\int_{w_{-}}^{w_{+}}dw^{\prime}e^{-3A(w^{\prime})}}\left[e^{-2A_{+}-\alpha_{+}^{(0)}}\bar{\tau}_{+(LL)}+e^{-2A_{--}\alpha_{-}^{(0)}}\bar{\tau}_{-(LL)}\right]+O(\mu).
{\calN}(\lambda_{i},t)\sim\intd\rho_{0}\,d\rho\,P_{\mathrm{creat}}(\rho_{0},\lambda_{i})\,P_{\mathrm{life}}(\rho,\lambda_{i})\,\int_{0}^{t}{\calV}(\rho,t^{\prime},\rho_{0},\lambda_{i})\dt^{\prime}\.
Y(T,U)=\int_{\calF}\frac{d^{2}\tau}{\Im\tau}\Gamma_{2,2}(T,U)\left(-6\left[{\overline{{\Omega}}}_{2}^{\phantom{2}}-\frac{1}{8\pi\Im\tau}\right]\frac{\overline{{\Omega}}}{\overline{{\eta}}^{24}}-\frac{\overline{{j}}}{8}+126\right)\,
\frac{\partial}{\partial\theta^{a}}(\theta^{a_{1}}\ldots\theta^{a_{p}})=\left\lbrace\begin{matrix}{0,}&{\mathrm{if}\,\,a\not=a_{1},\ldots,a_{k},}\\{(-1)^{l-1}\theta^{a_{1}}\ldots\theta^{a_{l-1}}\theta^{a_{l+1}}\ldots\theta^{a_{p}},}&{\mathrm{if}\,\,a=a_{l}}\\\end{matrix}\right.,
\widehat{W}={\widehatM}_{\infty}=\left(\begin{array}{cccccc}{-1}&{0}&{0}&{2}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{-1}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}\\\end{array}\right)\;.
V_{II}(\beta,\phi)=\mu^{D}\sqrt{a}\sum_{s=1}^{\infty}\frac{(-1)^{s+1}}{2s}g^{s}\phi^{2s}\sum_{n=-\infty}^{\infty}\intd^{d}k\frac{1}{(an^{2}+c^{2}+\vec{k}~^{2})^{s}}.
T_{\theta}(\tau)={\frac{1}{2}}\left[{\frac{\Lambda_{R}^{m}(\tau)}{\eta_{R}(\tau)}}+{\frac{1}{\eta_{-R}(\tau)}}\right]+{\frac{1}{2}}\mathrm{Tr}S\left[{\frac{\eta_{R}(\tau)}{\eta_{R}({\frac{\tau}{2}})}}+(\tau\mapsto\tau+1)\right]\,.
\Omega^{ij}\;=\;\left(\begin{array}{ccccc}{k_{1}}&{0}&{0}&{-k_{3}}&{0}\\{0}&{k_{2}}&{0}&{0}&{k_{3}}\\{0}&{0}&{k_{3}}&{0}&{0}\\{-k_{3}}&{0}&{0}&{0}&{0}\\{0}&{k_{3}}&{0}&{0}&{0}\\\end{array}\right)\,,\,\Omega_{ij}\;=\;\left(\begin{array}{ccccc}{0}&{0}&{0}&{-q_{1}}&{0}\\{0}&{0}&{0}&{0}&{q_{1}}\\{0}&{0}&{q_{1}}&{0}&{0}\\{-q_{1}}&{0}&{0}&{q_{2}}&{0}\\{0}&{q_{1}}&{0}&{0}&{q_{3}}\\\end{array}\right)\,,
\nabla_{+}x^{i}-\frac12\Omega_{+}^{+2i}x^{-2}-\frac12\Omega_{+}^{-2i}x^{+2}-i\chi_{q}^{1}\gamma_{q\dot{q}}^{i}\theta_{\dot{q}}^{1-}-ih_{+\dot{q}}^{1-}\tilde{\gamma}_{\dot{q}q}^{i}\theta_{q}^{1+}+ih_{+q}^{2-}\gamma_{q\dot{q}}^{i}\theta_{\dot{q}}^{2+}+i\chi_{\dot{q}}^{2}\tilde{\gamma}_{\dot{q}q}^{i}\theta_{q}^{2-}=0,
P_{R}^{\frac{1}{2}}(k)=2^{\mu-\frac{1}{2}}\frac{\Gamma(\mu)}{\Gamma(3/2)}(\mu-\frac{1}{2})^{\frac{1}{2}-\mu}\frac{1}{m_{pl}^{2}}\frac{H^{2}}{|H^{\prime}|}|_{k=aH}
e^{-2\phi^{\prime}}=\frac{2G^{2}M^{2}\rho(\rho-2GM)}{Q_{m}^{2}\rho^{2}+Q_{e}^{2}(\rho-2GM)^{2}}=-\frac{2G^{2}M^{2}\rhox}{Q_{m}^{2}\rho^{2}+Q_{e}^{2}x^{2}}
N_{G}^{2}(a)=\left[\frac{mk(G)\Gamma\left(\frac{1-x}{h}\right)\Gamma\left(\frac{x}{h}\right)x}{\sqrt{2}\Gamma\left(\frac{1}{h}\right)h}\right]^{2Q\cdota-a^{2}}\prod_{i=1}^{r}\left[\frac{-\pi\mu_{i}}{\gamma\left(-e_{i}^{2}b^{2}/2\right)}\right]^{-\omega_{i}^{\vee}\cdota}
B(x,y,z)=\sum_{r>0}{\frac{1}{r}}\left({\frac{y}{x}}\right)^{r}d+d\sum_{r,r^{\prime}>0}B_{rr^{\prime}}\left({\frac{z}{x}}\right)^{r}\left({\frac{y}{z}}\right)^{r^{\prime}}\,.
S_{2}^{(0)}={\frac{1}{2}}x^{i}\Lambda_{ij}x^{j}+x_{ia}^{*}D_{1\alpha}^{i}c^{\alphaa}+y_{\mua}^{*}D_{2\alpha}^{\mu}c^{\alphaa}+\bar{x}_{i}D_{3\alpha}^{i}B^{\alpha}+\bar{y}_{\mu}D_{4\alpha}^{\mu}B^{\alpha}-\varepsilon^{ab}c_{\alphaa|b}^{*}D_{5\beta}^{\alpha}B^{\beta}.
\frac{1}{2}\mathrm{Tr}\left(F_{\mu\nu}{\calA}^{\mu\nu}+{\calA}_{\mu\nu}F^{\mu\nu}\right)=m\epsilon^{ab}\left(n_{\mu}A_{a\nu}\partial^{\mu}A_{b}^{\nu}-n_{\mu}A_{a\nu}\partial^{\nu}A_{b}^{\mu}\right)+~O(A^{3})
\mathrm{M}\left(\frac{a}{2}+\frac{1}{4},\frac{1}{2},\frac{{x_{M}}^{2}}{2}\right)=(2a+1)\mathrm{M}\left(\frac{a}{2}+\frac{5}{4},\frac{3}{2},\frac{{x_{M}}^{2}}{2}\right)
\widetilde{W}_{i}[\tilde{g}_{\mu\nu},\beta]=-{\frac{\eta_{i}}{2}}\int_{0}^{\infty}{\frac{dt}{t}}\sum_{l=-\infty}^{\infty}e^{-4\pi^{2}(l+w_{i})^{2}\beta^{-2}t}~\mathrm{Tr}~e^{-(L_{i}+m_{i}^{2}g_{\tau\tau})t}~~,~~i=s,d~~~.
\begin{array}{ll}{{\calL}_{B}=}&{\nabla_{\mu}\phi[\nabla^{\mu}\phi]^{*}e^{-2gM}+{\calP}\partial_{\mu}M\partial^{\mu}M-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}}\\{}&{+\frac{1}{6}G_{\mu\nuk}G^{\mu\nuk}+m\epsilon^{\mu\nu\alpha\beta}A_{\mu}\partial_{\nu}B_{\alpha\beta},}\\\end{array}
\psi_{2}=f_{2}(x_{2}-\frac{2i\gamma_{2}\hbar}{\theta}x_{1})e^{-\frac{3x_{2}^{2}}{8\gamma_{2}\hbar}-\frac{\gamma_{2}\hbarx_{1}^{2}}{2\theta^{2}}-\frac{ix_{1}x_{2}}{2\theta}},
1\simeq\frac{2\alpha}{\pi}\left[J(0,\hat{T}_{c})+\frac{\partialJ(0,\hat{T}_{c})}{\partial\hat{T}}(\hat{T}-\hat{T}_{c})+\frac{\partialJ(0,\hat{T}_{c})}{\partial\hat{m}_{T}^{2}}\hat{m}_{T}^{2}\right]_{.}
dl^{2}=R^{2}\frac{(\sum_{s=1}^{N}|\Pi^{s}|^{2}+R^{2})\delta_{ik}-\Pi^{*i}\Pi^{k}}{(\sum_{s=1}^{N}|\Pi^{s}|^{2}+R^{2})^{2}}\delta\Pi^{i}\delta\Pi^{*k}
(Y_{u})^{ab}=e^{\hat{K}/2}(\varepsilon_{lmn}+(1-e^{-2\pii/M})s_{lmn})(t_{33}^{1/2})^{la}(t_{33}^{1/2})^{mb}(t_{33}^{1/2})^{nc}\langleh_{u}\rangle_{c}.
\left.E_{\Pi}^{\\N}\right|=\left[\begin{array}{ccc}{e_{\mu}^{\m}}&{\frac{1}{2}\psi_{\mu}^{\Aa}}&{\frac{1}{2}\psi_{\mu}^{\\dot{A}a}}\\{0}&{-\delta_{B}^{\A}\delta_{b}^{\a}}&{0}\\{0}&{0}&{-\delta_{\dot{B}}^{\\dot{A}}\delta_{b}^{\a}}\\\end{array}\right]
M_{ij}^{-1}=\left(\begin{array}{ccc}{-(2k^{2}+\ell^{2})^{2}/4k^{2}}&{-\ell^{2}/2k^{2}}&{-(2k^{2}\ell+\ell^{3})/2k^{2}}\\{}&{}&{}\\{-\ell^{2}/2k^{2}}&{-1/k^{2}}&{-\ell/k^{2}}\\{}&{}&{}\\{-(2k^{2}\ell+\ell^{3})/2k^{2}}&{-\ell/k^{2}}&{-(k^{2}+\ell^{2})/k^{2}}\\\end{array}\right)\,.
V^{0}=\frac{F_{3}}{F_{2}}(\omega-p)+\frac{p}{F_{3}}~~~~~~~~~~~V^{4}=\frac{p}{F_{3}}~~~~~~~~~~~V^{r}=\pm\left(\frac{F_{3}^{2}}{F_{2}^{2}}(\omega-p)^{2}+2\frac{(\omega-p)p}{F_{2}}-1\right)^{\frac{1}{2}}
S={\frac{1}{2\kappa_{D}^{2}}}\intdx^{D}\sqrt{-g}\left[{\calR}-{\frac{4}{D-2}}(\partial\phi)^{2}-{\frac{1}{2\cdot(p+3)!}}e^{2a_{p+1}\phi}F_{p+3}^{2}+e^{-2a\phi}\Lambda\right].
\left(\phi,f(A_{1},A_{2},\dots,A_{N})\psi\right)=\int_{\Lambda}f(\lambda_{1},\,\lambda_{2},\dots,\lambda_{N})\langle\phi|F_{\lambda}\rangle\langleF_{\lambda}|\psi\rangle\;d\mu(\lambda_{1},\,\lambda_{2},\dots,\lambda_{N})
{\calL}_{red^{2}}={\frac{1}{2}}J_{i}{\frac{R_{ij}}{\partial_{t}^{2}-\vec{\partial}^{2}+M^{2}}}J_{j}+{\frac{1}{2}}J_{0}{\frac{1}{\vec{\partial}^{2}-M^{2}}}J_{0}+\bar{\Psi}^{R}(i\not{\!\partial}-m)\Psi^{R}~.
e^{-\intd^{4}x\frac{\xi}{2g^{2}}(D^{\mu}V_{\mu})^{2}}=\frac{1}{N}\int{\calD}a\,\delta(D^{\mu}V_{\mu}^{g}-a)e^{-\intd^{4}x\frac{\xi}{2g^{2}}a^{2}},
C=\sqrt{1+\frac{KNa_{2}}{4}(\xiA(0)B(0))^{2}\left(1+\frac{\Lambda^{2}}{m_{H}^{2}}\right)^{\frac{K}{2\pi}}}\simeq\sqrt{1+\frac{KNa_{2}}{4}(\xiA(0)B(0))^{2}\left(1+\frac{\Lambda^{2}}{m_{H}^{2}}\frac{K}{2\pi}\right)}.
\langle\bigl(iA^{a}(\tau_{1})+\Phi^{a}(\tau_{1})\bigr)\bigl(iA^{b}(\tau_{2})+\Phi^{b}(\tau_{2})\bigr)\rangle_{0}=\frac{g^{2}\delta^{ab}}{4\pi^{2}}\frac{\lvert{\dot{x}}^{(1)}\rvert\lvert{\dot{x}}^{(2)}\rvert-{\dot{x}}^{(1)}\cdot{\dot{x}}^{(2)}}{\lvertx^{(1)}-x^{(2)}\rvert^{2}}=\frac{g^{2}\delta^{ab}}{8\pi^{2}}.
{\calL}_{PV}=\intd^{4}\theta[Q_{+}^{reg~\dagger}e^{V}Q_{+}^{reg}+Q_{-}^{reg~\dagger}e^{V}Q_{-}^{reg}]+(\intd^{2}\theta\Phi\Lambda_{UV}Q_{+}^{reg}Q_{-}^{reg}+\mathrm{h.c.}).
\sum_{\rho=0}^{N}\left[\eta_{0\rho}\ddot{Q}_{\mu\rho}(t)+\omega_{0}^{2}\eta_{0\rho}Q_{\mu\rho}(t)+\sum_{k=1}^{N}\lambda_{k}\eta_{k\rho}Q_{\mu\rho}(t)\right]=0\;,
\overline{{Z}}_{ab}^{i_{1}i_{2}i_{3}j_{1}j_{2}j_{3}}=\frac{g}{3}W_{ab}\left(\sum_{\sigma\inS_{3}}(-)^{\sigma}g^{i_{1}j_{\sigma(1)}}g^{i_{2}j_{\sigma(2)}}g^{i_{3}j_{\sigma(3)}}\right).
(\omega^{2}-{\omega^{\prime}}^{2})\int_{0}^{\theta}\!\phi_{\omega}^{E}(\sigma)\:\phi_{\omega^{\prime}}^{E}(\sigma)\,d\sigma-\int_{0}^{\theta}\!\partial_{\sigma}(\phi_{\omega^{\prime}}^{E}(\sigma)\:\partial_{\sigma}\phi_{\omega}^{E}(\sigma)-\phi_{\omega}^{E}(\sigma)\:\partial_{\sigma}\phi_{\omega^{\prime}}^{E}(\sigma))\,d\sigma=0
\begin{array}{rl}{(+,-)\qquadA:}&{u=u_{A}+p_{A}^{+}\sigma^{+},\quadv=v_{A}+p_{A}^{-}\sigma^{-}}\\{(-,+)\qquadB:}&{u=u_{B}+p_{B}^{+}\sigma^{-},\quadv=v_{B}+p_{B}^{-}\sigma^{+}}\\{(-,-)\qquadC:}&{u=u_{C},\quadv=v_{C}+p_{C}^{+}\sigma^{+}+p_{C}^{-}\sigma^{-}}\\{(+,+)\qquadD:}&{v=v_{C},\quadu=u_{D}+p_{D}^{+}\sigma^{+}+p_{D}^{-}\sigma^{-}}\\{}&{}\\\end{array}
F^{k}\!_{i}F^{l}\!_{j}\left(g^{ij}\left\{\begin{array}{c}{a}\\{ak}\\\end{array}\right\}-g^{jb}\left\{\begin{array}{c}{i}\\{kb}\\\end{array}\right\}\right)\partial_{l}+F^{l}\!_{i}g^{ij}(\partial_{k}F^{k}\!_{j})\partial_{l}=0\;,
\left[\mathbb{S}^{\mu},\mathbb{S}^{\nu}\right]=\varepsilon^{\mu\nu}{}_{\alpha\beta}\dot{x}^{\alpha}\mathbb{S}^{\beta}-\frac{q}{m^{2}}\left(\,^{*}\mathcal{S}F\,^{*}\mathcal{S}\right)^{\mu\nu}.
m^{\left(n\right)}=m_{0}f_{n}\left(t\right),\quadf_{n}\left(t\right)=t^{-1}\frac{\left(1+t\right)^{n}-\left(1-t\right)^{n}}{\left(1+t\right)^{n}+\left(1-t\right)^{n}},\quadt=\sqrt{m_{0}^{2}}.
E_{K}=\underbrace{\underbrace{\frac{2\sqrt{2}m^{3}}{3\lambda}}_{Classical}+\underbrace{\hbarm\frac{\sqrt{3}}{2\sqrt{2}}}_{Discrete}}_{Finite}+\underbrace{\frac{1}{2}\hbarm\underbrace{\sum_{n}\sqrt{\frac{1}{2}q_{n}^{2}+2}}_{Continuous}}_{Divergent}
\tilde{\xi}^{j}=\zeta^{2j-2}\left(\xi^{j}+2\sum_{i=1}^{k}\frac{\partial\eta_{i}}{\partial\alpha^{j}}\frac{\eta_{i}}{\zeta}\right)=\zeta^{2j-2}\left(\xi^{j}+\frac{\partial}{\partial\alpha^{j}}\sum_{i=1}^{k}\frac{\eta_{i}^{2}}{\zeta}\right).
\bar{\psi}_{a}(w_{n},\alpha)=\prod_{j=a}^{\infty}(1+w_{n}\alpha\bar{q}^{2j-1}),~~\bar{\psi}_{b}(w_{n}^{-1},\beta)=\prod_{j=b}^{\infty}(1+w_{n}^{-1}\beta\bar{q}^{2j-1})
[Q_{i}^{j_{i}},Q_{i}^{k_{i}}]=i\frac{\theta\epsilon^{rs}}{w(\sigma)}\partial_{r}Q_{i}^{j_{i}}\partial_{s}Q_{i}^{k_{i}},~~~\left[Q_{i}^{{\tilde{j}}_{i}},Q_{i}^{{\tilde{k}}_{i}}\right]=i\frac{\theta\epsilon^{rs}}{w(\sigma)}\partial_{r}Q_{i}^{\tilde{j}}\partial_{s}Q_{i}^{\tilde{k}}.
\Omega_{s}=-\mathrm{d}x^{a}\wedge\mathrm{d}p_{a}+\Omega_{\mathrm{m}}\qquad\Omega_{\mathrm{m}}=\frac{s}2\frac{\epsilon^{abc}p_{a}\mathrm{d}p_{b}\wedge\mathrm{d}p_{c}}{(-p^{2})^{3/2}}\qquad(p^{2}<0)\,,
{\calH}[p]*\phi(\tau,\bar{\tau})=\sum_{N=0}^{\infty}p^{N}\sum_{\stackrel{kr=N,~r~\mathrm{odd}}{s~\mathrm{mod}~k}}\frac{1}{N}\phi\left(\frac{r\tau+s}{k},\frac{r\bar{\tau}+s}{k}\right)
\nu_{b}^{a}=\frac{1}{\lambda^{1/3}}\,\left(\begin{matrix}{\frac{1}{\sqrt{\tau_{2}}}}&{}&{}\\{}&{\sqrt{\tau_{2}}}&{}\\{}&{}&{\lambda}\\\end{matrix}\right)\left(\begin{matrix}{1}&{\tau_{1}}&{\omega_{1}}\\{}&{1}&{\omega_{2}}\\{}&{}&{1}\\\end{matrix}\right),\qquadX^{i}=m^{i}\sigma_{1}+n^{i}\sigma_{2}+p^{i}\sigma_{3}\,.
S_{eff}(A_{\mu})={\calN}\int{\calD}q~e^{-S(q,A_{\mu})}=\prod_{k=1}^{\infty}det\left(\delta_{\mu\nu}-\frac{1}{\eta}F_{\mu\nu}\right)^{-1}=\sqrt{det\left(\delta_{\mu\nu}-\frac{1}{\eta}F_{\mu\nu}\right)}.
\left(\frac{{\calA}}{r_{S}^{6}}\right)_{\beta_{H}\gg\beta_{S}}^{\mathrm{open/closed}}\:\simeq\:\frac{1}{80640\pi}\:\left(\frac{M_{PL}}{M}\right)^{4}\:\left(\frac{M_{PL}}{m_{0}}\right)^{2}\:\ll1
H^{2}+\frac{k}{a^{2}}=\frac{\sqrt{-\Lambda}}{3\sqrt{6}}\left(1+\frac{\alpha\Lambda}{2}-\frac{8\alpha\mu}{a^{4}}\right)\varrho+\frac{\mu(1+\frac{2}{3}\alpha\Lambda)}{a^{4}}-\frac{2\alpha\mu^{2}}{a^{8}}+{\calO}(\varrho^{2})
K_{m+\alpha}((m+\alpha)z^{\prime})\\sim\\sqrt{\frac{\pi}{2(m+\alpha)}}\frac{e^{-(m+\alpha)\eta^{\prime}}}{(1+z^{2})^{1/4}}\left\{1+\sum_{j=1}^{\infty}(-1)^{j}\frac{u_{j}([1+z^{2}]^{-1/2})}{(m+\alpha)^{j}}\right\}\,
\theta=c\,\tau=3\,\sqrt{{\displaystyle\frac{A}{1+c^{2}}}}\,\left(\!{\displaystyle\frac{1}{2}}\,{\displaystyle\frac{x}{A\,(x^{2}+A)^{(1/5)}}}-{\displaystyle\frac{3}{10}}\,{\displaystyle\frac{x\,\mathrm{hypergeom}\Big([{\displaystyle\frac{1}{2}},\,{\displaystyle\frac{1}{5}}],\,[{\displaystyle\frac{3}{2}}],\,-{\displaystyle\frac{x^{2}}{A}}\Big)}{A^{(6/5)}}}\!\right)
lA_{1}+nB_{1}=lU_{l}^{n}Str\left(\ldots\phi^{j_{l}}\ldots\underbrace{\partial_{a_{l}}\phi^{i_{2k}^{\prime}}\phi^{i_{2k-1}^{\prime}}}\ldots\right)+nU_{l}^{n}Str\left(\ldots\phi^{j_{l}}\ldots\underbrace{\phi^{i_{n}}\phi^{i_{2k-1}^{\prime}}}\ldots\partial_{a_{l}}\phi^{i_{2k}^{\prime}}\ldots\right)
K(P^{\prime\prime},P^{\prime})=i\int_{0}^{\infty}dT\;e^{-iT(P^{2}-m^{2})}\delta^{4}(P^{\prime\prime}-P^{\prime})={\frac{\delta^{4}(P^{\prime\prime}-P^{\prime})}{P^{2}-m^{2}-i\epsilon}}\,.
\Upsilon_{\sigma}(U)\Upsilon_{\tau}(U^{\dagger})=\sum_{\upsilon\subsetT_{\sigma}}\sum_{\upsilon^{\prime}\subsetT_{\tau},\upsilon\approx\upsilon^{\prime}}C_{\upsilon}\Upsilon_{\overline{{\tau\setminus\upsilon}}\\sigma\setminus\upsilon}(U,U^{\dagger}),
{\calH}=-\hat{K}\cdot\left({\calF}\frac{\delta}{\delta\hat{K}}\right)-\hat{K}\cdot\left(\frac{\delta}{\delta\hat{K}}\vee\frac{\delta}{\delta\hat{K}}\right)-\frac{1}{N^{2}}\hat{K}\cdot\left(\hat{K}\cdot\left(\wedge\frac{\delta}{\delta\hat{K}}\right)\right).
\left\langle0\right|T\left[\bar{\psi}^{\uparrow}(t,\theta)\psi^{\downarrow}(0)\right]\left|0\right\rangle=\frac{e^{i\theta}}{2\pi}(k(c)M)^{\frac{-2c^{2}}{1-c^{2}}}t^{\frac{-1-c^{2}}{1-c^{2}}},
\begin{array}{rcl}{\partial_{j}^{}M_{k}^{\alpha}(q,u)\!}&{=}&{\!\alpha_{j}^{}\,\Bigl(\,\zeta(u-v)\,+\,\zeta(v)\,+\,\zeta(\alpha(q))\,-\,\zeta(\alpha(q)+u)\Bigr)}\\{}&{}&{\times\,\Bigl(M_{k}^{\alpha}(q,u)-M_{k}^{\alpha}(q,v)\Bigr)~,}\\\end{array}
\psi={\frac{e^{i\varphi(x^{-})}e^{i\phi}}{2\sqrt{2\lambda+\partial_{-}^{2}f^{3}}}}\left[\begin{matrix}{2\lambda+\partial_{-}^{2}f^{3}}\\{\partial_{-}^{2}f^{1}+i\partial_{-}^{2}f^{2}}\\\end{matrix}\right]
\begin{array}{lclcl}{C_{r}^{(I)}}&{=}&{C_{-r}^{(I)}}&{=}&{{\displaystyle\frac{k}{2}~\sqrt{-\tau_{0}}{}~J_{\nu}(-r\tau_{0})}}\\{}&{}&{}&{}&{}\\{D_{r}^{(I)}}&{=}&{D_{-r}^{(I)}}&{=}&{{\displaystyle\frac{k}{2}~\sqrt{-\tau_{0}}{}~J_{-\nu}(-r\tau_{0})}}\\{}&{}&{}&{}&{}\\{C_{n}^{(I)}}&{=}&{D_{n}^{(I)}}&{=}&{0~~\mathrm{for}~n\neq\pmr}\\\end{array}
G^{\mu\nu}\,=\,\left(\begin{array}{ccccccc}{0}&{2}&{}&{}&{}&{}&{-iq_{1}}\\{2}&{0}&{}&{}&{}&{}&{i\bar{q}_{1}}\\{}&{}&{\ddots}&{}&{}&{}&{\vdots}\\{}&{}&{}&{0}&{2}&{}&{-iq_{r}}\\{}&{}&{}&{2}&{0}&{}&{i\bar{q}_{r}}\\{}&{}&{}&{}&{}&{0}&{2}\\{-iq_{1}}&{i\bar{q}_{1}}&{\ldots}&{-iq_{r}}&{i\bar{q}_{r}}&{2}&{\gamma-4\beta^{2}}\\\end{array}\right)\,,
\left\{\phi(x),\phi(y)\right\}=\phi^{\prime}(x)\phi^{\prime}(y)\epsilon(x-y)+{\frac{1}{\alpha}}\left\{\phi^{\prime}(y)-\phi^{\prime}(x)\right\}\delta(x-y)-{\frac{1}{\alpha^{2}}}\delta^{\prime}(x-y)
{\calL}=\frac{1}{2}(\partial_{\mu}\varphi)^{2}-\frac{1}{2}m^{2}\varphi^{2}-\frac{\lambda}{4!}\varphi^{4}+\frac{1}{2}\deltaZ(\partial_{\mu}\varphi)^{2}-\frac{1}{2}\deltam^{2}\varphi^{2}-\frac{1}{4!}\delta\lambda\varphi^{4},
B(x,y,z)=\sum_{r>0}{\frac{1}{r}}\left({\frac{y}{x}}\right)^{r}d+\sum_{r,r^{\prime}>0}\langle\widetilde0|{\frac{x^{-r}}{r}}\bar{c}_{r}^{i}B_{rr^{\prime}}\bar{c}_{-r}\cdot\bar{c}_{r^{\prime}}{\frac{y^{r^{\prime}}}{r^{\prime}}}\bar{c}_{-r^{\prime}}^{i}|\widetilde0\rangle\,,
E_{+}(X^{0})=2N\intd\tilde{\sigma}\frac{e^{3H(X^{0}-\tilde{\sigma})}+H^{2}L^{2}e^{-H(X^{0}-\tilde{\sigma})}}{\sqrt{e^{4H(X^{0}-\tilde{\sigma})}+e^{2H(X^{0}-\tilde{\sigma})}+H^{2}L^{2}}}.
(u^{AB}{}_{\Omega\Sigma}+v^{AB\Omega\Sigma})e_{\Omega\Sigma}^{\alpha}{\calB}_{CDEF}={\frac{2}{3}}\delta^{[A}{}_{[B}T^{C]}{}_{D]GH}+{\frac{2}{3}}\delta^{[A}{}_{[G}T^{C]}{}_{H]BD}
H_{\mathrm{eff}}=\frac{K}{2}\sum_{[x,i]}\left(\psi^{\dagger}(x)\vec{\sigma}\psi(x)\cdot\psi^{\dagger}(x+\hat{i})\vec{\sigma}\psi(x+\hat{i})+\rho(x)\rho(x+\hat{i})\right)
\begin{array}{ccc}{R_{BCD}^{A}}&{=}&{\tilde{R}_{BCD}^{A},}\\{}&{}&{}\\{R_{iBj}^{A}}&{=}&{-a_{k}D_{B}\partial^{A}a_{l}\eta_{ij}^{kl},}\\{}&{}&{}\\{R_{AjB}^{i}}&{=}&{-a^{k}D_{B}\partial_{A}a_{l}\eta_{kj}^{li},}\\{}&{}&{}\\{R_{jkl}^{i}}&{=}&{-a^{m}\partial^{A}a_{n}a_{r}\partial_{A}a_{s}(\eta_{mk}^{ni}\eta_{jl}^{rs}-\eta_{ml}^{ni}\eta_{jk}^{rs}).}\\\end{array}
\left\langle\Psi_{1}(t)\left|A\right|\Psi_{2}(t)\right\rangle=\int\frac{d^{d}{\bfq^{\prime\prime}}}{(2\pi)^{d}}\int\frac{d^{d}{\bfq^{\prime}}}{(2\pi)^{d}}\,\chi_{1}^{*}({\bfq^{\prime\prime}})\chi_{2}({\bfq^{\prime}})\,e^{-i(\omega_{\bfq^{\prime}}-\omega_{\bfq^{\prime\prime}})\,t}\,\left\langle\psi_{{\bfq}^{\prime\prime}}\left|A\right|\psi_{{\bfq^{\prime}}}\right\rangle\;.
H_{FP}\equiv\sumn\bigg[{\frac{1}{\lambda}}\big(a_{n+1}^{\ast}-a_{n-1}^{\ast}\big)+a_{n}^{\ast}+\sum_{p=1}^{n-1}a_{p}^{\ast}a_{n-p}^{\ast}\bigg]a_{n}+\mathrm{~n<0~terms}+{\frac{1}{N^{2}}}\sum_{i,j}ija_{{i}+{j}}^{\ast}a_{i}a_{j}\.
\begin{array}{rcl}{G(x_{1},x_{2})}&{=}&{\displaystyle-\frac{g^{2}}{2!}\intd^{4}x_{3}\intd^{4}x_{4}~A,}\\{A}&{=}&{\displaystyle<0|T\left(\varphi_{1}\varphi_{2}(\chi\star\varphi\star\pi+\pi\star\varphi\star\chi)_{3}\right.}\\{}&{}&{\displaystyle\times\left.(\chi\star\varphi\star\pi+\pi\star\varphi\star\chi)_{4}\right)|0>,}\\\end{array}