harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Z(u)={\frac{\left(-1+{e^{u}}\right)\,\left(1+{e^{u}}\right)\,\left({e^{2\,\eta}}+{e^{2\,u}}\right)\,{z_{1}}}{2\,{e^{2\,u}}\,\left(1+{e^{2\,\eta}}\right)}},
{H}_{\mathrm{lc}}=\int_{I\!\!R\setminus\{0\}}{\frac{dp_{+}}{|p^{+}|}}\int_{I\!\!R^{d-2}}d^{d-2}\overline{{p}}\;{\calF}^{\mathrm{\scriptscriptstyleT}}(p^{+},\overline{p})\otimes{\calF}^{\mathrm{\scriptscriptstyleL}}(q)\;\;\;,
\psi^{(+)}({\bfk},{\bfr})=\frac{1}{2\pi}\,e^{i{\bfk}\cdot{\bfr}}-\frac{i}{4}\int\,d^{3}r^{\prime}\,H_{0}^{(+)}(k|{\bfr-r^{\prime}}|)V({\bfr^{\prime}})\psi^{(+)}({\bfk},{\bfr^{\prime}})\,,
S_{parent}^{\mathrm{V}}=-\frac{T_{p}}{2}\intd^{p+1}{\xi}\left[\sqrt{-G}\left({\Phi}^{(p-1)/2}G^{ij}h_{ij}-(p-1){\Phi}^{(p+1)/2}\right)+{\Lambda}_{ij}\left(G^{ij}-g^{ij}\right)\right],
\tilde{\Gamma}_{t}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})\equiv{\Gamma}_{t}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})-g_{t}\;.
a_{n}(\omega)=\sum_{p=0}^{n}(-1)^{p}{\frac{\mu^{p}}{p!}}\bar{a}_{n},(\omega)~~~b_{n}(\omega)=\sum_{p=0}^{n}(-1)^{p}{\frac{\mu^{p}}{p!}}\bar{b}_{n}(\omega).
R^{(1)}M_{1}M_{2}K_{1}R^{(2)}K_{2}\tilde{M}_{1}^{-1}\tilde{M}_{2}^{-1}=M_{2}M_{1}K_{2}R^{(3)}K_{1}\tilde{M}_{2}^{-1}\tilde{M}_{1}^{-1}R^{(4)}
\epsilon_{\mu\nu}^{1,2}=\frac12\left[\left(e_{\mu}^{1}e_{\nu}^{1}-e_{\mu}^{2}e_{\nu}^{2}\right)\pm\left(e_{\mu}^{1}e_{\nu}^{2}+e_{\nu}^{1}e_{\mu}^{2}\right)\right],
\Gamma_{local}=\intd^{d}x\sqrt{-g}\,\left[-\Lambda_{(d)}+M_{(d)}^{d-2}\,R_{(d)}-\sum_{l=0}^{k}\left(\alpha^{(l)}R_{(d)}\Box^{l}R_{(d)}+\beta^{(l)}R_{\mu\nu(d)}\Box^{l}R_{(d)}^{\mu\nu}\right)\right]
\begin{array}{llll}{S[A,e]}&{=}&{\intd^{4}x\,tr\left(e\wedgee\wedgeF[A]\right);}&{\mathrm{~4~dimensions},}\\{S[A,e]}&{=}&{\intd^{3}x\,tr\left(e\wedgeF[A]\right);}&{\mathrm{~3~dimensions},}\\{S[A]}&{\stackrel{?}{=}}&{\intd^{2}x\,tr\left(F[A]\right);}&{\mathrm{~2~dimensions},}\\\end{array}
M_{pl}^{2}=M_{X}^{3}e^{-4k_{1}y_{1}-2c}\left({\frac{1}{\displaystylek_{0}}}-{\frac{2k_{1}}{\displaystyle{(2k_{1}+k_{0})k_{0}}}}e^{-2k_{0}y_{1}}\right)~,~\,
\begin{array}{rcl}{q(t)}&{=}&{\frac{1}{\sqrt{2m\omega}}\left[\alpha\,e^{-i\omega(t-t_{0})}+\alpha^{*}\,e^{i\omega(t-t_{0})}\right]\,}\\{}&{}&{}\\{p(t)}&{=}&{-\frac{im\omega}{\sqrt{2m\omega}}\left[\alpha\,e^{-i\omega(t-t_{0})}-\alpha^{*}\,e^{i\omega(t-t_{0})}\right]\.}\\\end{array}
S=\intd\tau\left(\lambda_{A}\bar{\lambda}_{\dot{B}}\Pi_{\tau}^{A\dot{B}}+\lambda_{A}\lambda_{B}\Pi_{\tau}^{AB}+\bar{\lambda}_{\dot{A}}\bar{\lambda}_{\dot{B}}\,\Pi_{\tau}^{\dot{A}\dot{B}}\right)\qquad,
\frac{1}{2}\eta_{B}(0)=-\frac{1}{12}\frac{r_{b}(r_{b}^{5}+10mr_{b}^{4}+65r_{b}^{3}m^{2}+168m^{3}r_{b}^{2}+168r_{b}m^{4}+48m^{5})}{(r_{b}+2m)^{4}(r_{b}+m)^{2}}
{\widetildeG}_{\underbrace{{\scriptstyle{2\dots2}}}_{m}\underbrace{{\scriptstyle{1\dots1}}}_{n-m}}(1,\dots,n)=(-i)^{n-1}\langleT[\hat{\phi}(1)\cdots\hat{\phi}(m)]{\widetildeT}[\hat{\phi}(m+1)\cdots\hat{\phi}(n)]\rangle\,.
{\calT}_{\alpha}=(i\hbar)^{-1}[\Omega,\bar{\calP}_{\alpha}]=T_{\alpha}+\frac{1}{2}\left(C^{\beta}U_{\beta\alpha}^{\gamma}\bar{\calP}_{\gamma}(-1)^{\varepsilon_{\alpha}+\varepsilon_{\gamma}}+\bar{\calP}_{\gamma}U_{\alpha\beta}^{\gamma}C^{\beta}(-1)^{\varepsilon_{\alpha}+(\varepsilon_{\alpha}+\varepsilon_{\beta}+1)\varepsilon_{\gamma}}\right).
G_{\beta}(\tau,\tau^{\prime})={\frac{1}{\beta}}\sum_{n}\int{\frac{d^{d}k}{(2\pi)^{d}}}e^{-i(\omega_{n}\tau+{\bfk}\cdot{\bfx})}G_{\beta}(\omega_{n},{\bfk}),\quad\omega_{n}=2n\pi/\beta,~n\in\mathrm{Z}.
{\frac{\mathrm{d}^{2}\xi}{\mathrm{d}t^{2}}}+\left(\omega^{2}+3\lambda\varphi^{2}\right)\xi=3\lambda\varphi\left(\langle\xi^{2}\rangle-\xi^{2}\right)+\lambda\left(\langle\xi^{3}\rangle-\xi^{3}\right)\,.
\begin{array}{c}{[K_{\mu},K_{\nu}]=0\,,~~~~~~~~~[M_{\mu\nu},K_{\lambda}]=i(\eta_{\mu\lambda}K_{\nu}-\eta_{\nu\lambda}K_{\mu})\,,}\\{{}}\\{\{S^{a\alpha},\bar{S}_{b\beta}\}=2\delta^{a}{}_{b}\gamma^{\mu}{}^{\alpha}{}_{\beta}K_{\mu}\,,}\\{{}}\\{{}[M_{\mu\nu},S^{a}]=i\textstyle{\frac{1}{2}}\gamma_{[\mu}\gamma_{\nu]}S^{a}\,,}\\{{}}\\{{}[{K}_{\mu},S^{a\alpha}]=0\,.}\\\end{array}
V\left(\prod_{i=1}^{M}a_{-m_{i}}\prod_{j=1}^{N}b_{-n_{j}}|0\rangle,z\right)=:\prod_{i=1}^{M}{\frac{i}{(m_{i}-1)!}}{\frac{d^{m_{i}}}{dz^{m_{i}}}}X^{a}(z)\prod_{j=1}^{N}{\frac{i}{(n_{j}-1)!}}{\frac{d^{n_{j}}}{dz^{n_{j}}}}X^{b}(z):\Omega^{N}\,,
\begin{aligned}{|G|^{3}\sum_{r\inG}d_{r}^{-2}\biggl\{}&{{}\lambda_{0}^{r}\mu_{0}^{r}\nu_{0}^{r}\chi_{r}(abcdef)+\lambda_{0}^{r}\mu_{1}^{r}\nu{1}^{r}\chi_{r}(abde)}\\{+}&{{}\mu_{0}^{r}\nu_{1}^{r}\lambda_{1}^{r}\chi_{r}(cdfa)+\nu_{0}^{r}\lambda_{1}^{r}\mu_{1}^{r}\chi_{r}(efbc).\biggr\}}\\\end{aligned}
\pi_{b}^{a}\pi_{a}^{b}=\frac{2P_{z}P_{\bar{z}}}{4\pi^{2}}\frac{(z_{1}-z_{2})(\bar{z}_{1}-\bar{z}_{2})}{(z-z_{1})(\bar{z}-\bar{z}_{1})(z-z_{2})(\bar{z}-\bar{z}_{2})}
s_{\Gamma}=\int\left\{\partialc\frac{\delta}{\deltaA}+B\frac{\delta}{\delta\bar{c}}+\frac{\delta\Gamma}{\delta\underline{{Y}}}\frac{\delta}{\delta\underline{{\varphi}}}+\frac{\delta\Gamma}{\delta\underline{{\varphi}}}\frac{\delta}{\delta\underline{{Y}}}+\underline{{q}}\frac{\delta}{\delta\underline{{\hat{\varphi}}}}\right\}+\chi\partial_{\xi}+\chi_{A}\partial_{\xi_{A}}
[(\phi^{\dagger}\phi)^{n}]=Z_{n}(\phi^{\dagger}\phi)^{n}+Z_{n-1}^{0}(\phi^{\dagger}\phi)^{n-1}+Z_{n-2}^{2}(\phi^{\dagger}\phi)^{n-2}(\phi^{\dagger}\partial^{2}\phi)+...\;.
{L^{+}}^{\iota^{p}}=\left(\begin{array}{cc}{K_{+1^{\iota^{p}}}}&{0}\\{E^{\iota^{p}}}&{K_{+2}^{\iota^{p}}}\\\end{array}\right),\qquad{L^{-}}^{\iota^{p}}=\left(\begin{array}{cc}{K_{-1}^{\iota^{p}}}&{^{\iota^{p}}F}\\{0}&{K_{-2}^{\iota^{p}}}\\\end{array}\right)
\pi^{+}(x^{+},x^{-})={\frac{-i}{4L}}{\sqrt{4\pi}}\Bigl(\sum_{n=0}^{\infty}b_{n}{\frac{\mathrm{{sgn}}(n+z)}{u_{n}}}\mathrm{e}^{-ik_{n}x^{-}}-\sum_{n=1}^{\infty}d_{n}^{\dagger}{\frac{{\mathrm{sgn}}(n-z)}{v_{n}}}\mathrm{e}^{+ik_{n}x^{-}}\Bigr)\.
{\cal{M}}_{IV}=\left(\begin{matrix}{1}&{1}\\{0}&{1}\\\end{matrix}\right)\cdot\left(\begin{matrix}{1}&{1}\\{0}&{1}\\\end{matrix}\right)\cdot\left(\begin{matrix}{1}&{1}\\{0}&{1}\\\end{matrix}\right)\cdot\left(\begin{matrix}{2}&{1}\\{-1}&{0}\\\end{matrix}\right)=A\cdotA\cdotA\cdotB
\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}(x)}\\{\beta_{\lambdam}(x)}\\\end{array}\right)=-\left(\lambda^{2}-\frac14\right)\,\left(\begin{array}{c}{\alpha_{\lambdam}(x)}\\{\beta_{\lambdam}(x)}\\\end{array}\right).
\frac{\partial\Phi_{n}^{*}(z^{-1})}{\partial\bar{t}_{1}}=-\frac{S_{n}^{*}}{S_{n-1}^{*}}\frac{h_{n}}{h_{n-1}}(\Phi_{n}^{*}(z^{-1})-z^{-1}\Phi_{n-1}^{*}).
\pi_{q}^{1+}=-\frac{i}{2}e^{+2}\nabla_{+}\lambda_{q}^{1}+e^{+}\lambda_{q}^{1},\\pi_{\dot{q}}^{2-}=-\frac{i}{2}e^{-2}\nabla_{-}\lambda_{\dot{q}}^{2}+e^{-}\lambda_{\dot{q}}^{2},
\left[-x\sum_{k=0}^{n_{c}-1}(-1)^{k}s_{k}(\phi^{2})x^{n_{c}-1-k}+m_{1}\cdotsm_{n_{f}}\right]^{2}-\prod_{i=1}^{n_{f}}(x+m_{i}^{2})=-\frac{\alpha_{n_{c}}^{2}}{\Lambda^{2}}x\prod_{a=1}^{n_{c}-1}(x-\alpha_{a})^{2}.
(-m_{\pi}^{2}+\partial_{\bot}^{2})\omega_{\pi}^{(1)}=\frac{1}{2L}\int_{-L}^{L}dx^{-}(\varphi_{\pi}^{3}+\varphi_{\pi}\varphi_{\sigma}^{2}+2v\varphi_{\pi}\varphi_{\sigma}).
\frac{\alpha_{U(1)crit}}{\alpha_{1}(\mu_{Planck})}=6,\mathrm{~}\frac{\alpha_{SU(2)multicr.}}{\alpha_{2}(\mu_{Planck})}=3,\mathrm{}\frac{\alpha_{SU(3)multicr.}}{\alpha_{3}(\mu_{Planck})}=3.
\langle0|a_{p}H_{\mathrm{zm}}a_{p}^{\dagger}|0\rangle=-\frac{9}{gp}\sum_{k=1}^{p-1}\frac{1}{k\left(p-k\right)}-\frac{12}{gp}\sum_{k>0}\frac{1}{k\left(k+p\right)}+\frac{6}{gp}\sum_{k=1}^{p-1}\frac{1}{k\left(p-k\right)}
\frac{\deltaC_{Q}(\kappa)}{\delta\sigma(x)}|_{\sigma=0}=-2\lambda^{2}\sqrt{g(x)}\Psi_{\kappa}(x)-2\intd\muC_{Q}(\mu)\Psi_{\kappa}(x)\Psi_{\mu}(x)\,\frac{\kappa^{2}}{\kappa^{2}-\mu^{2}}\sqrt{g(x)}
{\calG}=M_{ab}d{\bfx}_{a}\cdotd{\bfx}_{b}+\frac{16\pi^{2}}{e^{4}}(M^{-1})_{ab}(d\xi_{a}+{\bfW}_{ac}\cdotd{\bfx}_{c})(d\xi_{b}+{\bfW}_{bd}\cdotd{\bfx}_{d}),
S=\intd^{4}x\sqrt{g}\Bigl\{\frac{R}{2\kappa^{2}}-\frac{1}{\kappa^{2}}\partial_{\mu}\phi^{\prime}\partial^{\mu}\phi^{\prime}-\frac{1}{2\kappa^{2}}e^{-2\phi^{\prime}}F^{2}\Bigr\}
\psi_{R_{1}f_{1},R_{2}f_{2}}^{\alpha_{1}^{1}\alpha_{1}^{2},\alpha_{2}^{1}\alpha_{2}^{2}}(x_{1},x_{2})=-\psi_{R_{2}f_{2},R_{1}f_{1}}^{\alpha_{2}^{1}\alpha_{2}^{2},\alpha_{1}^{1}\alpha_{1}^{2}}(x_{2},x_{1}).
\widehat{L}^{2}={\frac{1}{2}}\Bigl(\widehat{\ell}_{+}\widehat{\ell}_{-}+\widehat{\ell}_{-}\widehat{\ell}+\Bigr)+{\widehat{\ell}_{z}}^{2}=-f_{A}^{2}\;{\frac{\partial^{2}}{\partial\zeta_{A}\partial\zeta_{A}^{c}}}\;.
(\tilde{q}_{1},p_{1})S\left(\begin{array}{c}{\tilde{q}_{2}}\\{p_{2}}\\\end{array}\right)=(q_{1},\tilde{p}_{1})S\left(\begin{array}{c}{q_{2}}\\{\tilde{p}_{2}}\\\end{array}\right)=Q_{1}P_{2}-Q_{2}P_{1}\;.
\Omega=\left(\begin{array}{ccccccc}{0}&{\lambda_{1}}&{}&{}&{}&{}&{}\\{-\lambda_{1}}&{0}&{}&{}&{}&{}&{}\\{}&{}&{.}&{}&{}&{}&{}\\{}&{}&{}&{.}&{}&{}&{}\\{}&{}&{}&{}&{.}&{}&{}\\{}&{}&{}&{}&{}&{0}&{\lambda_{m}}\\{}&{}&{}&{}&{}&{-\lambda_{m}}&{0}\\\end{array}\right)
a_{n}^{\alpha\dot{1}(\dot{\beta})}=a_{n}^{\alpha\dot{2}(\dot{\beta})}=\frac{1}{\omega_{n}}{\calD}^{\alpha\dot{\beta}}\,\phi_{n}\,,\qquad\lambda_{n}^{\alpha(\dot{\beta})}=\frac{1}{\omega_{n}}{\calD}^{\alpha\dot{\beta}}\,\phi_{n}\,.
\left|\begin{array}{l}{R}\\{J}\\\end{array}\right\rangle=\sum_{J_{1},J_{2}}\left\langle\begin{array}{ll}{R_{1}}&{R_{2}}\\{J_{1}}&{J_{2}}\\\end{array}\right.\left|\begin{array}{l}{R}\\{J}\\\end{array}\right\rangle\left|\begin{array}{ll}{R_{1}}&{R_{2}}\\{J_{1}}&{J_{2}}\\\end{array}\right\rangle
({\hat{\gamma^{\prime}}}^{(\lambda,\mu,\alpha)})={\hat{P}}_{+}({\hat{\gamma}}_{(\lambda,\mu,\alpha)})=({\hat{\gamma}}^{(\lambda,\mu,\alpha)}),\quad({\hat{\gamma^{\prime}}}_{(\lambda,\mu,\alpha)})=({\hat{\gamma}}_{(\lambda,\mu,\alpha)}){\hat{P}}_{+}^{T}=({\hat{\gamma}}_{(\lambda,\mu,\alpha)}),
V\left(\Phi\right)=-m^{2}\mathrm{tr}\left({\Phi^{2}}\right)+\gamma\mathrm{tr}\left({\Phi^{3}}\right)+\lambda_{1}\mathrm{tr}\left({\Phi^{4}}\right)+\lambda_{2}\left[\mathrm{tr}\left({\Phi^{2}}\right)\right]^{2}.
\left(\begin{array}{cc}{\vec{m}_{1}}&{\vec{n}_{1}}\\\end{array}\right)\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\left(\begin{array}{c}{\vec{m}_{2}}\\{\vec{n}_{2}}\\\end{array}\right)=\vec{m}_{1}\cdot\vec{n}_{2}+\vec{n}_{1}\cdot\vec{m}_{2}.
{\calL}^{(1)}=\left(\frac{\kappa}{2\pi}\epsilon^{mn}A_{n}\right)\dot{A}_{m}+\pi_{a}\dot{z}_{a}+\pi_{a}^{*}\dot{z}_{a}^{*}+\Omega_{1}^{(0)}\dot{\alpha}+\Omega_{2}^{(1)}\dot{\beta}+\Omega_{3}^{(1)}\dot{\gamma}-{\calH}^{(1)}
S=S_{TMGT}^{[U(1)]}+S_{TMG}+\int_{\calM}d^{3}x~\sqrt{g}\,\left(g^{\mu\nu}[(\partial_{\mu}-iA_{\mu})\phi]^{*}[(\partial_{\nu}-iA_{\nu})\phi]-m^{2}~\phi^{*}\phi\right)
\mathrm{Tr}ZZ^{\dagger}=\lambda_{1}^{2}+\lambda_{2}^{2},\quad\mathrm{Tr}(ZZ^{\dagger})^{2}=\lambda_{1}^{4}+\lambda_{2}^{4},\quad\mathrm{det}ZZ^{\dagger}=\lambda_{1}^{2}\lambda_{2}^{2}
\frac{\partial\alpha^{l}}{\partial\bar{y}}+2\frac{\partial\tau}{\partialz}\alpha^{l}=\frac{\partial\alpha^{l+1}}{\partialz}\;,\frac{\partial\alpha^{l}}{\partial\bar{z}}-2\frac{\partial\tau}{\partialy}\alpha=\frac{\partial\alpha^{l+1}}{\partialy}
p^{2}-i\hbar\alphay\frac{dp}{dy}=E-A^{2}-\frac{B^{2}}{A^{2}}-\frac{4A(A-\alpha\hbar)y^{2}}{(y^{2}-1)^{2}}+\frac{2B(y^{2}+1)}{(y^{2}-1)}.
\alpha^{\prime}e^{-2\phi^{\prime}}\hat{R}^{2}=\alpha^{\prime}\frac{16G^{2}M^{2}(2GM-3\rho)}{\rho^{6}(\rho-2GM)}e^{-2\phi^{\prime}}=\alpha^{\prime}\frac{32G^{4}M^{4}}{\rho^{5}}\frac{(2GM-3\rho)}{Q_{m}^{2}\rho^{2}+Q_{e}^{2}x^{2}}
\frac{P_{D}}{P_{Q}}=\frac{5}{12}*\frac{4}{\alpha_{e}^{2}}*\frac{e^{2}}{4\pi\varepsilon_{0}\gammam_{e}^{2}}=\frac{5}{3\alpha_{e}^{2}}*\frac{W_{[E]}}{W_{[M]}}
+\frac{\tau_{3}}{4}\int\left[C_{4}\left(1+\frac{1}{2\pi^{2}\alpha^{\prime}}\sum_{i=1}^{3}{\tilde{b}}_{i}\right)+\frac{1}{2\pi^{2}\alpha^{\prime}}\sum_{i=1}^{3}A_{4}^{i}\right]
L_{+}={\frac{1}{2}}(w_{+}\epsilon_{ij}x_{i}\dot{x}_{j}-w_{+}^{2}x_{i}^{2})~;~~~L_{-}={\frac{1}{2}}(-w_{-}\epsilon_{ij}y_{i}\dot{y}_{j}-w_{-}^{2}y_{i}^{2})~,
\left[\bar{x}_{1},p_{1}\right]=i,\qquad\left[a_{n}^{1},{a_{m}^{1}}^{\dagger}\right]=\delta_{nm},\qquad\left[a_{n}^{2},{a_{m}^{2}}^{\dagger}\right]=\delta_{nm}\qquad\textrm{and}\quad\left[a_{0},{\bar{a}_{0}}\right]=1.
\frac{\partialF}{\partialx^{1}}=\phi_{1}^{3}+\sum_{j=2}^{3}\phi_{j}^{3}f_{1}^{j},\quad\quad\frac{\partialF}{\partialt}=\phi_{t}^{3}+\sum_{j=2}^{3}\phi_{j}^{3}f_{t}^{j}.
\prod_{i=1}^{n}\theta_{p_{i},u}(\tau,\frac{\sigma}{u})=\sum_{r=0}^{n-1}\bigl\{\sum_{D(n_{1},..,n_{n-1};r)}\prod_{i=1}^{n-1}\theta_{P(0;n-i)-2u\tilde{n}_{i},(n-i)(n-i+1)u}(\tau)\bigr\}\theta_{\bar{p}_{n}-2ur,nu}(\tau,\frac{\sigma}{u}).
\lbrackz^{k},P_{i}\rbrack=i\delta_{i}^{k}\qquad\lbrack\bar{z}^{\bar{l}},\bar{P}_{\bar{j}}\rbrack=i\delta_{\bar{j}}^{\bar{l}}\qquad\{\chi^{i\alpha},\bar{\chi}_{\beta}^{\bar{j}}\}=g^{i\bar{j}}\delta_{\beta}^{\alpha}
\chi_{\mu}(C_{m}^{\Lambda})=\sum_{\nu\inM(\Lambda)}S_{\nu}^{\Lambda}(q^{2\{g+m(g+\kappa_{\Lambda}+\kappa_{\mu})\}})\,q^{-m\beta_{\nu}(\mu)}\frac{D_{q}[\mu+\nu]}{D_{q}[\mu]}
\langle\overline{{\phi}}_{1},\overline{{\phi}}_{2}\rangle=\sum_{\gamma}\int_{R_{\gamma}}\dv\\overline{{\phi}}_{2}\phi_{1}^{*}=\int_{R_{0}}\dv\\overline{{\phi}}_{2}\sum_{\gamma}(\gamma\phi_{1}^{*})=\int_{R_{0}}\dv\\overline{{\phi}}_{2}(\overline{{\phi}}_{1})^{*},
(P_{12})_{ab}=\langleV_{a}^{++}(1)\vertV_{b}^{++}(2)\rangle=(D_{1}^{+})^{4}\left({\frac{\delta^{8}(\theta_{1}-\theta_{2})}{x_{12}^{2}}}\right)\delta^{(-2,2)}(u_{1},u_{2})\delta_{ab}\.
<\Psi_{np}^{j},\Psi_{n^{\prime}p^{\prime}}^{j^{\prime}}>=\int_{H_{3}^{+}}dh\left(\Psi_{np}^{j}(h)\right)^{\ast}\Psi_{n^{\prime}p^{\prime}}^{j^{\prime}}(h)=(2\pi)^{3}\delta_{n,n^{\prime}}\delta(p-p^{\prime})i\delta(j-j^{\prime}),
{\widetilde\phi}_{0.i}\approx0,\quad{\widetilde\Phi}_{0.i}\approx0,\quad{\widetilde\Phi}_{0.0}\approx0,\quad{\widetilde\phi}_{i.j}\approx0,\quad{\widetilde\Phi}_{i.j}-c^{2}\delta_{ij}\approx0.
\delta\{\chi^{ij}\}=\{\delta\chi^{ij}\}=\left(\begin{array}{ccccc}{ap+bq+rc}&{a}&{b}&{c}&{ap+bq+rc}\\{a}&{0}&{0}&{0}&{a}\\{b}&{0}&{0}&{0}&{b}\\{c}&{0}&{0}&{0}&{c}\\{ap+bq+rc}&{a}&{b}&{c}&{ap+bq+rc}\\\end{array}\right)
\int_{B_{1}}\left(\kappa_{1}\,b_{\mu\nu}^{0}(x)T^{\mu\nu}+\kappa_{1}\sum_{n=1}^{\infty}\frac{\psi_{n}(0)}{N_{0}}\,b_{\mu\nu}^{n}(x)T^{\mu\nu}-\frac{\kappa_{2}}{\sqrt{3}}\phi\,T_{\mu}^{\mu}\right)dx.
\begin{array}{lcr}{U(1)^{r}:X_{j}\longrightarrowX_{j}^{\prime}=\lambda^{q_{a}^{j}}X_{j}}\\{\qquadY_{j}\longrightarrowY_{j}^{\prime}=\lambda^{q_{a}^{j}}y_{j}}\\{\qquad\eta_{j}^{+}\longrightarrow\eta\prime_{j}^{+}=\eta_{j}^{+}}\\{\qquad\overline{{\eta}}_{j}^{+}\longrightarrow\overline{{\eta}}\prime_{j}^{+}=\overline{{\eta}}_{j}^{+};}\\\end{array}
d{\tilde{\rho}}_{K}\left(X\right)=\sum_{k=1}^{2K}{\overbrace{{\tilde{\rho}}_{1/2}\left(1\right)\otimes\cdots\otimes{\tilde{\rho}}_{1/2}\left(1\right)}^{k-1}\otimesd{\tilde{\rho}}_{1/2}\left(X\right)\otimes\overbrace{{\tilde{\rho}}_{1/2}\left(1\right)\otimes\cdots\otimes{\tilde{\rho}}_{1/2}\left(1\right)}^{2K-k}}\,
\begin{array}{l}{r_{t}+s_{xx}+2r_{x}s_{x}=0,}\\{s_{t}+2\mu_{2}r_{xx}+\mu_{1}s_{xx}+4(\mu_{2}+\mu_{5})r_{x}^{2}}\\{\quad+2(\mu_{1}+\mu_{4})r_{x}s_{x}+\mu_{3}s_{x}^{2}+\mu_{0}V(x)=0,}\\\end{array}
\def\arraystretch{1.1}\begin{array}{lcl}{\vec{\beta}_{1}=(1,0,0)}&{\mathrm{given~by}}&{\zeta}\\{\vec{\beta}_{2}=(2,0,-1)}&{\mathrm{\tt~"}}&{\zeta^{2}}\\{\vec{\beta}_{3}=(4,-1,-2)}&{\mathrm{\tt~"}}&{\zeta^{4}}\\{\vec{\beta}_{4}=(5,-1,-3)}&{\mathrm{\tt~"}}&{\zeta^{5}}\\{\vec{\beta}_{5}=(8,-2,-5)}&{\mathrm{\tt~"}}&{\zeta^{8}.}\\\end{array}
\Sigma_{\\\\nu}^{(2)\mu}=-{\frac{l^{2}}{2}}\left({\frac{y}{a^{4}}}+{\frac{l}{2a^{2}}}\right){\calS}_{\\nu}^{\mu}-{\frac{l^{3}}{24a^{2}}}\left(RR_{\\nu}^{\mu}-{\frac{1}{4}}\delta_{\nu}^{\mu}R^{2}\right)+{\frac{l^{3}}{a^{4}}}t_{\\nu}^{\mu}\,
\hat{\Sigma}(P)=\hat{M}^{-1}(P)\,\hat{\Sigma}_{F}(P)\,\hat{M}^{-1}(P)\;\;\;\;\;\;\;\hat{\Sigma}_{F}(P)=\mathrm{diag}\left(\Sigma_{F}(P)\,-\Sigma_{F}^{*}(P)\right)\,.
p_{a}^{\theta}:=\frac{\overrightarrow{\partial}L}{\partial{\dot{\theta}^{a}}}=\epsilon^{2}\theta^{a},\;\;p_{a}:=\frac{\partialL}{\partial\dot{x}^{a}}=\frac{2}{N}(\dot{x}_{a}-Mp^{\thetaa}),
\mathrm{Re}\,\Pi_{00}^{\mathrm{space}-\mathrm{like}}(k_{0},k)={\frac{2}{\pi}}\int_{0}^{k}d\omega\,{\frac{\omega\mathrm{Im}\,\Pi_{00}(\omega,k)}{\omega^{2}-k_{0}^{2}}}={\frac{2}{\pi}}\int_{0}^{1}dv\,{\frac{v\mathrm{Im}\,\Pi_{00}(vk,k)}{v^{2}-{\frac{k_{0}^{2}}{k^{2}}}}}
\!\!\!\!\!\!\!\!B_{N}(\lambda)+\frac{N}{2}\frac{\mathrm{d}V^{(\alpha)}(\lambda)}{\mathrm{d}\lambda}=\frac{\lambda}{2c_{N}}A_{N}(\lambda)-(-1)^{N}\frac{\alpha}{\lambda}-\frac{\alpha}{2}(T_{B}(N-1,\lambda)-T_{B}(N,\lambda))\.
T_{a}^{L}=\left(\begin{array}{ccc}{N_{a}}&{E_{a}^{\dagger}}&{F_{a}^{\dagger}}\\{E_{a}}&{P_{a}}&{G_{a}^{\dagger}}\\{F_{a}}&{G_{a}}&{b_{a}}\\\end{array}\right)~,~~~~~v=\left(\begin{array}{c}{u}\\{w}\\{z}\\\end{array}\right)~,
\left[\,{\frac{1}{2m_{p+1}(p+1)!}}\,{\frac{\partial^{2}}{\partial\sigma^{\mu_{1}\dots\mu_{p+1}}\,\partial\sigma_{\mu_{1}\dots\mu_{p+1}}}}\,\right]\,K\left[\,\sigma-\sigma_{0}\;\Omega_{p+1}\,\right]=-i{\frac{\partial}{\partial\Omega_{p+1}}}\,K\left[\,\sigma-\sigma_{0}\;\Omega_{p+1}\,\right]
\displaystyle|\mathrm{phys}{\rangle}={\Pi}_{n=1}^{N}exp\left[i{\int}_{P_{n}~({\Gamma}_{n})}^{Q_{n}}{\sum}_{i=1,2}\hat{A}_{i}^{(n)}({\bfx})dx^{i}\right]|0{\rangle}~,
H_{LC}=\int_{0}^{l}\mathcal{H}_{LC}=\int_{0}^{l}d\sigma\,\frac{G_{+-}}{2\,\pi_{-}}\left(G^{ij}\pi_{i}\pi_{j}+\frac{1}{(2\pi\alpha^{\prime})^{2}}G_{ij}Z^{i\,\prime}Z^{j\,\prime}\right)
\frac{\lambda}{4}L\upsilon^{2}\;\mathrm{tr}\left[(\nabla^{2}+m^{2})^{-1}\right]=\frac{\lambda}{4}\left(\int_{0}^{L}dx_{\bot}\upsilon^{2}\right)\;\mathrm{tr}\left[(\nabla^{2}+m^{2})^{-1}\right].
n_{{\bf{k}}}=n^{\beta}({\bf{k}})\frac{N}{\langleN\rangle}+\sum_{{\bf{q}}}a_{{\bf{k}}-{\bf{q}}/2}^{\dagger}({\bf{q}})a_{{\bf{k}}-{\bf{q}}/2}({\bf{q}})-\sum_{{\bf{q}}}a_{{\bf{k}}+{\bf{q}}/2}^{\dagger}({\bf{q}})a_{{\bf{k}}+{\bf{q}}/2}({\bf{q}})
\left(U(g)f\right)(z)=\epsilon(g;z)(\alpha+\overline{{\beta}}\overline{{z}})^{-\frac{1}{2}}(\overline{{\alpha}}+\betaz)^{-\frac{1}{2}}f\left(\frac{\alphaz+\overline{{\beta}}}{\betaz+\overline{{\alpha}}}\right)
\{f,g\}_{0}^{\mathrm{can}}=\sum_{i=1}^{N}\left(\frac{\partialf}{\partialx^{i}}\frac{\partialg}{\partialx^{i+N}}-\frac{\partialf}{\partialx^{i+N}}\frac{\partialg}{\partialx^{i}}\right)+\sum_{\alpha=1}^{M}\epsilon_{\alpha}\frac{\partial_{r}f}{\partial\theta_{\alpha}}\frac{\partial_{l}g}{\partial\theta_{\alpha}},\;\;\epsilon_{\alpha}=\pm1,
\Theta_{\Lambda}\left[\begin{array}{c}{a}\\{b}\\\end{array}\right](v^{I}+\taup^{I}+q^{I}|\tau)=e^{-i\pi\tauq^{2}-2\pii\,p\cdot(v+b)+2\piia\cdotq}\Theta_{\Lambda}\left[\begin{array}{c}{a}\\{b}\\\end{array}\right](v^{I}|\tau).
{\hat{T}}_{\\nu}^{\mu}[T]=-{\frac{1}{2}}e^{\Phi}\left({\frac{\Lambda\rho_{\perp}}{\sqrt{-g}}}\right)V(T)\sqrt{-A}\left[(A^{-1})^{\alpha\mu}g_{\alpha\nu}-{\frac{1}{8}}g_{\nu}^{\mu}(A^{-1})^{\lambda\sigma}g_{\lambda\sigma}\right]\,.
S_{g}=\intd^{D}x\sqrt{|g_{D}|}e^{-\varphi}\left[R_{D}+\left(\nabla\varphi\right)^{2}+\frac{1}{4}\nablah^{ab}\nablah_{ab}-\frac{1}{4}h_{ab}{F_{\mu\nu}}^{a}F^{\mu\nub}\right]
\langle\psi(q_{1})\psi(q_{2})\prod_{i}k_{i}\cdot\psi(z_{i})\,\epsilon\cdot\psi(z_{i})\rangle\proptoS(q_{1},z_{1})S(z_{1},z_{2})S(z_{2},z_{3})S(z_{3},q_{2})+\cdots.
<\Psi_{1}|\Psi_{2}>=\int\prod_{\alpha}d\bar{z}_{\alpha}d\mu(\bar{\xi}_{\alpha},\xi_{\alpha})\bar{\Psi}_{1}(z,\bar{\xi})U^{\dagger}(z,\bar{\xi})U(z,\xi)\Psi_{2}(z,\xi)e^{-W(\bar{\xi},\xi)}.
D_{+i}(q)=\frac{i}{q^{2}+i{\epsilon}}\left(\frac{n_{+}q_{i}}{q_{-}+i{\epsilon}\mathrm{sgn}(q_{+})}-\frac{n^{2}q_{+}q_{i}}{(q_{-}+i{\epsilon}\mathrm{sgn}(q_{+}))^{2}}\right).
S_{ij}=\left(\begin{array}{cc}{x}&{y}\\{y}&{y^{2}/x}\\\end{array}\right)\;\;x\neq0\;\mathrm{on~}\S{SO(N-1)}^{(A)},\;\;\;S_{ij}=\left(\begin{array}{cc}{y^{2}/z}&{y}\\{y}&{z}\\\end{array}\right)\;\;z\neq0\;\mathrm{on~}\S{SO(N-1)}^{(B)}.
q_{nk}^{{\mathrm{(in)}}l}=\frac{1}{\sqrt{2\omega_{lk}^{{\mathrm{out}}}}}\left(\alpha_{nk}^{l}e^{-\imath\omega_{lk}^{{\mathrm{out}}}t}+\beta_{nk}^{l}e^{\imath\omega_{lk}^{{\mathrm{out}}}t}\right).
\Omega=\frac{k\psi^{2}}{8\pi}\inttr(\tilde{u}^{-1}\delta\tilde{u}\partial_{x}(\tilde{u}^{-1}\delta\tilde{u})+4i\vec{p_{l}}\vec{H}\tilde{u}^{-1}\delta\tilde{u}\tilde{u}^{-1}\delta\tilde{u}-d\vec{p_{l}}\vec{H}\tilde{u}^{-1}\delta\tilde{u}-
|{\bar{\varphi}}_{i}\,\rangle=\left[1+\left(E_{i}-H+i\epsilon\,\right)^{-1}H_{\mathrm{\scriptsize~I}}\right]|i\,\rangle+i\epsilon\,\left(E_{i}-H+i\epsilon\,\right)^{-1}\left(\Xi-1\right)|i\,\rangle.
{\calL}_{\mathrm{SCS}}=-\frac{1}{2}\epsilon^{ij}\left(A_{i}^{(0)}+\sum_{n=1}^{\infty}A_{i}^{(n)}\right)\left(\dot{A}_{j}^{(0)}+\sum_{n=1}^{\infty}\dot{A}_{j}^{(n)}\right)+A_{0}^{(0)}\epsilon^{ij}\partial_{i}\left(A_{j}^{(0)}+\sum_{n=1}^{\infty}A_{j}^{(n)}\right)
\partial_{i}B^{i(1)}\equiv+\sqrt{L_{4}L_{5}}\partial_{i}((1/2!)\epsilon^{imn054}F_{mn4})\equiv-\sqrt{L_{4}L_{5}}\partial_{i}\,^{*}\!F^{i05}=-g^{\prime}\sqrt{\frac{L_{5}}{L_{4}}}\delta^{(3)}({\bfx}-{\bfb}),
X_{\vec{b}}^{\vec{a}}:={\frac{1}{b_{1}!b_{2}!b_{1}^{\prime}!b_{2}^{\prime}!}}\intd^{4}\alphae^{-i\sum\alpha_{j}b_{j}}\;q^{\vec{a}}\left({\frac{\partial}{\partialq}}\right)^{\vec{b}}X(\alpha_{1},\alpha_{2},\alpha_{1}^{\prime},\alpha_{2}^{\prime})
xy^{2}=\left[x\prod_{a=1}^{n_{c}}(x-\phi_{a}^{2})+2\Lambda^{2n_{c}+2-n_{f}}m_{1}\cdotsm_{n_{f}}\right]^{2}-4\Lambda^{2(2n_{c}+2-n_{f})}\prod_{i=1}^{n_{f}}(x+m_{i}^{2}).
g(z)=\left(\begin{array}{cc}{1}&{0}\\{X^{-}}&{1}\\\end{array}\right)\left(\begin{array}{cc}{:e^{-\fracu{k-2}}:}&{0}\\{0}&{:e^{\fracu{k-2}}:}\\\end{array}\right)\left(\begin{array}{cc}{1}&{X^{+}}\\{0}&{1}\\\end{array}\right)
\frac{\langle\langle\psi^{\prime}|e^{-\hat{H}\tau_{f}}\hat{O}_{1}(\tau_{1})\hat{O}_{2}(\tau_{2})\ldots\hat{O}_{n}(\tau_{n})e^{\hat{H}\tau_{i}}|\psi\rangle\rangle}{\langle\langle\psi^{\prime}|e^{-\hat{H}(\tau_{f}-\tau_{i})}|\psi\rangle\rangle}\,.
0=\delta\psi_{\mu}^{i}=\partial_{\mu}\epsilon^{i}+\frac14\omega_{\mu}^{ab}\Gamma_{ab}\epsilon^{i}-\frac1{8}e_{\mu}{}^{a}F_{abc}^{1}\Gamma^{bc}(\Gamma^{\prime})^{i}{}_{j}\epsilon^{j}\,.
\begin{array}{rl}{\mathrm{Bekenstein-Verlinde~bound}:}&{S_{BV}=\frac{2\pi}{n}E{\calR},}\\{\mathrm{Bekenstein-Hawking~bound}:}&{S_{BH}=(n-1)\frac{V}{4G_{n+1}{\calR}},}\\{\mathrm{Hubble~bound}:}&{S_{H}=(n-1)\frac{HV}{4G_{n+1}}.}\\\end{array}
\chi_{\tilde{w}(\mu)}(C_{m}^{\Lambda})=\sum_{\nu\inM(\Lambda)}S_{\nu}^{\Lambda}(q^{2\{g+m(g+\kappa_{\Lambda}+\kappa_{\mu})\}})\,q^{-m\beta_{w^{-1}(\nu)}(\mu)}\frac{D_{q}[\mu+w^{-1}(\nu)]}{D_{q}[\mu]}\;.
\widehat{V}^{<i>}(z)\equiv\frac{\left(\widehat{F}^{<i>}(z)\right)^{p_{i}m_{<j>}}}{\left(p_{i}m_{<j>}\right)!}=\prod_{i\in<i>}\frac{\left(\widehat{F}^{i}(z)\right)^{m_{<j>}}}{m_{<j>}!}\,.
[\hat{\calH}_{0}^{\pm}(n),\hat{\calH}_{0}^{\pm}(m)]_{-}=\pm\frac{2\pi}{2\pi}{L}(n-m)\hat{\calH}_{0}^{\pm}(n+m)\pm\frac{1}{3}(\frac{2\pi}{2\pi}{L})^{2}n(n^{2}-1)\delta_{n,-m},\,
\pi_{n+1}^{*}(x)-\pi_{n}^{*}(x)\,(x-\check{S}_{n^{n+1}}^{-1}\,\check{S}_{n^{n}(n+1)}+\check{S}_{(n-1)^{n}}^{-1}\,\check{S}_{(n-1)^{n-1}n})+\check{S}_{(n-1)^{n}}^{-1}\,\check{S}_{n^{n+1}}\,\pi_{n-1}(x)=0
\left(\begin{array}{cc}{\delta}&{\beta}\\{\gamma}&{-\delta^{t}}\\\end{array}\right)=\left(\begin{array}{cc}{\alpha}&{\beta}\\{\beta}&{\alpha}\\\end{array}\right)+\left(\begin{array}{cc}{\sigma}&{0}\\{0}&{-\sigma}\\\end{array}\right)+\left(\begin{array}{cc}{0}&{0}\\{\epsilon}&{0}\\\end{array}\right)\;,
\varepsilon_{0}=\frac{m^{3}}{3\pi}-\frac{m^{3}}{2\pi}\frac{1}{\Gamma(-1/2)}\sum_{n=1}^{m}\frac{B_{2n}}{(2n)!}\left(\frac{2eB}{m^{2}}\right)^{2n}\Gamma(2n-3/2)+O\left[\left(\frac{2eB}{m^{2}}\right)^{2(m+1)}\right].
G_{N}^{(\sigma_{j}i)}(z,z^{\prime}|\cdots,x_{\sigma(j+1)},x_{\sigma(j)},\cdots)=R_{\sigma(j),\sigma(j+1)}^{V^{\sigma(j)},V^{\sigma(j+1)}}G_{N}^{(\sigmai)}(z,z^{\prime}|\cdots,x_{\sigma(j)},x_{\sigma(j+1)},\cdots),
[\delta_{1},\delta_{2}]\chi_{i}=2iD_{\mu}\chi_{i}\overline{{\eta}}_{1}^{j}\gamma^{\mu}\eta_{2j}+3aM_{ij}\chi_{j}\overline{{\eta}}_{1}^{k}\eta_{2k}+\frac{a}{2}\gamma^{\mu\nu}\chi_{i}M_{kj}\overline{{\eta}}_{1}^{k}\gamma_{\mu\nu}\eta_{2j},
V\Big[\begin{array}{cc}{\alpha_{1}+b_{3}}&{\alpha_{2}+b_{4}}\\{\beta_{1}}&{\beta_{2}}\\\end{array}\Big]=0,\quad\mathrm{for}\quad\beta_{1}=F+b_{1}+b_{2}\quad\mathrm{or}\quad\beta_{2}=F+b_{1}+b_{2},
\langleT_{\mu\nu}^{l}(x_{1})T_{\rho\sigma}^{l}(x_{2})\rangle=\tilde{C}_{T}\left[g_{\mu\rho}(x_{12})g_{\nu\sigma}(x_{12})+g_{\mu\sigma}(x_{12})g_{\nu\rho}(x_{12})-\frac{2}{D}\delta_{\mu\nu}\delta_{\rho\sigma}\right]\frac{1}{(x_{12}^{2})^{l}},