harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left({\frac{\dot{a}_{0}}{a_{0}}}\right)^{2}={\frac{32\pi^{2}G_{5}^{2}}{9c^{6}}}(\varrho^{2}c^{4}+2\sigma\varrhoc^{2})+{\frac{c^{2}a_{R\,0}^{\prime\prime}}{a_{0}}}+{\frac{8\piG_{5}}{3c^{2}}}\left(\Lambda+{\frac{4\piG_{5}}{3c^{4}}}\sigma^{2}\right)-{\frac{kc^{2}}{a_{0}^{2}}},
\Phi_{1}^{(1)}=\pi_{e},\quad\,\Phi_{2}^{(1)}=\pi_{\chi},\,\quad\Phi_{3\mu}^{(1)}=\pi_{\mu}-\frac{i}{2}\xi_{\mu},\quad\,\Phi_{4\mu}^{(1)}=\pi_{\mu}^{b}
p_{\perp}:=\sqrt{p^{2}-p_{3}^{2}}=\sqrt{E_{p}^{2}-M^{2}-p_{3}^{2}},\quads=\pm\sqrt{1+{\frac{p_{3}^{2}}{M^{2}}}},\quad\epsilon_{3}:=\mathrm{sign}(sp_{3}),
\Psi_{+}=e^{-\frac{xy^{2}}{2}}\left(\begin{array}{c}{0}\\{1}\\\end{array}\right)\quad\mathrm{and}\quad\Psi_{-}=e^{+\frac{xy^{2}}{2}}\left(\begin{array}{c}{1}\\{0}\\\end{array}\right)\;,
\hat{\mu}_{i}^{3}=\left(\bar{z}_{+}^{M}\,\,\bar{z}_{-}^{M}\right)T_{i}\,\Big(\begin{array}{cc}{\mathrm{e}^{V}}\\\end{array}\Big)\left(\begin{array}{c}{z_{+}^{M}}\\{z_{-}^{M}}\\\end{array}\right)=0\.
S_{J,\Phi}=\intd^{2D}X\,Tr\left(-\frac{i}{3}\mathcal{J}\star\mathcal{J}\star\mathcal{J}-\mathcal{J}\star\mathcal{J}\mathbf{+\,}i\mathcal{J}\star\Phi\star\bar{\Phi}-V_{\star}\left(\Phi\star\bar{\Phi}\right)\right).
\oint_{\bar{w}}d\bar{z}\Phi_{1}(x,\bar{x};z,\bar{z})(\bar{j}(\bar{y};\bar{w})\Phi_{1}(y,\bar{y};w,\bar{w}))=-2(\bar{x}-\bar{y})\delta^{2}(x-y)-(\bar{x}-\bar{y})^{2}\partial_{\bar{x}}\delta^{2}(x-y)=0.
\Omega_{1}^{(2)}(x)=\intd^{2}y\lambda_{i_{2}}^{(2)}(x,y)L_{i_{2}}^{(2)}(y)=2\partial^{1}\alpha_{9}^{(2)}+2\partial^{2}\alpha_{10}^{(2)}+\alpha_{12}^{(2)}-\alpha_{13}^{(2)}=0.
\ddot{X}^{\mu}-X^{\mu}+\Gamma_{\rho\sigma}^{\mu}(\dot{X}^{\rho}\dot{X}^{\sigma}-X^{\rho}X^{\sigma})+H_{\rho\sigma}^{\mu}(\dot{X}^{\rho}X^{\sigma}-\dot{X}^{\sigma}X^{\rho})=0,
A_{ab}^{mn}\ddot{c}_{b}^{n}+\frac{\partialA_{ab}^{mn}}{\partialc_{c}^{r}}\dot{c}_{c}^{r}\dot{c}_{b}^{n}+\frac{\partialA_{ab}^{mn}}{\partialc_{c}^{r*}}\dot{c}_{c}^{r*}\dot{c}_{b}^{n}-\frac{\partialA_{cb}^{rn}}{\partialc_{a}^{m*}}\dot{c}_{c}^{r*}\dot{c}_{b}^{n}=0,
S_{eff}=\int{d^{5}\sigmad\tau[\mu\dot{C}+\mu_{\rho}\dot{C}^{\rho}+P^{\mu\nu}\dot{B}_{\mu\nu}+\Pi_{a}\dot{X}^{a}+\hat{\mu}^{i}\dot{\hat{C}}_{i}+\widehat{\mu}\dot{\widehat{C}}+\frac{1}{\sqrt{W}}\hat{\delta}\mu]}
V(\lambda,\kappa,0)\otimesV(\lambda_{0},\kappa_{0},0)=\bigoplus_{(\lambda+\lambda_{0}^{\prime},\kappa+\kappa_{0},-s)\inD^{+}}\bigoplus_{s\geq0}m_{\lambda_{0}^{\prime},s}V(\lambda+\lambda_{0}^{\prime},\kappa+\kappa_{0},-s)
S_{eff}^{(bm)}=\kappaL^{2}\betat\left[(1+\sigma_{0})^{1/2}(1+\sigma_{1})-\left(\frac{\lambda_{0}}{2}\sigma_{0}+\lambda_{1}\sigma_{1}\right)\right]-\frac{2\pi(d-3)L^{2}\zeta^{\prime}(-2)\lambda_{0}}{\lambda_{1}\beta^{2}t^{2}}.
C_{2}=\sum_{i,j=1}^{m}E_{ij}E_{ji}-\sum_{\bar{k},\bar{l}=m+1}^{m+n}E_{\bar{k}\bar{l}}E_{\bar{l}\bar{k}}+\sum_{i=1}^{m}\sum_{\bar{k}=m+1}^{m+n}(E_{\bar{k}i}E_{i\bar{k}}-E_{i\bar{k}}E_{\bar{k}i})-{\frac{m-n}{mn}}Y^{2}.
J_{0}(z)=\frac{1}{\gamma^{2}}\langle\,t_{0}\,\partial_{z}g(z,\bar{z})\,g^{-1}(z,\bar{z})\,\rangle=0,~~~~~\bar{J}_{0}(\bar{z})=\frac{1}{\gamma^{2}}\langle\,t_{0}\,g^{-1}(z,\bar{z})\,\partial_{\bar{z}}g(z,\bar{z})\,\rangle=0.
G_{s}(j-i)=<S_{i}^{z}\mathrm{e}^{(S_{i}^{z}+\cdots+S_{j}^{z})}S_{j}^{z}>_{\mathrm{VBS}}\stackrel{N\rightarrow\infty}{=}\frac{4}{(q^{2}+1+q^{-2})^{2}}\;.
\begin{array}{c}{*(c^{i_{1}}\dotsc^{i_{2k}})*=(-1)^{(r+1)q+k}\pi^{i_{1}}\dots\pi^{i_{2k}}\,\*(c^{i_{1}}\dotsc^{i_{2k+1}})*=(-1)^{r(q+1)+k}\pi^{i_{1}}\dots\pi^{i_{2k+1}}\,}\\{*(\pi^{i_{1}}\dots\pi^{i_{2k}})*=(-1)^{(r+1)q+k}c^{i_{1}}\dotsc^{i_{2k}}\,\*(\pi^{i_{1}}\dots\pi^{i_{2k+1}})*=(-1)^{r(q+1)+k+1}c^{i_{1}}\dotsc^{i_{2k+1}}\.}\\\end{array}
\begin{array}{ll}{{[}J_{i},J_{j}]=i{\epsilon_{ij}}^{k}J_{k}}&{i,j,k\,=\,1,2,3}\\{{[}J_{i},S_{j}]=i{\epsilon_{ij}}^{k}S_{k}}&{}\\{{[}S_{i},S_{j}]=i{\epsilon_{ij}}^{k}J_{k}\,(C_{2}-2\{\vec{J}\,}^{2})\\}&{}\\{{[}C_{2},J_{i}]=[C_{2},S_{i}]=0}&{\mathrm{with~}\{\vec{J}\,}^{2}=J_{1}^{2}+J_{2}^{2}+J_{3}^{2}}\\\end{array}
I_{\rho\alpha}=\int\frac{\partialc_{t}^{\mu}}{\partialx^{\rho}}\frac{\partialc_{t}^{\sigma}}{\partialx^{\alpha}}\frac{\partialc_{t}^{\nu}}{\partialt}\frac{\partialc^{\delta}(c_{t},x_{0},\tau)}{\partialc_{t}^{\sigma}}\frac{\partialc^{\beta}(c_{t},x_{0},\tau)}{\partial\tau}dtd\tau[G_{\beta\delta}(c(c_{t},x_{0},\tau)),G_{\mu\nu}(c_{t})]
{\calL}_{\psi}=iJ\frac{{\psi}^{*}\dot{\psi}-\dot{\psi^{*}}\psi}{1-\vert\psi\vert^{2}}-J^{2}\frac{2\partial_{i}\psi^{*}\partial_{i}\psi}{(1-\vert\psi\vert^{2})^{2}},
\langleT_{zz}(sz,s\bar{z})T_{zz}(0,0)\rangle=\frac{\pi}{3\cdot16}\int_{0}^{\infty}d\mu^{2}\,\tilde{c}(\mu^{2})\int\frac{d^{2}q}{(2\pi)^{2}}\frac{e^{\frac{i}{2}(q\bar{z}+\bar{q}z)}}{q\bar{q}+s^{2}\mu^{2}}\,\frac{\bar{q}^{4}}{s^{4}}.
S_{7}=\int\!\!d^{3}x~\left(m_{ab}\epsilon^{\mu\nu\alpha}H_{a\mu\nu\alpha}\varphi_{b}\right),\quadS_{8}=\int\!\!d^{3}x~\left(m_{abc}\epsilon^{\mu\nu\alpha}H_{a\mu\nu\alpha}\varphi_{b}\varphi_{c}\right)
\Sigma_{\mathrm{fix}}={\textstyle{\frac{1}{2}}}({\mathrm{e}}^{-\phi}|p|-{\mathrm{e}}^{\phi}|q|)_{\mathrm{fix}}\,=0\qquad\Longleftrightarrow\qquad{\mathrm{e}}_{\mathrm{fix}}^{-2\phi}={\frac{|q|}{|p|}}\.
L~=~\frac{1}{2}(\partial_{\mu}\phi^{ai})^{2}~+~\frac{u}{4!}\left(\phi^{ai}\phi^{ai}\right)^{2}~+~\frac{v}{4!}\left[\left(\phi^{ai}\phi^{bi}\right)^{2}~-~\phi^{ai}\phi^{ai}\phi^{bj}\phi^{bj}\right]
F[L,\beta,\nu]=\frac{8}{\sqrt{\pi\alpha^{\prime}}}\int_{0}^{\infty}\frac{dl}{l^{3/2}}\,e^{-L^{2}l/4\pi^{2}\alpha^{\prime}}\Theta_{2}\left(\nu\left\vert\frac{i\beta^{2}}{\pi\alpha^{\prime}l}\right.\right)\prod_{n=1}^{\infty}\left(\frac{1+e^{-nl}}{1-e^{-nl}}\right)^{8}
\mathrm{Q{\&}S~frame:}\qquad({\theta_{1}^{+\alpha,\dot{\alpha}}})^{\prime}=({\theta_{2}^{+\alpha,\dot{\alpha}}})^{\prime}=({\theta_{3}^{+\alpha,\dot{\alpha}}})^{\prime}=({\theta_{4}^{+\alpha,\dot{\alpha}}})^{\prime}=0\;.
\begin{array}{cc}{G(i_{B})^{-1}=\left(\begin{array}{ccc}{0}&{i\tilde{\gamma}^{B}}&{0}\\{-i\gamma^{B}}&{0}&{0}\\{0}&{0}&{R_{i}^{~j}}\\\end{array}\right)=e^{\frac{\pi}{2}M_{B}}~~~~}&{~~~~M_{B}=\left(\begin{array}{ccc}{0}&{i\tilde{\gamma}^{B}}&{0}\\{-i\gamma^{B}}&{0}&{0}\\{0}&{0}&{\frac{2}{\pi}\mathrm{ln}R}\\\end{array}\right)}\\\end{array}
ds^{2}=-\frac{2Q^{2}}{\gamma^{4}}\left(1-\frac{4M}{r}\right)dt^{2}+\frac{2Q^{2}}{r^{2}}\left(1-\frac{4M}{r}\right)^{-1}dr^{2}+2Q^{2}d\Omega^{2}
g_{YM}^{2}=\frac{4\sqrt{3\pi}M_{s}}{M_{P}^{(4d)}}.\frac{\chi^{3/2}}{\sqrt{[(n^{1})^{2}+(m^{1})^{2}\chi^{2}][(n^{2})^{2}+9(m^{2})^{2}\chi^{2}][(n^{3})^{2}+4(\tilde{m}^{3})^{2}\chi^{2}]}},
D^{2}\\rightarrow\\partial_{t}^{2}-\left(\frac1r\partial_{r}r\partial_{r}+\frac{1}{r^{2}}\partial_{\varphi}^{2}+\partial_{z}^{2}\right)\+\\frac{2i\beta\Theta(r-R)}{r^{2}}\\partial_{\varphi}\+\\frac{\beta^{2}\Theta^{2}(r-R)}{r^{2}}\.
L=\frac{1}{8}\,T_{(\mu\nu)\sigma}T^{(\mu\nu)\sigma}+\frac{1}{4}T_{(\mu\nu)\sigma}T^{(\mu\sigma)\nu}+\frac{1}{2}T_{(\mu\nu)\sigma}\epsilon^{\mu\nu\alpha\beta}\partial_{\alpha}h_{\;\beta}^{\sigma}+h^{\alpha\beta}\Theta_{\alpha\beta}\,
E=\intd^{2}x\left({\frac{\kappa^{2}}{2e^{2}|\Phi|^{2}}}\left(1-{\frac{e^{2}}{\kappa^{2}}}|\Phi|^{2}\right)\,B^{2}+{\frac{1}{2}}\left(\tilde{D}_{i}\Phi\right)^{\ast}\tilde{D}_{i}\Phi+{\frac{1}{2}}m^{2}|\Phi|^{2}\right)\,.
H\alpha^{\prime}Pr.=\frac{(H^{4}r^{4}-2H^{2}r^{2}-H^{2}E^{2})H^{2}r\dot{r}+(3H^{4}r^{4}-2H^{2}r^{2}+H^{2}E^{2})HE}{2(1-H^{2}r^{2})(H^{4}r^{4}+H^{2}E^{2})}.
\begin{array}{lcr}{P(W_{2})=(y^{2}+x^{3}+z^{6}+\muxyz)+}\\{w(a_{0}y^{{\frac{1}{2}}}z^{{\frac{(2n+3)}{2}}}+a_{n-1}z^{{\frac{7}{2}}}x^{{\frac{(n-2)}{2}}}y^{{\frac{1}{2}}})+w^{2}\sum_{i=0}^{n-3}a_{i+1}z^{2(n-i)}x^{i}.}\\\end{array}
\Psi_{(m,m,n)}=\prod_{i<j}(z_{i}^{(1)}-z_{j}^{(1)})^{m}\prod_{i<j}(z_{i}^{(2)}-z_{j}^{(2)})^{m}\prod_{i,j}(z_{i}^{(1)}-z_{j}^{(2)})^{n}\;\Psi_{0}.
\alpha^{0}c^{0}+\frac{1}{2}\sum_{m\neq0}(\alpha^{m}+\alpha_{m})c^{m}e^{imx}=\beta^{0}c^{0}+\frac{1}{2}\sum_{m\neq0}(\beta^{m}+\beta_{m})c^{m}e^{imx}
S(\phi)=\frac{1}{2}\inte^{-\phi}\tilde{Q}_{B}(e^{\phi})e^{-\phi}\eta_{0}(e^{\phi})-\int_{0}^{1}dte^{-t\phi}\partial_{t}e^{t\phi}\left\{e^{-t\phi}\tilde{Q}_{B}(e^{t\phi}),e^{-t\phi}\eta_{0}(e^{t\phi})\right\}\.
\hat{\phi}(f)=\hat{\phi}_{\mathrm{\scriptsize~in}}(f)+\hat{\phi}_{\mathrm{\scriptsize~out}}(f)\:\:\:\:\mathrm{with}\:\:\:\:\hat{\phi}_{\mathrm{\scriptsize~in/out}}(f):=\Omega(E_{\mathrm{\scriptsize~in/out}}f,\hat{\phi})
\tilde{L}=-\frac{1}{8\pi}\left(\frac{a}{a^{2}+b^{2}}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}-\frac{ib}{a^{2}+b^{2}}\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\tilde{F}^{\mu\nu}\tilde{F}^{\rho\sigma}\right).
4\left(\ddot{\Phi}+3\frac{\dot{a}}{a}\dot{\Phi}-\dot{\Phi}^{2}\right)-6\left(\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}+\frac{k}{a^{2}}\right)=\kappa^{2}e^{\Phi}p_{T},
\Gamma^{(1)}=\intd^{3}x\left(f^{(0)}u^{4}+f^{(2)}u^{3}\left[\psi^{2}\right]+f^{(4)}u^{2}\left[\psi^{4}\right]+f^{(6)}u\left[\psi^{6}\right]+f^{(8)}\left[\psi^{8}\right]\right),
\langleN\rangle=\left\{\begin{array}{lc}{-\frac{1}{2}F,}&{-1<A^{-1}<\infty}\\{}&{}\\{\hphantom{-}\frac{1}{2}(2-F),}&{-\infty<A^{-1}<-1}\\{}&{}\\{\hphantom{-}\frac{1}{2}(1-F),}&{A=0}\\\end{array}\right\},\frac{1}{2}<F<1.
M_{BPS}^{2}=e^{-\phi_{\infty}}\left\{{\vec{P}}^{T}{\calM}_{+}{\vec{P}}+{\vec{Q}}^{T}{\calM}_{+}{\vec{Q}}+2\left[({\vec{P}}^{T}{\calM}_{+}{\vec{P}})({\vec{Q}}^{T}{\calM}_{+}{\vec{Q}})-({\vec{P}}^{T}{\calM}_{+}{\vec{Q}})^{2}\right]^{\frac{1}{2}}\right\}.
\left[{\calA}^{+}{\left(D^{+}\right)^{j-1}}\right]_{I_{j}}=\left(A^{+}{\left(D^{+}\right)^{j-1}}\right)_{\beta_{1}}\left(\left(A^{+}\right)^{2}\left(D^{+}\right)^{j-1}\right)_{\beta_{2}}\dots\left(\left(A^{+}\right)^{N_{2}}\left(D^{+}\right)^{j-1}\right)_{\beta_{N_{2}}}.
\begin{array}{ll}{\Pi_{J_{0}}^{e}\equiv\Omega_{J_{0}}}&{\Pi_{\beta_{l}}^{r^{k}}\equiv\Omega_{J_{0}}\oplus\Omega_{J_{1}}}\\{\Pi_{J_{1}}^{e}\equiv\Omega_{J_{1}}}&{\Pi_{\gamma_{0}}^{s}\equiv(m+1)\Omega_{J_{0}}\oplusm\Omega_{J_{1}}}\\{\Pi_{\alpha_{l}}^{e}\equiv\Omega_{J_{0}}\oplus\Omega_{J_{1}}}&{\Pi_{\gamma_{1}}^{s}\equivm\Omega_{J_{0}}\oplus(m+1)\Omega_{J_{1}}}\\\end{array}
\hat{\Gamma}_{r}^{\,\alpha,\,\alpha_{0}}(q,p,\Phi_{c})\:=\(2\pi)^{\,4}\left\{\,\delta_{\Phi(p)}\delta_{\Phi(q)}\,L^{\,\alpha,\,\alpha_{0}}(\Phi)\,\vert_{\,\Phi=i\tilde{C}_{\alpha}^{\,\alpha_{0}}\!J(\Phi_{c})}\right\}_{r}\quad.
\tilde{\phi}^{I}\ast\tilde{\phi}^{J}(z,\bar{z})=\mathrm{e}^{\hbar\gamma^{\mu\nu}\frac{\partial}{\partialz_{\;b}^{\mua}}\frac{\partial}{\partialz_{\;a}^{\nub}}}\tilde{\phi}^{I}(z,\bar{z})\tilde{\phi}^{J}(z^{\prime},\bar{z^{\prime}})|_{z=z^{\prime},z^{\dagger}=z^{\dagger}}
x^{i}=i\sqrt{\frac{\alpha^{\prime}}{2}}{\Bigg[}\sqrt{\frac{2}{\omega_{0}}}(a_{0}^{i}e^{-i\omega_{0}\tau}-a_{0}^{i\dagger}e^{i\omega_{0}\tau})+\sum_{n=1}^{\infty}\frac{1}{\sqrt{\omega_{n}}}(a_{n}^{i}e^{-i\omega_{n}\tau}-a_{n}^{i\dagger}e^{i\omega_{n}\tau})(e^{in\sigma}+e^{-in\sigma}){\Bigg]},
\begin{array}{lcr}{\sum_{\nu=1}^{8}\{\{X^{\mu},X^{\nu}\},X^{\nu}\}=\sum_{\nu=1}^{8}\sum_{\stackrel{m_{i},n_{i}}{i=1,2,3}}X_{m_{1}n_{1}}^{\mu}(\sigma)X_{m_{2}n_{2}}^{\nu}(\sigma)X_{m_{3}n_{3}}^{\nu}(\sigma)\{\{L_{m_{1}n_{1}},L_{m_{2}n_{2}}\},L_{m_{3}n_{3}}\},}\\\end{array}
\sum_{k=1}^{2}\left[\alpha_{ik}(t)\beta_{jk}(t)-\beta_{ik}(t)\alpha_{jk}(t)\right]=\sum_{k=1}^{2}\left[\alpha_{ik}^{\ast}(t)\beta_{jk}^{\ast}(t)-\beta_{ik}^{\ast}(t)\alpha_{jk}^{\ast}(t)\right]=0\;.
\psi_{i\alpha}(x)~=~\int{\frac{d^{3}p}{(2\pi)^{3/2}}}~{\sqrt{\frac{m_{i}}{\omega_{i}}}}~\left[b_{i}({\bfp})~u_{i}^{(\alpha)}({\bfp})~e^{i({\bfpx}-\omega_{i}t)}~+~d_{i}^{\dagger}({\bfp})~v_{i}^{(\alpha)}({\bfp})~e^{-i({\bfpx}-\omega_{i}t)}\right]
\begin{array}{ll}{\bar{\bfs}X^{i*}=\psi^{i*}}&{{\bfs}X^{i*}=\xi^{i*}}\\{\bar{\bfs}\psi^{i}=-\frac{i}{2}\partial_{+}X^{i}}&{{\bfs}\psi^{i*}=-\frac{1}{2}F^{i*}}\\{\bar{\bfs}\xi^{i*}=\frac{1}{2}F^{i*}}&{{\bfs}\xi^{i}=-\frac{i}{2}\partial_{-}X^{i}}\\{\bar{\bfs}F^{i}=i\partial_{+}\xi^{i}}&{{\bfs}F^{i}=-i\partial_{-}\psi^{i}\,}\\\end{array}
E-E_{0}\ge\,\frac{v^{2}}{2}\,(\Psi-\Psi_{0})=\frac{v^{2}}{2}\intd^{2}x\,\,(F_{12}-\frac{\rho_{e}}{\kappa})=\frac{v^{2}}{4\kappa^{2}}\intd^{2}x\,f^{2}\,(v^{2}-f^{2}).
e^{\textstyle-J\alpha_{-}\Phi(\sigma,\tau)}=c_{J}\sum_{m=-J}^{J}\,{\frac{(-1)^{J-m}\,\lambda_{m}^{(J)}(\varpi)}{\sqrt{\lfloor\varpi\rfloor}\sqrt{\lfloor\varpi+2m\rfloor}}}\psi_{m}^{(J)}(x_{+})\,{\overline{\psi}_{m}^{(J)}}(x_{-})
G(x^{\prime},t^{\prime};x^{\prime},t^{\prime})=\frac{1}{2}\int\frac{dk^{\prime\prime}}{2\pi}\frac{1}{\sqrt{k^{2}+4}}\frac{ik^{\prime\prime}+2\sigma_{1}}{ik^{\prime\prime}-2\sigma_{1}}e^{-2ik^{\prime\prime}x^{\prime}}.
E_{A}=\left(\begin{array}{cc}{E_{1}}&{E_{2}}\\\end{array}\right)\;{\calE}_{A}\;;\;{\calE}_{A}=\left(\begin{array}{c}{\nu_{A}^{c}}\\{1}\\\end{array}\right)\;{\frac{1}{\sqrt{1+\vert\nu_{A}\vert^{2}}}}\;,
\left[-{\frac{d^{2}}{dx^{2}}}+{\frac{D}{x^{2}}}+{\frac{F(1-2x)}{x^{2}(1-x)}}+{\frac{(1-2x)^{2}}{2x^{2}(1-x)^{2}}}+{\frac{1}{4x^{2}(1-x)^{2}}}\right]f_{Jl}(x)={\frac{(E^{2}-m^{2})}{C^{2}}}f_{Jl}(x)
1-S=\left(\begin{array}{cc}{\frac{1}{2V_{oo}}\sqrt{1-3V_{oo}}\left[\sqrt{1+V_{oo}}-\sqrt{1-3V_{oo}}\right]}&{0}\\{0}&{\frac{1}{2V_{ee}}\sqrt{1-V_{ee}}\left[\sqrt{1+3V_{ee}}-\sqrt{(1-V_{ee})}\right]}\\\end{array}\right)\,,
{\calL}_{\mathrm{eff}}=\int\,d^{2}\thetad^{2}\bar{\theta}\,{\calK}[X^{a},\bar{X}^{a}]\,\,+\,\,\int\,d^{2}\theta\,{\calW}(X^{a})\,\,+\,\,\int\,d^{2}\bar{\theta}\,\bar{{\calW}}(\bar{X}^{a})
\prod_{n=0}^{\infty}\left[(1-e^{-\alpha\epsilonn+(\pi-\alpha)\epsilon+i\epsilonm\beta_{12}})(1-e^{-\alpha\epsilonn+(\pi-\alpha)\epsilon-i\epsilonm\beta_{12}})(1-e^{-\alpha\epsilonn-\alpha\epsilon+i\epsilonm\beta_{12}})(1-e^{-\alpha\epsilonn-\alpha\epsilon-i\epsilonm\beta_{12}})\right].
\left\{\begin{matrix}{h_{\mu\nu}(x)}\\{h_{\mu\nu}(y)}\\\end{matrix}\right\}=h_{\mu\nu}(X)\pm{\frac{1}{2}}r_{\alpha}\partial_{\alpha}h_{\mu\nu}(X)+{\frac{1}{8}}r_{\alpha}r_{\beta}\partial_{\alpha}\partial_{\beta}h_{\mu\nu}(X)+\cdots.
u^{j}(t,r)\,=\,i\,\frac{\sqrt{2}}{M_{0}^{2}}\,\sum_{n\not=0}^{}\,e^{-i\omega_{n}t}\,\frac{\alpha_{n}^{j}}{\omega_{n}}\,u_{n}(r),\quadj\,=\,2,\dots,D-1\,{,}
A_{p+1}=(-1)^{p}\Big(H^{-1}dt+\frac{r_{0}^{\frac{d-2}{2}}}{h^{\frac{d-2}{2}}}\sum_{i=1}^{n}l_{i}\mu_{i}^{2}d\phi_{i}\Big)\wedgedy^{1}\wedge\cdots\wedgedy^{p}
g_{B}^{\ast\mu\nu}\frac{\deltaL_{GR}}{\deltag_{B}^{\ast\mu\nu}}=\left(-\frac{s}{k_{B}^{2}}\Big(R_{B}-\frac{4\sigma_{B}}{k_{B}^{2}}\Big)+2s\frac{\omega_{B}}{\lambda_{B}}\nabla^{2}R_{B}\right)\sqrt{-g_{B}}\equiv0.
\frac{1}{g_{ph}^{2}}+\frac{2t_{2}(A)}{8\pi^{2}}\mathrm{log}\left(g_{ph}\right)=\frac{1}{g_{h}^{2}}-\sum_{i}\frac{2t_{2}(i)}{8\pi^{2}}\mathrm{log}\left(Z_{i}\right),
{}~~~~~~~~~~~~~\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{w(x_{13},x_{1}u,x_{58}|\sigma+j_{2}+j_{3})w(x_{2}x_{67},x_{2}v,x_{6}x_{24}|\sigma)s(\sigma,j_{1})}{w(x_{5}x_{13},\omegax_{1}u,\omegax_{1}x_{58}|\sigma+j_{2})w(x_{67},x_{2}v,x_{24}|\sigma+i_{3})}\Bigg\}_{0}~~
T_{\mu\nu}^{\mathrm{reg}}=\frac12(\theta_{\mu\nu}^{\mathrm{reg}}+\theta_{\nu\mu}^{\mathrm{reg}})=\frac14(\psi_{\varepsilon}^{+}\gamma_{\mu}\stackrel{\leftrightarrow}{D}_{\nu}\psi_{\varepsilon}+\psi_{\varepsilon}^{+}\gamma_{\nu}\stackrel{\leftrightarrow}{D}_{\mu}\psi_{\varepsilon})
|\phi_{(ML)}>=\prod_{(\vec{p})^{+}}\;|\phi_{M\vec{p},L}>,\;|\phi_{M\vec{p},L}>=(\beta_{\vec{p}}{}^{2}|1>+(-)\sqrt{2}\;\alpha_{\vec{p}}\beta_{\vec{p}}|2>+\alpha_{\vec{p}}{}^{2}|3>),
{\frac{d^{2}\psi^{(-)}}{dr^{2}}}+\left[E+{\frac{A(A+1)}{\mathrm{cosh}^{2}\,r}}-{\frac{B(B-1)}{\mathrm{sinh}^{2}\,r}}-\left(A-B\right)^{2}\right]\psi^{(-)}(r)=0.
i\hbar\frac{\partial\Psi^{b}}{\partialt}=-\frac{\hbar^{2}}{2m}~[~\sum_{A}(\delta_{d}^{b}\frac{\partial}{\partialx^{A}}+\omega_{~d,\mu}^{b}f_{A}^{\mu})(\delta_{c}^{d}\frac{\partial}{\partialx^{A}}+\omega_{~c,\nu}^{d}f_{A}^{\nu})-(H^{2})_{~c}^{b}~]~\Psi^{c},
H_{nm}^{\{{\bfR}\}}(0,0)=\frac{\mu_{nm}!|n-m|!(-1)^{\frac{m+n}2}}{2^{\frac{|m-n|}2}\left(\frac{|m-n|}2\right)!}\left[R_{11}^{n}R_{22}^{m}\left(r^{2}-1\right)^{\mu_{mn}}\right]^{\frac12}C_{\mu_{mn}}^{\frac{|m-n|+1}2}\left(\fracr{\sqrt{r^{2}-1}}\right).
\frac{1}{16\piG}\int_{S^{3}(r_{+})}\nabla_{\mu}\tilde{K}_{\nu}^{i}d\Sigma^{\mu\nu}+\frac{1}{16\piG}\int_{S^{3}(r_{c})}\nabla_{\mu}\tilde{K}_{\nu}^{i}d\Sigma^{\mu\nu}=\frac1G\int_{\Sigma_{t}}T_{\nu\rho}\tilde{K}^{i\rho}d\Sigma^{\nu},
\tau_{\{\mu}^{\vphantom{\dagger}}\tau_{\nu\}}^{\dagger}=\tau_{\{\mu}^{\dagger}\tau_{\nu\}}^{\vphantom{\dagger}}=\delta_{\mu\nu};\qquad\tau_{[\mu}^{\dagger}\tau_{\nu]}^{\vphantom{\dagger}}=i\hat{\eta}_{\mu\nu}^{+};\qquad\tau_{[\mu}^{\vphantom{\dagger}}\tau_{\nu]}^{\dagger}=i\hat{\eta}_{\mu\nu}^{-}.
I_{H}^{BH}={\frac{\beta}{2r_{+}^{2}}}\left[r_{+}^{4}(1-r_{+}^{2})+r_{+}^{2}\sum_{i=1}^{3}q_{i}(2+r_{+}^{2})+3r_{+}^{2}\sum_{i<j}^{3}q_{i}q_{j}+5\prod_{i=1}^{3}q_{i}\right]\,
S_{sb}\approx{\frac{\pir_{0}^{2}}{G}}\left(1-{\frac{r_{0}}{r_{\mathrm{e}}}}\right)^{3/8}\approx{\frac{1}{16\piGT^{2}}}\left(1-{\frac{r_{0}}{r_{\mathrm{e}}}}\right)^{3/8}\left(1-{\frac{r_{0}^{\prime}}{r_{\mathrm{e}}}}\right)^{-1}.
\sum_{l=1}^{\infty}(w-z)^{l-1}\langle\varphi,V(\psi,\zeta)\;V(J_{-l}^{a}\chi,z)\;\Omega\rangle-\sum_{m=-\infty}^{0}(w-\zeta)^{m-1}\langle\varphi,V(J_{-m}^{a}\psi,\zeta)\;V(\chi,z)\;\Omega\rangle.
<O>=\frac{\int_{\phi}O(\phi)e^{-\int{\calL}_{M}}}{\int_{\phi}e^{-\int{\calL}_{M}}}=<O>_{c}+\beta_{M}\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}c_{ij}\alpha_{M}^{i}\frac{1}{M^{2j}}<O{\calL}_{j}(\phi)>_{c}+...
W_{1}^{+}(0,\lambdaA,N)=ND_{1}(\lambdaA,N)=e^{-\frac{\lambdaA}{2}}\sum_{h}\frac{(\lambdaA)^{h}}{2^{h}\;h!}N^{2(1-h)}=N^{2}e^{-\frac{\lambdaA}{2}+\frac{\lambdaA}{2N^{2}}},
\frac{\partial}{\partialx_{k}}\frac{\partial\phi}{\partialx_{0}}=-\sum\xi_{j}\frac{\partial}{\partialx_{k}}\frac{\partialR}{\partial\xi_{j}}=-\sum\xi_{j}\frac{\partial}{\partialx_{j}}\frac{\partialR}{\partial\xi_{k}}=\frac{\partial}{\partialx_{0}}\frac{\partialR}{\partial\xi_{k}}
V_{2\xi}\=\\frac{\alpha}{2P_{L}r}\,\\\\\\U_{4\xi}\=\V_{2\xi}+U_{g\xi}\,\\\\\\T_{4\xi}\=\2x^{T2}\dot{U}_{g\xi}\,\\\\\\U_{g\xi}\=\-\frac{\xi}{2}\\frac{\alpha}{2P_{L}r}\.
I(g,A)=-\frac{k}{2\pi}\intd^{2}z\frac{1}{w^{2}}\left|A+\frac{1}{2w^{2}}(x\partialw-w\partialx)\right|^{2}+\frac{k}{2\pi}\intd^{2}z\frac{|\partialw|^{2}}{w^{2}}
S=-\frac{1}{2\pi}\intd^{2}\sigma[\partial_{\alpha}X^{\mu}\partial^{\alpha}X_{\mu}-i\bar{\psi}^{\mu,j}\rho^{\alpha}\partial_{\alpha}\psi_{\mu,j}+i\bar{\phi}^{\mu,k}\rho^{\alpha}\partial_{\alpha}\phi_{\mu,k}]\,
h(T_{1},k)\sime^{-\frac{2}{3}\pik^{2}T_{1}}\Bigl[\Theta_{3}(ikT_{1},2iT_{1})\Theta_{3}(ikT_{1},6iT_{1})+\Theta_{2}(ikT_{1},2iT_{1})\Theta_{2}(ikT_{1},6iT_{1})
S_{\mathrm{int}}^{\prime}[\phi,\varphi]=g^{2}\int_{t_{i}}^{t}d^{4}x\left[-{\frac{1}{24}}\varphi^{4}-{\frac{1}{6}}\phi\varphi^{3}-{\frac{1}{4}}\phi^{2}\varphi^{2}-{\frac{1}{6}}\phi^{3}\varphi\right]
\int\frac{d^{d}q_{1}}{N_{d}\pi^{d/2}}\int\frac{d^{d}q_{2}}{N_{d}\pi^{d/2}}\frac{\partial}{\partialq_{1}^{\mu}}\frac{q_{1}^{\mu}}{(q_{1}^{2}+1)^{a}(q_{2}^{2}+1)^{b}((q_{1}+q_{2})^{2}+1)^{c}}=0
A_{(b)}=\frac{g^{4}}{\alpha^{4}}\intd^{26}k~\frac{1}{(P_{1}+P_{4})^{2}-\frac{1}{\alpha^{\prime}}}\,\frac{1}{k^{2}-\frac{1}{\alpha^{\prime}}}\,\frac{1}{(P_{1}+P_{4}-k)^{2}-\frac{1}{\alpha^{\prime}}}\,\frac{1}{(P_{1}+P_{4})^{2}-\frac{1}{\alpha^{\prime}}},
{\calL}=\frac{p_{11}}{2}\left[\intdpdq\dot{Y}^{i}(p,q)\dot{Y}^{i}(p,q)-\frac{1}{P_{11}}\intdpdq[\partial_{q}Y^{i}\partial_{p}Y^{j}-\partial_{q}Y^{j}\partial_{p}Y^{i}]\right],
S_{\mathrm{NR}}^{p=2}=\intdtd^{2}\sigma\left[{\frac{1}{2}}\mu_{0}{\frac{dX_{k}}{dt}}{\frac{dX^{k}}{dt}}+{\frac{\alpha}{4}}\left\{X_{i},X_{j}\right\}_{\mathrm{PB}}\left\{X^{i},X^{j}\right\}_{\mathrm{PB}}\right]\,
{\calL}_{APEGT}^{0}=-{\frac{1-\rho^{2}}{4}}(f_{\mu\nu}^{i})^{2}-{\frac{1}{2}}\rho*B_{\mu\nu}^{i}f^{\mu\nu}{}^{i}-{\frac{1}{4}}(B_{\mu\nu}^{i})^{2},
\deltaS={\frac{1}{2}}\int_{{\calM}^{2}}d^{2}\xi\sqrt{-g}\deltag^{mn}(\xi)\left(\partial_{m}{\hat{X}}^{\underline{{m}}}\partial_{n}{\hat{X}}_{\underline{{m}}}-{\frac{1}{2}}g_{mn}g^{kl}\partial_{k}{\hat{X}}^{\underline{{m}}}\partial_{l}{\hat{X}}_{\underline{{m}}}\right)-
\frac{\theta^{2}V_{0}}{6}\{(2-\frac{16ma^{2}V_{0}}{3})a-\frac{4ma^{3}V_{0}}{3}+(\frac{1}{2}-a+\frac{8ma^{3}V_{0}}{3})\frac{8mV_{0}a^{2}}{n^{2}\pi^{2}}+\frac{n^{2}\pi^{2}a}{3}-\frac{64m^{2}a^{2}V_{0}}{n^{4}\pi^{4}}\}.
{\calL}=-\frac{1}{4}G_{\mu\,\nu}^{a\;2}+\sum_{i=1}^{N_{f}}\bar{\psi}_{i}i\gamma^{\mu}(x)\nabla_{\mu}\psi_{i}+\tilde{G}\sum_{i=1}^{n_{f}}(\bar{\psi_{i}}\psi_{i})^{2}\;,
g_{b}(z)=\frac{(x^{10}z^{2};x^{4},x^{8})_{\infty}(x^{6}z^{-2};x^{4},x^{8})_{\infty}}{(x^{12}z^{2};x^{4},x^{8})_{\infty}(x^{8}z^{-2};x^{4},x^{8})_{\infty}}.
\frac{1}{2}\\frac{\partial^{2}\theta}{\partialp^{A}\partial\tilde{q}_{B}}\\frac{\partial^{2}\theta}{\partialp_{A}\partial\tilde{q}^{B}}\+\\frac{\partial^{2}\theta}{\partialp^{A}\partialq_{A}}=0,\A,B=1,2,
{\calL}_{MCSP}=3D-{\frac{1}{4}}A_{\mu\nu}A^{\mu\nu}+{\frac{\mu}{4}}\epsilon_{\mu\nu\lambda}A^{\mu\nu}A^{\lambda}+{\frac{m^{2}}{2}}A_{\mu}A^{\mu},~~A_{\mu\nu}=3D\partial_{\mu}A_{\nu}=-\partial_{\nu}A_{\mu}.
A^{\prime}=2\sum_{n=1}^{\infty}\frac{1}{n}\left\{g^{\nu\rho}\left[\alpha_{-n}^{\mu},\alpha_{n}^{\sigma}\right]_{+}-g^{\mu\sigma}\left[\alpha_{-n}^{\rho},\alpha_{n}^{\nu}\right]_{+}\right\}\tag{B-4}
\sigma^{Si}(y^{i})={\frac{1}{2}}(1+(-1)^{L})k^{y^{i}}|y^{i}|+\sum_{j_{i}=1}^{L}(-1)^{j_{i}+L}k^{y^{i}}|y^{i}-y_{j_{i}}^{i}|~.~\,
\left[\widehat\lambda\stackrel{\ast}{,}\widehat{A}_{\mu}\right]=\frac{1}{2}\left\{\widehat{\lambda}^{I}\stackrel{\ast}{,}\widehat{A}_{\mu}^{J}\right\}\left[T_{I},T_{J}\right]+\frac{1}{2}\left[\widehat\lambda^{I}\stackrel{\ast}{,}\widehat{A}_{\mu}^{J}\right]\left\{T_{I},T_{J}\right\}.
\Psi_{I}(t,{\bfr})=\sum_{\bfk}\mathrm{\Large~e^{i~{\bf~k\cdot~r}~}~e^{-~i~v\left|{\bf~k}\right|~t}~}\psi_{{\bfk},+}a_{{\bfk},+}+\sum_{\bfk}\mathrm{\Large~e^{i~{\bf~k\cdot~r}~}~e^{~i~v\left|{\bf~k}\right|~t}~}\psi_{{\bfk},-}a_{{\bfk},-}
{\tilde{F}}_{3}=dC_{2}-{\frac{\kappa_{10}^{2}}{g_{10}^{2}}}~\mathrm{Tr}_{V}\left[A_{1}\wedgedA_{1}-i{\frac{2}{3}}A_{1}\wedgeA_{1}\wedgeA_{1}\right],\quad{\tilde{F}}_{2}=dC_{\mu9},\quadF_{2}=dA_{1},\quade^{\chi}={\sqrt{G_{99}}}\quad,
P\left(x_{1},...,x_{6}\right)=ax_{1}^{2}x_{2}^{2}x_{5}^{2}x_{6}^{2}+bx_{1}^{3}x_{2}^{3}x_{5}^{3}x_{6}^{5}+cx_{1}^{2}x_{2}^{2}x_{3}x_{4}x_{5}^{2}x_{6}^{4}+d\prod_{i=1}^{6}x_{i}
U_{\mu\nu}^{ab}(x,y):=-{\frac{\Gamma(D/2-1)}{4\pi^{D/2}}}\,\delta^{ab}\,\partial_{\mu}^{x}\partial_{\nu}^{y}\int_{0}^{1}\!\!ds\int_{0}^{1}\!\!dt\,x\cdoty\,[(sx-ty)^{2}]^{1-D/2}\,.
S[A_{\mu}]=\int_{0}^{\beta}dx_{5}\intd^{4}x\\frac{1}{2e^{2}}[\mathrm{Tr}\F_{\mu\nu}F_{\mu\nu}+\frac{1}{c^{2}}\mathrm{Tr}\\partial_{5}A_{\mu}\partial_{5}A_{\mu}].
U_{\mathrm{eff}}=\frac{B^{2}}{g_{0}^{2}}-\int_{V}\frac{d^{3}x}{V}\int_{0}^{\beta}\frac{dx_{4}}{\beta}\int_{M^{2}(T)}^{\infty}dm^{2}\left[D_{ii}^{\beta}(x,x|B,m)-D_{ii}^{\beta}(x,x|0,m)\right].
S=\intd^{d-1}x\sqrt{|g|}\k\\left[R-{\textstyle\frac{1}{4}}k^{2}F_{(2)}^{2}+{\textstyle\frac{1}{2}}\left({\calD}\phi\right)^{2}-{\textstyle\frac{1}{2}}m^{2}k^{-2}\right]\,,
\left[(1-\psi^{N})\frac{\partial^{N-1}}{\partial\psi^{N-1}}-\sum_{l=1}^{N-2}b_{l}^{(N)}\psi^{l+1}\frac{\partial^{l}}{\partial\psi^{l}}\right]\omega_{i}=0,\;\;\;for\;all\;i=0\cdotsN-2.
[T_{A},Q_{\hat{\alpha}}]=t_{A\hat{\alpha}}^{~~\hat{\beta}}Q_{\hat{\beta}},\quad\{Q_{\hat{\alpha}},S_{\hat{\beta}}\}=h_{\hat{\alpha}\hat{\beta}}^{A}T_{A},\quad[T_{A},S_{\hat{\alpha}}]=g_{A\hat{\alpha}}^{~~\hat{\beta}}S_{\hat{\beta}}.
\rho_{s}^{(2)}(z,m)=\frac{z}{2}\left(J_{0}^{2}(z)+J_{1}^{2}(z)\right)-\frac{2z}{(z^{2}+m^{2})^{2}}\frac{[zJ_{1}(z)I_{0}(m)+mJ_{0}(z)I_{1}(m)]^{2}}{I_{0}^{2}(m)-I_{1}^{2}(m)}\:.
\langle\Psi|\Psi^{\prime}\rangle=\int_{{\calM}_{L}\times{\calE}_{L}}{\calD}^{e}e\;{\calD}^{e}\widetilde{x}\left(\mathrm{Vol}_{e}{\calD}_{L}\right)^{-1}\overline{{\Psi}}\Psi^{\prime}\;\;\;.
\langle0|a_{p}(6{\textstyle\sum{}_{4}}){\frac{1}{E-g\sum_{2}}}(6{\textstyle\sum_{4}}){\frac{1}{E-g\sum_{2}}}(6{\textstyle\sum_{4}})a_{p}^{\dagger}|0\rangle={\frac{54}{pg^{2}}}I\hbar^{5},
\mu^{\prime\prime}+[k^{2}-V(\eta)]=0,~~~V(\eta)=(\frac{g}{a})(\frac{a}{g})^{\prime\prime}\equiv\frac{a^{\prime\prime}}{a}+\frac{\phi^{2}}{4}-\frac{\phi^{\prime\prime}}{2}-{\calH}\phi^{\prime}
\left({\frac{\dot{a}_{0}}{a_{0}}}\right)^{2}={\frac{8\piG_{4}}{3}}\varrho+{\frac{4\piG_{4}}{3c_{\mathrm{grav}}^{4}\sqrt{I}}}\dot{\Phi}^{2}+{\frac{c_{\mathrm{grav}}^{2}}{3}}\Lambda_{\mathrm{eff}}+{\frac{{\calC}c_{\mathrm{grav}}^{2}}{a_{0}^{4}}}-{\frac{kc_{\mathrm{grav}}^{2}}{a_{0}^{2}}},
\tilde{D}_{da,bc}^{li,jk}\equiv{\frac{1}{N}}\sum_{\tilde{m}=0}^{P-1}\tilde{B}(l-j-Q\tilde{m})\mathrm{e}^{i{\frac{2\pi}{P}}(d-a)\tilde{m}}\delta_{ab}\delta_{cd}\delta_{li}\delta_{jk},
{\cal{\pi}}_{0i}(p_{0}=0)=\frac{8v^{2}}{\kappa}\int\frac{d^{2}k}{(2\pi)^{2}}\epsilon_{0ij}k^{j}T\sum_{n}\frac{1}{k_{0}^{2}+w_{M}^{2}}\frac{1}{k_{0}^{2}+w_{\sigma}^{2}(p)}.