harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
N^{\calP}~=~\sum_{m_{1}=-j_{1}}^{j_{1}}\cdots\sum_{m_{p}=-j_{p}}^{j_{p}}\left[~\delta_{(\sum_{n=1}^{p}m_{n}),0}~-~\frac12~\delta_{(\sum_{n=1}^{p}m_{n}),1}~-~\frac12~\delta_{(\sum_{n=1}^{p}m_{n}),-1}~\right],
\pi_{\mu\nu}=-\tau_{\mu\alpha}\tau_{\nu}^{\phantom{\nu}\alpha}/4+\tau_{\phantom{\alpha}\alpha}^{\alpha}\tau_{\mu\nu}/12+q_{\mu\nu}\tau_{\alpha\beta}\tau^{\alpha\beta}/8-q_{\mu\nu}(\tau_{\phantom{\alpha}\alpha}^{\alpha})^{2}/24.
P(\chi,\chi;z)=\sum_{\lambda\in\Lambda}\langle\chi|\gamma_{\lambda}|\chi\rangle3^{-{\frac{3}{2}}\lambda^{2}}z^{-\lambda^{2}}e^{\sum_{{n,m\geq0\atopn+m>0}}A_{mn}b_{-m}\cdot\overline{b}_{-n}z^{-m-n}}|\lambda\rangle\,,
{\frac{1}{2}}(h_{1}-h_{2})^{\prime}=\left[{\frac{d^{\prime}}{d}}+{\frac{12/r^{2}-(2{f^{\prime}}_{1}^{2}/f_{1}^{2}+{f^{\prime}}_{K}^{2}/f_{K}^{2})}{f^{\prime}/f+6/r}}-{\frac{4}{r^{6}d}}{\frac{Q_{K}^{2}/f_{K}^{2}-Q_{5}^{2}/f_{1}^{2}}{f_{1}^{\prime}/f_{1}-f_{K}^{\prime}/f_{K}}}\right]h_{2}.
N_{11}=\left(\begin{matrix}{{\frac{{\partial^{2}}V}{\partial{T_{1}}^{2}}}}\\\end{matrix}\right)_{0}\;,\;N_{12}=\left(\begin{matrix}{{\frac{{\partial^{2}}V}{\partial{T_{1}}\partial{T_{2}}}}}\\\end{matrix}\right)_{0}\;,\;N_{22}=\left(\begin{matrix}{{\frac{{\partial^{2}}V}{\partial{T_{2}}^{2}}}}\\\end{matrix}\right)_{0}
(-\partial_{t}^{2}+\partial_{r}^{2}-\frac{15}{4r^{2}})\varphi_{(2,0)}=-\frac{2\delta(r)}{\sqrt{2}\pi^{3}r^{\small{5/2}}}\partial_{\mu}A_{(0,0)}^{i}\partial^{\mu}A_{(1,0)}^{i}.
\langle{\calA}_{a}\rangle_{0n}={}^{\parallel}\nabla_{a}\omega_{\bar{I}J}\langlez^{\bar{I}}z^{J}\rangle_{0n}-k_{\bar{I}ba}\langle\lambda^{\bar{I}}\lambda^{b}\rangle_{0n}-a_{I\bar{J}a}\langle\lambda^{\bar{J}}\lambda^{I}\rangle_{0n}+a^{\prime}{}_{I^{\prime}\bar{J}^{\prime}a}\langle\nu^{I^{\prime}}\nu^{\bar{J}^{\prime}}\rangle_{0n}
S=\frac{M_{5}^{3}}{2}\int_{{\calM}_{5}}d^{5}x\sqrt{-g}\left[R-\frac{1}{2}\partial_{\alpha}\phi\,\partial^{\alpha}\phi-\frac{3}{2}e^{-2\phi}\alpha^{2}\right]-3\sum_{i=1}^{2}M_{5}^{3}\alpha_{i}\int_{{\calM}_{4}^{(i)}}d^{4}\xi_{(i)}\sqrt{-g}\,e^{-\phi}\,.
\mp{\frac{R}{\sqrt{\lambda}}}\left(\lambda^{\prime}-\lambda\varphi^{\prime}\right)\varepsilon_{ul}-{\bfQ}_{i}\gamma^{i}\varepsilon_{lu}+{\frac{1}{3}}({\bfP}_{ij}\mp2i{\bfQ}^{ij})\gamma^{ij}\varepsilon_{lu}=0,
M_{NN}=Nm_{0}\,\frac{\sqrt{2}}{3}\left(\,\sqrt{1-\sqrt{1-\left(\frac{g^{2}N}{4\pim_{0}^{2}\alpha_{0}}\right)^{2}}}+\,\frac{4\pim_{0}^{2}\alpha_{0}}{g^{2}N}\,\sqrt{1+\sqrt{1-\left(\frac{g^{2}N}{4\pim_{0}^{2}\alpha_{0}}\right)^{2}}}\;\right)
F_{n}^{\mathcal{O}|\ldots\mu_{i}\mu_{j}\ldots}(\ldots\theta_{i},\theta_{j}\ldots)=F_{n}^{\mathcal{O}|\ldots\mu_{j}\mu_{i}\ldots}(\ldots\theta_{j},\theta_{i}\ldots)S_{\mu_{i}\mu_{j}}(\theta_{i,j})\,.
\chi_{2}^{i}(F,G;{\bfp^{\prime},q})=-\frac{q^{i}}{12\pi^{2}}\tilde{f}({\bfp^{\prime}})\tilde{g}({\bf-p^{\prime}-q})\frac{\mu^{2b-3}}{2}F(2b-3,b-1/2;2b-2;1-\mu^{2}),
\partial^{\mu}\partial_{\mu}F+F^{\prime\prime}-9\frac{K^{\prime}}{K}F^{\prime}+\bigg[16\bigg(\frac{K^{\prime}}{K}\bigg)^{2}-4\frac{K^{\prime\prime}}{K}+\sum_{i}\frac{2\Lambda_{i}}{3K}\delta(z-z_{i})\bigg]F=0
H_{0}^{x}={\frac{\pi}{2}}\left[{\frac{1}{2M_{x}}}p_{0}^{2}+\sum_{k=1}^{\infty}\left({\frac{1}{M_{x}}}p_{k}^{2}+M_{x}k^{2}x_{k}^{2}\right)\right]\;\;\;;\;\;\;M_{x}={\frac{1}{4\alpha^{\prime}}}\;\;.
F_{k_{1}\dotsk_{n}}^{(a)}\left(\vartheta_{1},\dots,\vartheta_{n}\right)=\big\langle0\big|V_{a}(0)\big|B_{k_{1}}\left(\vartheta_{1}\right)\dotsB_{k_{n}}\left(\vartheta_{n}\right)\big\rangle_{0}^{in}\,,
G^{0}\,=\,G^{0}\,(1-\Sigma\,G^{0}\,)^{-1}\,=\,G_{1}^{0}\,(1-\Sigma_{1}\,G_{1}^{0}\,)^{-1}G_{2}^{0}\,(1-\Sigma_{2}\,G_{2}^{0}\,)^{-1}.
\mathcal{D}_{g}\left(\begin{array}{ccc}{p_{1}}&{\ldots}&{p_{N}}\\{r_{1}}&{\ldots}&{r_{N}}\\\end{array}\right)=\sum_{q\in\mathcal{I}}\mathcal{D}_{h}\left(\begin{array}{cccc}{p_{1}}&{\ldots}&{p_{l}}&{q}\\{r_{1}}&{\ldots}&{r_{l}}&{r}\\\end{array}\right)\mathcal{D}_{g-h}\left(\begin{array}{cccc}{\overline{{q}}}&{p_{l+1}}&{\ldots}&{p_{N}}\\{-r}&{r_{l+1}}&{\ldots}&{r_{N}}\\\end{array}\right)
f_{\mu\rho}(\psi,\overline{{\psi}})=\left(\,\overline{{\psi_{A}^{a}}}\,g^{AB}\,(\partial_{\mu}\psi_{bB})+\psi_{bB}\,g^{AB}\,(\partial_{\mu}\overline{{\psi_{A}^{a}}}\,)\right)\,{(e_{\rho})_{a}}^{b},
M_{pl}^{2}={M_{X}^{3}}\left({\frac{k_{0}}{\displaystyle{k_{0}^{2}-(k_{1}-k_{c})^{2}}}}e^{-2\sigma(y_{0})}+{\frac{k_{1}}{\displaystyle{k_{1}^{2}-(k_{0}+k_{c})^{2}}}}e^{-2\sigma(y_{1})}\right)~.~\,
J(s)=\sqrt{\pi}\sum_{p}\sum_{n_{2}}\int_{0}^{\infty}\frac{dt}{t}\frac{t^{s}}{\sqrt{t}}e^{-[t(\frac{\tilde{n}_{2}^{2}}{|\tau|^{2}}+\tilde{t}_{1}^{2})+\frac{\pi^{2}}{t}(p+\frac{i\tilde{t}_{1}t}{\pi})^{2}]}\equiv\sum_{p}J_{p}(s).
V_{D}=\frac{\tau_{p}(g_{s})}{16N(2\pil_{s}^{2})^{2}}\mathrm{Tr}\left([\Phi_{1},\Phi_{1}^{\dagger}]^{2}+[\Phi_{2},\Phi_{2}^{\dagger}]^{2}+[\Phi_{3},\Phi_{3}^{\dagger}]^{2}\right),
{R^{\lambda}}_{\mu\nu\rho}=\partial_{\rho}{\Gamma_{\mu\nu}}^{\lambda}-\partial_{\nu}{\Gamma_{\mu\rho}}^{\lambda}+{\Gamma_{\mu\nu}}^{\alpha}\diamondsuit{\Gamma_{\rho\alpha}}^{\lambda}-{\Gamma_{\mu\rho}}^{\alpha}\diamondsuit{\Gamma_{\nu\alpha}}^{\lambda}.
C_{rs}^{mn}=i[\pi_{rt}^{m},F_{ts}^{n}]=i[\pi_{ts}^{n},F_{rt}^{m}]=[A_{rt}^{m},(A^{n\dagger})_{ts}]\stackrel{_{\textstyle=}}{_{_{N}}}A_{rt}^{m}(A^{n\dagger})_{ts}
A_{\mu_{1}\cdots\mu_{N}}(k_{i0},{\bfk}_{i},T)=M^{2\epsilon}\,T\sum_{Q_{0}=2\piniT}\int\mathrm{d}^{3-2\epsilon}{\bfQ}F_{\mu_{1}\cdots\mu_{N}}(Q_{0},{\bfQ},k_{i0},{\bfk}_{i}).
G(\beta,L)=\frac{3\mu}{8\pim}\left[(e^{m\beta}-1)^{-1}+(e^{mL}-1)^{-1}+2\sum_{n_{1},n_{2}=1}^{\infty}e^{-m\sqrt{\beta^{2}n_{1}^{2}+L^{2}n_{2}^{2}}}\right],
A_{M}\tilde{A}_{N}=R_{MN,M^{\prime}N^{\prime}}\tilde{A}_{N^{\prime}}A_{M^{\prime}},\;\;\;\;\;\;A_{M}^{\dagger}\tilde{A}_{N}^{\dagger}=\tilde{A}_{N^{\prime}}^{\dagger}A_{M^{\prime}}^{\dagger}R_{M^{\prime}N^{\prime},MN}
S_{fi}^{(2)}=-\int\intdt_{1}dt_{2}\,\theta(t_{2}-t_{1})\intd{\bfr_{2}}\,\Psi_{{\bfp^{\prime}}}^{(+)\,\dagger}({\bfr_{2}})\,\tau_{3}\,H_{int}({\bfr_{2}},t_{2})\,H_{int}({\bfr_{2}},t_{1})\,\Psi_{{\bfp}}^{(+)}({\bfr_{2}})\;,
\tilde{\epsilon}_{\mu}^{a}(q)=(2\pi)^{D}\delta(q-p_{a})\epsilon_{\mu}^{a}+g(2\pi)^{D}\delta(q-p_{b}-p_{c})(t_{G}^{a})^{bc}2\frac{\epsilon^{b}\cdotp_{c}\epsilon_{\mu}^{c}}{(p_{b}+p_{c})^{2}}+{\calO}(g^{2}),
f^{(i)}=f_{\,\,{\hat{\alpha}}{\hat{\beta}}}^{(i)}{\hat{e}}^{\hat{\alpha}}\wedge{\hat{e}}^{\hat{\beta}}=2{\hat{e}}^{5}\wedge{\hat{e}}^{i}+\varepsilon_{ijk}{\hat{e}}^{j}\wedge{\hat{e}}^{k}\,,
\langleS_{\mu\nu}^{La}(x)S^{Lb}{\rho\sigma}(y)\rangle=[(M_{\mu,\rho}^{ac,bd}\eta_{c\nu}\eta_{d\sigma}-M_{\mu,\sigma}^{ac,bd}\eta_{c\nu}\eta_{d\rho})-(\mu\leftrightarrow\nu)]
L_{Weyl}^{pzm}=\intd^{3}{x}\left[-\partial_{i}a_{-}\partial_{+}a_{i}+\frac12(\partial_{+}a_{-})^{2}-\frac14(\partial_{i}a_{j}-\partial_{j}a_{i})^{2}+J_{pzm}^{\perp}a_{\perp}+J_{pzm}^{-}a_{-}\right]\;,
T_{ab}\equiv{\frac{1}{2}}\partial_{a}\phi\partial_{b}\phi-{\frac{1}{4}}\bar{\gamma}_{ab}(\nabla\phi)^{2}+{\frac{1}{4}}q\Phi_{h}(\bar{\gamma}_{ab}\Box\phi-\nabla_{a}\nabla_{b}\phi)-{\frac{1}{2}}\bar{\gamma}_{ab}U(\phi)=0~~.
ds_{11}^{2}=\left(\frac{\pi^{4}}{15}Q\right)^{-1/3}l_{p}^{\prime2}U^{2}(-d\tau^{2}+dx_{6}^{2}+\cdots+dx_{9}^{2}+dx_{11}^{2})+\left(\frac{\pi^{4}}{15}Q\right)^{2/3}l_{p}^{\prime2}(\frac{4dU^{2}}{U^{2}}+d\Omega_{(4)}^{2})
B(T,X)=\frac{\left[bt^{2}+e^{2b(x^{2}-r^{2})}(1-bt^{2})-4b^{2}r^{2}t^{2}\right]^{2}\left[bt^{2}+e^{2b(x^{2}-r^{2})}(1-bt^{2})\right]}{e^{4b(x^{2}-r^{2})}\left[1-4b^{2}x^{2}t^{2}\right]}
\gamma^{0}=\left(\begin{array}{cc}{\sigma_{3}}&{0}\\{0}&{-\sigma_{3}}\\\end{array}\right),\quad\gamma^{1}=\left(\begin{array}{cc}{i\sigma_{1}}&{0}\\{0}&{-i\sigma_{1}}\\\end{array}\right),\quad\gamma^{2}=\left(\begin{array}{cc}{i\sigma_{2}}&{0}\\{0}&{-i\sigma_{2}}\\\end{array}\right).
\Gamma\left[A\right]=\displaystyle-i\int_{0}^{\infty}\frac{dT}{T}{\calN}\int_{x(T)=x(0)}{\calD}x\:e^{\displaystylei\int_{0}^{T}d\tau\left[-\frac{\dot{x}^{2}}{4}-eA\dot{x}-m^{2}\right]},
\mu_{1,2}^{2}=m_{3/2}^{2}+V_{o}+{\tilde{\mu}}^{2},\;\;\mu_{3}^{2}=-Bm_{3/2}{\tilde{\mu}},\;\;{\tilde{\mu}}=e^{\frac{1}{2}K_{o}}\mu\frac{W_{o}^{*}}{|W_{o}|},
\widehatF_{3/2,5/2}=-{\frac{1}{40}}\simeq-.0250,\,\,\,\,\,\,\,\,\,\widehatF_{3/2,9/2}={\frac{71}{15120}}\simeq.0046,\,\,\,\,\,\,\,\,\,\widehatF_{5/2,7/2}=-{\frac{239}{7560}}\simeq-.0316.
\begin{array}{rlll}{\alpha_{1}}&{u_{2}-u_{3}}&{e_{1}}&{B_{2}^{3}}\\{\alpha_{2}}&{u_{3}-u_{4}}&{e_{2}}&{B_{3}^{4}}\\{\alpha_{3}}&{u_{4}}&{e_{3}}&{B_{4}^{0}}\\{\alpha_{4}}&{-\frac{1}{2}\left(u_{1}+u_{2}+u_{3}+u_{4}\right)}&{e_{4}}&{S^{++++}}\\\end{array}
\delta^{4}(\rho_{\mathrm{T}}-\rho_{\mathrm{T}}^{\prime}+\rho_{\mathrm{B}}-\rho_{\mathrm{B}}^{\prime})=\frac{1}{(2\pi)^{4}}\intd^{4}xe^{ix\cdot(\rho_{\mathrm{T}}-\rho_{\mathrm{T}}^{\prime}+\rho_{\mathrm{B}}-\rho_{\mathrm{B}}^{\prime})}
D_{z}=\left(\begin{array}{cc}{(\mathbf{p}_{1})^{\dagger}\mathbf{x}^{1}+(\mathbf{p}_{1}^{\prime})^{\dagger}\mathbf{x}^{2}-(\mathbf{a}_{1})}&{0}\\{0}&{(\mathbf{p}_{2})^{\dagger}\mathbf{x}^{1}+(\mathbf{p}_{2}^{\prime})^{\dagger}\mathbf{x}^{2}-(\mathbf{a}_{2})}\\{-\lambda_{1}A_{1}}&{-\lambda_{2}A_{2}}\\\end{array}\right)\,,
M_{i}=\left(\begin{array}{cc}{a_{r}}&{-1}\\{1}&{0}\\\end{array}\right)\left(\begin{array}{cc}{a_{r-1}}&{-1}\\{1}&{0}\\\end{array}\right)\left(\begin{array}{cc}{a_{1}}&{-1}\\{1}&{0}\\\end{array}\right)
\begin{array}{rlrlrlrlrl}{y_{2}y_{4}}&{y_{8}y_{10}}&{y_{1}y_{4}}&{y_{8}y_{9}}&{y_{4}y_{11}}&{y_{3}y_{12}}&{y_{5}y_{8}}&{y_{6}y_{7}}&{y_{6}y_{2}}&{y_{12}y_{1}}\\{y_{2}y_{3}}&{y_{7}y_{10}}&{y_{1}y_{3}}&{y_{7}y_{9}}&{y_{11}y_{8}}&{y_{12}y_{7}}&{y_{4}y_{5}}&{y_{3}y_{6}}&{y_{5}y_{2}}&{y_{11}y_{1}}\\\end{array}
\begin{array}{l}{\psi_{1z}=p\varphi_{1}\quad\quad,}\\{\varphi_{1\overline{{z}}}=-\overline{{p}}\psi_{1}\quad\,\,,}\\\end{array}\qquad\begin{array}{l}{\psi_{2z}=\overline{{p}}\varphi_{2}\quad\quad,}\\{\varphi_{2\overline{{z}}}=-p\psi_{2}\qquad.}\\\end{array}
(1-\eta^{2})^{-M/2}e^{-{\frac{\eta^{2}}{1-\eta^{2}}}x^{2}}H_{M-1}({\frac{x}{\sqrt{1-\eta^{2}}}})=\sum_{n=0}^{\infty}{\frac{(-\eta^{2}/4)^{n}}{n!}}H_{2n+M-1}(x).
\sum_{a=1}^{2}z_{a}^{\dagger}z_{a}=1~,\quad\sum_{a=1}^{2}z_{a}^{\dagger}\psi_{a}+h.c.=0~,\quad\sum_{a=1}^{2}\left(z_{a}^{\dagger}f_{a}+f_{a}^{\dagger}z_{a}+\psi_{a}^{\dagger}\psi_{a}\right)=0.
4\pi\rho^{4}\Theta.\nabla\rho=W+K(W^{2}+\frac{E}{2})-\frac{1}{2}V=\rho\mathrm{\Large~a}+V(\rho\mathrm{\Large~a}_{K}+\frac{1}{2})+K{\Big(}\rho^{2}(\mathrm{\Large~a}^{2}-\mathrm{\Large~a}_{K}^{2})-\frac{1}{2}(1+3\rho\mathrm{\Large~a}_{K}){\Big)}.
U_{n,\hat{q},j,m_{j}}^{\pm}=\left(\begin{array}{l}{u_{n,\hat{q},j,m_{j}}^{(1)\pm}}\\{u_{n,\hat{q},j,m_{j}}^{(2)\pm}}\\\end{array}\right)\sim\left(\begin{array}{r}{f_{n,q,j}^{\pm}\,\Psi_{q,j,m_{j}}^{\pm}}\\{h_{n,q,j}^{\pm}\,\Psi_{q,j,m_{j}}^{\pm}+g_{n,q,j}^{\pm}\,\Psi_{q,j,m_{j}}^{\mp}}\\\end{array}\right)\,,
V=-\frac{1}{2}\phi^{2}+g\kappa\phi^{3}+\frac{1}{3g^{2}\kappa^{2}\gamma^{2}}\left(1-3g^{2}\kappa^{2}\beta\gamma\phi^{2}-(1-2g^{2}\kappa^{2}\beta\gamma\phi^{2})^{3/2}\right)
{\calH}={\frac{(\Pi^{1})^{2}}{2(1+{\frac{ae^{2}}{\pi}})}}+{\frac{1}{2}}\Pi_{\Sigma}^{2}+{\frac{e}{\sqrt\pi}}A_{0}\Pi_{\Sigma}+\partial_{1}A_{0}\Pi^{1}+{\frac{e^{2}}{2\pi}}A_{1}^{2}+{\frac{1}{2}}\Sigma^{2}-{\frac{e}{\sqrt\pi}}\Sigma^{\prime}A_{1}.
S_{\mathrm{DBI}}^{(\mathrm{M10})}=-\int_{R^{9+1}}|{\hat{k}}|^{3}\sqrt{|\mathrm{det}\left({\hat{\Pi}}+|{\hat{k}}|^{-1}{\hat{\calF}}\right)|}\,{\hat{R}}({\hat{T}},\partial{\hat{T}},\dots)\,,
Z=\sum_{N_{+},N_{-}}\frac{e^{i\thetaN_{+}}}{(N_{+}!)^{N_{c}}}\frac{e^{i\thetaN_{-}}}{(N_{-}!)^{N_{c}}}\int{\calD}a{\calD}b\Lambda^{2N_{c}(N_{+}+N_{-})}w_{a}w_{b}w_{ab}
Q_{\mathrm{non-crit},W_{3}}=\oint\frac{dz}{2\pii}c_{1}\left(T_{M}+T_{G}+\frac{1}{2}T_{\mathrm{gh}}\right)+c_{2}\left(W_{3,M}+W_{3,G}+\frac{1}{2}W_{3,\mathrm{gh}}\right)
F=i\frac{dg_{1}\wedged\overline{{g}}_{1}}{(1-g_{1}\overline{{g}}_{1})^{2}}+i\frac{dg_{2}\wedged\overline{{g}}_{2}}{(1-g_{2}\overline{{g}}_{2})^{2}}=-\frac{1}{2}(R_{2}^{1}+R_{4}^{3}),
K_{\mu}^{(j)}(\theta_{1})\K_{\nu}^{(i)}(\theta_{2}){}~=~\sum_{\mu^{\prime},\nu^{\prime}}\left(S_{ji}^{({\widehatG}/H)}(\theta,q)\right)_{\mu,\nu}^{\mu^{\prime},\nu^{\prime}}~K_{\nu^{\prime}}^{(i)}(\theta_{2})\K_{\mu^{\prime}}^{(j)}(\theta_{1})\.\nonumber
{\widehatT}{}_{l}[\varphi]\equiv\frac{2l-1}{2\pii}\int_{|z|<1}\varphi(z,\bar{z})(1-z\bar{z})^{2l-2}dzd\bar{z}=\varphi(0,0)+\frac{1}{2l}\left.\bigtriangleup\varphi(z,\bar{z})\right|_{\scriptstylez=\bar{z}=0}+{\calO}(l^{-2})
\Omega_{(-2)}\left[W\right]~=~\int\,\left[\,4\,{\bar{C}}^{i}\,\wedge\,{\bar{C}}^{j}\,\partial_{i}\partial_{j}{\calW}\,+\,2i\,{\bar{C}}^{i}\,\wedge\,d\theta^{i^{*}}\,\right]
\langle\varphi^{2}(x)\rangle^{(b)}=\frac{-1}{2^{D-2}\pi^{(D+1)/2}\Gamma\left(\frac{D-1}{2}\right)}\int_{0}^{\infty}dk\,k^{D-2}\int_{0}^{\infty}d\omega\frac{\bar{K}_{\omega}(\lambdaa)}{\bar{I}_{\omega}(\lambdaa)}I_{\omega}^{2}(\lambda\xi),\quad0<\xi<a.
\sqrt{-\hat{g}_{5}}\hat{R}_{5}=\sqrt{-g_{4}}\Big\{R_{4}-6g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac14e^{-6\varphi}{\calF}_{\mu\nu}{\calF}^{\mu\nu}-2\nabla_{\mu}\left(g^{\mu\nu}\partial_{\nu}\varphi\right)\Big\},
{\calZ}(t,A,\beta)~\sim~{\frac{t^{4}}{\beta^{\frac{9}{2}}}}~e^{{\bar{f}}(\beta)A}~A^{-{\frac{7}{2}}}\quad\mathrm{with}\quad{\bar{f}}(\beta)={\frac{\pi}{\sqrt{2}}}\beta\Bigl[(1+\dots)+e^{-{\frac{\pi\sqrt{2}}{\beta}}}(4+\dots)\Bigr]
K_{\rho\sigma}^{\mu}=-\frac{\sqrt{\left|g\right|}}{4\kappa^{2}}\left(\Gamma_{\rho\sigma}^{\mu}-\frac{1}{2}g_{\rho\sigma}\Gamma_{\alpha\beta}^{\mu}g^{\alpha\beta}-\Gamma_{\alpha(\rho}^{\alpha}\delta_{\sigma)}^{\mu}+\frac{1}{2}\Gamma_{\beta\alpha}^{\alpha}g^{\beta\mu}g_{\rho\sigma}\right).
\bar{\bfN}=-\sqrt{1-\bar{U}^{2}}\frac{~\partial}{\partialU}-\frac{\bar{U}}{\sqrt{1-\bar{U}^{2}}}\left(T\frac{~\partial}{\partialT}+V\frac{~\partial}{\partialV}+X\frac{~\partial}{\partialX}+Y\frac{~\partial}{\partialY}+Z\frac{~\partial}{\partialZ}\right)
{\cal{L}}=\frac{1}{2}\,{\cal{D}}_{\mu}\Phi{\cal{D}}^{\mu}\Phi^{*}+\beta^{2}\,(1-{\cal{R}})-\frac{\lambda}{2}\,(\Phi^{*}\,\Phi-v^{2})^{2}+\frac{1}{16\piG}\,R
[a_{0},a_{n}]={\frac{1}{A}}{\frac{1}{n}}a_{0}a_{n}\hbar+{\frac{1}{2A}}\sum_{k\ne0,n}{\frac{\epsilon(n)}{\sqrt{|kn(n-k)|}}}a_{k}a_{n-k}\hbar+O(\hbar^{2}),\,n\ne0,
\lambda(z)=\frac{1}{(r^{2};t^{2})_{\infty}(t^{2}r^{-2};t^{2})_{\infty}}\frac{(r^{2}u^{2};t^{2},q^{2n})_{\infty}(t^{2}r^{-2}u^{2};t^{2},q^{2n})_{\infty}}{(r^{2}q^{2n}u^{-2};t^{2},q^{2n})_{\infty}(t^{2}r^{-2}q^{2n}u^{-2};t^{2},q^{2n})_{\infty}}\frac{\phi(u^{2})}{\phi(u^{-2})},
{\calU}=\left[\begin{matrix}{{\calU}^{L}(x^{L})}&{0}\\{0}&{{\calU}^{R}(x^{R})}\\\end{matrix}\right],~~~~~~~({\calU}^{L\dagger}{\calU}^{L}={\calU}^{R\dagger}{\calU}^{R}={\bf1}),
A_{0}=(4\pit)^{d/2}K(t)\,\sum_{i}\mathrm{tr}\left(T_{i}\right)\,e^{-m_{i}^{2}t}\left[\sum_{j=1}^{4}\delta_{ij}\left(\mathrm{tr}P+\xi^{d/2}\right)\,+\sum_{j=5}^{8}\delta_{ij}\right]
\begin{array}{lcl}{\phi_{a}}&{=}&{(\phi_{+},\phi_{-},H,\chi)}\\{Y_{a}}&{=}&{(Y_{+},Y_{-},Y_{H},Y_{\chi})}\\\end{array}\quad\begin{array}{lcl}{\hat{\phi}_{a}}&{=}&{(\hat{\phi}_{+},\hat{\phi}_{-},\hat{H},\hat{\chi})}\\{q_{a}}&{=}&{(q_{+},q_{-},q_{H},q_{\chi})}\\\end{array}
Z_{\mathrm{SS}}={\frac{1}{4}}{\frac{1}{\eta^{4}{\bar{\eta}}^{4}}}\left[(\Theta_{3}^{4}-\Theta_{4}^{4}-\Theta_{2}^{4})({\bar{\Theta}}_{3}^{4}-{\bar{\Theta}}_{4}^{4}-{\bar{\Theta}}_{2}^{4})\right]\quad,
F_{1}(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dk[\frac{M_{1}^{(+)}(k)}{M_{o}^{(-)}(k)}-\frac{M_{o}^{(+)}(k)M_{1}^{(-)}(k)}{[M_{o}^{(-)}(k)]^{2}}]e^{ikx}.
a^{\mu}(x,\tau)=\sum_{s=\mathrm{polarizations}}\int\frac{d^{4}k}{2\kappa}\left[\varepsilon_{s}^{\mu}a(k,s)e^{i(k\cdotx+\sigma\kappa\tau)}+\varepsilon_{s}^{\mu*}a^{*}(k,s)e^{-i(k\cdotx+\sigma\kappa\tau)}\right]
i\hbar\,\partial_{t}\,\alpha_{p}(t)=\sum_{j=0}^{p}\sum_{m_{0}\cdotsm_{p}}\alpha_{m_{0}\cdotsm_{p}}\,\,\hat{a}_{m_{0}}(t)\otimes\cdots\otimes\left[\hat{H},\hat{a}_{m_{j}}(t)\right]\otimes\cdots\otimes\hat{a}_{m_{p}}(t)
\sum_{n=1}^{\infty}\epsilon(n)=\int_{1}^{\infty}{dn}\,\epsilon(n)+{\frac{1}{2}}\,\epsilon(1)-{\frac{1}{12}}\,\epsilon^{\prime}(1)+{\frac{1}{720}}\,\epsilon^{\prime\prime\prime}(1)-{\frac{1}{30240}}\,\epsilon^{(v)}(1)+\cdots.
T_{\mu\nu}\equiv\nabla_{\mu}\nabla_{\nu}\phi-{\frac{1}{2}}g_{\mu\nu}[V(\phi)+2\Box\phi]+{\frac{1}{2}}(\partial_{\mu}\psi\partial_{\nu}\psi-{\frac{1}{2}}g_{\mu\nu}\partial_{\alpha}\psi\partial^{\alpha}\psi)=0
4\pi\frac{\partialz_{B}^{\prime}}{\partialP_{n}^{\prime}}=(\frac{1}{z_{B}^{\prime}-z_{n}^{\prime}}-\frac{1}{z_{B}^{\prime}})\frac{{\calP}(z_{B}^{\prime}+z_{1})}{\prod_{C\neqB}(z_{B}^{\prime}-z_{C}^{\prime})}
{\calL}_{5}\equiv{\calL}_{6}\left(\varphi,\nabla_{\mu}\varphi,\tilde{\nabla}_{\mu\nu}\varphi;\tilde{F}^{(1)},\tilde{F}^{(2)},\tilde{\tilde{C}}^{(1)},\tilde{\tilde{C}}^{(2)};\tilde{\tilde{D}}^{(2)},\tilde{\tilde{\tilde{G}}}^{(2)},\tilde{\tilde{G}}^{(3)}\right),
\begin{array}{l}{J_{\pm}\Psi_{j,m}^{\zeta}(w)=[j\pmm+1]\Psi_{j,m\pm1}^{\zeta}(w),}\\{{}}\\{{\calK}\Psi_{j,m}^{\zeta}(w)=q^{m}\Psi_{j,m}^{\zeta}(w),}\\\end{array}
G_{0}^{\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}{2}}g^{\mu\nu}g^{\alpha\beta}+{\calO}(p)}{p^{2}-i\epsilon}}e^{-ip(x-x^{\prime})}~.
\sigma=\theta\qquad\implies\qquad2\left.\sqrt{F(r)}\right|_{r=a}=\left.{\frac{\partial}{\partialr}}\left(r\sqrt{F(r)}\right)\right|_{r=a}\qquad\implies\qquad\left.{\frac{\partial}{\partialr}}\left({\frac{F(r)}{r^{2}}}\right)\right|_{r=a}=0.
ds^{2}=-\widetilde{A}(r,\theta)^{2}dt^{2}+\widetilde{B}(r,\theta)^{2}d\phi^{2}+\widetilde{C}(r,\theta)(\frac{dr^{2}}{1+\frac{r^{2}}{l^{2}}}+r^{2}d\theta^{2})
h_{ij}=\langle\psi_{i}|\psi_{j}\rangle=\langle\psi_{i}|\tilde{\psi}_{l}\rangle\tilde{h}^{lm}\langle\tilde{\psi}_{m}|\psi_{j}\rangle=B_{il}^{*}\tilde{h}^{lm}B_{mj}=(B^{*}\tilde{h}^{-1}B)_{ij}.
ds^{2}=-\left(1-\left(\frac{r_{0}}{r}\right)^{p-1}+\alpha^{2}r^{2}\right)dt^{2}+\frac{dr^{2}}{\left(1-\left(\frac{r_{0}}{r}\right)^{p-1}+\alpha^{2}r^{2}\right)}+r^{2}d\Omega_{p}^{2}
z^{l}F^{\prime}(\frac{1}{z})+\mu\frac{z^{l+2}}{1-z}F(\frac{1}{z})=zF^{\prime}(z)-\mu\frac{1}{1-z}F(z)+\muF(1)[\frac{(-1)^{l}}{1-z}+(1-z)^{l+1}+\frac{(-z)^{l}}{1-z}]+
\langle0|q|0\rangle=\left(\begin{array}{ccccc}{q_{1}}&{}&{}&{}&{}\\{}&{\ddots}&{}&{0}&{}\\{}&{}&{q_{\tilde{N}_{c}}}&{}&{}\\\end{array}\right),\qquad\langle0|{\bar{q}}|0\rangle=\left(\begin{array}{ccc}{\bar{q}_{1}}&{}&{}\\{}&{\ddots}&{}\\{}&{}&{\bar{q}_{\tilde{N}_{c}}}\\{}&{}&{}\\{}&{0}&{}\\\end{array}\right),
\begin{array}{l}{2n-2(n_{1}+\cdots+n_{j})+\beta_{ij}^{(k)}\geqa_{1}^{(j)}\geq\cdots\geqa_{n_{j}-n_{j+1}}^{(j)}\geq0,~~~~1\leqj\leqk-2,}\\{n-(n_{1}+\cdots+n_{k-1})\geqb_{1}\geq\cdots\geqb_{n_{k-1}}\geq0.}\\\end{array}
\begin{array}{c}{(\dot{x},\dot{y})=i(a\alpha\zeta^{\alpha-1},b\beta\zeta^{\beta-1})=(A\zeta^{\alpha-1},B\zeta^{\beta-1})}\\{(i\dot{y}^{*},-i\dot{x}^{*})=(iB^{*}(\zeta^{*})^{\beta-1},-iA^{*}(\zeta^{*})^{\alpha-1})}\\\end{array}
t_{R}^{(n)}=s_{R}^{(n)}\left\{\theta(s_{1}-t){\frac{t}{s_{1}}}+[\theta(s_{2}-t)-\theta(s_{1}-t)]{\frac{s_{2}-t}{s_{2}-s_{1}}}+\cdot\cdot\cdot+\theta(t_{n}-s){\frac{t-s_{n}}{s-s_{n}}}\right\}.
\phi(x)|_{x^{+}=0}=\frac{1}{\sqrt{4\pi}}\sum_{n=1}^{\infty}\frac{1}{\sqrt{\tilde{n}}}\left[a_{n}e^{-ik_{n}^{+}x^{-}}+a_{n}^{\dagger}e^{ik_{n}^{+}x^{-}}\right],
\Psi_{n,m}=\xi^{p_{1}}\eta^{p_{2}}L_{n}^{(-1/2+p_{1})}(\sqrt{-2E}\xi^{2})L_{m}^{(-1/2+p_{2})}(\sqrt{-2E}\eta^{2})e^{-\sqrt{-E/2}(\xi^{2}+\eta^{2})}
D_{mm^{\prime}}^{j}(\psi)=\frac{1}{\sqrt{(j+m^{\prime})!}}~\frac{1}{\sqrt{(j-m^{\prime})!}}~(\frac{\partial}{\partial\xi_{1}^{*}})^{j+m}~(\frac{\partial}{\partial\xi_{2}^{*}})^{j-m}~<\xi\midU(\psi)\midj,m^{\prime}>
{\calL}_{\mathrm{even/even}}=t_{8}^{r_{1}\cdotsr_{8}}t_{8}^{s_{1}\cdotss_{8}}\,W_{r_{1}r_{2}s_{1}s_{2}}W_{r_{3}r_{4}s_{3}s_{4}}W_{r_{5}r_{6}s_{5}s_{6}}\,H_{r_{7}r_{8}m}H_{s_{7}s_{8}}{}^{m}\,.
\hat{\calR}_{mn}={\calD}_{[m}\psi_{n]}+\frac{\kappa}{2}\gamma_{ab[m}\psi_{n]}v^{ab}+2i\kappa\gamma_{[m}\vec{\tau}\psi_{n]}\vec{t}-\frac{\kappa}{\sqrt{3}}\gamma^{a}\psi_{[n}\hat{F}_{m]a}
r_{0}\int[\partial_{r}\sigma_{V}(r_{0})-\partial_{r}\sigma(r_{0},\theta)]d\theta=\int_{|z|\leqr_{0}}\left(-2\pi_{z}^{\bar{z}}\pi_{\bar{z}}^{z}e^{-2\sigma(z)}(e^{-\eta(z)}-1)\right)d^{2}z=O(r_{0}^{-2\alpha}),
\begin{array}{llll}{\mathrm{spin-0}\;:\;\;}&{|0,0>}&{=}&{(q^{2}+1+q^{-2})^{-\frac{1}{2}}(q^{-1}|+->-|00>+q|-+>)}\\{}&{|1,1>}&{=}&{(q^{2}+q^{-2})^{-\frac{1}{2}}(q|0+>-q^{-1}|+0>)}\\{\mathrm{spin-1}\;:\;\;}&{|1,0>}&{=}&{(q^{2}+q^{-2})^{-\frac{1}{2}}(|-+>+(q-q^{-1})|00>)-|+->)}\\{}&{|1,-1>}&{=}&{(q^{2}+q^{-2})^{-\frac{1}{2}}(q|-0>-q^{-1}|0->)\;.}\\\end{array}
{\calL}_{F}=\bar{\psi}i\gamma^{\mu}(\partial_{\mu}-ieA_{\mu})\psi-m\bar{\psi}\psi+{\frac{g}{2}}\mid\bar{\psi}\gamma^{\mu}\psi\mid^{2}-{\frac{pe^{2}}{4}}\midA_{\mu\nu}\mid^{2}+{\frac{qe^{2}}{2}}\epsilon_{\mu\nu\lambda}A^{\mu}A^{\nu\lambda}.
G=\int\d^{2}x\left(\epsilon^{0}\omega_{0}+\epsilon^{3}\omega_{3}+\frac{1}{\sqrt{\kappa}}\epsilon^{(1)i}\tilde{\omega}_{i}^{(1)}+\sum_{n=2}^{\infty}\epsilon^{(n)i}\tilde{\omega}_{i}^{(n)}\right),
\gamma_{{\scriptscriptstyle2}s{\scriptscriptstyle-}\!{\scriptscriptstyle1}\,{\scriptscriptstyle2}s}\gamma_{123456}=\Omega_{{\scriptscriptstyle++}}+(-1)^{s}\Omega_{{\scriptscriptstyle+-}}+(-1)^{\frac{1}{2}s(s+1)}\Omega_{{\scriptscriptstyle-+}}+(-1)^{\frac{1}{2}s(s-1)}\Omega_{{\scriptscriptstyle--}}\,.
I[\Phi,\Phi^{\dagger}]=\intd^{2D}X\left[a\,\overline{{\mathcal{D}}}^{\mu\nu}\Phi^{\dagger}\star\mathcal{D}_{\mu\nu}\Phi+b\,\mathcal{D}_{\mu\nu}\Phi^{\dagger}\star\overline{{\mathcal{D}}}^{\mu\nu}\Phi-V_{\star}\left(\Phi\star\Phi^{\dagger}\right)\right].
\begin{array}{l}{M_{cd}^{ab}\equivR(\theta-\theta^{\prime})_{ef}^{ab}\;U_{eg}(\theta)R(\theta+\theta^{\prime})_{hd}^{gf}\;U_{hc}(\theta^{\prime})=}\\{N_{cd}^{ab}\equivU_{ae}(\theta^{\prime})\;R(\theta+\theta^{\prime})_{fg}^{eb}\;U_{fh}(\theta)\;R(\theta-\theta^{\prime})_{cd}^{hg}}\\\end{array}
\psi_{1}(x,t)=\sum_{n}a_{n}\phi_{n}(x)e^{-{\frac{i}{\hbar}}\epsilon_{n}t},\quad\psi_{2}(x,t)=\sum_{n}b_{n}^{\dagger}\phi_{n}(x)e^{{\frac{i}{\hbar}}\epsilon_{n}t},
\sigma(\mathrm{even})={\frac{4}{k}}{\frac{J_{0}^{2}(a)}{J_{0}^{2}(a)+N_{0}^{2}(a)}}+{\frac{8}{k}}\sum_{n=1}^{\infty}{\frac{J_{n}^{2}(a)}{J_{n}^{2}(a)+N_{n}^{2}(a)}}
{\calL}_{\mathrm{WZ}}^{(p)}=\epsilon^{i_{1}\ldotsi_{p+1}}\sum_{k=0}^{(p-1)/2}\,\,a_{p,k}C_{i_{1}\ldotsi_{p-2k}}(F^{k})_{i_{p-2k+1}\ldotsi_{p}}\partial_{i_{p+1}}T\,,
R=\sum_{n=1}^{N}x_{n}^{2}(S_{n}^{2}-T_{n})=\sum_{\Omega(n,m,m^{\prime})}\frac{x_{n}^{2}}{(x_{n}^{2}-x_{m}^{2})(x_{n}^{2}-x_{m^{\prime}}^{2})}\,
\frac{(-1)^{l}}{2^{n-l}}\int_{0}^{1}{\calD}_{n}\eta\;\sum_{\{m_{i}\}}\frac{(2(n-l))!}{m_{1}!\dotsm_{l}!}V^{(m_{1})}(x_{1})\dotsV^{(m_{l})}(x_{l}),
\begin{array}{lll}{\eta_{1}(x,z)=\beta_{1}(y)\eta(x),}&{\psi_{m1}(x,z)=\gamma\beta_{1}(y)\psi_{m}(x)}&{\psi_{51}(x,z)=0,}\\{\eta_{2}(x,z)=\beta_{2}(y)\eta(x),}&{\psi_{m2}(x,z)=\gamma\beta_{2}(y)\psi_{m}(x),}&{\psi_{52}(x,z)=0,}\\\end{array}
S=\intd^{4}xd^{2}\thetad^{2}\bar{\theta}\,\Phi\star\bar{\Phi}+\intd^{4}x\left[\intd^{2}\theta\left(\fracm2\Phi\star\Phi+\fracg3\Phi\star\Phi\star\Phi\right)+{\mathrm{c.~c.}}\right].
[m^{2}{\bfn}{\bf\nabla}_{\bfp}-(pn){\bfp}{\bf\nabla}_{\bfp}-\imath\frac{m{\vec{\sigma}}\cdot({{\bfp}\times{\bfn}})}{2(p_{0}+m)}]A_{\lambda}({\bfp};{\bfn})=0,
\intd^{8}z\,\frac{W^{2}\bar{W}^{2}}{(\Phi\bar{\Phi})^{2(m+l+k+1)}}(\nabla^{2}W^{2})^{m}({\bar{\nabla}}^{2}\bar{W}^{2})^{l}\left(-(\bar{Q}Q+\bar{\tilde{Q}}\tilde{Q})\right)^{k}
{\tilde{D}}=1+{\frac{k_{11}^{2}T_{6}}{4\pi^{2}}}\left({\frac{1}{[(x-x_{0})^{2}+(x^{11}-x_{0}^{11})^{2}]^{3/2}}}+{\frac{1}{[(x-x_{0})^{2}+(x^{11}+x_{0}^{11})^{2}]^{3/2}}}\right)\.
p(t)=p_{0},\quadx(t)=x_{0}e^{i[-2\varepsilon^{-1}{\mit\Lambda}K^{2}p_{0}+\hbar\varepsilon^{-2}({\mit\Lambda}K)^{2}+\hbar^{-1}(V(qK,\varepsilon^{-1}{\mit\Lambda})-V(K,{\mit\Lambda}))]t}
T^{00}={\frac{2cosh^{2}(\gamma)}{g^{2}}}\left[C^{2}csch^{2}(Cr)\Big(1-Cr\;coth(Cr)\Big)^{2}+{\frac{\Big(C^{2}r^{2}csch^{2}(Cr)-1\Big)^{2}}{2r^{2}}}\right]
S={\frac{1}{2}}\intd^{10}x\ee^{-2\phi}\biggl(4(\partial^{a}\phi+{\frac{1}{2}}\omega_{b}{}^{ba})^{2}+\omega_{a}{}^{bc}\omega_{b}{}^{ca}-{\frac{3}{4}}H_{abc}H^{abc}\biggr)\,
\Psi_{(1)}=\sqrt{\frac{V}{2}}\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right),\;\;\;\Psi_{(2)}=\sqrt{\frac{V}{2}}\left(\begin{matrix}{\frac{1}{\tau}}&{0}\\{0}&{-\tau}\\\end{matrix}\right).
\left[1+\frac{\partial}{\partial\lambda}+\frac{\partial^{2}}{\partial\lambda^{2}}+\cdots\right]\lambda^{\sum_{j=0}^{\infty}jn_{j}}\Bigg\vert_{\lambda=0}=\left(1-\frac{\partial}{\partial\lambda}\right)^{-1}\lambda^{\sum_{j=0}^{\infty}jn_{j}}\Bigg\vert_{\lambda=0}=\left(\sum_{j=0}^{\infty}jn_{j}\right)!,