harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
P_{N}=\left\{\left(\begin{array}{cccc|ccc}{\times}&{\times}&{}&{}&{}&{}&{}\\{\times}&{\times}&{}&{}&{}&{}&{}\\{\times}&{\times}&{\times}&{\times}&{\times}&{.}&{\times}\\{\times}&{\times}&{\times}&{\times}&{\times}&{.}&{\times}\\\hline{\times}&{\times}&{}&{}&{\times}&{.}&{\times}\\{.}&{.}&{}&{}&{.}&{.}&{.}\\{\times}&{\times}&{}&{}&{\times}&{.}&{\times}\\\end{array}\right)\right\}
\sum_{{\lambda}=1}^{2}{\epsilon}_{\mu}^{({\lambda})}(k){\epsilon}_{\nu}^{({\lambda})}(k)=-g_{{\mu}{\nu}}+\frac{n_{\mu}k_{\nu}+n_{\nu}k_{\mu}}{k_{-}}-n^{2}\frac{k_{\mu}k_{\nu}}{k_{-}^{2}}.
\begin{array}{l}{\Phi_{2}\bar{\Phi}_{2}-q^{2}\bar{\Phi}_{2}\Phi_{2}=1,}\\{\Phi_{1}\bar{\Phi}_{1}-q^{2}\bar{\Phi}_{1}\Phi_{1}=1+(q^{2}-1)\bar{\Phi}_{2}\Phi_{2},}\\{\Phi_{2}\Phi_{1}=q\Phi_{1}\Phi_{2},}\\{\Phi_{2}\bar{\Phi}_{1}=q\bar{\Phi}_{1}\Phi_{2},}\\\end{array}
\begin{array}{llll}{(1)}&{O3^{-}}&{(1/2D5-1/2NS)~or~(1/2NS-1/2D5)}&{\widetilde{O3^{+}},}\\{(2)}&{O3^{+}}&{(1/2D5-1/2NS)~or~(1/2NS-1/2D5)}&{\widetilde{O3^{-}},}\\{(3)}&{\widetilde{O3^{-}}}&{(1/2D5-1/2NS)~or~(1/2NS-1/2D5)}&{O3^{+},}\\{(4)}&{\widetilde{O3^{+}}}&{(1/2D5-1/2NS)~or~(1/2NS-1/2D5)}&{O3^{-},}\\\end{array}
I=-\intd^{4}x\left[\frac{1}{2}B_{i}^{\alpha}{\cal{L}}_{\alpha\beta}E_{i}^{\beta}+\sqrt{1+B_{i}^{\alpha}({\cal{L}}^{T}M{\cal{L}})_{\alpha\beta}B_{i}^{\beta}+\frac{1}{4}(\epsilon^{ijk}B_{j}^{\alpha}{\cal{L}}_{\alpha\beta}B_{k}^{\beta})^{2}}-1\right].
\intdq_{3}\intd\mathrm{~\left|~{\bf~q}~\right|~}\,K(\mathrm{~\left|~{\bf~p}~\right|~},p_{3};\mathrm{~\left|~{\bf~q}~\right|~},q_{3};M){\bff}(\mathrm{~\left|~{\bf~q}~\right|~},q_{3};M)=\Lambda(M)\,\,\,{\bff}(\mathrm{~\left|~{\bf~p}~\right|~},p_{3};M),
\lambda_{0}(\frac{1}{u}-i\int\frac{d^{d}k}{(2\pi)^{d}}\frac{\mu^{2}+\lambda_{0}k^{2}-2\muk^{2}}{k^{2}(k^{2}-\lambda_{0})(k^{2}-\mu)^{2}})=(c_{cr}^{2}-c^{2})
\alpha_{2}^{(1)}-\alpha_{2}^{(2)}\tau\rho+i\alpha_{1}^{(2)}\rho-i\alpha_{1}^{(1)}=0,\\\\beta_{2}^{(1)}-\beta_{2}^{(2)}\tau\rho+i\beta_{1}^{(2)}\rho-i\beta_{1}^{(1)}=0.
\left\langle{\calO}_{\alpha_{1}}\ldots{\calO}_{\alpha_{n}}\cdot\widetilde{\calO}_{\widetilde{\beta}_{1}}\ldots\widetilde{\calO}_{\widetilde{\beta}_{s}}\right\rangle=\int_{{\bfR}M}\iota^{\ast}\alpha_{1}\wedge\ldots\wedge\iota^{\ast}\alpha_{n}\wedge\widetilde{\beta}_{1}\wedge\ldots\wedge\widetilde{\beta}_{s},
({\calH}_{V})_{cl}^{\it{approx}}=\sum_{{\vec{n}},{\vec{m}}}\sum_{s,s^{\prime}}\kappa_{\vec{m}}\left[([a_{\vec{n}}^{s}]_{cl}^{*}\bar{u}_{\vec{n}}^{s}+[b_{-{\vec{n}}}^{s^{\dagger}}]_{cl}^{*}\bar{v}_{-{\vec{n}}}^{s})([a_{\vec{m}}^{s^{\prime}}]_{cl}\gamma^{0}u_{\vec{m}}^{s^{\prime}}-[b_{-{\vec{m}}}^{s^{\prime}\dagger}]_{cl}\gamma^{0}v_{-{\vec{m}}}^{s^{\prime}})\right]F({\vec{n}}-{\vec{m}},X),
\begin{array}{c}{\chi(s;k)={\calY}_{l+1/2}(\bar{z}_{0}){\calJ}_{l+1/2}(z_{2})-{\calJ}_{l+1/2}(\bar{z}_{0}){\calY}_{l+1/2}(z_{2}),}\\{\rho(s;k)={\calY}_{l+1/2}^{\prime}(\bar{z}_{0}){\calJ}_{l+1/2}(z_{2})-{\calJ}_{l+1/2}^{\prime}(\bar{z}_{0}){\calY}_{l+1/2}(z_{2}),}\\\end{array}
\left(-{\frac{1}{2}}V{\frac{\partial}{\partialQ^{a}}}\V^{-1}G^{ar}{\frac{\partial}{\partialy^{r}}}-{\frac{1}{2}}V{\omega(y)}^{-1}{\frac{\partial}{\partialy^{r}}}\{\omega(y)}V^{-1}G^{rs}{\frac{\partial}{\partialy^{s}}}+{\frac{1}{2}}C_{oa}^{ar}{\frac{\partial}{\partialy^{r}}}+V\right)\\phi=0.
ds^{2}\sim\frac{1}{8\pi^{2}g_{YM}^{2}}2^{1/3}\left(\frac{u^{2}{\alpha^{\prime}}^{2}}{L^{2}}\right)^{1/3}\left(\frac{4}{3}du^{2}+u^{2}\left(\sigma_{1}^{2}+\sigma_{2}^{2}+\frac{4}{3}\sigma_{3}^{2}\right)\right)\,
Z=\sum_{n}{\frac{1}{n!}}\prod_{i=1}^{n}\left\{\int_{0}^{\infty}{\frac{dT_{i}}{T_{i}}}\oint_{x_{i}^{\mu}(0)=x_{i}^{\mu}(T)}{\calD}x_{i}^{\mu}(-1)^{\omega_{x_{i}}}e^{-imT_{i}}\right\}\langle\prod_{i=1}^{n}W_{x_{i}}(A)\rangle
L_{0}=-\left(\dot{z}-A_{\mu}\dot{x}^{\mu}\right)^{2}-\left(\dot{w}-A_{\mu}\dot{u}^{\mu}\right)^{2}+\eta_{\mu\nu}\left(\dot{x}^{\mu}\dot{x}^{\nu}+\dot{u}^{\mu}\dot{u}^{\nu}\right),
V_{eff}=\frac{4}{\kappa^{4}}\lbrack(e^{\frac{\alphaG}{2}}),_{\bar{A}}G^{\bar{A}D}(e^{\frac{\alphaG}{2}}),_{D}-\frac{3}{4}e^{\alphaG}\rbrack=\frac{e^{\alphaG}}{\kappa^{4}}\lbrack\alpha^{2}G^{A}G_{A}-3\rbrack\,.
\deltaB_{\mu\nu}^{*a}=-\frac{1}{2}\partial_{\left[\mu\right.}A_{\left.\nu\right]}^{a},\;\delta\eta_{a}^{*}=-\partial_{\mu}A_{a}^{*\mu},\;\delta\eta_{a}^{*\mu\nu}=\frac{1}{2}\partial^{\left[\mu\right.}H_{a}^{*\left.\nu\right]},
e^{-2\phi}\approxe^{-2\phi_{0}}\frac{\Lambda}{\Psi},\,A_{\varphi}\approx\frac{e^{\phi_{0}}B\rho^{2}}{2\Lambda},\,B_{\varphi}\approx\frac{e^{-\phi_{0}}E\rho^{2}}{2\Psi},
\tilde{\calR}_{2}~=~\left(\begin{matrix}{-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}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}\\\end{matrix}\right)
\frac{1}{2\pii}\sum_{(mn)}\int_{C_{mn}}\biggl\{dz_{m}\frac{\partialf_{nm}}{\partialt^{r}}\biggr|_{z_{n}}b_{z_{m}z_{m}}-{d\bar{z}_{m}}\frac{\partial\bar{f}_{nm}}{\partialt^{r}}\biggr|_{z_{n}}b_{\bar{z}_{m}\bar{z}_{m}}\biggr\}\,
N_{e}\Sigma=\left(-\frac12N_{A}+\frac12N_{A^{*}}+\frac32N_{b}+\frac32N_{\bar{c}}-\frac12N_{c}+\frac12N_{{c}^{*}}\right)\Sigma\,+\left(m\partial_{m}+\frac12g\partial_{g}\right)\Sigma\,\,.
S=\int_{0}^{1}\left[-\frac{1}{2e}\left(\dot{x}^{\mu}+g_{1}Q^{\mu}-i\psi^{\mu}\chi\right)^{2}-i\psi_{\mu}\dot{\psi}^{\mu}+g_{2}\left(\Lambda-\frac{\alpha}{2}\right)\right]d\tau\;,
\left(\begin{matrix}{{\calH}_{2}\;,-{\calH}_{1}}\\\end{matrix}\right)^{\prime}={\frac{|S_{0}|}{S_{0}}}\left(\begin{matrix}{{\calH}_{2}\;,-{\calH}_{1}}\\\end{matrix}\right)\Lambda^{-1}\,\qquad\left(\begin{matrix}{{\calH}_{1}}\\{{\calH}_{2}}\\\end{matrix}\right)^{\prime}={\frac{|S_{0}|}{S_{0}}}\,\Lambda\left(\begin{matrix}{{\calH}_{1}}\\{{\calH}_{2}}\\\end{matrix}\right)\.
i{\tilde{A}}^{+}=\left[\begin{array}{ll}{<\psi_{3}^{(0)}|d\psi_{3}^{(0)}>}&{<\psi_{4}^{(0)}|d\psi_{3}^{(0)}>}\\{<\psi_{3}^{(0)}|d\psi_{4}^{(0)}>}&{<\psi_{4}^{(0)}|d\psi_{4}^{(0)}>}\\\end{array}\right]=i{\tilde{A}}_{a}dx^{a}.
G_{N}(t^{\prime},x^{\prime};t,x)=\int\frac{dw}{2\pi}\frac{1}{2\bar{k}}e^{-iw(t^{\prime}-t)}(e^{i\bar{k}|x^{\prime}-x|}+e^{-i\bar{k}(x^{\prime}+x)}),{}~~\bar{k}=\sqrt{w^{2}-m_{a}^{2}}.
V_{\mathrm{grav}}=\epsilon_{\mu\nu}\:\Big(\bar{\partial}X^{\nu}-\frac{1}{4}\bar{S}^{a}\gamma_{ab}^{\nu\rho}\bar{S}^{b}k_{\rho}\Big)\Big({\partial}X^{\mu}-\frac{1}{4}S^{a}\gamma^{\mu\sigma}S^{b}k_{\sigma}\Big)e^{ikX}:
\breve{\omega}_{1}\simeq-{\frac{m_{0}a}{\breve{R}^{4}}}x^{2}\int_{-\infty}^{\infty}dv{\frac{1}{(1+v^{2})^{{\frac{5}{2}}}}}=-{\frac{4m_{0}a}{3\breve{R}^{4}}}x^{2}\,\,,
|g_{1},g_{2}\rangle\otimes{\frac{1}{\sqrtn_{\nu}}}\sum_{i}|e_{i}^{\nu}\otimese_{i}^{*\nu}\rangle\rightarrow|g_{1},g_{2}\rangle\otimes{\frac{1}{\sqrtn_{\nu}}}\sum_{ij}|e_{j}^{\nu}\otimese_{i}^{*\nu}\rangleD_{ji}^{(\nu)}(g_{1}g_{2}),
I_{m}=\intd^{4}x\sqrt{{\tilde{g}}}\left[-\frac{1}{2}D_{\mu}\Phi(D^{\mu}\Phi)^{*}-\frac{1}{2}D_{\mu}\Sigma(D^{\mu}\Sigma)^{*}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}H_{\mu\nu}H^{\mu\nu}-V(|\Phi|,|\Sigma|)\right],
\mu^{\prime\prime\rho}=-\left(\partial^{\rho}\eta^{*\alpha(\sigma)}\right)B_{\alpha(\sigma)}+\eta^{*\alpha(\sigma)}\partial^{\rho}B_{\alpha(\sigma)}+\varepsilon^{\alpha\beta\gamma\delta}A_{\alpha\beta}^{*\;\;(\sigma)}\partial^{\rho}A_{\gamma\delta(\sigma)}^{*}.
\begin{array}{lrr}{D_{+}=\frac\partial{\partial\theta_{+}}-i\theta_{+}\partial_{+}}&{}&{}\\{D_{-}=\frac\partial{\partial\theta_{-}}-i\theta_{-}\partial_{-}}&{}&{}\\{\partial_{\pm}=\partial_{0}\pm\partial_{1}}&{}&{}\\\end{array}
ds^{2}=2e^{2\phi}d\tilde{u}(d\hat{v}+\omega_{i}d\tilde{x}^{i})+\xi^{2}d\tilde{u}^{2}-\sum_{i=1}^{i=3}d\tilde{x}^{i}d\tilde{x}^{i}-(d\hat{x})^{2}-\sum_{i=5}^{i=8}d\tilde{x}^{i}d\tilde{x}^{i}\,
X_{\mu\nu}^{\bot}X_{\rho\sigma}^{\bot}=\frac{1}{2}\left(\,\pi_{\mu\nu}\pi_{\sigma\rho}-\pi_{\mu\sigma}\pi_{\nu\rho}-\pi_{\mu\rho}\pi_{\nu\sigma}\,\right)Q_{0}^{2}+\frac{i}{4}\left(\,\pi_{\mu\sigma}\varepsilon_{\nu\rho\lambda}+\pi_{\nu\rho}\varepsilon_{\mu\sigma\lambda}\,\right)e^{(0)\lambda}Q_{0}.
\rho_{m_{0}\rightarrowm_{1}+\cdots+m_{N}}^{D}\!=\!\left(\!\prod_{i}\!\int\!\frac{d\mu_{i}}{2\pi}\!\right)2\pi\delta(\sum_{i}\mu_{i})\rho_{m_{0}\rightarrow\!m_{1\mu_{1}}\!+\cdots+m_{N\mu_{N}}}^{D-1};\,m_{i\mu}^{2}\!\equiv\!m_{i}^{2}\!+\!\mu^{2},
S=\intd^{5}x\sqrt{-G}e^{-\frac{3}{4}b\phi}\left[R+\left(\frac{3}{4}b^{2}-\frac{4}{3}\right)\left(\nabla\phi\right)^{2}\right]-\int_{x^{5}=0}d^{4}x\sqrt{-g}T.
\frac{\partial}{\partialA_{p}^{c}}{\calA}_{n(b)}^{m(a)}=\frac{1}{2}\delta_{n}^{m}g^{2}f_{cbe}f_{d}^{~ae}A_{p}^{d}+\frac{1}{2}\delta_{n}^{m}g^{2}f_{ebd}f_{c}^{~ad}A_{p}^{e}
K_{c}=\left(\frac{(ik)^{2}+2ik\sqrt{1+\sigma_{0}}\sqrt{1+\sigma_{1}}+2(\sigma_{0}+\sigma_{1})}{(ik)^{2}-2ik\sqrt{1+\sigma_{0}}\sqrt{1+\sigma_{1}}+2(\sigma_{0}+\sigma_{1})}\right)\left(\frac{ik-2}{ik+2}\right).
R_{\;B}^{A}\;=\;\left(\begin{array}{cc}{\gamma^{\mu\nu}\,R_{\mu\nu}^{(1)}\;_{\;B}^{A}}&{k\,\gamma^{\mu}\,\gamma^{5}\,e^{-\sigma}\,P_{\mu}\;_{\;B}^{A}}\\{k^{*}\,\gamma^{\mu}\,\gamma^{5}\,e^{-\sigma}\,Q_{\mu}\;_{\;B}^{A}\quad}&{\gamma^{\mu\nu}\,R_{\mu\nu\;B}^{(2)\,A}\,+\,(k^{*}k-1)\,L_{\;B}^{A}}\\\end{array}\right)\;.
\frac{1}{2^{N}}<\frac{\langleO_{N}^{\dagger}\,(O_{N/2}O_{N/2})\rangle}{\sqrt{\langleO_{N}^{\dagger}\,O_{N}\rangle}{\sqrt{\langle(O_{N/2}O_{N/2})^{\dagger}\,(O_{N/2}O_{N/2})\rangle}}}<\frac{1}{2^{N/2}}
\frac{\partial\Gamma_{\Lambda}}{\partial\Lambda}=-\frac{1}{2}\int\frac{d^{D}q}{(2\pi)^{D}}\frac{\delta(q-\Lambda)}{\gamma(q,\Lambda)}\left[\hat{\Gamma}.\left(1+G.\hat{\Gamma}\right)^{-1}\right](\mathbf{q},\mathbf{q^{\prime}})
\prod_{i=0}^{r}\left[\frac{-\pi\mu_{i}}{\gamma(-e_{i}^{2}b^{2}/2)}\right]^{n_{i}}=\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]^{2h(1+b^{2})},
{\calD}_{\mu}{}^{\pm}{\calF}_{\lambda\nu}=\partial_{\mu}{}^{\pm}{\calF}_{\lambda\nu}+[{}^{\pm}{\calA}_{\mu},{}^{\pm}{\calF}_{\lambda\nu}]-{}^{\pm}{\calF}_{\lambda\sigma}\Gamma^{\sigma}{}_{\nu\mu}-{}^{\pm}{\calF}_{\sigma\nu}\Gamma^{\sigma}{}_{\lambda\mu}
H_{R}(NN)=N_{F}^{R}(NN)+\frac{1}{12}=\sum_{n=1}^{\infty}n({d_{-n}}^{-j}{d_{n}}_{j}+{d_{-n}}^{j}{d_{n}}_{-j})+\frac{1}{12}\;\;\;\;\;\;\;(j=2)
\{\tilde{\Theta}_{\alpha}^{a},\tilde{\Theta}_{\beta}^{b}\}=\{\tilde{\Theta}_{\alpha}^{a},\tilde{\Psi}_{l}\}=\{\tilde{\Theta}_{\alpha}^{a},\tilde{H}\}=0~~;~~\{\tilde{\Psi}_{l},\tilde{\Psi}_{n}\}\approx0~;~\{\tilde{\Psi}_{l},\tilde{H}\}\approx0.
{\calH}_{S}=3D{\frac{1}{2}}\Pi_{i}^{2}+{\frac{1}{2}}\partial_{i}A_{j}\partial=_{i}A_{j}+({\frac{m^{2}}{2}}+{\frac{\mu^{2}}{8}})A_{i}A_{i}=20-{\frac{\mu}{2}}\epsilon_{ij}\Pi_{i}A_{j}+{\frac{1}{2m^{2}}}\Pi_{\theta}^{2}+{\frac{m^{2}}{2}}\partial_{i}\theta\partial_{i}\theta.
\frac{1}{3}\intp_{1}=-\frac{1}{48\pi^{2}}\intd\tau\intd\vec{\sigma}\cdot\vec{\nabla}(\frac{2\vec{\nabla}V\cdot\vec{\nabla}V}{V^{3}})=-\frac{1}{6\pi^{2}}\intd\vec{\sigma}\cdot\vec{\nabla}(\frac{\vec{\nabla}V\cdot\vec{\nabla}V}{V^{3}})~~,
\begin{array}{lll}{{\widetildeT}_{\alpha{\dot{\alpha}}}}&{=}&{\frac{1}{12}D_{\alpha}\phi{\bar{D}}_{\dot{\alpha}}{\bar{\phi}}K_{\phi{\bar{\phi}}}-\frac{i}{6}\partial_{\alpha{\dot{\alpha}}}\phiK_{\phi}+\frac{i}{6}\partial_{\alpha{\dot{\alpha}}}{\bar{\phi}}K_{\bar{\phi}}\,}\\{{\widetildeJ}}&{=}&{-\frac{1}{4}{\bar{D}}^{2}(K-q\phiK_{\phi})+3{\mathcalW}-q\phi{\mathcalW}_{\phi}\.}\\\end{array}
\frac{c_{2r-2}^{-\nu}}{c_{2r}^{-\nu}}=\frac{1}{\displaystyleh^{-2}[s^{2}-(-\nu+2r-2)^{2}]-\frac{1}{\displaystyleh^{-2}[s^{2}-(-\nu+2r-4)^{2}]-\frac{1}{\displaystyleh^{-2}[s^{2}-(-\nu+2r-6)^{2}]\cdot\cdot\cdot}}}
S_{\mathrm{P5h}}=-T_{\mathrm{P5h}}\intd^{6}\xi\e^{-2{\phi}}k^{-1}\sqrt{|\mathrm{det}\(\partial_{i}X^{\mu}\partial_{j}{X}^{\nu}g_{\mu\nu}-k^{2}F_{i}F_{j})|}+\mathrm{WZ}\,,
\sum_{k=0}^{\infty}\frac{q^{2k^{2}+(2\mu_{a}-1)k}}{(q)_{2k+\mu_{a}}}=\frac{Q(q^{2})}{Q(q)}=q^{-1/24}\chi_{1,2}^{(4)}(q)\equivq^{-1/24}\chi_{1/16}(q).
T_{00}={\frac{1}{4}}F_{ij}^{a}F^{aij}+{\frac{1}{2}}F_{0i}^{a}F^{a0i}+{\frac{A^{2}}{2}}D_{i}\Phi^{a}D^{i}\Phi^{a}+{\frac{A^{2}}{2}}D_{0}\Phi^{a}D^{0}\Phi^{a}
\begin{array}{ll}{N(t)|n;t\bigr>=n|n;t\bigr>,}&{\Lambda(t)|n;t\bigr>=2\hbar\sqrt{\kappa}\Bigl(n+\frac{1}{2}\Bigr),}\\{\sum_{n}|n;t\bigr>\bigl<n;t|=\mathrm{I},}&{\bigl<n;t|m;t\bigr>=\delta_{nm}.}\\\end{array}
+\sum_{i<j}\xi_{i}\xi_{j}\xi_{i}^{*}\xi_{j}^{*}+\cdots+\sum_{i_{1}<\cdots<i_{n}}\xi_{i_{1}}\cdots\xi_{i_{n}}\xi_{i_{1}}^{*}\cdots\xi_{i_{n}}^{*}+\cdots+\xi_{1}\cdots\xi_{p}\xi_{1}^{*}\cdots\xi_{p}^{*}
\begin{array}{c}{[P_{\mu}~,~P_{\nu}]=[P_{\mu}~,~Q_{\alpha}]=[P_{\mu}~~\overline{{Q}}_{\dot{\alpha}}]=\{Q_{\alpha}~,~Q_{\beta}\}=\{\overline{{Q}}_{\dot{\alpha}}~,~\overline{{Q}}_{\dot{\beta}}\}=0}\\{\{Q_{\alpha}~,~\overline{{Q}}_{\dot{\alpha}}\}=2\sigma_{\alpha\dot{\alpha}}^{\mu}P_{\mu}}\\\end{array}
S=T\intd^{3}x\Phi[-\frac{1}{2}\gamma^{ab}L_{a}^{\mu}L_{b\mu}-\frac{\varepsilon^{abc}}{\sqrt{-\gamma}}(L_{a}^{\mu}L_{b}^{\nu}L_{c\mu\nu}+\frac{9}{10}L_{a}^{\mu}L_{b}^{\alpha}L_{c\mu\alpha}-\frac{1}{5}L_{a}^{\alpha}L_{b}^{\beta}L_{c\alpha\beta})]
\mathrm{Tr}e^{iF_{9i}}-2\mathrm{Tr}_{A}e^{iF_{9i}}=n_{i}-2\mathrm{tr}F_{9i}^{2}+2\Big(\mathrm{tr}F_{9i}\Big)^{2}+{\frac{2}{3}}\mathrm{tr}F_{9i}^{4}-{\frac{2}{3}}\mathrm{tr}F_{9i}\mathrm{tr}F_{9i}^{3}+\dots
\int\frac{d(Reu_{j})d(Rev_{j})d(Imu_{j})d(Imv_{j})d\mu_{j}d\nu_{j}d\overline{\mu}_{j}d\overline{\nu}_{j}}{|u_{j}-v_{j}-\mu_{j}\nu_{j}|^{2}}Z_{n-1}A_{1}
S_{\mathrm{YM}}=-\,\frac{1}{4g^{2}}\intd^{4}x\,F_{\alpha\beta}^{a}F_{a}^{\alpha\beta}+\frac{\theta_{\mathrm{YM}}}{32\pi^{2}}\intd^{4}x\,F_{\alpha\beta}^{a}{\widetilde{F}}_{a}^{\alpha\beta}
F_{{\sfGI}\,ij}^{a}({\bfr})=\partial_{j}A_{{\sfGI}\,i}^{a}({\bfr})-\partial_{i}A_{{\sfGI}\,j}^{a}({\bfr})-g\epsilon^{abc}A_{{\sfGI}\,i}^{b}({\bfr})A_{{\sfGI}\,j}^{c}({\bfr}).
\tilde{\quad\bar{g}^{\gamma\nu}}(x)\partial_{y}(a^{-2}(y)\bar{h}_{\mu\nu}(x))=-{\kappa_{5}}\left(T^{\gamma}{}_{\mu}-{\frac{1}{3}}\delta^{\gamma}{}_{\mu}\,T^{\lambda}{}_{\lambda}\right)({\bfx}).\qquad(\mathrm{at}\quady=0+)
M=\left(\begin{matrix}{G^{-1}}&{-G^{-1}C}&{-G^{-1}a^{T}}\\{-C^{T}G^{-1}}&{G+C^{T}G^{-1}C+a^{T}a}&{C^{T}G^{-1}a^{T}+a^{T}}\\{-aG^{-1}}&{aG^{-1}C+a}&{I+aG^{-1}a^{T}}\\\end{matrix}\right),
a=\left(\begin{array}{llll}{a_{11}}&{a_{12}}&{0}&{0}\\{a_{21}}&{a_{22}}&{0}&{0}\\{0}&{0}&{a_{11}}&{a_{12}}\\{0}&{0}&{a_{21}}&{a_{22}}\\\end{array}\right)\simeq\left(\begin{array}{ll}{a_{11}}&{a_{12}}\\{a_{21}}&{a_{22}}\\\end{array}\right)\otimes\underline{{1}}
S_{grav}=-\intd^{5}x\sqrt{h}\Big({\frac{1}{2}}R+{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}\Big)+{\frac{1}{6\sqrt{6}}}\intd^{5}x\epsilon^{\mu\nu\rho\sigma\tau}F_{\mu\nu}F_{\rho\sigma}A_{\tau}.
G^{\mu}(k)=F^{\mu}(k)-\frac{1}{2\pi}\intd^{2}k^{\prime}(g^{{\mu}{\nu}}+{\epsilon}^{{\mu}{\nu}}){\lambda}_{{\nu}{\sigma}}(k-k^{\prime})\left[F^{\sigma}(k^{\prime})-{\epsilon}^{{\sigma}{\rho}}F_{\rho}(k^{\prime})\right].
\gamma^{\mu\rho\nu}\partial_{\rho}\Psi_{\nu}^{A}-\gamma^{\mu\nu}\gamma^{5}\partial_{5}\Psi_{\nu}^{A}+k\epsilon\gamma^{\mu\nu}\gamma^{\hat{5}}\Psi_{\nu}^{A}-\sqrt{2}gR\epsilon(\sigma_{3})_{B}^{A}\gamma^{\mu\nu}\Psi_{\nu}^{B}=0,
\tilde{\Phi}^{J}=\phi^{J}+\theta\;(\psi_{+}^{J}+\psi_{-}^{J})\qquad\mathrm{and}\qquad\tilde{\Sigma}_{J}=(\psi_{+}^{J}-\psi_{-}^{J})+2\theta\;\partial_{x}\phi^{J}\.
H_{1}=1+\sum_{i}{\frac{Q_{i}}{[|\vec{x}-\vec{x}_{0\,i}|^{2}+{\frac{4Q}{(p-3)^{2}}}|\vec{z}-\vec{z}_{0}|^{p-3}]^{\frac{2}{p-3}}}},\\\H_{2}={\frac{Q}{|\vec{z}-\vec{z}_{0}|^{5-p}}}.
Q^{xA}=0,\quad\bar{Q}_{A}^{l}=\left(\begin{array}{ccc|cc}{\bar{a}_{1}}&{}&{}&{}&{}\\{}&{\ddots}&{}&{}&{}\\{}&{}&{\bar{a}_{r}}&{}&{}\\\hline{}&{}&{}&{}&{}\\{}&{}&{}&{}&{}\\{}&{}&{}&{}&{}\\\end{array}\right).
\delta_{\kappa_{1}}p_{1}^{\mu}=2\bar{\kappa}_{1}\left(\begin{array}{c}{\lambda_{2}^{2}\gamma^{\mu}\,\gamma\cdotp_{1}\-\gamma\cdot\lambda_{2}\,\,\gamma\cdotp_{1}\,\lambda_{2}^{\mu}}\\{+\lambda_{2}\cdotp_{1}\,\,\gamma\cdot\lambda_{2}\,\,\gamma^{\mu}\,}\\\end{array}\right)\dot{\theta}_{1}.
{\frac{1}{a}}\cdot\left(\begin{array}{cccc}{1}&{-{\frac{b}{a}}}&{\left({\frac{b}{a}}\right)^{2}}&{\dots}\\{{\frac{c}{a}}}&{\left({\frac{d}{a}}-{\frac{2bc}{a^{2}}}\right)}&{\dots}&{}\\{\left({\frac{c}{a}}\right)^{2}}&{\dots}&{}&{}\\{\vdots}&{}&{}&{}\\\end{array}\right)\left(\begin{array}{l}{1}\\{z}\\{{z}^{2}}\\{\vdots}\\\end{array}\right).
g_{\tilde{r}\tilde{r}}(G_{\mu}^{\mu}-2G_{t}^{t})=16\piG\frac{m}{f}(L_{M}+g^{tt}\frac{\partialL_{M}}{\partialg^{tt}}-g^{\tilde{r}\tilde{r}}\frac{\partialL_{M}}{\partialg^{\tilde{r}\tilde{r}}}-g^{\theta\theta}\frac{\partialL_{M}}{\partialg^{\theta\theta}}-g^{\varphi\varphi}\frac{\partialL_{M}}{\partialg^{\varphi\varphi}})
\frac{2{\dot{m}}(\tau,a)}{{\kappa}^{2}{r_{h}}^{2}}{\sim}-\frac{1}{f}\left[\frac{d}{da}\left({\sqrt{f+{\dot{a}}^{2}}}\right)-\frac{1}{2}\frac{\dot{f}}{f^{2}}\left(\dot{a}+\sqrt{f+\dot{a}^{2}}\right)^{2}\right]^{5}\left(\dot{a}+\sqrt{\dot{a}^{2}+f}\right)
\int{\calD}G^{+}exp\Bigg\{-i\int_{X}\epsilon^{\mu\nu\rho\sigma}G_{\mu\nu}^{+ab}\bigg(R_{\rho\sigmaab}^{+}-(\partial_{\rho}\omega_{\sigmaab}^{+}-\partial_{\sigma}\omega_{\rhoab}^{+}+[\omega_{\rho}^{+},\omega_{\sigma}^{+}]_{ab})\bigg)\Bigg\}.
ds^{2}=-\left(1+{\frac{\hat{r}^{2}}{\ell^{2}}}\right)\,d\hat{t}^{2}+\left(1+{\frac{\hat{r}^{2}}{\ell^{2}}}\right)^{-1}\,d\hat{r}^{2}+\hat{r}^{2}\,d\hat{\phi}^{2},
u^{{m(k)}}\deltau_{{m}}^{{(n)}}(\Gamma_{{(k)(n)}})_{{b}}^{{a}}=D(v^{-{1}}\deltav)_{{b}}^{{a}}+(C\Gamma^{{(k)}}(v^{-{1}}\deltav)\Gamma_{{(k)}}C^{-{1}})_{{b}}^{{a}}\qquad
\Gamma_{\nu}(p)=-\int\frac{d^{3}q}{(2\pi)^{3}}D(p-q)\,\gamma_{\mu}\,S\left[-\left(m+\frac{\delta}{2}\right)iw+q\right]\Gamma_{\nu}(q)\,S\left[\left(m+\frac{\delta}{2}\right)iw+q\right]\gamma_{\mu}.
{\hat{\Pi}}=D{\hat{X}}^{\hat{\mu}}D{\hat{X}}^{\hat{\nu}}{\hat{g}}_{{\hat{\mu}}{\hat{\nu}}}=\partial{\hat{X}}^{\hat{\mu}}\partial{\hat{X}}^{\hat{\nu}}\left({\hat{g}}_{{\hat{\mu}}{\hat{\nu}}}+|{\hat{k}}|^{-2}{\hat{k}}_{\hat{\mu}}{\hat{k}}_{\hat{\nu}}\right)\,.
tu=-1/2(\eta^{\mu\alpha}\eta^{\nu\beta}+\eta^{\nu\alpha}\eta^{\mu\beta}-\eta^{\mu\nu}\eta^{\alpha\beta})k_{\mu}^{1}k_{\nu}^{2}k_{\alpha}^{3}k_{\beta}^{4}\equiv-1/2K^{\mu\nu;\alpha\beta}k_{\mu}^{1}k_{\nu}^{2}k_{\alpha}^{3}k_{\beta}^{4}
\Omega_{2}=\hat{R}_{MNPQ}\hat{R}^{MNPQ}-4\hat{R}_{MN}\hat{R}^{MN}+\hat{R}^{2}\stackrel{\mathrm{4D}}{=}\frac{1}{4}\hat{R}_{MN}^{\\\\AB}\hat{R}_{PQ}^{\\\\CD}\eta^{MNPQ}\eta_{ABCD}.
\hat{H}_{0}=\int_{(\infty)}\frac{d^{3}\vec{k}}{(2\pi)^{3}2\omega(\vec{k}\,)}\frac{1}{2}\omega(\vec{k}\,)\left[a^{\dagger}(\vec{k}\,)a(\vec{k}\,)+a(\vec{k}\,)a^{\dagger}(\vec{k}\,)\right]\,
S={\frac{l}{2\kappa^{2}}}\intd^{4}x\sqrt{-h}\left[\PsiR-{\frac{\omega(\Psi)}{\Psi}}\Psi^{|\alpha}\Psi_{|\alpha}\right]+\intd^{4}x\sqrt{-h}{\calL}^{A}+\intd^{4}x\sqrt{-h}\left(1-\Psi\right)^{2}{\calL}^{B}\.
\begin{array}{l}{\displaystyle-\frac{1}{r}\frac{d}{dr}\left(r\frac{d\varphi_{a}}{dr}\right)+\frac{4}{r^{2}}(6a_{0}^{2}-6a_{0}+1)\varphi_{a}=0,}\\{\displaystyle-\frac{1}{r^{3}}\frac{d}{dr}\left(r^{3}\frac{d\varphi_{h}}{dr}\right)+\frac{3}{r^{2}}a_{0}^{2}\varphi_{h}=0.}\\\end{array}
\dot{S}=\intdV{\frac{\eta}{2kT}}\Bigl\{\Bigl({\frac{\partialv_{i}}{\partialx_{k}}}-{\frac{\partialv_{k}}{\partialx_{i}}}\Bigr)^{2}+p{\frac{\partialv_{l}}{\partialx_{l}}}+{\frac{\partial}{\partialx_{i}}}\Bigl(v_{k}{\frac{\partialv_{i}}{\partialx_{k}}}\Bigr)\Bigr\}\.
\mathrm{\bf~W}^{\prime}(\rho)+\mathrm{\bf~W}^{2}(\rho)=\frac{(m+\frac12)(m+\frac12-\sigma_{3})}{\rho^{2}}+\frac{\sigma_{1}}{\rho}+\frac{{\bfI}}{8(m+1)^{2}},
\begin{array}{ccc}{J_{1}}&{=}&{\displaystyle\frac{1}{q+q^{-1}}\left(q^{2}J_{+}+q^{-2}J_{-}-{\calT}_{+}-{\calT}_{-}\right)}\\{J_{2}}&{=}&{\displaystyle-\frac{1}{q+q^{-1}}\left(q^{-2}J_{+}+q^{2}J_{-}-{\calT}_{+}-{\calT}_{-}\right)}\\{J_{3}}&{=}&{\displaystyleq^{2J_{0}}+q^{-2J_{0}}}\\\end{array}
\qquad\qquad\qquad\qquad=\intdx\left(\pi\left(x,t\right)\frac{\delta}{\delta\phi\left(x,t\right)}-\left(\left(m^{2}-\nabla_{x}^{2}\right)\phi\left(x,t\right)\right)\,\frac{\delta}{\delta\pi\left(x,t\right)}\right)
\begin{array}{rcl}{x^{i}x^{j}}&{=}&{qx^{j}x^{i}~,~~~~~\mathrm{for}~i<j~~\mathrm{and}~~i\not=-j~,}\\{x^{i}x^{-i}}&{=}&{x^{-i}x^{i}+\lambdaq^{i-2}L_{i-1}}\\{}&{=}&{q^{-2}x^{-i}x^{i}+\lambdaq^{i-2}L_{i}~,~~~~~\mathrm{for}~i>0~,}\\\end{array}
e^{-1}{\calL}=-{\frac{1}{2}}R-{\frac{1}{4}}G_{IJ}F_{\mu\nu}{}^{I}F^{\mu\nuJ}-{\frac{1}{2}}g_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{-1}}{48}}\epsilon^{\mu\nu\rho\sigma\lambda}C_{IJK}F_{\mu\nu}^{I}F_{\rho\sigma}^{J}A_{\lambda}^{k}\.
\Theta_{[1/2]}\,=\,\left(\begin{array}{cc}{0}&{-\,1}\\{1}&{0}\\\end{array}\right)\,\,,\quad\Theta_{[1]}\,=\,\left(\begin{array}{ccc}{0}&{0}&{1}\\{0}&{-\,1}&{0}\\{1}&{0}&{0}\\\end{array}\right)\quad.
S_{DBI}\sim\intdx_{4}^{2}\,\,d\Omega_{2}\,\,e^{-\Phi_{D}}\sqrt{g}\left[1+\frac{1}{2}g^{\mu\nu}g_{mn}\partial_{\mu}x^{m}\partial_{\nu}x^{n}-\frac{1}{2}g^{\mu\tau}g^{\nu\rho}F_{\mu\nu}F_{\tau\rho}\right].
\widetilde{\Phi}_{c}^{2}\simeq\frac{4!N\widetilde{h}^{2}\widetilde{m}^{2}}{(4\pi)^{2}}\left(\frac{N\widetilde{h}^{4}}{(4\pi)^{2}}-\lambda\right)^{-1}\simeq\frac{48\widetilde{m}^{2}}{\widetilde{h}^{2}},\\\\\R_{c}\simeq12\widetilde{m}^{2}.
\Gamma({\bfp},{\bfq};E)=K({\bfp},{\bfq})+\int\frac{d^{2}{\bfk}}{(2\pi)^{2}}K({\bfp},{\bfk})\frac{i}{E-\frac{{\bfk}^{2}}{m}+i\epsilon}\Gamma({\bfk},{\bfq};E).
S_{\mathrm{eff}}=\int\!d^{2}x\,\Bigl({\frac{1}{2}}\,\partial_{\alpha}\chi\partial_{\alpha}\chi+{\frac{1}{2}}\,\partial_{\alpha}\chi^{\prime}\partial_{\alpha}\chi^{\prime}+{\frac{v_{N-1}}{2}}\,\chi^{2}-{\frac{\tilde{\delta}v}{2}}\,\left(\chi^{2}+\chi^{2}\right)+{\frac{\pi\tilde{V}}{N}}\,\left(\chi^{2}+\chi^{2}\right)^{2}\Bigr).
\varphi_{1}(z;r)=\frac{(x^{2}rz;x^{4})_{\infty}}{(x^{4}rz;x^{4})_{\infty}},~~~~\varphi_{2}(z^{2})=\frac{(x^{8}z^{2};x^{8})_{\infty}}{(x^{6}z^{2};x^{8})_{\infty}}.
|z)=e_{q}^{zJ_{q}^{+}}|-j>=e^{zT_{J}^{+}}|-j>=\sum_{m=-j}^{j}\left[\begin{array}{c}{2j}\\{j+m}\\\end{array}\right]_{q}^{1/2}z^{j+m}|m>=e^{zT_{z}^{+}}e^{-zJ^{+}}|z),
\Gamma_{\mu_{1}\mu_{2}\mu_{3}}^{\mathrm{UV}}=\left(\frac{17}{6}-\frac{3\xi}{2}\right)\frac{g^{2}C_{G}}{\left(4\pi\right)^{2}}\left(\frac{1}{2\epsilon}+\mathrm{ln}\left(M\right)\right)i\left(\eta_{\mu_{1}\mu_{2}}\left(k_{1}-k_{2}\right)_{\mu_{3}}+\eta_{\mu_{2}\mu_{3}}\left(k_{2}-k_{3}\right)_{\mu_{1}}+\eta_{\mu_{3}\mu_{1}}\left(k_{3}-k_{1}\right)_{\mu_{2}}\right).
\begin{array}{ll}{\hat{q}}&{=\frac{2}{(k^{+}+1)}\ell_{3}^{+}}\\{}&{}\\{\hat{h}}&{=\frac{1}{(k^{+}+1)}\{\ell^{+}(\ell^{+}+1)-(\ell_{3}^{+})^{2}\}}\\\end{array}
\stackrel{\alpha}{\omega}_{0,p}[\beta]\lhd\stackrel{P}{\calO}{}\!\!_{0,p}^{(\pm)}\;=\;\sum_{s\in{\calS}^{p-3}}\stackrel{P}{K}{}\!\!_{0,p}^{(\pm)}\left(\begin{array}{c}{\!\!\alpha\!\!}\\{\!\!\beta,s\!\!}\\\end{array}\right)\stackrel{\alpha}{\omega}_{0,p}\!\!\lbrack\beta+s\rbrack\;\;,
S_{G}=\frac{1}{2(\kappa_{3})^{2}}\sum_{n=1}^{N}\intd^{3}x\frac{1}{2}\epsilon^{ijk}\left(e_{i}^{a}(n)R_{jk}^{a}(n)+\bar{e}_{i}^{a}(n)\bar{R}_{jk}^{a}(n)\right),
\int_{-\infty}^{\infty}dx_{1}\mathrm{e}^{iq_{1}x_{1}}D_{n}(\rho)D_{n^{\prime\prime}}(\rho^{\prime\prime})=\frac{1}{\sqrt{2|eH|}}\mathrm{e}^{iq_{1}(p_{2}^{\prime\prime}+p_{2})/(2eH)}I_{nn^{\prime\prime}}(\hat{q}_{1},\hat{q}_{2}),
\left|\alpha_{k}\right|=\left|\beta_{k}\right|=2^{n_{1}-n_{s}}\left|\frac{\Gamma\left(-n_{s}^{\pi}\right)}{\Gamma\left(-n_{1}^{\pi}\right)}\right|z_{S}^{-1/2-m+n_{1}}\left(\frac{g_{S}}{g_{1}}\right)^{l}\left(\frack{k_{s}}\right)^{-n_{1}+n_{s}}\mathrm{~for~}n_{1}>0
\langle\sigma\rangle=0=-i8N_{c}gm_{q}\int(p^{2}-m_{q}^{2})^{-1}d\!\!\!\,{^{-}}^{4}p+3ig^{\prime}\int(p^{2}-m_{\sigma}^{2})^{-1}d\!\!\!\,{^{-}}^{4}p.
U(r)=-\frac{1}{\pir^{2}}\int_{0}^{+\infty}\,\omega^{4}\alpha_{1}(i\omega)\,\alpha_{2}(i\omega)\,e^{-2\,\omegar}\Bigl[1+\frac{2}{\omegar}+\frac{5}{(\omegar)^{2}}+\frac{6}{(\omegar)^{3}}+\frac{3}{(\omegar)^{4}}\Bigr]d\omega
\sigma_{i}^{+}\rightarrow(\sqrt{Q}{P})^{i-1}\,\sigma_{i}^{+},\;\;\;\;\;\;\sigma_{i}^{-}\rightarrow(\sqrt{P}{Q})^{i-1}\,\sigma_{i}^{-},\;\;\;\;\;\;\sigma_{i}^{z}\rightarrow\sigma_{i}^{z}
u^{\mu}({\bfp},0_{t})={\frac{N}{m}}\left(\begin{matrix}{E_{p}}\\{p_{1}}\\{p_{2}}\\{p_{3}}\\\end{matrix}\right)\quad,\quad{\bfB}^{(\pm)}({\bfp},0_{t})={\bf0}\quad,\quad{\bfE}^{(\pm)}({\bfp},0_{t})={\bf0}\,\,.
\intdt\;{\calL}=\intdt\;\left[\,\frac{1}{2}D_{t}X_{i}^{a}D_{t}X_{i}^{a}-\frac{1}{4}(\epsilon^{abc}X_{i}^{b}X_{j}^{c})^{2}+\frac{i}{2}\tilde{\psi}^{a}D_{t}\psi^{a}+\frac{1}{2}X_{i}^{a}K_{i}^{a}\right],
G_{n}(\sigma)=-\sum_{k=1}^{n+1}{\frac{\Gamma\mathrm{\large(}{\frac{n-k}{1-{\frac{1}{\beta}}}}+k\mathrm{\large)}}{\Gamma\mathrm{\large(}k+1\mathrm{\large)}\Gamma\mathrm{\large(}{\frac{n-k}{1-{\frac{1}{\beta}}}}+1\mathrm{\large)}}}\;\sigma^{k}\;\chi_{n+1-k}\;,\;n\ge1
\sigma^{Ti}(\tau^{i})=\pm(\sum_{j_{i}=1}^{L-1}(-1)^{j_{i}}k^{\tau^{i}}|\tau^{i}-\tau_{j_{i}}^{i}|+{\frac{1}{2}}(1+(-1)^{L+1})k^{\tau^{i}}\tau^{i})~.~\,