harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calR}^{sh}\,=\,\mathrm{e}^{\nu(N\otimes\xi-\xi\otimesN)-\Omega(N\otimes\xi+\xi\otimesN)}\left(1\!\mathrm{l}\otimes1\!\mathrm{l}-2\Omega\mathrm{e}^{\frac{1}{2}(\Omega+\nu)\xi}c_{-}\otimes\mathrm{e}^{-\frac{1}{2}(\Omega-\nu)\xi}c_{+}\right)\,.
G_{\psi}^{ij}(\nu)\equiv(-1)^{i+j}\Sigma((-1)^{i}\bar{R}_{i})\Sigma((-1)^{j}\bar{R}_{j})\bar{\gamma}_{i}^{-2}\bar{\gamma}_{j}^{-2}\sum_{n=1}^{n_{\mathrm{max}}}\frac{1}{\chi_{n}-\nu^{2}}\frac{s_{n}^{\prime}((-1)^{i}\bar{R}_{i})s_{n}^{\prime}((-1)^{i}\bar{R}_{j})}{\int_{-\bar{R}_{1}}^{\bar{R}_{2}}dr^{\prime\prime}\Sigma(r^{\prime\prime})s_{n}^{2}(r^{\prime\prime})}\,.
j_{\mu}^{(-)}=\sum_{k,k^{\prime}}(\bar{u}_{k^{\prime}}\gamma_{\mu}u_{k}b_{k^{\prime}}^{\dagger}b_{k}-\bar{v}_{k}\gamma_{\mu}v_{k^{\prime}}d_{k^{\prime}}^{\dagger}d_{k})+2j_{\mu}^{\mathrm{mix}},
\psi_{SE1}(x,{\bfp};s_{1},s_{2})=\int{\frac{xdp^{-}}{2\pi}}{\frac{\bar{u}(xP^{+},{\bfp};s_{1})\Sigma_{1}(p)\Phi(p)\gamma^{+}v((1-x)P^{+},-{\bfp};s_{2})}{p^{2}-m^{2}+i\epsilon}}
S=\sum_{f}\sum_{n}\Bigl[i\lambda_{n}\bar{\phi}_{f}^{n}\phi_{f}^{n}-i\lambda_{n}\bar{\chi}_{f}^{n}\chi_{f}^{n}+m_{f}(\bar{\phi}_{f}^{n}\phi_{f}^{n}+\bar{\chi}_{f}^{n}\chi_{f}^{n})+ih_{f}(\bar{\phi}_{f}^{n}\chi_{f}^{n}+\bar{\chi}_{f}^{n}\phi_{f}^{n})\Bigr].
\Psi_{\mathrm{T}}^{(\xi)}=\intd^{4}x\,\mathrm{tr}\Bigr\{\chi^{\mu\nu}F_{\mu\nu}+2\bar{\phi}D^{\mu}\psi_{\mu}-\xi\Bigr(\mathrm{\large~\frac{1}{2}~}\chi^{\mu\nu}\lambda_{\mu\nu}+2\eta[\bar{\phi},\phi]\Bigr)\Bigr\}.
C_{1}^{(0)}(g,q_{1}^{+}=xk^{+},k^{+})\equivC_{1}^{(0)}(g,x,k)=-{\frac{g\phi_{c}(k)\sqrt{x(1+x)}}{{\itv}({\frac{k^{2}}{\mu^{2}}})x(1+x)+{\frac{g}{2}}(2x+1)}}.
A_{+}=-\frac{m}{{\partial}_{-}}\tilde{\Sigma}+\frac{{\partial}_{+}}{m^{2}-n_{-}{\partial}_{+}^{\;\;2}}\left(C-n_{-}x^{-}B\right)+\frac{n_{+}m^{2}}{(m^{2}-n_{-}{\partial}_{+}^{\;\;2})^{2}}B
\mathrm{Solv}\left(\frac{\mathrm{USp}(3,3)}{\mathrm{SU}(3)\times{U}(1)}\right)\subset\mathrm{Solv}\left(\frac{\mathrm{SU}(3,3)}{\mathrm{SU}(3)\times\mathrm{SU}(3)\times\mathrm{U}(1)}\right)~.
{\calS}_{B}\,\sim\,\int\!d^{2}w\,|w|^{-s_{1}}\Lambda^{t_{1}}{\calO}_{1}(w,z)\,+\,\int\!d^{2}y\,|y|^{-s_{2}}\Lambda^{t_{2}}\,{\calO}_{2}(y,z)\,,
\left(\begin{matrix}{\displaystyle{z_{1}}}\\{\displaystyle{z_{2}}}\\\end{matrix}\right)=\zeta\left(\begin{matrix}{\displaystyle{\xi}}\\{\displaystyle{1}}\\\end{matrix}\right),\quad\left({dz^{*}dz}\right)^{2}=|\zeta|^{2}d\zeta^{*}d\zetad\xi^{*}d\xi\quad,
[\Phi_{ij}(x^{-},{\bfx}^{\perp}),\partial_{-}\Phi_{lk}(y^{-},{\bfy}^{\perp})]=\frac{\mathrm{i}}{2}\delta_{ik}\delta_{jl}\delta(x^{-}-y^{-})\delta({\bfx}^{\perp}-{\bfy}^{\perp}).
\frac{2\pi}{k}\sum_{n_{2}=-\infty}^{+\infty}\,\left[\psi\left(\frac{2\pin_{2}}{k}\right)\,\sum_{n_{1}=-\infty}^{+\infty}\delta\left(A_{1}-\frac{2\pin_{2}}{k}-2\pin_{1}\right)\right]=\psi(A_{1})\\\.
\langleB|S_{\mu}^{\nu}|B\rangle\to\frac{1}{2}\frac{\alpha^{2}-\beta^{2}}{(24)^{2}}\frac{1}{(2M)^{4}f^{2}}\left(\begin{matrix}{-1}&{0}&{0}&{0}\\{0}&{1/3}&{0}&{0}\\{0}&{0}&{1/3}&{0}\\{0}&{0}&{0}&{1/3}\\\end{matrix}\right)
\Phi^{\alpha}\rightarrowV\Phi^{\alpha}=\left(\begin{array}{c}{-f^{\alpha}(r)e^{i\theta}+f^{\beta}(r)}\\{f^{\alpha}(r)e^{i\theta}+f^{\beta}(r)}\\\end{array}\right)=\left(\begin{array}{c}{\phi^{1}(r,\theta)}\\{\phi^{2}(r,\theta)}\\\end{array}\right).
\Sigma(q)=\int\frac{d^{4}r}{\left(2\pi\right)^{4}}\phi_{0}(r)\;e^{-iqr},\qquad\tilde{\pi}_{0}(q)=\int\frac{d^{4}r}{\left(2\pi\right)^{4}}\pi_{0}(r)\;e^{-iqr}.
\tilde{z}_{\pm}=-Q_{\pm}^{-1}\displaystyle\displaystyle\frac{\partial}{\partial\overline{{z}}_{\pm}}-z_{\pm}~,~~~\tilde{p}_{\pm}=\displaystyle\displaystyle\frac{\partial}{\partialz_{\pm}}-Q_{\pm}^{-1}\displaystyle\frac{\partialp_{\pm}}{\partialz_{\pm}}\displaystyle\displaystyle\frac{\partial}{\partial\overline{{z}}_{\pm}}~.
\Gamma={\textstyle\frac{\sqrt{|g|}}{\sqrt{|g+{\calF}|}}}\sum_{n=0}^{\infty}{\frac{1}{2^{n}n!}}\gamma^{j_{1}k_{1}\dotsj_{n}k_{n}}{\calF}_{j_{1}k_{1}}\dots{\calF}_{j_{n}k_{n}}J_{(p)}^{(n)}\,,
\begin{array}{ccc}{L_{1}}&{L_{0}}&{L_{-1}}\\{Z_{rs}}&{R_{rs}}&{\widetilde{Z}_{rs}}\\{528\mathrm{\scriptsize~\mathrm{~\Abelian~}}}&{GL(32;R)}&{528\mathrm{\scriptsize~\\mathrm{~Abelian~}}}\\{\mathrm{\scriptsize~generators}}&{\mathrm{\scriptsize~algebra}}&{\mathrm{\scriptsize~\mathrm{~generators}}}\\\end{array}\,.
\begin{array}{rl}{\Gamma_{4V}^{(0)\mathrm{{pert}}}}&{C^{(4)}\left[L_{(0,2)}-L_{(0,3)}\right]+C^{(5)}\left[L_{(0,3)}-L_{(0,1)}\right]+C^{(6)}\left[L_{(0,1)}-L_{(0,2)}\right]}\\{\mathrm{}}&{C^{(4)}\left[\frac{3}{2}\left(L_{(0,2)}-L_{(0,3)}\right)\right]+\frac{1}{2}\left(C^{(5)}-C^{(6)}\right)\left[-2L_{(0,1)}+L_{(0,2)}+L_{(0,3)}\right]}\\\end{array}
G^{(n)}({\bf{r}}_{1},\ldots,{\bf{r}}_{n})\equiv\langle\phi({\bf{r}}_{1})\ldots\phi({\bf{r}}_{n})\rangle=\left.\frac{\delta^{(n)}{\cal{G}}\{J\}}{\deltaJ({\bf{r}}_{1})\ldots\deltaJ({\bf{r}}_{n})}\right|_{J=0}\;.
\overrightarrow{\not\!\!D}_{L}\equiv\frac{1}{2}V_{a}^{\mu}\gamma^{a}\left(\vec{\partial}_{\mu}+\frac{1}{2}\omega_{\mu}^{cd}I\sigma_{cd}-\frac{i}{2}\tilde{g}\vec{\tau}\tilde{\vec{A}}_{\mu}+\frac{i}{2}\tilde{g}^{\prime}\tilde{B}_{\mu}\right)
(\bar{C}_{\alpha},{\calT}_{\beta})_{\Omega}=\frac{1}{2}[{\calT}_{\alpha},{\calT}_{\beta}],\quad({\calT}_{\beta}^{\dagger},\bar{C}_{\alpha}^{\dagger})_{\Omega}=\frac{1}{2}[{\calT}_{\beta}^{\dagger},{\calT}_{\alpha}^{\dagger}],
d\star\widehat{\mathcal{H}}_{+}-\mathrm{i}\,Q\wedge\,*\widehat{\mathcal{H}}_{+}=\mathrm{i}\,\mathcal{F}_{[5]}\,\wedge\,\widehat{\mathcal{H}}_{+}-P\wedge\star\widehat{\mathcal{H}}_{-}
<\psi^{l}(x,y)H(x_{1},y_{1})\psi^{l}(x,y)>={\calA}(l)\sum_{n_{1}=0}^{r}\sum_{n_{2}=n_{1}}^{r}\sum_{m_{1}=0}^{s}\sum_{m_{2}=0}^{s}\sum_{m_{3}=0}^{s-m_{1}}c_{m_{1},m_{2},m_{3}}^{n_{1}n_{2}}a_{m_{1},m_{2},m_{3}}^{n_{1}n_{2}}(l).
\left\{\left[\frac{g(\nu)}{4\pi}\right]^{2}\Gamma_{TT}^{\left[\left.r,1\right)\right.}(k^{2})\right\}_{MS}=\left[\frac{g(\nu)}{4\pi}\right]^{2}(k^{2})\left\{K_{0}^{(r)}(k^{2})-\left[\frac{1}{2}N_{C}\left(\frac{13}{3}\right)-\frac{2}{3}N_{f}\right]\left(\frac{1}{\epsilon}\right)\right\}_{\epsilon\rightarrow0}\,,
B_{14}^{ML}+B_{12}^{ML}=-\frac{2^{8-4\omega}T^{8-4\omega}}{(2\omega-3)(2\omega-4)}\left[\beta^{8-4\omega}+e^{-2i\pi\omega}\left(1-(1-\beta^{2})^{4-2\omega}\right)\right].
\left(\frac{\alpha(\lambda_{\beta})}{\delta(\lambda_{\beta})}\right)^{N}=t^{-N}\prod_{\atop{\alpha=1\atop\alpha\neq\beta}}^{M}\left\{\frac{a(\lambda_{\beta}-\lambda_{\alpha})}{a(\lambda_{\alpha}-\lambda_{\beta})}\frac{c(\lambda_{\alpha}-\lambda_{\beta})}{c(\lambda_{\beta}-\lambda_{\alpha})}\right\}\;;\;\beta=1,\cdots,M\leqN\;,
S_{m\rightarrow\infty}=\intd^{2}xdud^{2}\theta_{+}^{+}\;[iX^{+Y}\partial_{++}X_{Y}^{+}+iP_{++}^{-Y}D^{++}X_{Y}^{+}+{\frac{1}{2}}\Lambda_{+}^{i}(\delta^{ij}D^{++}+({V}^{++})^{ij})\Lambda_{+}^{j}]\;.
\begin{array}{ccc}{ds_{\mathrm{str}}^{2}=-dt^{2}+e^{-4U}d\vec{x}^{2}\,}&{}&{e^{-4U}=2\,\chi^{T}L\chi=e^{4\phi}\,}\\{\nonumber}\\{\nonumber}\\{{\calM}={\bf1}_{28}+2e^{4U}\left(\begin{array}{cc}{A^{2}\,nn^{T}}&{AB\,np^{T}}\\{AB\,pn^{T}}&{A^{2}\,pp^{T}}\\\end{array}\right),}&{}&{\vec{H}_{m}=\partial_{m}\vec{\chi}\.}\\\end{array}
dX^{0}=\sum_{i=1}^{g}\left(\lambda_{i}\omega_{i}+\bar{\lambda}_{i}\bar{\omega}_{i}\right)+\mathrm{exact}~~,~~dX^{9}=\sum_{i=1}^{g}\left(\gamma_{i}\omega_{i}+\bar{\gamma}_{i}\bar{\omega}_{i}\right)+\mathrm{exact}
x=(0,\,0,\,(\frac{\textrm{det}c^{2}}{m_{1}\Lambda_{1}^{4}\Lambda_{2}^{4}})^{\frac{1}{3}},\,e^{i\pi/3}(\frac{\textrm{det}c^{2}}{m_{1}\Lambda_{1}^{4}\Lambda_{2}^{4}})^{\frac{1}{3}},\,e^{i2\pi/3}(\frac{\textrm{det}c^{2}}{m_{1}\Lambda_{1}^{4}\Lambda_{2}^{4}})^{\frac{1}{3}})
\frac{\ddot{T}}{1-e^{-2\Phi}\dot{T}^{2}}+3H\dot{T}+\dot{\Phi}\dot{T}\frac{1-2e^{-2\Phi}\dot{T}^{2}}{1-e^{-2\Phi}\dot{T}^{2}}+e^{2\Phi}\frac{1}{V}\frac{dV}{dT}=0.
\mathrm{S_{-}}\equiv\left.\frac{\partial}{\partialb}\mathrm{S}(g)\right|_{g=I}=\sum_{i=1}^{n}\frac{\partial}{\partialz_{i}}\,\,rm{S_{+}}\equiv\left.\frac{\partial}{\partialc}\mathrm{S}(g)\right|_{g=I}=-\sum_{i=1}^{n}(z_{i}^{2}\frac{\partial}{\partialz_{i}}+z_{i}).
A^{\mu}(x)=\sum_{k}\sum_{r=0}^{3}\frac{1}{(2Vw_{k})^{1/2}}(\epsilon_{r}^{\mu}({\bfk})a_{r}({\bfk})e^{-ikx}+\epsilon_{r}^{\mu}({\bfk})a_{r}^{\dagger}({\bfk})e^{ikx})
\psi_{in,out}=\frac{1}{\left(2\pi\right)^{3/2}}\intd^{3}p\left\{a_{in,out}^{-}\left({\bfp}\right)e^{-ipx}\psi^{a}\left({\bfp}\right)+b_{in,out}^{+}\left({\bfp}\right)e^{ipx}\psi^{b}\left({\bfp}\right)\right\}
\begin{array}{ll}{\phi_{qI}(x,y)=U_{R}(y){\tilde{\phi}}_{qI}(x,y),}&{\phi_{\overline{q}}^{I}(x,y)=U_{R}(y){\tilde{\phi}}_{\overline{q}}^{I}(x,y),}\\{F_{qI}(x,y)=U_{R}^{\dagger}(y){\tilde{F}}_{qI}(x,y),}&{F_{\overline{q}}^{I}(x,y)=U_{R}^{\dagger}(y){\tilde{F}}_{\overline{q}}^{I}(x,y).}\\\end{array}
\qquad\left.-6gA_{\mu51}\frac{1}{\theta}\sigma_{1}\bar{\psi}_{1i1j2k}\gamma^{\mu}\psi_{1i1j2k}-3gA_{\mu51}\frac{1}{\theta}\sigma_{2}\bar{\psi}_{2i1j1k}\gamma^{\mu}\psi_{2i1j1k}\right)\quad
\Psi_{\mathrm{DW}}=\frac{i}{e^{2}}\intd^{4}x\,\sqrt{g}\,\mathrm{tr}\Bigr\{\frac{1}{2}\chi^{\mu\nu}F_{\mu\nu}+\frac{i}{4}\chi^{\mu\nu}B_{\mu\nu}-\psi^{\mu}D_{\mu}\bar{\phi}+\frac{i}{2}\eta[\bar{\phi},\phi]\Bigr\},
[\,A_{{\sfGI}\,i}^{b}({\bf{r}})\,{\textstyle\frac{\lambda^{b}}{2}}\,]=V_{\cal{C}}({\bf{r}})\,[\,A_{i}^{b}({\bf{r}})\,{\textstyle\frac{\lambda^{b}}{2}}\,]\,V_{\cal{C}}^{-1}({\bf{r}})+{\textstyle\frac{i}{g}}\,V_{\cal{C}}({\bf{r}})\,\partial_{i}V_{\cal{C}}^{-1}({\bf{r}})\,;
\begin{array}{l}{\displaystyle(P_{20})_{ij,pq}={\frac{1}{2}}(\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp})-{\frac{1}{3}}f_{ijl}f_{lpq}}\\{\displaystyle(P_{8})_{ij,pq}={\frac{1}{3}}f_{ijl}f_{lpq}\quad.}\\\end{array}
\displaystyleg_{j,0}(x,w)={\calF}_{\xi}^{-1}\left[P.V.\left[\int_{0}^{\infty}t^{\nu-j-2}\\tilde{d}_{-1-j}(x,t;\xi/\vert\xi\vert,t;0)\dt-M_{j}(x)\right]\vert\xi\vert^{-(\nu-1)}\right](w).
I_{5}=\frac{-16\pi^{\frac{D}{2}}\Gamma(\frac{D}{2}-1)}{(4-D)^{2}}\left[(n_{0}^{2}+i\eta^{\prime})^{2-\frac{D}{2}}+(-n_{0}^{2}+i\eta^{\prime})^{2-\frac{D}{2}}\right],\eta^{\prime}>0.
\begin{array}{llcl}{}&{(1~0~0~0~0~0~0~-1)}&{}\\{\mid\mathrm{gnd}>}&{(1~0~0~0~0~0~0~-1)}&{=}&{\beta_{i}^{+}}\\{\mid\mathrm{1st}>}&{(1~0~0~0~0~0~1~-1)}&{=}&{\mathrm{SU(8)}}\\{}&{(0~0~0~0~0~0~0~0)}&{=}&{\mathrm{U(1)}}\\{\mid\mathrm{2nd}>}&{(0~0~0~0~0~0~1~0)}&{=}&{\beta_{i}^{-}.}\\\end{array}
\mathrm{Tr}e^{iF}={\frac{1}{2}}\left(\mathrm{tr}e^{iF}\right)^{2}-{\frac{1}{2}}\mathrm{tr}e^{2iF}={\frac{1}{2}}\left(\mathrm{tr}\,\mathrm{cos}F\right)^{2}-{\frac{1}{2}}\mathrm{tr}\,\mathrm{cos}2F.
\delta\psi=iD^{\mu}\psi\overline{{\epsilon}}_{1}\gamma_{\mu}\epsilon_{2}+a\sigma^{\mu\nu}\psi\overline{{\epsilon}}_{1}\sigma_{\mu\nu}\epsilon_{2}=K^{\mu}D_{\mu}\psi+\frac{1}{4}D^{\mu}K^{\nu}\gamma_{\mu\nu}\psi
f(r)\approxf_{n}^{max}\kappa^{n}\left(R-\xi\right)^{n}=f_{n}^{max}\,n^{n}\,\left(1+{\frac{(1/2)-\kappa\xi}{n}}\right)^{n}\approxf_{n}^{max}\,n^{n}\,e^{[(1/2)-\kappa|\xi|]}\,.
d_{[\alpha_{1}],[\alpha_{2}],(\alpha_{3})}^{\epsilon}=ie^{-\frac{i\pi\epsilon}{4}}\left[\beta_{2}C_{[\alpha_{1}],[\alpha_{2}],(\alpha_{3})}^{\epsilon}-\beta_{1}C_{[\alpha_{1}],[\alpha_{2}],(\alpha_{3})}^{-\epsilon}\right].
{\calL}^{(4)}=f_{\pi}^{2}\mathrm{tr}\left(K_{1}\partial_{\mu}U\partial_{\rho}U^{+}\partial_{\mu}U\partial_{\rho}U^{+}+K_{2}\partial_{\mu}U\partial_{\mu}U^{+}\partial_{\rho}U\partial_{\rho}U^{+}\right).
{\frac{\kappa^{2}}{8}}\epsilon_{~\mu}^{\alpha~\rho\sigma}\epsilon_{~\nu}^{\beta~\lambda\gamma}F_{\rho\sigma}F_{\lambda\gamma}\epsilon^{ij}\epsilon^{lk}={\calT}_{\mu\nu}^{\alpha\beta}\epsilon^{ij}\epsilon^{lk}\\,
P_{N}=\left(\begin{array}{cccccc}{0}&{1}&{}&{}&{}&{}\\{}&{0}&{1}&{}&{}&{}\\{}&{}&{\ddots}&{\ddots}&{}&{}\\{}&{}&{}&{}&{0}&{1}\\{1}&{}&{}&{}&{}&{0}\\\end{array}\right),\quadQ_{N}=\left(\begin{array}{cccccc}{1}&{}&{}&{}&{}&{}\\{}&{\omega_{N}}&{}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}&{}\\{}&{}&{}&{}&{\omega_{N}^{N-2}}&{}\\{}&{}&{}&{}&{}&{\omega_{N}^{N-1}}\\\end{array}\right),
\tilde{D}_{\alpha\beta}^{\mu\nu}(k)=\frac1{k^{2}}\left[P_{\alpha\beta}^{\mu\nu}+\frac1{k^{2}}\tilde{\Pi}_{\alpha\beta}^{\mu\nu}(k)+\left(\frac1{k^{2}}\right)^{2}\tilde{\Pi}_{\rho\sigma}^{\mu\nu}(k)\tilde{\Pi}_{\alpha\beta}^{\rho\sigma}(k)+\dots\right]
\left(\begin{array}{cc}{\hat{K}}&{g}\\{-g}&{\hat{F}}\\\end{array}\right)\longrightarrow\left(\begin{array}{cc}{\Lambda\hat{K}\Lambda^{t}}&{\Lambdag\Lambda^{t}}\\{-\Lambdag\Lambda^{t}}&{\Lambda\hat{F}\Lambda^{t}}\\\end{array}\right).
u_{\underline{{m}}}^{(0)}\tilde{\sigma}^{\underline{{m}}~\underline{{\mu}}\underline{{\nu}}}=v_{A}^{~\underline{{\mu}}}v_{A}^{~\underline{{\nu}}},\qquadu_{\underline{{m}}}^{i}\tilde{\sigma}^{\underline{{m}}~\underline{{\mu}}\underline{{\nu}}}=-v_{A}^{~\underline{{\mu}}}\Gamma_{AB}^{i}v_{A}^{~\underline{{\nu}}}.
\begin{array}{c}{\lambda_{1}\rightarrowe^{i(\phi_{1}-\phi_{2}-\phi_{3})/2}\lambda_{1},}\\{\lambda_{2}\rightarrowe^{i(-\phi_{1}+\phi_{2}-\phi_{3})/2}\lambda_{2},}\\{\lambda_{3}\rightarrowe^{i(-\phi_{1}-\phi_{2}+\phi_{3})/2}\lambda_{3},}\\{\lambda_{4}\rightarrowe^{i(\phi_{1}+\phi_{2}+\phi_{3})/2}\lambda_{4}.}\\\end{array}
E^{\alpha}={\calD}\theta^{\alpha}-{\frac{i}{3!R}}\bar{\theta}\gamma^{a}{\calD}\theta(\gamma_{a}\theta)^{\alpha}+{\frac{i}{2\cdot3!R}}\bar{\theta}\gamma^{ab}{\calD}\theta(\gamma_{ab}\theta)^{\alpha}-{\frac{2}{5!R^{2}}}{\calD}\theta^{\alpha}(\bar{\theta}\theta)^{2},
E_{n}=(b-\frac{aq}{2p})^{2}-(b-\frac{aq}{2p})^{2}\frac{a^{2}}{(a+n\frac{p}{\sqrt{2}})^{2}}+\sqrt{2}(ar-\frac{aq^{2}}{4p})n+(\frac{pr}{2}-\frac{q^{2}}{8})n^{2}.
B_{0i}(T=0)=-\frac{2v^{2}}{\pi\kappa}\epsilon_{0ij}p^{j}\frac{\partial}{\partialm_{\sigma}^{2}}\left\{\frac{1}{m_{\sigma}^{2}-M^{2}}\left[\left(M\frac{e^{-\tilde{p}M}}{(\tilde{p})^{2}}+\frac{e^{-\tilde{p}M}}{(\tilde{p})^{3}}\right)-(M\rightarrowm_{\sigma})\right]\right\}
(v^{2}-f^{2})[-(f_{1})^{2}+\frac{1}{2}(f^{2})_{11}]-2f^{2}\frac{C-\frac{1}{4}[(f^{2})_{1}]^{2}}{v^{2}-f^{2}}+\frac{1}{2}f^{2}(f^{2})_{11}+[\frac{1}{2}(f^{2})_{1}]^{2}=0\,.
S=\frac{1}{4N^{2}}\sum_{x\in\left(T_{n}\right)^{d}}\left(F_{\mu\nu}\left(x\right)\right)_{a}^{a^{\prime}}\diamond\left(F_{\mu\nu}\left(x\right)\right)_{a^{\prime}}^{a}=-\frac{1}{4}Tr\left(\left[a_{\mu},a_{\nu}\right]^{2}\right),
\{D_{\alpha}^{i},D_{\beta}^{j}\}=\{\bar{D}_{\dot{\alpha}i},\bar{D}_{\dot{\beta}j}\}=0\;,\quad\{D_{\alpha}^{i},\bar{D}_{\dot{\beta}j}\}=-2i\delta_{j}^{i}(\sigma^{\mu})_{\alpha\dot{\beta}}\partial_{\mu}\;.
\epsilon({AdS})=\left[\sqrt{f^{1/2}+1}-\sqrt{f^{1/2}-1}\gamma_{\overline{{r}}}\right]e^{-{\frac{1}{2}}\gamma_{\overline{{\thetar}}}}e^{-{\frac{1}{2}}\gamma_{\overline{{\phi\theta}}}}e^{-{\frac{gt}{\sqrt{2}}}\gamma_{\overline{{0}}}}\epsilon_{0}.
f_{{\bf{q}}}=(\frac{\omega_{{\bf{q}}}+\epsilon_{{\bf{q}}}+\rho_{0}v_{{\bf{q}}}}{2\mathrm{~}\omega_{{\bf{q}}}})^{\frac{1}{2}}d_{{\bf{q}}/2}({\bf{q}})+(\frac{-\omega_{{\bf{q}}}+\epsilon_{{\bf{q}}}+\rho_{0}v_{{\bf{q}}}}{2\mathrm{~}\omega_{{\bf{q}}}})^{\frac{1}{2}}d_{-{\bf{q}}/2}^{\dagger}(-{\bf{q}})
\tau_{\mu}^{\dagger}\tau_{\nu}^{\vphantom{\dagger}}=\delta_{\mu\nu}+i\hat{\eta}_{\mu\nu}^{+};\qquad\qquad\tau_{\mu}^{\vphantom{\dagger}}\tau_{\nu}^{\dagger}=\delta_{\mu\nu}+i\hat{\eta}_{\mu\nu}^{-};
V_{\mathrm{eff.}}(r,u)=-4(8\pi^{2}\alpha^{\prime})^{-1/2}\int_{0}^{\infty}dle^{-r^{2}l/2\pi\alpha^{\prime}}{l}^{1/2}\eta(il)^{-21}{\frac{\mathrm{tanh}(u)e^{-u^{2}l/\pi}}{\Theta_{11}({\frac{ul}{\pi}},il)}}\quad.
{\bf\alpha}^{\mu}={\bfF}_{\mu}^{-1}f^{\mu}{}^{*}{\bf\alpha}{\bfF}_{\mu}+{\bfF}_{\mu}^{-1}d{\bfF}_{\mu}=\left(\begin{array}{cc}{0}&{-{\frac{1}{2}}T}\\{1}&{0}\\\end{array}\right)dw+\left(\begin{array}{cc}{-{\frac{1}{2}}\mu_{w}}&{-{\frac{1}{2}}(T\mu+\mu_{ww})}\\{\mu}&{{\frac{1}{2}}\mu_{w}}\\\end{array}\right)d\bar{w}.
I_{j,k,m}=\int_{0}^{\infty}\!\!\frac{q^{2j}dq}{\left(1+(2V^{\prime}+4\phi^{2}V^{\prime\prime})q^{2}+(K+\phi^{2}Z)q^{4}\right)^{k}\left(1+2V^{\prime}q^{2}+Kq^{4}\right)^{m}}
R(u_{1}-u_{2})t_{b}^{c}(u_{1})\otimest_{a}^{b}(u_{2})=\sum_{d}W\left(\left.\begin{array}{cc}{c}&{d}\\{b}&{a}\\\end{array}\right|u_{1}-u_{2}\right)t_{a}^{d}(u_{1})\otimest_{d}^{c}(u_{2}).
r^{2}(\lambda)={\frac{1}{(\widetilde{N}-\widetilde{M})^{2}}}\sum_{m,n=\widetilde{M}}^{\widetilde{N}}\left[{\frac{W_{\lambdamn}(N=16)}{W_{\lambdamn}(N=20)}}\,\,-\,\,1\right]^{2}\,,
\kappa_{1,\pm}^{2}=\left[\frac{1}{2}\left(\kappa_{1,1}^{2}+\kappa_{1,2}^{2}\right)-\kappa_{1,3}^{2}\right]\pmi\left(\kappa_{1,1}+\kappa_{1,2}\right)\sqrt{\kappa_{1,3}^{2}-\left[\frac{1}{2}\left(\kappa_{1,1}-\kappa_{1,2}\right)\right]^{2}}\,.
\alpha=\frac{1}{2}\left(\omega\ell+n\right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\beta=\frac{1}{2}\left(\omega\ell-n\right).
\Gamma\left(L_{\mathrm{cl}},z_{\mathrm{cl}},\bar{L}_{\mathrm{cl}},\bar{z}_{\mathrm{cl}};m^{\prime},\bar{m}^{\prime}\right)=-\bar{L}_{\mathrm{cl}}K_{z\bar{z}}L_{\mathrm{cl}}+\left(L_{\mathrm{cl}}W_{z}+\mathrm{h.c.}\right)
\left\{M_{ij}^{(r)},\M_{hk}^{(s)}\right\}=\left[\delta_{ih}M_{jk}^{(r)}+\delta_{jk}M_{ih}^{(r)}-\delta_{ik}M_{jh}^{(r)}-\delta_{jh}M_{ik}^{(r)}\right]\delta_{\left(r\right)\left(s\right)},
{\calW}^{(1)}={\frac{1}{8N}}\oint\oint\oint\oint\mathrm{Tr}[{\calP}(T_{x}^{a}T_{y}^{a}T_{z}^{b}T_{w}^{b})]D_{\mu\nu}(x-y)D_{\rho\sigma}(z-w)\;dx^{\mu}dy^{\nu}dz^{\rho}dw^{\sigma}\;\;,
{\calL}=-\pi_{i}^{T}\dot{A}_{i}^{T}-\frac{1}{2}{\pi_{i}^{T}}^{2}-\frac{1}{4}F_{ij}^{2}(A^{T})+\frac{1}{2}j_{0}\frac{1}{\partial^{2}}j_{0}+j_{i}A_{i}^{T}+{\calL}_{M}
J_{\mathrm{asym}}=-\intd^{2}x\biggl\{\epsilon_{ij}x^{i}\biggl[\dot{f}\partial_{j}f+f^{2}(\dot{\theta}+A_{0})(\partial_{j}\theta+A_{j})\biggr]+\frac{\rho_{e}}{2}x^{2}F_{12}\biggr\}.
\Phi^{\ell}(t,\bar{t},\psi)=\sum_{n=0}^{\infty}a_{n}t_{0n}^{\ell}(t,\bar{t},\psi)=\frac{e^{-i\ell\psi}}{(1-|t|^{2})^{\ell}}\sum_{n=0}^{\infty}b_{n}t^{n}\equive^{-i\ell\psi}\varphi(t,\bar{t})\,
\pi_{(\underline{\alpha})}^{2}=\left(\pi_{q}^{2-}\atop\pi_{\dot{q}}^{2+}\right)=\left(\nabla\theta_{q}^{2-}+\frac12\gamma_{q\dot{r}}^{i}\theta_{\dot{r}}^{2+}\Omega^{-2i}\atop\nabla\theta_{\dot{q}}^{2+}+\frac12\tilde{\gamma}_{\dot{q}r}^{i}\theta_{r}^{2-}\Omega^{+2i}\right).
K_{N}(\lambda,\lambda^{\prime})~=~(\lambda+\lambda^{\prime})\sqrt{\lambda\lambda^{\prime}}\prod_{f=1}^{\alpha}\sqrt{\lambda^{2}+m_{f}^{2}}\sqrt{{\lambda^{\prime}}^{2}+m_{f}^{2}}e^{-\frac{N}{2}(V(\lambda^{2})+V({\lambda^{\prime}}^{2}))}\frac{1}{N}\sum_{j=0}^{n-1}\frac{P_{j}(\lambda^{2})P_{j}({\lambda^{\prime}}^{2})}{h_{j}}~.
\tilde{\beta}=\frac{3N-M}{16\pi^{2}}\;,\;\;\tilde{d}_{T}=(\frac{M}{3}-N)(32\pi^{2}e)^{\frac{3(M-N)}{3N-M}}\left(\frac{M}{3\hat{h}(T)}\right)^{\frac{M}{3N-M}}(h(T))^{\frac{3M}{3N-M}}\;\;.
\Big(A_{\mu},Ta_{\mu}^{(b)}\Big)=-\Big(\varphi,\varepsilon_{\mu\nu}\partial_{\mu}a_{\nu}^{(b)}\Big)\;\;\;,\;\;\;\Big(h_{\mu}^{\prime},Ta_{\mu}^{(b)}\Big)=-\Big(\theta,\varepsilon_{\mu\nu}\partial_{\mu}a_{\nu}^{(b)}\Big)\;.
\begin{array}{c}{{\tilde{\calZ}}\equivH(z,\bar{z})+i\partial_{A}\partial_{\bar{B}}H(z,\bar{z})\theta^{A}{\bar{\theta}}^{\bar{B}}}\\{Q=\mathrm{e}^{i\alpha}Q_{+}+\mathrm{e}^{-i\alpha}Q_{-}.}\\\end{array}
Z=Z_{\mathrm{dir}}\intd\hat{X}_{j}e^{-DT\left(\frac{\hat{X}^{2}}{D}\right)}\left(1-2c_{1}DT^{\prime}\left(\hat{\frac{X^{2}}{D}}\right)+2D\zeta(2)\left[T^{\prime}\left(\frac{\hat{X}^{2}}{D}\right)\right]^{2}+\ldots\right)
\int_{M_{\delta}}dp\;p^{\alpha}\\leq\\frac{1}{\delta^{2}}\int_{0}^{\infty}dp\;p^{\alpha}\left|\chi\left(p\right)-\phi_{0}\right|^{2}\<\\frac{q_{0}^{2}\varepsilon\left(q_{0}\right)}{\delta^{2}}\.
\left\langle{{\xi}\left\vert{\xi^{\prime}}\right\rangle}\right.={\frac{\left({1+\xi^{*}\xi^{\prime}}\right)^{2J}}{\left({1+{\vert\xi\vert}^{2}}\right)^{J}\left({1+{\vert\xi^{\prime}\vert}^{2}}\right)^{J}}}\,
S_{\mathrm{matter}}=\intd^{2}\sigma\,\left[\frac{1}{4}\left(F_{rs}\right)^{2}+\frac{1}{2}\left(D_{r}Y^{I}\right)^{2}+\frac{g^{2}}{4}\left(\{Y^{I},Y^{J}\}\right)^{2}\right]\,,
e^{dx(t)}=e^{dt}(1+d(\sum_{f=1}^{\infty}e^{ft}\sum_{\sigma_{f}\inP_{f}}(-1)^{l(\sigma_{f})}(d+f)^{l(\sigma_{f})-1}S(\sigma_{f})\prod_{j=1}^{l(\sigma_{f})}\frac{b_{d_{j}}}{d_{j}})).
H_{c}=\int{d^{3}x}\left({-\frac{1}{2}\Pi_{i}^{a}\Pi_{a}^{i}+\Pi_{i}^{a}\partial^{i}A_{a}^{0}+\frac{1}{4}F_{ij}^{a}F^{aij}-gf_{abc}\Pi_{i}^{a}A_{b}^{0}A_{c}^{i}}+A_{0}^{a}J^{0}\right).
{}~~~~~~~~~~~~~\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{w(x_{4}x_{13},x_{1}x_{34},x_{3}x_{14}|\sigma+j_{2}+j_{3})w(x_{23},x_{34},x_{24}|\sigma)s(\sigma,j_{1})}{w(x_{13},\omegax_{34},\omegax_{14}|\sigma+j_{2})w(x_{4}x_{23},x_{2}x_{34},x_{3}x_{24}|\sigma+i_{3})}\Bigg\}_{0}~~
\varepsilon_{1}^{\alpha\beta}=\frac{1}{\sqrt{2}}\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{0}\\\end{array}\right)\,,\quad\varepsilon_{2}^{\alpha\beta}=\frac{1}{\sqrt{2}}\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}\\\end{array}\right)\,;
(u_{1},u_{2})=\left({\frac{\phi_{2}}{\phi_{1}}},{\frac{\phi_{3}}{\phi_{1}}}\right),\quad(v_{1},v_{2})=\left({\frac{\phi_{1}}{\phi_{2}}},{\frac{\phi_{3}}{\phi_{2}}}\right),\quad(w_{1},w_{2})=\left({\frac{\phi_{1}}{\phi_{3}}},{\frac{\phi_{2}}{\phi_{3}}}\right).
e^{-\nu}(\partial_{3}+iA_{3}+i\partial_{4}-A_{4}+{\frac{1}{2}}\partial_{3}\nu+{\frac{i}{2}}\partial_{4}\nu)\psi_{1}+e^{-\mu}(\partial_{1}+iA_{1}-i\partial_{2}+A_{2}+{\frac{1}{2}}\partial_{1}\mu-{\frac{i}{2}}\partial_{2}\mu)\psi_{2}=0,
\hat{C}^{\hat{a}\hat{b}}={\frac{1}{K}}\,\left[(1+{\frac{1}{12}}\hat{{\calH}}^{2})\,\hat{g}^{\hat{a}\hat{b}}-\,{\frac{1}{4}}(\hat{{\calH}}^{2})^{\hat{a}\hat{b}}\right]\,.
\Pi({\bf{x}},t)=(\frac{i}{2\sqrt{N_{0}}})\sum_{{\bf{q}}}exp(i{\bf{q.x}})(A_{{\bf{q}}}+B_{{\bf{q}}})[f_{{\bf{q}}}e^{-i\omega_{{\bf{q}}}t}-f_{-{\bf{q}}}^{\dagger}e^{i\omega_{{\bf{q}}}t}]
\delta=\frac{1}{2}\left(\ell\,m_{\mathrm{\scriptsize{eff}}}-\tilde{\beta}\right),\,\,\,\,\,\,\,\,\,\,\,\,m_{\mathrm{\scriptsize{eff}}}=m+\frac{1}{2\ell}\,,\,\,\,\,\,\,\,\,\,\,\,\,\tilde{\beta}=\frac{\ell\beta}{r_{ex}}.
\phi^{AB}=\frac{1}{\sqrt2}\left\{S^{i}\eta^{iAB}+iP^{i}\bar{\eta}^{iAB}\right\},\qquad\bar{\phi}_{AB}=\frac{1}{\sqrt2}\left\{S^{i}\eta_{AB}^{i}-iP^{i}\bar{\eta}_{AB}^{i}\right\}\,
\begin{array}{c}{\beta_{0}\\left[G(\beta_{0},L_{0},\mu\\beta/\beta_{0})-G(\beta_{0},L_{0},0)\right]=\beta_{0}\frac{\mu\\beta}{\beta_{0}}\frac{\partialG}{\partial\mu_{0}}(\beta_{0},L_{0},\mu_{0})}\\{=-\mu\\beta\\bar{N}(\beta_{0},L_{0},\mu_{0}),}\\\end{array}
e^{2\rho}=\begin{array}{l}{(1+{\frac{a}{\lambda}}e^{\lambday^{-}})^{-1}\;\;\;\mathrm{for}\;\;\;y^{+}<y_{0}^{+}}\\{(1+{\frac{a}{\lambda}}e^{\lambda(y^{-}-y^{+}+y_{0}^{+})})^{-1}\;\;\;\mathrm{for}\;\;\;y^{+}>y_{0}^{+},}\\\end{array}
S_{CS}=\sum_{i=1}^{3}\frac{1}{p_{i}}(r_{i}n_{i}^{2}-2\lambdan_{i}-q_{i}\lambda^{2})=\sum_{i=1}^{3}\left(\frac{r_{i}}{p_{i}}\tilde{n}_{i}^{2}-q_{i}s_{i}\lambda^{2}\right)\;(\mathrm{mod}\;1).
Q^{a}={\frac{1}{\sqrt{2p^{+}}}}\int_{0}^{\pi}d\sigmaS^{a}(\sigma),\qquad{Q}^{\dot{a}}={\frac{1}{\pi\sqrt{p^{+}}}}\int_{0}^{\pi}d\sigma\gamma_{\dot{a}b}^{I}\partialX^{I}{S}^{b}(\sigma)
\begin{array}{rclrclrcl}{\sigma_{ij}[y_{i}]}&{=}&{y_{j},\qquad}&{\sigma_{ij}[y_{j}]}&{=}&{y_{i},\qquad}&{\sigma_{ij}[y_{k}]}&{=}&{y_{k},}\\{\sigma_{ij}[h_{i}]}&{=}&{h_{j},\qquad}&{\sigma_{ij}[y_{h}]}&{=}&{h_{i},\qquad}&{\sigma_{ij}[h_{k}]}&{=}&{h_{k}.}\\\end{array}
0=\frac{\partialV}{\partial\mathrm{Im}F}=-i\left(\frac{N_{c}}{\gamma}-\frac{N_{f}}{\tilde{\gamma}}\right)\,\mathrm{ln}\,\left(\frac{F}{F^{*}}\right)-i\,\mathrm{ln}\,\left(\frac{\mathrm{det}F_{T}}{\mathrm{det}F_{T}^{\dagger}}\right)\.
T^{++}(-k)|0\rangle={\frac{\pi}{2L}}\sum_{n=1}^{k-1}\left[-\sqrt{n(k-n)}A_{ij}(-k+n)A_{ji}(-n)+\left({\frac{k}{2}}-n\right)B_{ij}(-k+n)B_{ji}(-n)\right]|0\rangle.
\tilde{C}_{\alpha}^{\,\alpha_{0}}(p):=\int_{\alpha_{0}}^{\alpha}d\alpha^{\prime}\,e^{\,-\alpha^{\prime}(\,p_{0}^{2}+\underline{{p}}^{2}+m^{2})}\quad,\quad0<\alpha_{0}\leq\alpha<\infty\quad.
S=\frac{1}{8\piG_{N}}\int_{\partialM}\Bigl[(K-K_{0})+{\textstyle{\frac{1}{4}}}n_{\mu}\Bigl(\sqrt{-g}e^{-\gamma\Phi}A_{\nu}^{\alpha}(LML)_{\alpha\beta}F^{\beta\mu\nu}\Bigr)\Bigr]\quad.
\begin{matrix}{K}\\\end{matrix}{im}{n\ell}\begin{matrix}{\tilde{K}}\\\end{matrix}{nk}{jm}=\begin{matrix}{K}\\\end{matrix}{mi}{\elln}\begin{matrix}{\tilde{K}}\\\end{matrix}{kn}{mj}=\delta_{j}^{i}\delta_{\ell}^{k}.
S={\frac{1}{4}}\int_{0}^{T}d\tau\dot{x}^{2}+{\frac{1}{2}}\int_{0}^{T}d\tau\psi\cdot\dot{\psi}-{\frac{1}{2}}\int_{0}^{T}d\tau\psi_{\mu}\psi_{\nu}w_{\mu\nu}(x(\tau))-ig\int_{0}^{T}d\tau\bar{z}\dot{x}\cdotA\,z,
\intd^{8}x\vert_{\theta_{j}^{\alpha}=\bar{\theta}_{j}^{\bar{\alpha}}=0}\left[(D_{+})^{8}(D_{-})^{8}f_{V}(W)+\intd\zeta(D_{-})^{8}(\bar{D}^{+})^{8}f_{T}(\tilde{L},\zeta)\right]+c.c.