harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\check{\tilde{R}}_{12}(u,v)\check{R}_{23}^{\iota_{1}\iota_{2}}(u)\check{\bar{R}}_{12}^{\iota_{1}}(v)=\check{\bar{R}}_{23}^{\iota_{1}\iota_{2}}(v)\check{R}_{12}^{\iota_{1}}(u)\check{R}_{23}(u,v),
\tilde{\Phi}=(-1)^{j_{1}^{2}-j_{1}^{3}+j_{2}^{2}-j_{2}^{3}+r_{2}^{2}-r_{2}^{3}}\sqrt{\frac{(2r_{2}^{1}+1)(2j_{2}^{1}+1)}{(8\pi^{2})^{2}(2j_{1}^{1}+1)}}\prod_{i=1}^{3}(-1)^{j_{1}^{i}-r_{2}^{i}}\sqrt{(2r_{2}^{i}+1)(2j_{1}^{i}+1)(2j_{2}^{i}+1)}.
Z={\calN}\int\prod_{\mu}\prod_{\nu}\prod_{\lambda}{\calD}a_{\mu}{\calD}B_{\nu}^{*}{\calD}B_{\lambda}{\calD}c{\calD}\bar{c}\mathrm{exp}\left(-\int_{0}^{\beta}d\tau\intd^{2}x{\calL}\right)\;,
\frac{dp^{\mu}}{d\tilde{\tau}}=\frac{q}{c}\tilde{g}^{\mu\nu}F_{\nu\rho}u^{\rho}-\frac{q^{2}}{2A^{2}m}\tilde{g}^{\mu\nu}(F_{\,\,\,\nu}^{\lambda}F_{\lambda\alpha,\beta}+F_{\,\,\,\alpha}^{\lambda}F_{\beta\nu,\lambda})u^{\alpha}u^{\beta}\,{.}
S=\intd^{6}x\left[{\frac16}F_{{\mu}{\nu}{\sigma}}F^{{\mu}{\nu}{\sigma}}+{\frac13}{\lambda}_{{\mu}{\nu}{\sigma}}{\calF}^{{\mu}{\nu}{\sigma}}+{\frac13}G^{{\mu}{\nu}{\sigma}}(F_{{\mu}{\nu}{\sigma}}-{\partial}_{[{\mu}}A_{{\nu}{\sigma}]})\right],
S=\intd^{2}x[(z\overline{\zeta}+\overline{z}\zeta)f+i(z\overline{\zeta}-\overline{z}\zeta)g]\partial_{-}[{\frac{z\overline{\zeta}+\overline{z}\zeta}{(1+{\frac{1}{2}}z\overline{z})^{2}}}]\,
A\left[j\right]\equiv\sum_{n=-\infty}^{\infty}F^{n}\left[j\right]=\sum_{n=-\infty}^{\infty}F_{\left(0\right)}^{n}\left[j\right]-i\Sigma\!\!\!\!\!\!\int_{n\lambda\mathbf{x}}V\left(\tilde{\varphi}_{n}\right)
\sum_{i,j,k=0,1...}\left(g^{2}\right)^{i}\left(g^{2}ln{\frac{\Lambda^{2}}{m_{0}^{2}}}\right)^{j}\left(g^{2}{\frac{\Lambda^{2}}{M^{2}}}\right)^{k}C_{ijk}\left(\Lambda^{2},M^{2},m_{0},p^{2}\right)
\begin{array}{l}{y^{0}=\displaystylea\frac{1+xx^{\prime}}{1-xx^{\prime}}~,~~~~~~iy^{n+1}=a\frac{2x^{n+1}}{1-xx^{\prime}}~,}\\{y^{i}=a\displaystyle\frac{2x^{i}}{1-xx^{\prime}}~,~~~~~~(i=1,~2,~\cdots,~n)~.}\\\end{array}
{\calL}=\sum_{i,\bar{j}=1}^{n}\left\{G_{i\bar{j}}\,\partial_{\mu}\Phi^{i}\,\partial_{\mu}\bar{\Phi}^{\bar{j}}+G^{i\bar{j}}\frac{\partial{W}}{\partial\Phi^{i}}\,\frac{\partial\bar{W}}{\partial\bar{\Phi}^{\bar{j}}}\right\}+\mathrm{fermions}\,,
\left[\overline{{\lambda_{\uparrowL-}^{xy}}}\:(\Lambda_{-}+\Lambda_{+})\:+\:\lambda_{\uparrowL-}^{xy}\:(\overline{{\Lambda_{-}}}+\overline{{\Lambda_{+}}})\right]\:\overline{{\lambda_{\uparrowL-}^{xy}}}\;P(x,y)\;=\;f(x,y)\:I_{\uparrow}\:P(x,y)\;,
H=\left(\begin{array}{cccc}{r_{5}}&{0}&{r_{3}+ir_{4}}&{r_{1}+ir_{2}}\\{0}&{r_{5}}&{-r_{1}+ir_{2}}&{r_{3}-ir_{4}}\\{r_{3}-ir_{4}}&{-r_{1}-ir_{2}}&{-r_{5}}&{0}\\{r_{1}-ir_{2}}&{r_{3}+ir_{4}}&{0}&{-r_{5}}\\\end{array}\right).
\intd^{2}{\bfr}\psi^{\ast}(r)\left\{\frac{1}{m}(-i{\bf\nabla}-\alpha{\bfA})^{2}+U(r)+\lambda_{\theta}(r)\delta^{2}({\bfr})-\frac{k^{2}}{m}\right\}\psi_{\theta}(r)=0,
\left[{\frac{g_{B}}{\pi}}(B-1)+B-{\frac{2\sqrt{2}}{\pi^{2}}}\right]X^{2}+\left({\frac{g_{B}}{\pi}}-B+{\frac{4\sqrt{2}}{\pi^{2}}}\right)X-{\frac{2\sqrt{2}}{\pi^{2}}}=0.
\left({\frac{\dot{a}_{0}}{a_{0}}}\right)^{2}={\frac{16\pi^{2}G_{5}^{2}}{9c^{6}}}(\varrho^{2}c^{4}+2\sigma\varrhoc^{2})+{\frac{{\calC}c^{2}}{a_{0}^{4}}}+{\frac{4\piG_{5}}{3c^{2}}}\left(\Lambda+{\frac{4\piG_{5}}{3c^{4}}}\sigma^{2}\right)-{\frac{kc^{2}}{a_{0}^{2}}},
{\calL}_{nonrel}=\kappaNF_{01}+i\psi^{*}D_{t}\psi-{\frac{1}{2\mu}}|D_{x}\psi|^{2}-{\frac{1}{2\mu}}N^{2}|\psi|^{2}+{\frac{1}{2\mu\kappa}}(|\psi|^{2})^{2}.
\deltam_{\xi}\simg^{2}\left.\xi\frac{\hat{p}}{p^{2}}(p^{2}-m^{2})^{2}\intd^{3}q\frac{1}{q^{4}((p-q)^{2}-m^{2})}\right\vert_{\hat{p}=m}=2i\pi^{2}g^{2}\xi\.
{\sfM}^{ab}[A^{\omega}]={\sfD}^{ac}[\omega]\cdot\left.\frac{\delta\chi^{c}[A^{\tilde{\Omega}}]}{\delta\tilde{\varphi}^{d}}\right|_{\varphi=0}\cdot\left.\frac{\delta\tilde{\varphi}^{d}}{\delta\varphi^{b}}\right|_{\varphi=0}.
F(A)=-\frac{w}{\kappa}(\mathrm{sign~}A)\left[B_{A^{p+1}}\left(\frac{-1}{p+1},1-\frac{2}{\kappa}\right)+2B_{A^{p+1}}\left(\frac{p}{p+1},-\frac{2}{\kappa}\right)\right]\!.
\dot{A}=\frac{\dot{\omega}}{2\,\omega}\,\mathrm{e}^{2\,i\,\int^{\eta}\omega\,d\eta^{\prime}}B\,,\qquad\dot{B}=\frac{\dot{\omega}}{2\,\omega}\,\mathrm{e}^{-2\,i\,\int^{\eta}\omega\,d\eta^{\prime}}A\,,
\Theta=-\frac{3N_{c}-N_{f}(1-\gamma)}{32\pi^{2}}\left(F_{\mu\nu}^{a}\right)^{2}+\frac{N_{c}N_{f}(1-\gamma)}{16\pi^{2}N_{c}^{2}}\left(B_{\mu\nu}\right)^{2},
\begin{array}{l}{\ell_{3}^{-}=\ell^{-};\;\ell^{+}=0}\\{\hat{h}=\frac{k^{+}}{k}\ell^{-}=\frac{\hat{q}}{2}}\\{\hat{h}=\hat{h}^{\prime},\;\hat{q}=-\hat{q}^{\prime}}\\\end{array}
\frac{Q_{1}}{Q_{2}}=b^{2}p_{2}^{2}\gamma(-ibp_{2})^{2}\gamma(\frac{3}{4}+\frac{ib}{2}p_{1+2+3})\gamma(\frac{3}{4}+\frac{ib}{2}p_{1+2-3})\gamma(\frac{1}{4}+\frac{ib}{2}p_{-1+2+3})\gamma(\frac{1}{4}+\frac{ib}{2}p_{-1+2-3}).
w(\epsilon)\,=\,\frac{1}{2}\,\sum_{n=1}^{\infty}\,\sqrt{n^{2}\,+\,\epsilon^{-1}}\,=\,\frac{1}{4}\,\left(\sum_{n=-\infty}^{+\infty}\,\sqrt{n^{2}\,+\,\epsilon^{-1}}\,-\,\frac{1}{\sqrt{\epsilon}}\right)\,{,}
S_{h}=\intd^{4}x\left[{\frac{1}{4\kappa}}h_{\mu\nu}(x)h_{\mu\nu}(x)+{\frac{C}{\kappa_{0}^{2}}}{\frac{\delta_{\mu\nu}}{4}}\Big\langle{\frac{1}{2}}\mathscr{A}_{\rho}^{2}(x)\Big\rangle_{YM}\partial_{\alpha}h_{\mu\alpha}(x)\partial_{\beta}h_{\nu\beta}(x)\right],
{\calA}(\theta)=2\,V\,(8\pi^{2}\alpha^{\prime})^{-1/2}\int_{0}^{\infty}{\frac{ds}{2s}}\,s^{-1/2}\Big(\mathrm{Z}_{\mathrm{NS}}(r;\theta)-\mathrm{Z}_{\mathrm{R}}(r;\theta)\Big)
\langle\xi,\eta\rangle=\sum_{z\inT({\bfH})}\sum_{j=1}^{2g}\,(\xi,\eta\,{{\calU}_{j}^{(z)}}^{\dagger})\,{\calU}_{j}^{(z)},\qquad\xi,\eta\inL^{2}({\bfH}).
\begin{aligned}{q_{i\,yy}=}&{{}-\left\{\frac{q_{i\,y}^{2}}{h-q_{i\,x}}\right\}_{\!x}+\frac{1}{Nh}(h-q_{i\,x})\sum_{k=1}^{N}\left\{\frac{q_{k\,y}^{2}}{h-q_{k\,x}}\right\}_{\!x}+{}}\\{}&{{}{}+2(h-q_{i\,x})\frac{\deltaU(q)}{\deltaq_{i}}-\frac{2}{Nh}(h-q_{i\,x})\sum_{k=1}^{N}(h-q_{k\,x})\frac{\deltaU(q)}{\deltaq_{k}},\qquad1\lei\leN,}\\\end{aligned}
(\sigma_{\mu})_{\alpha\dot{\alpha}}J^{\mu\alpha}=J_{\alpha\dot{\alpha}}^{\alpha}=i\,\frac{T_{G}}{4\pi^{2}}\,\mathrm{Tr}\left[\bar{G}_{\dot{\alpha}\dot{\beta}}\bar{\lambda}^{\dot{\beta}}\right]\,,\qquad\vartheta_{\mu}^{\mu}=\frac{T_{G}}{16\pi^{2}}\,\mathrm{Tr}\,\left[G_{\rho\sigma}G^{\rho\sigma}\right]\,.
A=\int_{\Sigma}d\xi^{1}\wedged\xi^{2}\left\{\big(\mathrm{det}[\partial_{a}(X+\deltay).\partial_{b}(X+\deltay)]\big)^{1/2}+\big(\mathrm{det}[\partial_{a}(X-\deltay).\partial_{b}(X-\deltay)]\big)^{1/2}\right\}
\varphi^{i}~\rightarrow~\varphi^{k}~~~,~~~\partial_{\mu}\varphi^{i}~\rightarrow~\partial_{\mu}\varphi^{k}={\frac{\partial\varphi^{k}}{\partial\varphi^{i}}}\,\partial_{\mu}\varphi^{i}~~~,~~~\pi_{i}~\rightarrow~\pi_{k}^{\prime}={\frac{\partial\varphi^{i}}{\partial\varphi^{k}}}\,\pi_{i}~~~,
W_{eff}=f\left(\tau,\frac{det(\tilde{Q}Q)^{1/(N_{f}-N_{c})}}{W^{\alpha}W_{\alpha}},m^{N_{f}(N_{c}-N_{f})/N_{C}}\mathrm{det}(\tilde{Q}Q)\right)W^{\beta}W_{\beta}~+~h.c.
L_{1}=\beta^{2}(1-\mathcal{R}),\quad\mathcal{R}=\sqrt{1-\frac{2\dot{u}^{2}+\dot{v}^{2}-u^{2}(u^{2}+2v^{2})}{\beta^{2}}-\frac{u^{2}(2\dot{u}v+\dot{v}{u})^{2}}{\beta^{4}}}.
\sum_{i}\tilde{\alpha}_{i}^{\prime2}-\tilde{\alpha}_{i}^{\prime}\sum_{j}\tilde{\alpha}_{j}^{\prime}+{\frac{1}{D-2}}\varphi^{\prime2}+\sum_{a}\alpha_{a}^{\prime2}={\frac{P^{2}}{2}}e^{\tilde{\Phi}}\.
ds^{2}={\frac{dr^{2}}{1-(r_{0}/r)^{3}}}+{\frac{r^{2}}{9}}\left(1-(r_{0}/r)^{3}\right)\left(\sigma^{a}-{\frac{1}{2}}\Sigma^{a}\right)^{2}+{\frac{r^{2}}{12}}\left(\Sigma^{a}\right)^{2}\.
y_{reg}=r^{1/2}M_{\nu}^{(3)}(z,h)\stackrel{\Rez\rightarrow-\infty}{\simeq}r^{1/2}\bigg[H_{\nu}^{(1)}(\omega)+O(h^{2})\bigg]\stackrel{r\rightarrow0}{\simeq}\bigg(\frac{2}{\pig}\bigg)^{\frac{1}{2}}e^{-\frac{g}{r}}e^{-i(\nu+1)\frac{\pi}{2}}\bigg[1+O(h^{2})\bigg]
\int_{g}\int_{g^{\prime}}dg\,dg^{\prime}\,\biggl({}_{\nu\bar{\nu}}\langleO_{2}|\times{}_{\eta\bar{\eta}}\langleO_{2}|\timesf^{*}(Q)\biggl)_{t=0}{g^{\prime}}^{\dagger}\,||\,(\deltaH+\deltaH^{\dagger})^{n}\,||\,g\biggl(f(Q)\times{|O_{2}\rangle}_{\eta\bar{\eta}}\times{|O_{k}\rangle}_{\nu\bar{\nu}}\biggr)_{t=0}
\delta_{\epsilon_{1}^{\prime}\epsilon_{1}}\,\delta_{\epsilon_{2}^{\prime}\epsilon_{2}}\,{\calM}^{F}(\vec{k},\vec{p})\,\equiv\,T_{a}^{L}(\vec{k},\vec{p})\,+\,T_{c,dir}^{L}(\vec{k},\vec{p})\,
{\calL}_{2}=\frac{c_{1}}{2}\Phi^{\mu\nu,\alpha}\Phi_{\mu\nu,\alpha}+\frac{c_{2}}{2}\Phi^{\mu}\Phi_{\mu}+c_{3}\Phi^{\mu}A_{\mu}+\frac{c_{4}}{2}A_{\mu}{}^{2}+\frac{c_{5}}{2}h^{\alpha\beta}h_{\alpha\beta}+\frac{c_{6}}{2}h^{2}+\frac{c_{7}}{2}B^{\alpha\beta}B_{\alpha\beta}
\left(b_{1}^{\dagger}\right)^{n_{1}}\left(b_{2}^{\dagger}\right)^{n_{2}}\left(b_{3}^{\dagger}\right)^{n_{3}}|0\,\\\\left(b_{1}^{\dagger}\right)^{n_{1}}\left(b_{2}^{\dagger}\right)^{n_{2}}\left(b_{3}^{\dagger}\right)^{n_{3}}b_{4}^{\dagger}|0\,
\frac{\partialF}{\partialy}=S\frac{\partialf}{\partialy}S^{-1}+\frac{\partialS}{\partial\bar{z}}S^{-1},\qquad\frac{\partialF}{\partialz}=S\frac{\partialf}{\partialz}S^{-1}-\frac{\partialS}{\partial\bar{y}}S^{-1}.
E_{A}{}^{M}=\left(\begin{matrix}{e^{z}}&{ie^{z}\lambda_{j}}&{-ie^{z}\bar{\lambda}^{j}}\\{0}&{-e^{z/2}(s^{-1})_{j}{}^{i}}&{0}\\{0}&{0}&{-e^{z/2}s_{i}{}^{j}}\\\end{matrix}\right)\,.
\frac{1}{{\partial}_{-}}{\equiv}\frac{\mathrm{sin}2{\theta}{\partial}_{+}+{\partial}^{-}}{{\partial}_{\bot}^{2}+n^{2}{\partial}_{+}^{2}},\quad{\partial}_{\bot}^{2}{\equiv}{\partial}_{1}^{2}+{\partial}_{2}^{2}.
\left(\begin{array}{c}{\chi^{(1)}}\\{\chi^{(2)}}\\\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{1}&{1}\\{-1}&{1}\\\end{array}\right)\left(\begin{array}{c}{\chi^{L}}\\{\chi^{R}}\\\end{array}\right)
T_{+}=\frac{1}{2}\sum_{i=1}^{N}\frac{\partial^{2}}{\partial\xi_{i}^{2}}+\frac{\lambda}{2}\sum_{i\neqj}^{N}\frac{1}{(\xi_{i}-\xi_{j})}\left(\frac{\partial}{\partial\xi_{i}}-\frac{\partial}{\partial\xi_{j}}\right)
\begin{array}{ll}{\mathrm{Sector}\\}&{\mathrm{Field~content}}\\{\mathrm{even}}&{[1,1],\,[3,3],\,[5,5],\,[7,7],\,[9,9],\,[11,11],\,[13,13],\,[15,15]}\\{\mathrm{odd}}&{[2,2],\,[4,4],\,[6,6],\,[8,8],\,[10,10],\,[12,12],\,[14,14]}\\{\mathrm{twisted}}&{[2,14],\,[4,12],\,[6,10],\,[8,8],\,[8,8],\,[10,6],\,[12,4],\,[14,2].}\\\end{array}
\{\check{A},\,\check{B}\}_{*}=\{\check{A},\,\check{B}\}+i\hbar\frac{\nu^{2}}{2!}\left(\frac{\partial^{2}\check{A}}{\partiala^{2}}\frac{\partial^{2}\check{B}}{\partiala^{*\,2}}-\frac{\partial^{2}\check{B}}{\partiala^{2}}\frac{\partial^{2}\check{A}}{\partiala^{*\,2}}\right)+\cdot\cdot\cdot,
\mathrm{d}\equiv\mathrm{d}z^{{\calA}}\frac{\partial~}{\partialz^{{\calA}}}=e^{{\calA}}D_{{\calA}}=e^{A}\partial_{A}+\mathrm{d}\theta^{i\alpha}D_{i\alpha}+\mathrm{d}\tilde{\theta}_{\alpha}^{\tilde{\imath}}{\tilde{D}}_{\tilde{\imath}}^{\alpha}
g_{00}=-H^{-\frac{1}{2}},g(r)=H^{-\frac{1}{2}},g_{S}(r)=H^{\frac{1}{2}}r^{2},g_{rr}(r)=H^{\frac{1}{2}},H=1+\frac{e^{\Phi_{0}}Q}{2r^{4}}
\frac{Z_{3}}{Z_{\lambda}\lambda}\,\langle\,\partialA_{x}^{a}\,\partialA_{y}^{b}\,\rangle\,=\,\widetildeZ_{3}\,\frac{1}{2}\,\Big(\langle\,i(\partialD_{r}c)_{x}^{a}\,\bar{c}_{y}^{b}\,\rangle\,+\,\langle\,c_{y}^{b}\,i(\partialD_{r}\bar{c})_{x}^{a}\,\rangle\Big)\,-\,i\widetildeZ_{1}\,\frac{1}{2}\,(1-\alpha)\,\langle\,\partialA_{x}^{a}\,(\bar{c}\timesc)_{y}^{b}\,\rangle\;.
\left(\frac{\deltaA}{\deltay(s)}\right)^{2}+\left(\frac{\deltaA}{\delta\vec{x}(s)}\right)^{2}=\frac{1}{y^{4}(s)}\left\{\left(\frac{dy}{ds}\right)^{2}+\left(\frac{d\vec{x}}{ds}\right)^{2}\right\}\;.
{\calL}=-\frac{a}{4}W_{\mu\nu}W^{\mu\nu}+\frac{\alpha}{2}\varepsilon_{\mu\nu\lambda}W^{\mu}W^{\nu\lambda}-\frac{1}{{2g}}B_{\mu}^{2}-\frac{{pe^{2}}}{4}F_{\mu\nu}F^{\mu\nu}\,+\frac{{qe^{2}}}{4}\varepsilon_{\mu\nu\lambda}A^{\nu\lambda}\,,
\ddot{\varphi}+3\,\frac{\dot{a}}{a}\,\dot{\varphi}-\frac{1}{U+3U^{2}}\left(2VU^{\prime}-UV^{\prime}-\frac12\,U^{\prime}\dot{\varphi}^{2}-3U^{\prime}U^{\prime\prime}\dot{\varphi}^{2}\right)=0.
\begin{array}{ccc}{SU(4)}&{\rightarrow}&{SU(2)\timesSU(2)\timesU(1)}\\{{\bf4}}&{\rightarrow}&{({\bf2,1})_{+1}\bigoplus({\bf1,2})_{-1}}\\{{\bf6}}&{\rightarrow}&{({\bf1,1})_{+2}\bigoplus({\bf1,1})_{-2}\bigoplus({\bf2,2})_{0}}\\\end{array}
\frac{d^{2}\phi}{dx^{2}}-\frac{(1+x^{2})}{x(1-x^{2})}\frac{d\phi}{dx}-\frac{l(l+4)}{4x^{2}(1-x^{2})}\phi+\frac{\lambda^{2}}{4x(1-x^{2})^{2}}\phi=0\,.
{\bff}=-\frac{1}{2}E^{2}\nabla\varepsilon+\frac{1}{2}\nabla\left[E^{2}\rho\left(\frac{\partial\varepsilon}{\partial\rho}\right)_{T}\right]+\frac{\varepsilon-1}{c}\frac{\partial}{\partialt}({\bfE}\times{\bfH}).
b^{(3)}=\frac{2}{3}\Pi_{\tilde{\mu}\tilde{\nu}}\Pi^{\tilde{\mu}}\Pi^{\tilde{\nu}}-\frac{2}{15}\Pi_{\alpha\beta}\Pi^{\alpha}\Pi^{\beta}-\frac{3}{5}\Pi_{\tilde{\mu}\alpha}\Pi^{\tilde{\mu}}\Pi^{\alpha}\quad.
S(g)^{-1}={\mathbf1}+\sum_{n=1}^{\infty}\frac{1}{n!}\int\!\!d^{4}x_{1}\ldotsd^{4}x_{n}\,\tilde{T}_{n}(x_{1},\ldots,x_{n})\,g(x_{1})\cdot\ldots\cdotg(x_{n})\,.
\left[\phi(\psi\otimes\overline{{\psi}};z,\bar{z})\;\left(|\chi\rangle_{l}\otimes|\overline{{\chi}}\rangle_{l}\right)\right]_{m}=\sum_{\alpha}\Bigl(V_{ml}^{\alpha}(\psi,z)\;|\chi\rangle_{l}\Bigr)\otimes\left(\bar{V}_{ml}^{\alpha}(\overline{{\psi}},\bar{z})\;|\overline{{\chi}}\rangle_{l}\right).
n_{\mathrm{eff}}^{2}(n,\eta=0)=n^{2}\biggl[1-\biggl(\frac{n}{n_{\mathrm{inf}}}\biggr)^{2}\biggr],\quadn_{\mathrm{inf}}\equiv2\pi\frac{\ell_{\mathrm{b}}}{\ell_{_\mathrm{C}}}.
\chi_{T}=\left(\frac{\partialJ}{\partial\Omega}\right)_{T}=2^{5/3}\pi^{1/3}\kappa^{-4/3}Nmr_{+}^{-1}\frac{r_{+}^{4}+4l^{2}r_{+}^{2}-l^{4}}{3r_{+}^{4}+4l^{2}r_{+}^{2}+l^{4}}\frac{3r_{+}^{2}+l^{2}}{3r_{+}^{2}-l^{2}}.
F_{A}=\left(\begin{matrix}{{\frac{1}{2}}F^{0}}&{0}\\{0}&{-{\frac{1}{2}}F^{0}}\\\end{matrix}\right)=\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right)h^{2}dz\wedged{\bar{z}}\quad.
\tilde{\omega}=4\pi\sum_{\rho,\rho^{\prime},i=1}^{g}(w_{i\rho}{\bar{w}}_{i+g\rho^{\prime}}-{\bar{w}}_{i\rho^{\prime}}w_{i+g\rho})dc_{\rho}\wedged{\bar{c}}_{\rho^{\prime}}.
R(u_{1},u_{2},u_{3})_{i_{1}i_{2}i_{3}}^{j_{1}j_{2}j_{3}}=R((\omegau_{3})^{-1},(\omegau_{2})^{-1},u_{1})_{-i_{3}-i_{2}j_{1}}^{-j_{3}-j_{2}i_{1}}.
K_{I}(\theta)=\sqrt{\ell^{\prime}}\;\;\frac{\textrm{sn}\left[\frac{\textbf{K}}{i\pi}\left(\theta-i\frac{\pi}{2}\right)\right]}{\textrm{cn}\left[\frac{\textbf{K}}{i\pi}\left(\theta-i\frac{\pi}{2}\right)\right]}\,\,\,.
S[X]=\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}d\taud\sigma\partial_{a}X^{M}\partial^{a}X^{N}g_{MN}-\frac{i}{2}\int_{\partial\Sigma}d\sigmaB_{\mu\nu}X^{\mu}\partial_{\sigma}X^{\nu}~.
\breve{R}^{\alpha\beta}(x)={\bfP}_{\Lambda_{1}}^{\alpha\beta}+\frac{1-xq^{\alpha+\beta}}{x-q^{\alpha+\beta}}\,{\bfP}_{\Lambda_{3}}^{\alpha\beta}+\frac{1-xq^{\alpha+\beta}}{x-q^{\alpha+\beta}}\frac{1-xq^{\alpha+\beta+2}}{x-q^{\alpha+\beta+2}}\,{\bfP}_{\Lambda_{2}}^{\alpha\beta}
\begin{aligned}{E_{a}\,,\,H-E_{a}-E_{b}\,,\,2H-\sum_{i=1}^{5}E_{a_{i}}\,,\,3H-2E_{a}-\sum_{i=1}^{6}E_{a_{i}}\,,}\\{4H-2E_{a}-2E_{b}-2E_{c}-\sum_{i=1}^{5}E_{a_{i}}\,,5H-2\sum_{i=1}^{6}E_{a_{i}}-\sum_{j=1}^{2}E_{a_{j}}\,,}\\{6H-3E_{a}-2\sum_{i=1}^{7}E_{a_{i}}\,.}\\\end{aligned}
\left(1-\frac{r_{h}}{r}\right)\partial_{r}^{2}R(r)+\left(\frac{2}{r}-\frac{r_{h}}{r^{2}}\right)\partial_{r}R(r)+\left(\frac{\omega^{2}}{\left(1-\frac{r_{h}}{r}\right)}-\frac{\nu(\nu+1)}{r^{2}}-m^{2}\right)R(r)=0,
\tilde{J}\,\delta(\epsilon-\sum_{i=1}^{N}\sqrt{m^{2}+N{(\sum_{a=1}^{N-1}{\hat{\gamma}}_{ai}{\vec{\pi}}_{a})}^{2}}\delta(\hat{T}-\tau)\prod_{a=1}^{N-1}\delta({\hat{T}}_{Ra})\delta({\hat{\epsilon}}_{Ra}),
w^{2}=p^{2}+\frac{1}{2}\,\biggl({\calE}^{\prime\prime}(u)+\frac{u^{4}}{\kappa^{2}}\biggr)\pm\frac{1}{2}\sqrt{\biggl({\calE}^{\prime\prime}(u)-\frac{u^{4}}{\kappa^{2}}\biggr)^{2}+16a^{2}(u)\,p^{2}}.
R_{\hat{U}m^{\prime}\hat{V}}^{m^{\prime}}=\partial_{\hat{V}}\Gamma_{\hat{U}m^{\prime}}^{m^{\prime}}-\partial_{m^{\prime}}\Gamma_{\hat{U}\hat{V}}^{m^{\prime}}+\Gamma_{\hat{U}m^{\prime}}^{V}\Gamma_{V\hat{V}}^{m^{\prime}}-\Gamma_{\hat{U}\hat{V}}^{V}\Gamma_{Vm^{\prime}}^{m^{\prime}},
\Phi_{0}=\sum_{N,M,P_{i}\geq0}\frac{\Gamma(3P_{1}+1)\;(-z_{1})^{P_{1}}z_{2}^{P_{2}}z_{3}^{P_{3}}(-x)^{M}y^{N}}{\left\{\begin{array}{c}{\Gamma(P_{1}+M-N+1)\Gamma(P_{2}-2P_{3}+N+1)\Gamma(P_{1}-2P_{2}+P_{3}+1)}\\{\times\Gamma(N-3M+1)\Gamma^{2}(P_{1}+1)\Gamma(M+1)}\\{\times\Gamma(P_{3}+M-N+1)\Gamma(P_{2}-P_{1}+1)}\\\end{array}\right\}}.
\left(\,\psi_{1}\,\psi_{2}\,\right)^{*}\,=\,\left(-1\right)^{F_{1}F_{2}}\,\psi_{1}^{*}\,\psi_{2}^{*}=(-1)^{p_{1}p_{2}}\psi_{2}^{*}\,\psi_{1}^{*}
V_{eff}(z)=\frac{3(5\alpha+4)k^{2}}{64\left({\frac{kz}{2}}+1\right)^{2}}+{\frac{l^{2}}{2R^{2}}}\left({\frac{kz}{2}}+1\right)^{{\frac{5}{2}}(\alpha-2)}-{\frac{3k}{4}}\delta(z)
\left(\hat{V}^{(B=0)}\right)^{-1}\partial_{\pm}\hat{V}^{(B=0)}={\frac{1\mpt}{1\pmt}}\left(\begin{array}{cc}{-E^{-1}\partial_{\pm}E}&{0}\\{0}&{E^{-1}\partial_{\pm}E}\\\end{array}\right)={\frac{1\mpt}{1\pmt}}\,P_{\pm}
\left(\begin{array}{ccc}{d_{j_{0}-1,j_{0}-1}}&{d_{j_{0}-1,j_{0}}}&{d_{j_{0}-1,j_{0}+1}}\\{d_{j_{0},j_{0}-1}}&{d_{j_{0},j_{0}}}&{d_{j_{0},j_{0}+1}}\\{d_{j_{0}+1,j_{0}-1}}&{d_{j_{0}+1,j_{0}}}&{d_{j_{0}+1,j_{0}+1}}\\\end{array}\right)=(-1)^{3j_{0}+1}\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{-1}&{0}\\{0}&{0}&{1}\\\end{array}\right),
\hat{K}_{\hat{a}\hat{b}\hat{c}}={\textstyle\frac{m}{4}}\left[\hat{k}_{\hat{a}}\left(i_{\hat{k}}\hat{C}\right)_{\hat{b}\hat{c}}+\hat{k}_{\hat{b}}\left(i_{\hat{k}}\hat{C}\right)_{\hat{a}\hat{c}}-\hat{k}_{\hat{c}}\left(i_{\hat{k}}\hat{C}\right)_{\hat{a}\hat{b}}\right]\,.
\left({\frac{\dot{a}}{a}}\right)^{2}={\frac{1}{\tilde{L}_{p}^{2}}}\left[(\tilde{E}^{2}-\tilde{P}_{\phi}^{2})\left({\frac{\tilde{L}_{p}}{a}}\right)^{2(p+1)}-\tilde{E}^{2}\left({\frac{\tilde{L}_{p}}{a}}\right)^{2p}+2\tilde{P}_{\phi}\left({\frac{\tilde{L}_{p}}{a}}\right)^{p+1}-\left({\frac{\tilde{L}_{p}}{a}}\right)^{2}\right],
{\calP}_{{\bfn}}=\frac{{\calP}_{0}(\beta)}{{\bfn}!}\left|H_{{\bfn}}^{\{\zeta^{-1}\eta\}}\left(\eta^{-1}[\beta-{\bfd}]\right)\right|^{2},\qquad{\calP}_{0}(\beta)=|{\calF}_{0}(\beta)|^{2},
\begin{array}{lc}{I(\alpha_{i})=\alpha_{i},}&{I(\alpha_{i}\alpha_{j})=\alpha_{j}\alpha_{i},}\\{I(\alpha_{i}\alpha_{j}\alpha_{k})=\alpha_{k}\alpha_{j}\alpha_{i},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{\cdots.}\\\end{array}
g(t,\eta,\bar{\eta})=e^{-itH}e^{i(\eta_{i}Q^{i}+\bar{\eta}^{i}\bar{Q}_{i})}e^{i(\lambda_{i}S^{i}+\bar{\lambda}^{i}\bar{S}_{i})}e^{izD}e^{i\omegaK}e^{i\phiJ_{1}}e^{i\thetaJ_{2}}e^{i\psiJ_{3}}\quad,
L=2t\int_{\Sigma}d^{2}z(\frac{1}{2}g_{IJ}\partial_{z}\phi^{I}\partial_{\bar{z}}\phi^{J}+i\psi_{z}^{\bar{i}}D_{\bar{z}}\chi^{i}g_{i\bar{i}}+i\psi_{\bar{z}}^{i}D_{z}\chi^{\bar{i}}g_{\bar{i}i}-R_{i\bar{i}j\bar{j}}\psi_{\bar{z}}^{i}\psi_{z}^{\bar{i}}\chi^{j}\chi^{\bar{j}})
\mathcal{B}=*B^{A}\left[\begin{array}{c}{v_{A}}\\{\overline{{v}}_{A}}\\\end{array}\right]-iB^{A}\left[\begin{array}{c}{v_{A}}\\{-\overline{{v}}_{A}}\\\end{array}\right]+b^{\alpha}\left[\begin{array}{c}{e_{\alpha}}\\{\overline{{e}}_{\alpha}}\\\end{array}\right]+i*b^{\alpha}\left[\begin{array}{c}{e_{\alpha}}\\{-\overline{{e}}_{\alpha}}\\\end{array}\right]
D_{i}^{\pm}=\frac{1}{2}(1\pmi)~,~~~~~D_{j}^{\pm}=\frac{1}{2}(1\pmj)~,~~~~~D_{l}^{\pm}=\frac{1}{2}(1\pml)~,~~~~~D_{(ij)l}^{\pm}=\frac{1}{2}(1\pm(ij)l)~,
W=2\kappa^{2}\int(\xi^{\prime})^{2}\;dr=2\kappa^{2}\left(\mathrm{tanh}(\betar)-\frac{1}{3}\mathrm{tanh}^{3}(\betar)\right)=2\kappa\sqrt{\beta}(\xi-\frac{\beta}{3\kappa^{2}}\,\xi^{3})\,.
\deltaa_{a}^{+}=(\epsilon_{b}^{\beta}\theta^{+b}\partial_{\beta}^{-}+\epsilon_{b}^{\beta}\theta^{-b}\partial_{\beta}^{+}-\epsilon^{+b}\partial_{b}^{-})a_{a}^{+}-\epsilon_{a}^{\alpha}A_{\alpha}^{+}\.
\langle\partial_{\mu}J_{5}^{\mu}\rangle={\frac{i\timesd}{768\pi^{2}}}F_{\alpha\betaAB}\epsilon^{\alpha\beta\mu\nu}F_{\mu\nu}\,^{AB}+{\frac{i}{32\pi^{2}}}Tr(\epsilon^{\alpha\beta\mu\nu}G_{\alpha\betaa}T^{a}G_{\mu\nub}T^{b})
{\calL}_{\sigmaJ^{a}}={\calL}_{\sigma}+J^{a}\phi^{a}=-{\frac{\sigma}{2}}\left(\phi^{a}-{\frac{J^{a}}{\sigma}}\right)^{2}-{\frac{Nm_{0}^{2}}{g_{0}}}\sigma+{\frac{N}{2g_{0}}}\sigma^{2}+{\frac{J^{a}J^{a}}{2\sigma}}.
\langle\,\partial_{-}f^{R}(\sigma)\partial_{-}f^{R}(\sigma^{\prime})\,\rangle\:_{\mathrm{{in}}}=\langle\,\partial_{-}f(\sigma)\partial_{-}f(\sigma^{\prime})\,\rangle\:_{\mathrm{in}}=-\,\frac{1}{4\pi}\,\frac{p^{\prime}(\sigma^{-})\,p^{\prime}(\sigma^{-})}{[\,p(\sigma^{-})-p(\sigma^{-})-i\epsilon\,]^{2}}\,.
{\calR}=\frac{\alpha}{2\theta}|0\rangle\langle0|dz\wedged\bar{z}\left(\begin{array}{rr}{1}&{0}\\{0}&{-1}\\\end{array}\right)+\frac{\alpha^{2}}{4}\left[{\tt\frac{1}{z}},{\tt\frac{1}{\bar{z}}}\right]dz\wedged\bar{z}\left(\begin{array}{rr}{1}&{0}\\{0}&{1}\\\end{array}\right)\,.
\mu^{\prime}=\alpha^{2}\Biggl(NK^{2}+\frac{1}{2}Nx^{2}H^{2}+\frac{(K^{2}-1)^{2}}{2x^{2}}+H^{2}K^{2}+\frac{\beta^{2}}{4}x^{2}(H^{2}-1)^{2}\Biggr)\,
R:=\left(\begin{array}{ccccc}{N-1}&{-1}&{.}&{.}&{-1}\\{-1}&{N-1}&{-1}&{}&{.}\\{.}&{-1}&{.}&{.}&{.}\\{.}&{}&{.}&{.}&{-1}\\{-1}&{.}&{.}&{-1}&{N-1}\\\end{array}\right)\;.
\begin{aligned}{\int_{\mathbb{R}^{1,3}}\sum_{a}N_{a}(n_{a}^{1}N_{a}^{2}+m_{a}^{1}M_{a}^{2})M_{a}^{3}B_{2}\wedgeF_{a},}\\{\int_{\mathbb{R}^{1,3}}\sum_{b}(n_{b}^{1}N_{b}^{2}+m_{b}^{1}M_{b}^{2})n_{b}^{3}C_{0}\wedgeF_{b}\wedgeF_{b}.}\\\end{aligned}
\left(\begin{array}{c}{\theta_{1}}\\{\theta_{2}}\\{\theta_{3}}\\{\vdots}\\{\theta_{N-2}}\\{\theta_{N-1}}\\\end{array}\right)={\frac{\pi}{N}}\left(\begin{array}{c}{N-1}\\{N-3}\\{N-5}\\{\vdots}\\{-(N-5)}\\{-(N-3)}\\\end{array}\right)\quad\mathrm{or}\quad\theta_{i}={\frac{\pi}{N}}\Bigl(N-(2i-1)\Bigr)\quad(i=1,\cdots,N-1).
\{x_{{i}_{1}},\ldots,x_{{i}_{M}}\}:=\frac{1}{\rho}\in^{r_{1}\ldotsr_{M}}\frac{\partialx_{i_{1}}}{\partial\varphi^{r_{1}}}\ldots\frac{\partialx_{i_{M}}}{\partial\varphi^{r_{M}}}\.
a\left(\partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha}\right)+\frac{ib}{2}\left(\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho}\right)\epsilon^{\rho\sigma\alpha\beta}=\frac{1}{2}\epsilon^{\alpha\beta\rho\sigma}\left(\partial_{\rho}B_{\sigma}-\partial_{\sigma}B_{\rho}\right).
\bar{\epsilon}_{1}\Gamma_{m}d\theta_{1}\bar{\theta}_{1}\Gamma^{m}d\theta_{1}=-{\frac{1}{2}}\bar{\epsilon}_{1}\Gamma_{m}\theta_{1}d\bar{\theta}_{1}\Gamma^{m}d\theta_{1}=-{\frac{1}{3}}d\Big(\bar{\epsilon}_{1}\Gamma_{m}\theta_{1}\bar{\theta}_{1}\Gamma^{m}d\theta_{1}\Big).
\tilde{\Psi}_{\mathrm{far}}(r)={\frac{A_{\mathrm{in}}}{r}}\left(r^{\sqrt{1-s}}+ir^{-\sqrt{1-s}}\right)+{\frac{A_{\mathrm{out}}}{r}}\left(r^{\sqrt{1-s}}-ir^{-\sqrt{1-s}}\right).
\begin{array}{lll}{d_{v}}&{=}&{4\sum_{n=1}^{k-1}n+k(2N-4k+3)=2kN-k(2k-1),}\\{d_{H}}&{=}&{[\frac{1}{2}\times2\sum_{n=1}^{2k-2}(n+1)(n+2)+\frac{1}{2}4k^{2}(2N-4k+2)+\frac{1}{2}(2k+2k+2+2)]}\\{}&{-}&{[4\sum_{n=1}^{k-1}n(2n+1)+k(2k+1)(N-2k+2)+k(2k-1)(N-2k+1)]}\\{}&{=}&{k.}\\\end{array}
D_{\mu\nu}=\overline{{P}}_{\mu\nu}\,\frac{1}{p^{2}+\Pi_{\mathrm{T}}}+Q_{\mu\nu}\,\frac{1}{p^{2}+\Pi_{\mathrm{L}}}+R_{\mu\nu}\,\frac{1}{p^{2}+\tilde{\Pi}_{\mathrm{T}}}+\xi\,\frac{p_{\mu}p_{\nu}}{p^{2}}
\langlex^{\prime}|\hat{V}|x^{\prime\prime}\rangle=\frac{\hbar^{2}}{2m}\left[(\delta^{ij}-g^{ij})\partial_{i}^{\prime}\partial_{j}^{\prime}-\frac{\partial_{i}^{\prime}(\sqrt{g}g^{ij})}{\sqrt{g}}\partial_{j}^{\prime}+2\lambdaR\right]\delta(\vec{x}^{\prime}-\vec{x}^{\prime\prime})\;,
\leq\;\prod_{a=1}^{N}\left(\xi\;8^{[1-\frac{g}{\pi+gN}]}\right)^{2n_{a}}\;\prod_{b=1}^{N}\Big[(2n_{b})!\Big]^{\frac{1}{p}-1}\;\prod_{c<d}^{N}\Big[(2n_{c}+2n_{d})!\Big]^{\frac{1}{q(N-1)}}\;.
I_{pk}\left(M_{p}\right)=\int_{M_{p}}^{\infty}\,d\omega\,\omega\left(\omega+M_{p}\right)^{-1/2}\left(\omega-M_{p}\right)^{-1/2}\left(\omega+\mu\right)^{-k}\left(\omega-\mu\right)^{k}e^{-2k\omega}.
\int{d^{3}\sigma\vec{\pi}^{2}(\tau,\vec{\sigma})}=\int{d^{3}\sigma\vec{\pi}_{\perp}^{2}(\tau,\vec{\sigma})}+\sum_{i\neqj}^{1...N}\frac{Q_{i}Q_{j}}{4\pi\mid\vec{\eta}_{i}(\tau)-\vec{\eta}_{j}(\tau)\mid},
{\cal{S}}=\intd^{4}x\intdy\sqrt{-G}\left\{2M^{3}R-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2}(\partial\pi)^{2}-V(\phi,\pi)-{\frac{1}{4g^{2}}}e^{-\lambda\pi/2\sqrt{3M^{3}}}F_{MN}F^{MN}\right\}\,,
\psi_{d=3}(\vec{x},\vec{y}):=\frac{1}{r^{8}}\left((\vec{x}^{2}-\vec{y}^{2})(\lambda_{1}\lambda_{2}-\mu_{1}\mu_{2})+2\vec{x}\cdot\vec{y}(\mu_{1}\lambda_{2}-\mu_{2}\lambda_{1})\right)
\mathcal{H}=\left(a^{+},b\right)\overline{{H}}\left(\begin{matrix}{a}\\{b^{\dagger}}\\\end{matrix}\right)\equiv\left({\hat{a}}^{+},{\hat{b}}\right)H_{d}\left(\begin{matrix}{{\hat{a}}}\\{{\hat{b}}^{\dagger}}\\\end{matrix}\right).
\begin{aligned}{h_{i}(u)}&{{}=\psi_{i1}(u),}\\{\eta_{\alpha\beta}}&{{}=\sum_{i=1}^{n}\psi_{i\alpha}(u)\psi_{i\beta}(u),}\\{\sum_{\beta}\eta_{\alpha\beta}\frac{\partialx^{\beta}(u)}{\partialu_{i}}}&{{}=\psi_{i1}(u)\psi_{i\alpha}(u),}\\{c_{\alpha\beta\gamma}(x(u))}&{{}=\sum_{i=1}^{n}\frac{\psi_{i\alpha}(u)\psi_{i\beta}(u)\psi_{i\gamma}(u)}{\psi_{i1}(u)}.}\\\end{aligned}