harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\widetilde{\calM}_{n}=\left(\begin{matrix}{2}&{-C}&{0}&{\ldots}&{0}&{-C}\\{-C}&{2}&{-C}&{\ldots}&{0}&{0}\\{0}&{-C}&{2}&{\ldots}&{0}&{0}\\{\vdots}&{\vdots}&{\vdots}&{}&{\vdots}&{\vdots}\\{0}&{0}&{0}&{\ldots}&{2}&{-C}\\{-C}&{0}&{0}&{\ldots}&{-C}&{2}\\\end{matrix}\right)
\frac{\lambda^{2}\sqrt{2}}{m^{3}}E_{\mathrm{TK1}}=\int_{0}^{\bar{\sigma}_{3}^{2}}\frac{\lambda_{1}d\lambda_{1}}{\sqrt{1-\lambda_{1}}}+\int_{\bar{\sigma}_{3}^{2}}^{\bar{\sigma}_{2}^{2}}\frac{\lambda_{2}d\lambda_{2}}{\sqrt{1-\lambda_{2}}}+\int_{\bar{\sigma}_{2}^{2}}^{1}\frac{\lambda_{3}d\lambda_{3}}{\sqrt{1-\lambda_{3}}}=\frac{4}{3}
\langlex^{\prime\prime}|(E-\hat{H}_{\pm})^{-1}|x^{\prime}\rangle=\frac{1}{\mathrm{i}\hbar}\int_{0}^{\infty}\mathrm{d}t\,\langlex^{\prime\prime}|\mathrm{e}^{-(\mathrm{i}/\hbar)t\hat{H}_{\pm}}|x^{\prime}\rangle\,\mathrm{e}^{(\mathrm{i}/\hbar)tE},\qquad\mathrm{Im}\,E>0.
I_{jk,i}D\Phi^{j}D\Phi^{k}D^{2}\Phi^{i}=\frac{1}{3}D\left[I_{jk,i}D\Phi^{j}D\Phi^{k}D\Phi^{i}\right]+\frac{2}{3}\left(I_{jk,i}+I_{ji,k}\right)D\Phi^{j}D\Phi^{k}D^{2}\Phi^{i}\quad,
D(h_{L,M})=\left\{\left(\begin{array}{c}{\varphi}\\{\Phi}\\\end{array}\right)\mid\varphi,\varphi^{\prime}\inAC_{\mathrm{loc}}(0,\infty),~\varphi,\varphi^{\prime\prime}\inL^{2};\quad\Phi=-|M|^{1/2}\varphi(0)\right\},
{\mathcalL}=+\frac{1}{2}\partial_{\mu}\theta\partial^{\mu}\theta+\frac{1}{4}\delta_{ij}\partial_{\mu}\pi^{i}\partial^{\mu}\pi^{j}+\frac{1}{2}\frac{1}{\Lambda}\epsilon_{ij}\pi^{j}\partial_{\mu}\theta\partial^{\mu}\pi^{i}-\frac{1}{24\Lambda^{2}}\left[\delta_{ij}\delta_{lm}-\delta_{il}\delta_{jm}\right]\pi^{i}\pi^{j}\partial_{\mu}\pi^{l}\partial^{\mu}\pi^{m}
V_{\theta}-V_{0}=-{\frac{1}{\calA}}\int_{0}^{\theta}a_{5/2}(f_{\theta^{\prime}},P_{\theta^{\prime}})d\theta^{\prime}={\frac{1}{\epsilon{\calA}}}\left[a_{5/2}^{D}(P_{\theta})-a_{5/2}^{D}(P_{0})\right].
\begin{array}{lll}{Z}&{=\int\\left(d\psid\bar{\psi}\right)(d\chi)\exp\left[-\sum_{n,m}\left(\psi(n)\chi(n,m)\bar{\psi}(m)+{\frac{\displaystyletr\left(\chi(n,m)\chi(m,n)\right)}{\displaystyle2J(n-m)}}\right)\right]}\\{}&{\phantom{aa}}\\{}&{=\int(d\chi)\exp\\left[-ln\Det\\chi\-\\sum_{n,m}{\frac{\displaystyletr\left(\chi(n,m)\chi(m,n)\right)}{\displaystyle2J(n-m)}}\right]}\\\end{array}
\begin{array}{rl}{\deltae}&{=-\partial_{\tau}(ae)-{i}\alpha(\tau)\psi,}\\{\delta\psi}&{=-a(\tau)\dot{\psi}-{\frac{3}{2}}\dot{a}\psi-{\frac{1}{4}}\alpha(\tau)\dot{e}-{\frac{1}{2}}\dot{\alpha}(\tau)e.}\\\end{array}
{\calL}_{s}=\frac{_1}{^2}\dot{\varphi}^{\alpha}M^{\alpha\beta}\left(\partial_{i}\varphi^{\mu}\right)\dot{\varphi}^{\beta}-W\left(\partial_{i}\varphi^{\mu}\right)+\frac{_1}{^2}\mu\left(\varphi^{\alpha}\varphi^{\alpha}-1\right),
k_{1}^{\mu}k_{2}^{\nu}T_{\rho\sigma\mu\nu}^{(3)abcd}=C_{3}M^{2}S_{\rho\sigma}^{(3)}+C_{3}k_{1}^{\lambda}\Gamma_{\rho\sigma\lambda}^{(3)}(p_{1},p_{2},q_{3})
\delta{\calE}=\delta{\calE}_{0}-\frac{3\hbar^{2}}{4m}\intd^{3}\vec{r}\,\varphi^{\dagger}[(\vec{\sigma}\cdot\vec{\epsilon})(\vec{\sigma}\cdot\vec{\nabla})+(\vec{\sigma}\cdot\vec{\nabla})(\vec{\sigma}\cdot\vec{\epsilon})]\varphi\,{,}
\left.\begin{array}{lll}{F_{1}^{(n)}=q_{3}^{(n)}+q_{1}^{(n)}\pi_{\phi},\,\,\,\,}&{F_{2}^{(n)}=q_{1}^{(n-1)},\,\,\,\,}&{F_{3}^{(n)}=-q_{2}^{(n-1)}-q_{3}^{(n)}\pi_{\phi},}\\{G_{1}^{(n)}=2q_{2}^{(n+1)}e^{\phi},\,\,\,\,}&{G_{2}^{(n)}=-2q_{3}^{(n+1)}e^{\phi},\,\,\,\,}&{G_{3}^{(n)}=2q_{1}^{(n)}e^{-2\phi}.}\\\end{array}\right.
\begin{array}{ccc}{\sigma^{1}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right)}&{\sigma^{2}=\left(\begin{matrix}{0}&{-i}\\{i}&{0}\\\end{matrix}\right)}&{\sigma^{3}=\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right)~.}\\\end{array}
H={\frac{p^{2}}{m}}-{\frac{p^{4}}{4m^{3}c^{2}}}+{\frac{Q^{2}}{2\pi\epsilon_{0}}}ln{\frac{r}{r_{0}}}+{\frac{Q^{2}}{2\pi\epsilon_{0}c^{2}}}[{\frac{\bfr.p}{mr}}({\frac{\bfr.p}{mr}}+c(1+ln{\frac{r}{r_{0}}}))-{\frac{p^{2}}{2m^{2}}}].
\intd^{4}x\left({\frac{1}{2}}(\partial\phi)^{2}+{\frac{m^{2}}{2}}\phi^{2}+{\frac{\rho_{1}(\Lambda_{0})}{2}}\phi^{2}+{\frac{\rho_{2}(\Lambda_{0})}{2}}(\partial\phi)^{2}+{\frac{\rho_{3}(\Lambda_{0})}{4!}}\phi^{4}\right)
\left[{\frac{dn_{d}(\omega,D)}{d\omega}}\right]_{\mathrm{div}}=r_{d}{\frac{\Gamma\left(1-\fracD2\right)}{(4\pi)^{D/2}}}{\frac{m^{D-4}}{\kappa}}\int_{\Sigma}\left[2\left(m^{2}+{\frac{R}{12}}-{\frac{R_{ii}}{8}}\right)-{\frac{\omega^{2}}{\kappa^{2}}}P\right]~~~.
\sum_{j=1}^{r+s}\alpha_{j}\!H_{j}~=~H_{\alpha}~~,~~\sum_{j=1}^{r}\alpha_{j}\!H_{j}~=~(H_{\alpha})_{\mathfrak{a}}~~,~~\sum_{j=r+1}^{r+s}\alpha_{j}\!H_{j}~=~(H_{\alpha})_{\mathfrak{b}}~,
\langle\hat{\Phi}^{\dagger}(p)\hat{\Phi}(p)\rangle={\frac{1}{2\;E(p)\;\mathrm{th}{\frac{\betaE(p)}{2}}}}\;,\qquad\langle\hat{\Pi}^{\dagger}(p)\hat{\Pi}(p)\rangle={\frac{E(p)}{2\;\mathrm{th}{\frac{\betaE(p)}{2}}}}\;,
\delta\omega^{k}=D\phi^{k}+{\tilde{\calL}}_{\xi}\omega^{k}+{\frac{\Lambda}{12}}\theta^{k}\,\qquad\delta\Sigma^{k}=2(\Sigma\times\phi)^{k}+{\tilde{\calL}}_{\xi}\Sigma^{k}+D\theta^{k}\,
\frac{G_{2}}{SO(4)},\,\,\,\frac{F_{4}}{Sp(3)\timesSp(1)},\,\,\,\frac{E_{6}}{SU(6)\timesSp(1)},\,\,\,\frac{E_{7}}{Spin(12)\timesSp(1)},\,\,\,\frac{E_{8}}{E_{7}\timesSp(1)}\,
\frac{1}{16}(\vec{\Theta}_{\hat{\rho}}\cdot\vec{e})(\vec{\Theta}_{\hat{\alpha}}\cdot\vec{e})\,\gamma_{\hat{\rho}\hat{\beta}}^{9}\gamma_{\hat{\beta}\hat{\beta}^{\prime}}^{\epsilon}\gamma_{\hat{\beta}^{\prime}\hat{\alpha}^{\prime}}^{9}\gamma_{\hat{\alpha}^{\prime}\hat{\alpha}}^{\epsilon}=-4\,.
\begin{array}{c}{{\calL}_{\pm}(A,B)=\int_{M}H_{[\mu\nu}^{a}K_{\lambda\rho]}^{b}}\\{mathrm{tr}(T^{a})\mathrm{tr}(T^{b})\d^{4}x}\\{\pm\frac12\int_{M}K_{[\mu\nu}^{a}K_{\lambda\rho]}^{b}\\mathrm{tr}(T^{a}T^{b})\d^{4}x.}\\\end{array}
\zeta(s;\bar{L}_{i}^{a},\bar{m}_{\mu})\equiv\sum_{n=0}^{\infty}\sum_{n_{1},n_{2}=-\infty}^{+\infty}\left[\left({\frac{n_{1}}{\bar{L}_{1}}}\right)^{2}+\left({\frac{n_{2}}{\bar{L}_{2}}}\right)^{2}+n+\alpha_{\mu}\right]^{-s},
M=\left(\begin{matrix}{x_{0}}&{x_{1}}&{x_{2}}&{}&{\ldots}&{x_{q-1}}\\{x_{q-1}}&{x_{0}}&{x_{1}}&{x_{2}}&{\ldots}&{x_{q-2}}\\{x_{q-2}}&{x_{q-1}}&{x_{0}}&{x_{1}}\\{}&{x_{q-2}}&{x_{q-1}}&{x_{0}}&{\ddots}&{\vdots}\\{\vdots}&{\vdots}&{}&{\ddots}&{\ddots}&{x_{1}}\\{x_{1}}&{x_{2}}&{\ldots}&{\ldots}&{x_{q-1}}&{x_{0}}\\\end{matrix}\right).
\begin{array}{lcl}{x_{a}\starx_{b}}&{=}&{\sum_{\alpha=0}^{\infty}(x_{a}\starx_{b})_{\alpha}}\\{}&{=}&{x_{a}x_{b}+\sum_{{\alpha}=1}^{\infty}\frac{1}{\alpha!}(\frac{i}{2})^{\alpha}\prod_{i=1}^{\alpha}\theta^{\mu_{i}\nu_{i}}(\prod_{j=1}^{\alpha}\partial_{\mu_{j}}x_{a})(\prod_{k=1}^{\alpha}\partial_{\nu_{k}}x_{b})}\\\end{array}
\psi_{-2\dot{q}}^{1-}=-\frac{i}{2}\nabla_{-}\chi_{\dot{q}}^{1},\\psi_{+2\dot{q}}^{1-}=\frac{i}{4}\Omega_{+}^{-2i}\tilde{\gamma}_{\dot{q}q}^{i}\chi_{q}^{1},
\begin{array}{l}{\left[M_{i_{1}j_{1}}(x_{1}^{-},y_{1}^{-};x^{+}),M_{i_{2}j_{2}}(x_{2}^{-},y_{2}^{-};x^{+})\right]}\\{{}~~~~~~~~~~~~~~~~~={\frac{1}{2N}}~\delta_{j_{1}i_{2}}\delta(y_{1}^{-}-x_{2}^{-})M_{i_{1}j_{2}}(x_{1}^{-},y_{2}^{-};x^{+})}\\{{}~~~~~~~~~~~~~~~~~-{\frac{1}{2N}}~\delta_{j_{2}i_{1}}\delta(y_{2}^{-}-x_{1}^{-})~M_{i_{2}j_{1}}(x_{2}^{-},y_{1}^{-};x^{+})}\\\end{array}
t_{0}=\left(\begin{array}{cr}{0}&{-1}\\{1}&{0}\\\end{array}\right),~~~~t_{1}=\left(\begin{array}{cr}{0}&{~1}\\{1}&{~0}\\\end{array}\right),~~~~t_{2}=\left(\begin{array}{cr}{1}&{0}\\{0}&{-1}\\\end{array}\right).
g_{\mu\nu}=v_{0}h_{\mu\nu}+\kappav_{0}(\varphih_{\mu\nu}+\varphi_{\mu\nu})\;,\qquad\mathrm{and}\qquadI_{\mu}^{\\nu}=\left[\delta_{\mu}^{\\nu}+\kappa\frac{\varphi_{\mu\alpha}h^{\alpha\nu}}{(1+\kappa\varphi)}\right]\;.
\lambda_{RL}^{-1}(k,\mu)+\frac{2}{\pi}\sum_{i=0}^{L}\frac{k^{2(L-i)}\mu^{2i+1}}{2i+1}=\lambda_{RL}^{-1}(k,\mu_{0})+\frac{2}{\pi}\sum_{i=0}^{L}\frac{k^{2(L-i)}\mu_{0}^{2i+1}}{2i+1}.
W_{\nu_{1},\nu_{2}}\simy_{1}^{\frac{1}{3}}y_{2}^{\frac{1}{3}}(y_{1}^{\frac{2}{3}}+y_{2}^{\frac{2}{3}})^{-\frac{1}{4}}exp(-(y_{1}^{\frac{2}{3}}+y_{2}^{\frac{2}{3}})^{\frac{3}{2}})~.
\deltaA_{\mu\nu}=\epsilon_{\nu}^{*}\psi_{\mu}-\epsilon_{\mu}\psi_{\nu}^{*}\,\delta\psi_{\mu}=i\epsilon_{\nu}A_{\mu\nu}\,\delta\psi_{\mu}^{*}=i\epsilon_{\nu}^{*}A_{\nu\mu}\,
F(z)\;=\;F(0)\,+\,\sum_{m=1}^{\infty}\,\left\{\frac{b_{m}}{z-a_{m}}+\frac{b_{m}}{a_{m}}\right\}\,+\,\sum_{m=1}^{\infty}\,\left\{\frac{b_{-m}}{z-a_{-m}}+\frac{b_{-m}}{a_{-m}}\right\}\,.
\sumq^{\frac{1}{2}p^{2}}=\sum_{n}q^{\frac{6}{2}\left(2n+1+\frac{4m_{1}+3m_{2}}{12}\right)^{2}}=\sum_{n}q^{12\left(n+\frac{4m_{1}+3m_{2}+12}{24}\right)^{2}}=\theta_{4m_{1}+3m_{2}+12,12}\:(\tau).
{\bfQ}_{\mu}^{\nu}=\frac{1}{4\pi}\biggl[\frac{1}{4}{\bfF}_{\alpha\beta}{\bfF}^{\alpha\beta}\delta_{\mu}^{\nu}-{\bfF}_{\mu\sigma}{\bfF}^{\nu\sigma}\biggr]=\frac{1}{8\pi}\biggl[-{\bfF}_{\mu\sigma}{\bfF}^{\nu\sigma}-(*{\bfF})_{\mu\sigma}(*{\bfF})^{\nu\sigma}\biggr],
I_{CS}=\frac{1}{8\pi\alpha}\int_{\Omega}\epsilon_{\mu\nu\lambda}\left(A_{\mu}^{a}\partial_{\nu}A_{\lambda}^{a}+\frac{1}{3}\epsilon^{abc}A_{\mu}^{a}A_{\nu}^{b}A_{\lambda}^{c}\right)\;,
+{\frac{1}{24}}~{\frac{i}{\sqrt{2}}}~e^{-\phi/2}\Gamma^{M_{1}M_{2}M_{3}}\varepsilonF_{M_{1}M_{2}M_{3}}-{\frac{1}{192}}~{\frac{i}{\sqrt{2}}}~e^{\phi/4}\Gamma^{M_{1}M_{2}M_{3}M_{4}}\varepsilonF_{M_{1}M_{2}M_{3}M_{4}}
\left.+\left(\alpha(2-\alpha)(\frac43+\frac23\alpha+\alpha^{2})+2\alpha(2-\alpha)(k_{1}^{2}+k_{2}^{2})-(k_{1}^{2}-k_{2}^{2})^{2}\right)\left(\frac{g^{2}}{4\pi}\right)^{3}\zeta_{3}+O(g^{8})\right),
H_{p}=1+\sum_{i}{\frac{Q_{i}}{[|\vec{x}-\vec{x}_{0\,i}|^{2}+4Q|\vec{z}-\vec{z}_{0}|]^{\frac{5}{2}}}},\\\H_{p+2}={\frac{Q}{|\vec{z}-\vec{z}_{0}|}}.
H_{0}=\intd^{2}x_{\perp}dx^{-}\{{\frac{\imath}{2}}\bar{\xi}\gamma^{-}\stackrel{\leftrightarrow}{\partial}_{-}\xi+{\frac{1}{2}}(F_{12})^{2}-{\frac{1}{2}}a_{+}\partial_{-}\partial_{k}a_{k}\}
ds_{11}^{2}(r\rightarrow\infty)\simdx_{0}^{2}+d\widehat{\rho}^{2}+\widehat{\rho}^{2}d\phi^{2}+d\widetilde{r}^{2}+\widetilde{r}^{2}d\Omega_{6}-d\widetilde{\tau}^{2}
{\calS}(x,y)=-\,{\frac{ig^{2}}{2\pi^{2}\alpham^{2}}}\qquad{\frac{1}{y+i\epsilon\,\,\mathrm{sign}(x)}}\qquad{\frac{1}{\bar{y}+i\epsilon\,\,\mathrm{sign}(\bar{x})}}.
[A,\lambda,\mu,\kappa]:=\left(\begin{matrix}{1}&{0}&{0}&{\mu}\\{\lambda}&{1}&{\mu}&{\kappa}\\{0}&{0}&{1}&{-\lambda}\\{0}&{0}&{0}&{1}\\\end{matrix}\right)\left(\begin{matrix}{a}&{0}&{b}&{0}\\{0}&{1}&{0}&{0}\\{c}&{0}&{d}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right)\in\Gamma_{2}\,.
\begin{array}{ccc}{e_{1a}^{\mu}(x)}&{=}&{\beta(x)e_{2a}^{\mu}(x),}\\{a_{1\mu}(x)}&{=}&{\beta(x)a_{2\mu}(x),}\\{\phi_{1}(x)}&{=}&{\beta(x)\phi_{2}(x),}\\\end{array}
T_{0}(z^{\prime})t(z)=\frac{\frac{c_{1}}{2}\omega(z)+\frac{c_{2}}{2}\Omega(z)}{(z^{\prime}-z)^{4}}+\frac{T_{0}(z)+2t(z)}{(z^{\prime}-z)^{2}}+\frac{\partial_{z}t(z)}{z^{\prime}-z}+\cdots\:\:\:.
E=E_{\mathrm{bh}}{\frac{\kappaL^{2}}{a}}={\frac{m}{\kappaa}},\\\\\\T={\calT}{\frac{\kappaL^{2}}{a}}={\frac{\kappaL}{4\pia}}\left[(n+1){\frac{a_{H}}{L}}+(n-1){\frac{kL}{a_{H}}}\right],
\sigma=\int_{P_{\mathrm{i}}}^{P_{\mathrm{f}}}d\sigma=\int_{\lambda_{\mathrm{i}}}^{\lambda_{\mathrm{f}}}d\lambda\,\frac{dI}{d\lambda}=\int_{\lambda_{\mathrm{i}}}^{\lambda_{\mathrm{f}}}d\lambda\,L=\int_{\lambda_{\mathrm{i}}}^{\lambda_{\mathrm{f}}}d\lambda\,\sqrt{g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}}=(\mathrm{extremum}).
\frac{1}{3!}\sum_{i,j,k}\left.\frac{\partial^{3}V}{\partial\psi_{i}\partial\psi_{j}\partial\psi_{k}}\right|_{\tilde{\psi}}(\psi_{i}-\tilde{\psi}_{i})(\psi_{j}-\tilde{\psi}_{j})(\psi_{k}-\tilde{\psi}_{k}),
\hat{G}_{k}=\int\frac{d^{\,4}Q_{k}}{(2\pi)^{4}}\,e^{-iQ_{k}\cdot(\xi_{i_{k}}-\xi_{j_{k}})}\left[1+\sum_{l\,(\neqk)}c_{kl}(\xi_{i_{l}}-\xi_{j_{l}})\cdot\partial_{X}+...\right]\hat{G}(X;Q_{k})).
\psi_{\varkappa,n}^{c}=\gamma^{2}\psi_{-\varkappa,n}^{\ast}\;,\;\;\epsilon_{\varkappa,n}^{c}=-\epsilon_{-\varkappa,n}\,,\;\left(\psi_{\varkappa,n}^{c},\psi_{\varkappa^{\prime},n^{\prime}}^{c}\right)=\delta_{\varkappa,\varkappa^{\prime}}\delta_{n,n^{\prime}},\;\;\varkappa=\pm\;.
\tilde{G}(\Lambdax_{1},A_{1}S_{1}^{\dagger},B_{1}S_{2}^{\dagger};\Lambdax_{2},A_{2}S_{2}^{\dagger},B_{2}S_{1}^{\dagger})=\tilde{G}(x_{1},A_{1},B_{1};x_{2},A_{2},B_{2}).
{\calL}_{\mathrm{MCS}}=-\frac{1}{4e^{2}}F_{\mu\nu}F^{\mu\nu}+\frac{\kappa}{2}\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}+|D_{\mu}\phi|^{2}+\frac{1}{2e^{2}}(\partial_{\mu}{\calN})^{2}-V(|\phi|,{\calN})
\begin{array}{rcl}{{\hat{\calF}}}&{=}&{{\calH}^{(2)}}\\{}&{}&{}\\{{\hat{A}}}&{=}&{A-(2\pi\alpha^{\prime})\frac{e^{2\phi}(i_{k}C^{(1)})}{|k|^{2}+e^{2\phi}(i_{k}C^{(1)})^{2}}{\calG}^{(1)}\,,}\\\end{array}
<W(A,\gamma)>^{(1)}=\oint_{\gamma}ds\oint_{\gamma}dt\dot{\gamma}^{a}(s)\dot{\gamma}^{b}(t)\epsilon_{abc}{\frac{(\gamma^{c}(s)-\gamma^{c}(t))}{|\gamma(s)-\gamma(t)|^{3}}}=\mathrm{GSL}(\gamma)
S\bigl(K^{\otimes(p-2n)}\otimesK_{[0,1]}^{\otimesn}\bigr)T=\sum_{m=n}^{[p/2]}M_{n,m}^{(p)}\,S\bigl(K_{[p-2m]}\otimesK_{[0,1]}^{\otimesm}\bigr)T
\psi_{ij}(q^{+},q^{+}-P^{+};P^{-})\equiv\frac{\sigma_{ij}(q^{+},q^{+}-P^{+};P^{-})}{P^{--}(\frac{m_{i}^{2}}{q^{+}}+\frac{m_{j}^{2}}{P^{+}-q^{+}})-v_{i}(q^{+})+v_{j}(q^{+}-P^{+})}.
\begin{array}{c}{(q^{\frac{\sigma^{3}}{2}}\otimes\sigma^{+}/z+\sigma^{+}/w\otimesq^{-{\frac{\sigma^{3}}{2}}})\downarrow\otimes\downarrow=}\\{=(q^{-1/2}/z)\downarrow\otimes\uparrow+(q^{1/2}/w)\uparrow\otimes\downarrow}\\\end{array}
S=\intd^{4}x\,{\sqrt{-g}}\,{\frac{M^{2}}{\alpha_{1}^{2}}}\lbrackR+{\frac{{\vec{\alpha}}_{2}}{M^{2}}}\,{\vec{R}}^{(2)}\,+\,{\frac{{\vec{\alpha}}_{4}}{M^{4}}}\,{\vec{R}}^{(4)}\,+\,...\rbrack.
\langleV_{\mu}(x)V_{\nu}(y)\rangle_{0}=e^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\left(g_{\mu\nu}+\frac{k_{\mu}k_{\nu}}{m_{V}^{2}}\right)\frac{1}{k^{2}+m_{V}^{2}}e^{ik\cdot(x-y)},
\langleb_{\nu},t^{\prime}\mid\mida_{\mu},t\rangle=\sum_{\stackrel{\alpha}{t_{\alpha}\leqt}}\langleb_{\nu},t^{\prime}\mid\psi_{F\alpha}^{t_{\alpha}}\rangle_{t^{\prime}}\langle\psi_{F\alpha}^{t_{\alpha}}\mida_{\mu},t\rangle_{t}
Q^{m}=\frac{-2J}{\sqrt{2m(m+1)}}(\sum_{k=0}^{m-1}u_{k}^{*}u_{k}-mu_{m}^{*}u_{m}){\Big\vert}_{u_{0}=\frac{1}{\sqrt{1+\vert\xi\vert^{2}}},u_{m}=u_{0}\xi_{m}}.
\tilde{d}_{\gamma}(\tilde{t})=\tilde{a}\int_{\tilde{t}(t_{i})}^{\tilde{t}(t)}{\frac{d\tilde{t}^{\prime}}{\tilde{a}(t^{\prime})}}={\frac{M_{o}^{2}}{C}}\tilde{a}^{3}\left(|1-\delta|\right)\\,
\begin{array}{rcl}{\lrcorner\,:\:\bigwedge{}^{k}\,T^{*}\otimes\bigwedge{}^{n}\,T^{*}}&{\to}&{\bigwedge{}^{n-k}\,T^{*}}\\{\left(L_{(k)},M_{(n)}\right)}&{\mapsto}&{\displaystyle\frac{1}{n!}\,\left(\begin{array}{c}{n}\\{k}\\\end{array}\right)\,L^{a_{1}\ldotsa_{k}}M_{a_{1}\ldotsa_{n}}e^{a_{k+1}}\ldotse^{a_{n}}\,}\\\end{array}
W_{ct1}=-{\frac{1}{16\piG}}{\frac{d^{2}(d-1)}{(d-3)^{2}(d-2)}}(l^{*})^{3}\int_{\partialM_{r}}\left({\calR}_{ij}^{2}-{\frac{1}{d-1}}{\calR}^{2}\right)
Im(\bar{Y}_{\omega}^{\perp})=-i\pie^{-2\omegab}(\frac{7\omegab}{128}-\frac{1}{16}-\frac{1}{32\omegab})(\frac{r_{g}}{b})^{2}+2\pii\omega^{2}b^{2}K_{o}(2\omegab)(\frac{r_{g}}{b})^{2},
\left|\frac{8E}{M_{0}^{2}}\right|=8\,\frac{7.6\cdot10^{-3}}{a^{2}M_{0}^{2}}=8\,\frac{7.6\cdot10^{-3}}{R_{c}^{2}M_{0}^{2}}\simeq0.1\,{.}
{\calL}=-\frac{1}{4}F_{ij}^{a}F_{ij}^{a}+\frac{1}{2}F_{i0}^{a}F_{i0}^{a}+j_{i}^{a}A_{i}^{a}-j_{0}^{a}A_{0}^{a}+{\calL}_{\mathrm{gauge}}-{\bar{\psi}}\left(m-i{\gamma}\cdot\partial\right)\psi
{\cal{A}}={\cal{A}}_{\mu}dx^{\mu}={\frac{Q_{e}r}{{\rho}^{2}}}[dt-Asin^{2}\thetad\varphi]+{\frac{Q_{m}cos\theta}{{\rho}^{2}}}[Adt-(r^{2}+A^{2})d\varphi]
\frac{1}{2}\left(\sum_{i=1}^{T}n_{i}\frac{a_{i}^{\prime}}{a_{i}}\right)^{2}-\frac{1}{2}\sum_{i=1}^{T}n_{i}\left(\frac{a_{i}^{\prime}}{a_{i}}\right)^{2}-\sum_{i=1}^{T}\frac{n_{i}\Lambda_{i}}{2a_{i}^{2}}=\frac{1}{2}\phi^{2}-{\calV}
\hat{\psi}_{A}={\frac{\sqrt{3}}{2^{\frac{1}{4}}}}\sigmaa^{\frac{3}{2}}\psi_{A}~,~\hat{\bar{\psi}}_{A^{\prime}}={\frac{\sqrt{3}}{2^{\frac{1}{4}}}}\sigmaa^{\frac{3}{2}}\bar{\psi}_{A^{\prime}}~,
\psi_{\mu}(w)\rightarrow{\psi^{\prime}}_{\mu}(z)=\biggl(\frac{dz}{dw}{\biggr)}^{\frac{1}{2}}\psi_{\mu}(z(w))=e^{w/2}\psi_{\mu}(z(w))=\sqrtz\psi_{\mu}(z)\.
ds^{2}={\frac{r^{2}}{L^{2}}}\left[-(1-{\frac{r_{0}^{4}}{r^{4}}})dt^{2}+d\vec{x}^{2}\right]+{\frac{L^{2}}{r^{2}}}(1-{\frac{r_{0}^{4}}{r^{4}}})^{-1}dr^{2}+L^{2}d\Omega_{5}^{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}
(g_{ik})|_{{\calS}_{0}}=\left(\begin{array}{ccc}{g_{11}^{(0)}~~}&{~~0~~}&{0}\\{0~~}&{~~g_{11}^{(0)}~~}&{~~0}\\{0}&{0}&{1}\\\end{array}\right),~~~~~g_{11}^{(0)}=1+(\partial_{1}a)^{2}+(\partial_{2}a)^{2}+(\partial_{1}c)^{2}+(\partial_{2}c)^{2}.
2\omega=\frac{\partial{\calG}^{-1}({\calP}_{0})}{\partial{\calP}_{0}}|_{{\calP}_{0}=\omega({\calP}_{1})}=\frac{dM({\calP}_{0})}{d{\calP}_{0}}\frac{dI(M)}{dM}|_{{\calP}_{0}=\omega}~.
\mathsf{B}(p)=\mathsf{I\otimesI}+i\hbar\,\mathsf{r}(p)-\frac{1}{2}\bigg(1-\sqrt{1-\frac{\hbar^{2}}{p^{2}}}\bigg)(\mathsf{I\otimesI}+\mathsf{T}_{2}\otimes\mathsf{T}_{2}).
\int{\partialX\bar{\partial}X^{\ast}+\psi\bar{\partial}\psi^{\ast}+\bar{\psi}\partial\bar{\psi}^{\ast}+FF^{\ast}+W^{\prime}(X)F+\psi\bar{\psi}W^{\prime\prime}(X)+W^{\prime}(X^{\ast})F^{\ast}+\psi^{\ast}\bar{\psi}^{\ast}W^{\prime\prime}(X^{\ast})}
I(\gamma_{1},...,\gamma_{n})=\oint_{\gamma_{1}}dx_{1}^{\mu_{1}}\oint_{\gamma_{2}}dx_{2}^{\mu_{2}}\;.....\oint_{\gamma_{n}}dx_{n}^{\mu_{n}}\left\langleA_{\mu_{1}}(x_{1})......A_{\mu_{n}}(x_{n})\right\rangle\;,
\left(\begin{matrix}{1}&{0}&{0}&{\xi}\\{0}&{1}&{0}&{\eta}\\{\xi}&{\eta}&{1}&{(\xi^{2}+\eta^{2})/2}\\{0}&{0}&{0}&{1}\\\end{matrix}\right)\left(\begin{matrix}{A_{1}}\\{A_{2}}\\{A_{u}}\\{0}\\\end{matrix}\right)=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{\xi}&{\eta}&{1}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right)\left(\begin{matrix}{A_{1}}\\{A_{2}}\\{A_{u}}\\{0}\\\end{matrix}\right).
|{\cal{\psi}}_{0}[\bar{\chi},\chi]|^{2}=\sum_{n}~\left(\frac{2i}{\pi}\right)^{n}~\frac{1}{n!^{2}}~\int\left(\prod_{i=1}^{n}dx_{i}dy_{i}\right)\prod_{j=1}^{n}{\bar{\chi}}_{\alpha_{j}}(x_{j})~{\chi}_{\beta_{j}}(y_{j})~F\left([x_{j},y_{j}]\right)_{\alpha_{k}\beta_{k}}
{\textstyle\sum_{n}}\equiv{\frac{1}{n!}}\sum_{k_{1},k_{2}\cdots,k_{n}\ne0}{\frac{\delta_{k_{1}+k_{2}+\cdots+k_{n},0}}{\sqrt{|k_{1}k_{2}\cdotsk_{n}|}}}:a_{k_{1}}a_{k_{2}}\cdotsa_{k_{n}}:.
\begin{array}{c}{{[}\hat{\xi},\hat{\eta}{]}_{L}=-{\frac{1}{2}}\sum_{n}\mathrm{tr}_{L}\left({[}\hat{\xi},\hat{\eta}{]}I_{n}\right)I_{n}=}\\{={\frac{1}{2}}\sum_{n}\mathrm{tr}_{L}\left(\hat{\xi}\widehat{I_{n}\eta}\right)I_{n}=\sum_{n}\omega^{n}(\xi,\eta)I_{n}}\\\end{array}
\delta~(b)=~\frac{1}{2}\int_{0}^{\infty}~\frac{d^{NN}k}{(2\pi)^{(}NN)}\sum_{i}\int_{0}^{\infty}~\frac{dt}{t}~e^{-2\pi\alpha^{\prime}t(k^{2}+M_{i}^{2})}~~,
\langle\phi^{2}\rangle_{\mathrm{reg}}^{(1)}=(4\pi)^{-2}\left[-(\gamma-1)\bar{\tau}_{1}^{(1)}+\bar{\tau}_{2}^{(1)}(\bar{\tau}_{1}^{(1)})^{-1}+\bar{\tau}_{3}^{(1)}(\bar{\tau}_{1}^{(1)})^{-2}+...\right]
{\calE}_{M}^{A}=\left(\begin{array}{cc}{e^{-\psi/2}}&{e^{-\psi/2}\Psi_{m}}\\{0}&{\epsilone_{m}^{a}}\\\end{array}\right),\quad\eta_{AB}=\left(\begin{array}{cc}{1}&{0}\\{0}&{\epsilon\eta_{ab}}\\\end{array}\right),
\Gamma_{WZ}[\widetilde{\phi}_{1}\widetilde{\phi}_{2}]\=\\Gamma_{WZ}[\widetilde{\phi}_{1}]+\Gamma_{WZ}[\widetilde{\phi}_{2}]-\frac{i}{4\pi}\int_{\Sigma}Tr(\phi_{1}^{-1}d\phi_{1}\,\phi_{2}d\phi_{2}^{-1})\.
A_{1}=\left({\frac{\omega-m\Omega_{H}}{4\piT_{H}}}\right)^{2},~~B_{1}=-{\frac{\rho_{-}^{2}}{\rho_{+}^{2}}}\left({\frac{\omega-m\Omega_{H}\rho_{+}^{2}/\rho_{-}^{2}}{4\piT_{H}^{\mathrm{BTZ}}}}\right)^{2}.
S_{GF}^{(0)}=-\frac{1}{4}\intdx^{4}\sqrt{-\bar{g}}\Bigl[\bar{g}^{\mu\nu}\bar{g}^{\lambda\sigma}f_{\mu\lambda}^{(0)}f_{\nu\sigma}^{(0)}+\frac{I_{2}}{I_{1}}\bar{g}^{\mu\nu}a_{\mu}^{(0)}a_{\nu}^{(0)}\Bigr].
{\calA}_{fi({\bfA}^{2})}^{(2)}=\intd{\bfr\,}\mathrm{e}^{-i{\bfp}^{\prime}\cdot{\bfr}}\left[\frac{m}{w_{{\bfp}}}\frac{e^{2}}{2m}{\bfA}^{2}({\bfr})\right]\mathrm{e}^{i{\bfp}\cdot{\bfr}}=\frac{m}{w_{{\bfp}}}\frac{e^{2}}{2m}\,\tilde{{\calC}}({\bfq})
\left(\begin{matrix}{\displaystyle{z_{1}}}\\{\displaystyle{z_{2}}}\\\end{matrix}\right)=i\zeta\left(\begin{matrix}{\displaystyle{1}}\\{\displaystyle{\xi}}\\\end{matrix}\right),\quad\left({dz^{*}dz}\right)^{2}=|\zeta|^{2}d\zeta^{*}d\zetad\xi^{*}d\xi\quad,
\frac{1}{2}\frac{\partial}{\partial\phi}\left(\frac{H^{3/2}(\phi)}{2\pi}\frac{\partial}{\partial\phi}\left(\frac{H^{3/2}(\phi)}{2\pi}\pi_{1}(\phi)\right)\right)+\frac{\partial}{\partial\phi}\left(\frac{V^{\prime}(\phi)}{3H(\phi)}\,\pi_{1}(\phi)\right)+3H(\phi)\cdot\pi_{1}(\phi)=\lambda_{1}\,\pi_{1}(\phi)\.
\partial_{\alpha}[(\delta^{\alpha\beta}G_{AB}+\epsilon^{\alpha\beta}B_{AB}){\partial_{\beta}X^{B}}]-\frac{1}{2}\partial_{A}(\delta^{\alpha\beta}G_{BC}+\epsilon^{\alpha\beta}B_{BC})\partial_{\alpha}X^{B}\partial_{\beta}X^{C}=0.
{\frac{{\calW}^{(2)}\pi^{2\omega}e^{2i\pi\omega}}{g^{4}C_{F}C_{A}(2T)^{4-4\omega}(LT)^{2}}}={\frac{1}{(\omega-1)^{2}}}+{\frac{9-4\gamma}{2(\omega-1)}}+{\frac{39}{2}}-9\gamma+2\gamma^{2}+{\frac{\pi^{2}}{6}}+{\calO}(\omega-1)\,
{\calL}(A+h)\!={\calL}(A)+\frac{\partial{\calL}(A)}{\partialA}h+\frac{1}{2}\partial_{\mu}hW_{\mu\nu}(A)\partial_{\nu}h+hN_{\mu}(A)\partial_{\mu}h+\frac{1}{2}hM(A)h+{\calO}(h^{3}),
\frac{p_{\alpha}(p+k_{1})_{\beta}(p+k)_{\gamma}}{p^{2}(p+k_{1})^{2}(p+k)^{2}}=\frac{p_{\alpha}(p+k_{1})_{\beta}}{p^{4}(p+k_{1})^{2}}\left[(p+k)_{\gamma}-\frac{2(kp)p_{\gamma}}{p^{2}}\right]+\ldots\;.
\intd^{4}xe^{iq\cdotx}\left\langleJ_{\mathrm{\tiny{CFT}}}^{\mu}(x)J_{\mathrm{\tiny{CFT}}}^{\nu}(0)\right\rangle_{\mathrm{\tiny{CFT}}}=\left(q^{\mu}q^{\nu}-q^{2}\eta^{\mu\nu}\right)\Pi(q^{2}),
<\phi_{\mathcal{A}}^{N_{1}N_{2}N_{3}N_{4}\,\Lambda_{1}}\\ldots\\phi_{\mathcal{A}}^{N_{10}N_{8}N_{5}N_{1}\,\Lambda_{5}}>\=\\lambda\\{\mathcal{A}}_{N_{1},\ldots,N_{10},\Lambda_{1},\ldots,\Lambda_{5}}.
\deltaR_{\mu\nu}=-{\frac{1}{2}}{\bar{\nabla}}^{2}h_{\mu\nu}+{\bar{R}}_{\sigma(\nu}h_{~\mu)}^{\sigma}-{\bar{R}}_{\sigma\mu\rho\nu}h^{\sigma\rho}+{\bar{\nabla}}_{(\nu}{\bar{\nabla}}_{|\rho|}{\hat{h}}_{~\mu)}^{\rho}.
\partial_{\chi_{(A)}}\Gamma_{Y_{1}}(0)\;\;\Gamma_{\varphi_{1}\varphi_{1}}(0)+\partial_{\chi_{(A)}}\Gamma_{Y_{1}\varphi_{1}}(0)\;\;\Gamma_{\varphi_{1}}(0)=-\partial_{\xi_{(A)}}\Gamma_{\varphi_{1}}(0)
2\sum_{r=0}^{\infty}\theta^{-r}[a^{\dagger},[a,\phi_{r}]]+\sum_{r_{0}\geqr_{1}\geq\geq0}\theta^{1-\sum_{i}r_{i}}\left(V^{(r_{0}+1)}(\phi_{0}){\frac{\phi_{1}}{(r_{0}-r_{1})!}}{\frac{\phi_{2}}{(r_{1}-r_{2})!}}\cdots\right)_{S}=0,
M_{1}=-{\frac{1}{4\pi}}\intd^{2}x\left\lbrace{\frac{(a-1)}{2}}\,\partial_{\mu}\theta\;\partial^{\mu}\theta+\theta\left\lbrack(a-1)\partial_{\mu}A^{\mu}+\epsilon^{\mu\nu}\partial_{\mu}A_{\nu}\right\rbrack\right\rbrace
\widehat{F}_{\mu\nu}=F_{\mu\nu}+\frac{1}{4}\theta^{\rho\sigma}\bigg(2\left\{F_{\mu\rho},F_{\nu\sigma}\right\}-\left\{A_{\rho},D_{\sigma}F_{\mu\nu}+\partial_{\sigma}F_{\mu\nu}\right\}\bigg)+{\calO}\left(\theta^{2}\right).
\int\mathrm{d}x\,\frac{\delta^{R}\Gamma}{\delta\Phi_{c}^{a}(x)}\,\frac{\delta^{L}\Gamma}{\deltaK_{a}(x)}\;+\;\sum_{r\geq2}\frac1{r!}\,\frac{\partial^{R}\Gamma}{\partial\xi^{B}}\,f_{A_{1}\cdotsA_{r}}^{B}\xi^{A_{r}}\cdots\xi^{A_{1}}=0\;.
F_{a_{1}\dotsa_{n}}^{\Psi}\left(\vartheta_{1},\dots,\:\vartheta_{n}\right)_{0}=\,_{0}\langle0|\Psi(0)|a_{1}\left(\vartheta_{1}\right)\dotsa_{n}\left(\vartheta_{n}\right)\rangle_{0}^{\mathrm{in}}\,,
x_{\pm}=8\frac{\tilde{\omega}_{c}^{4}}{\tilde{\omega}^{2}}\left(1-\frac{1}{2}\left(\frac{\tilde{\omega}}{\tilde{\omega}_{c}}\right)^{2}\pm\sqrt{1-\left(\frac{\tilde{\omega}}{\tilde{\omega}_{c}}\right)^{2}}\right)
G_{4}^{\mu\nu\alpha\beta}(x)=\int{\frac{dp^{4}}{(2\pi)^{4}}}\left({\frac{1}{2}}(g^{\mu\alpha}g^{\nu\beta}+g^{\mu\beta}g^{\nu\alpha})-{\frac{1}{3}}g^{\mu\nu}g^{\alpha\beta}+{\calO}(p)\right)D(p^{2},m_{0}^{2},\Gamma)e^{-ip(x)},
e^{-1}{\calL}^{\mathrm{fermionic}}={\textstyle{\frac{1}{16}}}\left\{\bar{\Omega}^{iI}\stackrel{\leftrightarrow}{\rlap{D}/}\big(F_{IJ}\Omega_{i}^{J}\big)+\big(\bar{F}_{IJ}\Omega^{iJ}\big)\stackrel{\leftrightarrow}{\rlap{D}/}\Omega_{i}^{I}\right\}\,
\psi_{+2q}^{1,2+}=-\frac{i}{2}\nabla_{+}\chi_{q}^{1,2},\quad\psi_{-2q}^{1,2+}=\frac{i}{4}\Omega_{-}^{+2i}\gamma_{q\dot{p}}^{i}\chi_{\dot{p}}^{1,2},
dK^{(1)}-2K^{(1)}\frac{da}{a}=-4\pi\frac{da}{a}\left(z^{(0)}+\frac{z^{(1)}}{\mua^{2}}\right)\left(1+\frac{K^{(0)}}{2\pi}+\frac{3K^{(1)}}{K^{(0)}}\right),
\frac{(2\pi)^{D/2}}{GN\sqrt{2}}=\frac{\Lambda^{D-2}(1-(eH/\Lambda^{2})^{D/2-1})}{D/2-1}+\frac{(eH)^{(D-1)/2}}{\rho}\int_{\rho^{2}/eH}^{\infty}\frac{ds}{s^{1/2}}e^{-s}.
4\langle\varphi|\psi\rangle=||\psi+\varphi||^{2}-||\psi-\varphi||^{2}+\mathrm{i}\,||\psi+\mathrm{i}\,\varphi||^{2}-\mathrm{i}\,||\psi-\mathrm{i}\,\varphi||^{2}
\sum_{l=1}^{N}\frac{w_{pR(r)}(n+l)}{w_{R(q)R(r)}(m+l)w_{pq}(k+l)}=\bar{R^{\prime}}_{pqr}\frac{w_{pR(q)}(n-m)w_{qR(r)}(n-k)}{w_{R(p)R(r)}(m-k)}