harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L^{(+)}=\left(\begin{array}{cc}{q^{-\left(\begin{array}{cc}{1}&{0}\\{0}&{0}\\\end{array}\right)}}&{(q^{-1}-q)X^{+}}\\{0}&{q^{-\left(\begin{array}{cc}{0}&{0}\\{0}&{1}\\\end{array}\right)}}\\\end{array}\right)
E[f^{(0)}+f^{(1)}]=E[f^{(0)}]+\frac{\deltaE}{\deltaf}[f^{(0)}]\cdotf^{(1)}+\frac{1}{2}\frac{\delta^{2}E}{\deltaf^{2}}[f^{(0)}]\cdot(f^{(1)})^{2}+\cdots,
X^{\mu}=x^{\mu}\!+\!\sqrt2\theta\psi^{\mu}=x_{0}^{\mu}\!+\!Y^{\mu},\,\,\,\,D={\frac{\partial}{\partial\theta}}\!-\!\theta{\frac{\partial}{\partial\tau}},\,\,\,\,\,\intd\theta\,\theta=1\,,
g_{\mu\rho}\hat{\Gamma}_{\nu\lambda}^{\rho}\frac{u^{\lambda}}{\hat{\gamma}}=\hat{\Gamma}_{\mu,\nu\lambda}\frac{u^{\lambda}}{\hat{\gamma}}=F_{[g]\mu\nu}=\frac{\partialA_{[g]\nu}}{\partialx^{\mu}}-\frac{\partialA_{[g]\mu}}{\partialx^{\nu}}
\begin{array}{rcl}{}&{=}&{\displaystyle(1-xx^{\prime})^{n+1}\sum_{i=0}^{n}\frac{\partial}{\partialx^{i}}\left[(1-xx^{\prime})^{1-n}\frac{\partial\Phi(x)}{\partialx^{i}}\right]}\\{}&{=}&{0~.}\\\end{array}
l_{z}\rightarrow\frac{1}{2P}e^{3P\tau}\simt,\;\;\;\lambda\rightarrow\frac{2\pi}{k}e^{P\tau}\approx\frac{2\pi}{k}a,\;\;\;g_{ij}^{(0)}\rightarrowe^{2P\tau}\;\mathrm{diag}\;(1,1,1).
((K_{\nu})^{L})_{1,r}=I(L,r)=\frac{1}{2\pii}\cdot\oint\frac{dx}{x}\frac{G_{r}^{(2)}}{x^{\frac{L}{2}}}=\frac{1}{2\pii}\cdot\oint\frac{dx}{x}\frac{G_{r-1}^{(3)}}{x^{\frac{L}{2}}}=J(L,r-1).
\tilde{I}={\frac{\omega_{n-1}\beta}{16\piGl^{2}}}\left[l^{2}r_{+}^{n-2}-r_{+}^{n}+{\frac{(2n-3)q^{2}l^{2}}{r_{+}^{n-2}}}-{\frac{2(n-1)}{n}}l^{2}r_{e}^{n-2}-{\frac{2(n-1)^{2}}{n}}{\frac{q^{2}l^{2}}{r_{e}^{n-2}}}\right].
S=\frac{1}{2}\int_{-\pi}^{\pi}\int_{t_{1}}^{t_{2}}dxdt[(\partial_{t}\phi_{i}(x,t))^{2}-(\partial_{x}\phi_{i}(x,t))^{2}-m^{2}(\phi_{i}(x,t))^{2}+F_{ij}\partial_{t}\phi_{i}(x,t)\partial_{x}\phi_{j}(x,t)].
\begin{array}{c}{P^{s}(b,c)\,\vec{\hat{U}}_{\upsilon}^{(s)\,a}(b,c)=\hat{P}^{(s)\,a}(b,c)\,\vec{\hat{U}}_{\upsilon}^{(s)\,a}(b,c)=\vec{\hat{U}}_{\upsilon}^{(s)\,a}(b,c)\,;}\\{\hat{P}^{(s)\,f}(b,c)\,\vec{\hat{U}}_{\upsilon}^{(s)\,a}(b,c)=\vec{0},\qquad\quada\neqf.}\\\end{array}
\begin{array}{lcll}{\mathrm{floor}}&{\mathrm{SU(2/1)}}&{\mathrm{field}}&{\mathrm{helicity}}\\{\mid\mathrm{gnd}>}&{(0~+\frac{5}{2})}&{A_{\mu}}&{-1}\\{\mid\mathrm{2nd}>}&{(1~+\frac{5}{2})}&{2\Psi_{\mu}}&{-\frac{3}{2}}\\{\mid\mathrm{3rd}>}&{(0~+\frac{7}{2})}&{e_{\mu}^{a}}&{-2.}\\\end{array}
\hat{V}_{\mu\nu\rho}^{(1\,x,y)\,AB}=-2\cdot2\,\mathrm{Tr}\,T^{A}T^{B}\,(Y_{R}^{(d)}-Y_{L}^{(d)})\,V_{\mu\nu\rho}^{(1)}=4\,\mathrm{Tr}\,T^{A}T^{B}\,V_{\mu\nu\rho}^{(1)}\,.
\partial_{t}(g_{\mu\nu}\dot{q}^{\nu})-\frac{1}{2}\partial_{\lambda\nu}\dot{q}^{\lambda}\dot{q}^{\nu}=g_{\mu\nu}\ddot{q}^{\nu}+\bar{\Gamma}_{\lambda\nu\mu}\dot{q}^{\lambda}\dot{q}^{\nu}=0.
\begin{array}{l}{\left\{Q_{\gamma}^{k},Q_{\rho}^{l}\right\}=}\\{-\left[\begin{array}{lr}{u^{lk}(C_{\rho\beta}\delta_{\gamma}^{\alpha}+\muC_{\gamma\beta}\delta_{\rho}^{\alpha})}&{}\\{}&{C_{\gamma\rho}(u^{ki}\delta_{j}^{l}+\lambdau^{li}\delta_{j}^{k})}\\\end{array}\right].}\\\end{array}
\langle\psi^{(0)}|\hat{x}^{3}V^{\prime}-\frac{3}{\mu}\hat{x}^{2}\hat{p}^{2}+\frac{3i}{\mu}\hat{x}\hat{p}|\psi^{(0)}\rangle=0,\qquad\langle\psi^{(0)}|1+2i\hat{x}\hat{p}|\psi^{(0)}\rangle=0.\nonumber
Q_{f}=f_{\mu_{1}\dots\mu_{r}}\Pi^{\mu_{1}}\psi^{\mu_{2}}\dots\psi^{\mu_{r}}+\frac{i}{r+1}(-1)^{r+1}f_{[\mu_{1}\dots\mu_{r};\mu_{r+1}]}\psi^{\mu_{1}}\dots\psi^{\mu_{r+1}}.\nonumber
H=\intd\sigma^{1}\left[\Lambda^{\mu}\partial_{1}A_{0}^{\mu}+\frac1\phi\Lambda^{2}+\lambda_{\phi}\pi_{\phi}+\lambda_{\Lambda}^{\mu}p_{\Lambda}^{\mu}+\lambda_{0}^{\mu}p_{0}^{\mu}+\lambda_{1}^{\mu}(p_{1}^{\mu}-\Lambda^{\mu})\right],
<q_{2}|\hat{U}(t_{2},t_{1})|q_{1}>=\sum_{n=0}^{\infty}\,\psi_{n}^{*}(q_{2})\,e^{-\frac{i}{\hbar}(t_{2}-t_{1})\hbar\omega(n+1/2)}\,\psi_{n}(q_{1})\,
G_{\mu\nu}=\eta_{\mu\nu}+\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{i}-2\bar{\lambda}(\Gamma_{(\mu}+\Gamma_{i}\partial_{(\mu}\phi^{i})\partial_{\nu)}\lambda+\bar{\lambda}\Gamma^{m}\partial_{\mu}\lambda\bar{\lambda}\Gamma_{m}\partial_{\nu}\lambda,
\Pi_{\alpha}^{N*}\Pi_{\alpha}^{N}=-\partial_{\alpha}^{*}\partial_{\alpha}+\frac{3Z_{\alpha}^{*}Z_{\beta}}{4Z_{\gamma}^{*}Z_{\gamma}}\partial_{\alpha}^{*}\partial_{\beta}+\frac{3}{4Z_{\gamma}^{*}Z_{\gamma}}Z_{\alpha}\partial_{\alpha}-\frac{1}{16Z_{\gamma}^{*}Z_{\gamma}}(2c-1)(2c+3)
\Gamma_{a,s}^{(o)}(l_{1s})=\{z\rightarrowz,\theta\rightarrow(-1)^{2l_{1s}}\theta\},\quad\Gamma_{b,s}(l_{2s})=\Gamma_{a,s}^{(o)}(1/2-l_{2,s})\Gamma_{s}^{(o)}.
A_{2-\mu}~=~\frac{2\pi^{2\mu}}{(\mu-1)^{2}\Gamma^{2}(\mu)}~~~,~~~A_{3-\mu}~=~-\,\frac{2\pi^{2\mu}(\mu^{2}-4\mu+2)}{(\mu-1)^{2}(\mu-2)\Gamma^{2}(\mu)}
{\tilde{\hat{\calY}}}_{J_{1}}^{R_{1}}{\tilde{\hat{\calY}}}_{J_{2}}^{R_{2}}=\sum_{R,J}\left\langle\begin{array}{l}{R}\\{J}\\\end{array}\right.\left|\begin{array}{ll}{R_{1}}&{R_{2}}\\{J_{1}}&{J_{2}}\\\end{array}\right\rangle\left\{\begin{array}{lll}{R}&{R_{1}}&{R_{2}}\\{\frac{P}{2}}&{\frac{P}{2}}&{\frac{P}{2}}\\\end{array}\right\}[D(\frac{P}{2})]^{-1}{\tilde{\hat{\calY}}}_{J}^{R},
{\calC}=\int_{0}^{\infty}{\frac{dt}{t}}~{\frac{1}{8\pi^{2}\alpha^{\prime}t}}\sum_{\alpha,\beta}{\frac{(q_{\alpha}-q_{\beta})B}{2\pi}}\left[U_{\alpha\beta}+T_{\alpha\beta}\right]e^{-{\frac{t}{2\pi\alpha^{\prime}}}~X_{\alpha\beta}^{2}}~,
\left[\,\bar{\psi}_{n}^{(K)}\,,\,\Phi_{\chi_{k}\otimes\chi_{\bar{k}}}^{\beta}\,\right]\=\,Phi_{\chi_{k}\otimes\bar{k}\,\left(\delta_{1}\,\left(\bar{\psi}^{(K)}\right)_{n}\right)\,\chi_{\bar{k}}}^{\beta}\,
\frac{A^{n+1}}{V^{n}}\le\frac{4\,\pi}{(n-2)}\,\,\mathrm{Vol(M)}\,\frac{2^{\frac{1}{n-2}}(\frac{1}{n})^{\frac{n}{2}}(n-2)^{\frac{n+1}{2}}}{\left(1-(\frac{2}{n})^{\frac{n}{n-2}}\right)^{n}}.
\tilde{C}_{[\alpha_{1}],[\alpha_{2}],(\alpha_{3})}^{\epsilon}=i\epsilon\left[\left(\beta_{1}^{2}+\beta_{2}^{2}\right)C_{[\alpha_{1}],[\alpha_{2}],(\alpha_{3})}^{\epsilon}-2\beta_{1}\beta_{2}C_{[\alpha_{1}],[\alpha_{2}],(\alpha_{3})}^{-\epsilon}\right],
\left[{\frac{dn_{d}(\omega,\mu)}{d\omega}}\right]_{\mathrm{div}}=r_{d}{\frac{1}{(4\pi)^{2}\kappa}}\int_{\Sigma}\left[2b+a\left({\frac{\omega^{2}}{\kappa^{2}}}P-{\frac{R}{6}}+{\frac{R_{ii}}{4}}\right)\right]~~~.
{\frac{dn^{(\lambda)}}{d\omega}}\simeq{\frac{2\omega^{D-2}}{(4\pi)^{(D-1)/2}}}\sum_{n=0}^{\infty}\left[{\frac{a_{n}(\lambda)}{\Gamma\left({\frac{D-1}{2}}-n\right)}}\omega^{-2n}+{\frac{b_{n}(\lambda)}{\Gamma\left({\frac{D-2}{2}}-n\right)}}\omega^{-(2n+1)}\right]~~~.
\partial^{\mu}\sigma_{\mu}^{\alpha\dot{\alpha}}\phi_{\alpha\alpha_{2}\ldots\alpha_{2j_{1}}}=0\;,\qquad\partial^{\mu}\sigma_{\mu}^{\alpha\dot{\alpha}}\bar{\phi}_{\dot{\alpha}\dot{\alpha}_{2}\ldots\dot{\alpha}_{2j_{2}}}=0
\begin{array}{ll}{(O4^{-}-----\widetilde{O4^{+}})~~\&~~(\widetilde{O4^{+}}-----O4^{-})}&{~~~~~N+\widetilde{N}=0}\\{(O4^{+}-----\widetilde{O4^{-}})~~\&~~(\widetilde{O4^{-}}-----O4^{+})}&{~~~~~N+\widetilde{N}=1}\\\end{array}
\hat{g}_{11}=\frac{1}{\tilde{g}_{22}}\Big(\tilde{g}_{11}\tilde{g}_{22}-\tilde{g}_{12}^{2}\Big)-\frac{\tilde{B}_{12}^{2}}{\tilde{g}_{22}}=\frac{1}{\tilde{g}_{22}}=H^{-1/2}D
J_{S}=-i\int\frac{d^{2}p}{2w_{p}}\sum_{\lambda,\lambda^{\prime}=1}^{2}\epsilon_{ij}\,\epsilon_{\lambda^{\prime}}^{i}(p)\epsilon_{\lambda}^{j}(p)\,\bigl[a_{\lambda^{\prime}}^{\dagger}(p)a_{\lambda}(p)+b_{\lambda^{\prime}}^{\dagger}(p)b_{\lambda}(p)\bigr].
S=\frac{T}{2\pi}\intdx\sqrt{H^{-1}(U)+(\partial_{x}U)^{2}+{U^{2}}/{9}(\partial_{x}\psi)^{2}+{U^{2}}/{9}(\partial_{x}\psi)(\partial_{x}\phi_{1})+{U^{2}}/{6}(\partial_{x}\phi_{1})^{2}},
\left\langle\phi^{2}\right\rangle\stackrel{\mathrm{approx.}}{\longrightarrow}N\left(\frac{c_{d}}{1+r}-4\frac{uc_{d}^{2}}{(1+r)^{3}}+\cdots\right)=N\frac{c_{d}}{1+r}C_{2}\left(\frac{uc_{d}}{(1+r)^{2}}\right),
{\bfN}({\bfp},{\bfs}):=ip_{0}{\nabla}_{\bfp}-\frac{{\bfs}\times{\bfp}}{p_{0}+m},\quad{\bfJ}({\bfp},{\bfs}):=-i{\bfp}\times{\bf\nabla}_{\bfp}+{\bfs}:={\bfL}({\bfp})+{\bfs},
\begin{array}{c}{{{\calH}}_{0}=g^{A\bar{B}}\partial_{A}K\partial_{\bar{B}}K-ig_{A\bar{B}}\theta^{A}{\bar{\theta}}^{\bar{B}}+}\\{+i\theta^{C}K_{C;A}{\tilde{g}}^{A\bar{B}}K_{\bar{B};\bar{D}}{\bar{\theta}}^{\bar{D}}}\\\end{array}
\frac{e^{2i\theta_{\alpha}}z_{l}^{2}-q^{2n_{0}+1}}{q^{2n_{0}+1}e^{2i\theta_{\alpha}}z_{l}^{2}+1}=-\prod_{m=1}^{2j}\frac{qz_{l}-z_{m}}{z_{l}-qz_{m}},\qquadl=1,\cdots,2j.
E_{TE}(a)=\frac{\hbarc}{a}\,\left\{-\frac{{n_{0}}^{3}\,a^{4}\Omega^{4}}{24\,\pi\,c^{4}\,\left(1+s\right)}-\frac{n_{0}a^{2}\Omega^{2}}{4\pic^{2}(1+s)}+\mathcal{O}(s+1)^{0}\right\}.
\frac{\partial\Phi_{n}^{*}(z^{-1})}{\partialt_{-1}}=-\frac{S_{n}^{*}}{S_{n-1}^{*}}\frac{h_{n}}{h_{n-1}}(\Phi_{n}^{*}(z^{-1})-z^{-1}\Phi_{n-1}^{*}),
\frac{1}{4\pi^{2}g^{2}N^{3}}\sum_{\sigma}\left(\partialA^{\mu}(\sigma)\times\partialA^{\nu}(\sigma)\right)^{2}\rightarrow\frac{1}{4\pi^{2}g^{2}N}\int_{T^{2}}d^{2}\sigma\left(\partialA^{\mu}(\sigma)\times\partialA^{\nu}(\sigma)\right)^{2},
Q_{\calM}=\left(\begin{array}{ccccccc}{1}&{0}&{-2}&{1}&{0}&{0}&{\zeta_{1}+\zeta_{2}+\zeta_{3}}\\{0}&{1}&{0}&{-2}&{1}&{0}&{\zeta_{1}+\zeta_{4}+\zeta_{5}}\\{0}&{0}&{1}&{0}&{-2}&{1}&{\zeta_{2}+\zeta_{4}+\zeta_{6}}\\\end{array}\right).
S_{\mathrm{eff}}\left[A,\psi,\bar{\psi}\right]=\intd^{4}x{\calL}_{\mathrm{eff}}=S\left[A,\psi,\bar{\psi}\right]+S_{\mathrm{gf}}\left[A_{\perp}^{3}\right]+\frac{1}{2}M^{2}\sum_{a=1,2}\intd^{4}xA_{\mu}^{a}\left(x\right)A^{a,\mu}\left(x\right)\.
\partial_{\mu}\partial^{\mu}\,\hat{\xi}\,^{5}(x)=\frac{(1+\gamma)^{-1}}{6\,M_{*}^{3}}\,\left[\frac{S_{\mu}^{\mu}(x)}{2}+\ell\,T_{55}(0)-\frac{1+\gamma}{1-\gamma}\,\ell\,\int_{0}^{y_{m}}\,dy\,\partial^{\mu}T_{\mu5}\,\right]\,.
V=\frac{e^{2}}{r}\left[1-\frac{1}{2}\left(\frac{\rho\Omega}{r}\right)^{2}+\frac{3}{8}\left(\frac{\rho\Omega}{r}\right)^{4}-\frac{5}{16}\left(\frac{\rho\Omega}{r}\right)^{6}+...\right].
0=\alpha^{\prime}g^{\mu\nu}h_{\mu\nu}(p_{0})=\left(\alpha^{\prime}G^{\mu\nu}-\frac{\left(\thetaG\theta\right)^{\mu\nu}}{(2\pi)^{2}\alpha^{\prime}}\right)h_{\mu\nu}(p_{0})\approx-\frac{\left(\thetaG\theta\right)^{\mu\nu}}{(2\pi)^{2}\alpha^{\prime}}h_{\mu\nu}(p_{0})~.
<\dot{x}(\tau_{1})\cdot\dot{x}(\tau_{2})>=[<\dot{x}(\tau_{1})\cdot\dot{x}(\tau_{2})>_{o}+\dot{x}^{cl}(\tau_{1})\cdot\dot{x}^{cl}(\tau_{2})<1>_{o}]e^{-S^{cl}},
\sigma_{abs}^{\phi_{b}}=\frac{9\pi^{3}r_{1}^{6}r_{5}^{6}}{64(r_{1}^{2}+r_{5}^{2}-\sqrt{r_{1}^{4}+r_{5}^{4}-r_{1}^{2}r_{5}^{2}})^{2}}\omega(\omega^{2}+16\pi^{2}T_{L}^{2})(\omega^{2}+16\pi^{2}T_{R}^{2})\frac{e^{\frac{\omega}{T_{H}}}-1}{(e^{\frac{\omega}{T_{L}}}-1)(e^{\frac{\omega}{T_{R}}}-1)},
{\calF}=\frac{1}{2}(\frac{2}{3}\lbrack\lambda^{\dagger},\lambda\rbrack+\frac{1}{\kappa^{2}}\lbrack\bar{\chi}_{A},\chi^{A}\rbrack+\frac{1}{\kappa^{2}}\lbrack\phi^{A},\bar{\phi}_{A}\rbrack),
{}~~~~~~~~~~~~~\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{w(x_{3},x_{13},x_{1}|d,h+\sigma)w(x_{4},x_{24},x_{2}|a,g+\sigma)}{w(x_{4},x_{14},x_{1}|e,c+\sigma)w(x_{3}/\omega,x_{23},x_{2}|f,b+\sigma)}\Bigg\}_{0}
\begin{aligned}{\big(W^{(8)},Q,Q\big)}&{{}\rightarrow\tilde{a}=\frac{1}{\tilde{m}-1}\,,}\\{\big(W^{(5)},Q,Q\big)}&{{}\rightarrow\tilde{b}=\frac{-2}{\tilde{n}-1}\,.}\\\end{aligned}
\partial_{I}X^{M}=\partial_{I}x^{M}+{\alpha^{\prime}}^{\frac{3}{2}}\nabla_{I}\phi^{M}-\frac{1}{3}{\alpha^{\prime}}^{3}\partial_{I}x^{S}R^{M}{}_{PSQ}\phi^{P}\phi^{Q}+{\calO}({\alpha^{\prime}}^{\frac{9}{2}})\,.
{\bf\bar{F}}_{\mu}=\left(\begin{array}{cc}{(\bar{f}_{\bar{w}}^{\mu})^{-1/2}}&{0}\\{-{\frac{d}{d\bar{w}}}(\bar{f}_{\bar{w}}^{\mu})^{-1/2}}&{(\bar{f}_{\bar{w}}^{\mu})^{1/2}}\\\end{array}\right).
C\,T^{(\kappa\lambda)\sigma}=-e\left(2m+n-4d\right)D^{\lambda\sigma\kappa}+\frac{e}{2}\left(2a+c\right)\epsilon_{\mu\nu}{}^{\kappa\lambda}D^{\nu\sigma\mu}-\frac{e}{2}E\,\epsilon_{\mu}{}^{\sigma\kappa\lambda}D^{\mu}-\frac{e}{2}F\,G^{\sigma\kappa\lambda}\,.
{\frac{2}{3}}\delta^{[A}{}_{[B}T^{C]}{}_{D]GH}+{\frac{2}{3}}\delta^{[A}{}_{[G}T^{C]}{}_{H]BD}=(u^{AB}{}_{\Omega\Sigma}+v^{AB\Omega\Sigma})e_{\Omega\Sigma}^{\alpha}{\calB}_{CDEF}
\begin{array}{rl}{S=\int\;dt\;\mathrm{Tr}}&{\left[\sum_{j=1}^{3}{\frac{1}{2}}(D_{0}y^{j})^{2}+{\frac{1}{4}}\sum_{j,k=1}^{3}[y^{j},y^{k}]^{2}-\right.}\\{}&{\left.-{\frac{1}{2}}\left({\frac{\mu}{3}}\right)^{2}\sum_{j=1}^{3}(y^{j})^{2}-i{\frac{\mu}{3}}\sum_{j,k,l=1}^{3}\epsilon_{jkl}\mathrm{Tr}\;y^{j}y^{k}y^{l}\right]}\\\end{array}
\overline{{\Gamma}}(m\rho)=-{\frac{1}{75}}{\frac{1}{m^{2}\rho^{2}}}-{\frac{17}{735}}{\frac{1}{m^{4}\rho^{4}}}+{\frac{232}{2835}}{\frac{1}{m^{6}\rho^{6}}}-{\frac{7916}{148225}}{\frac{1}{m^{8}\rho^{8}}}+\cdots
\int_{|k|<\frac{\Lambda}{s}}\phi_{s}(4)\phi_{s}(3)\phi_{s}(2)\phi_{s}(1)\biggl<\int_{\frac{\Lambda}{s}<|k|<\Lambda}\phi_{f}(8)\phi_{f}(7)\phi_{f}(6)\phi_{f}(5){\calP}(k_{8},k_{7},\cdots,k_{1})\biggr>_{0f},
q_{b}(a,b,r)=-\frac{1}{2\pir^{n}S_{D}}\sum_{l=0}^{\infty}D_{l}\int_{m}^{\infty}\frac{z^{3}dz}{\sqrt{z^{2}-m^{2}}}\Omega_{b\nu}(az,bz)F_{\nu}^{(q)}\left[G_{\nu}^{(b)}(bz,zr)\right],\quadq=\varepsilon,\,p,\,p_{\perp}.
Z[T;x_{1},\ldots,x_{K};R_{1},\ldots,R_{K}]~=~\intdA_{\mu}~e^{-S[A]}~\prod_{i=1}^{K}\mathrm{Tr}{\calP}e^{i\int_{0}^{1/T}d\tauA_{0}^{R_{i}}(\tau,x_{i})}\;,
\Psi_{\sigma}^{(n)}(w^{\prime},x^{4}w_{2n}|\zeta^{\prime},x^{2}\zeta_{2n})J(w^{\prime}x^{4}w_{2n}|\zeta^{\prime},x^{2}\zeta_{2n})=\sigma\Psi_{\sigma}^{(n)}(w_{2n},w^{\prime}|\zeta)J(w_{2n},w^{\prime}|\zeta)\prod_{j=1}^{2n-1}\left(\frac{\zeta_{j}}{\zeta_{2n}}\right)^{\frac{1}{r}},
\overline{{\xi}}\partial_{\tau}\xi-(\xi+\overline{{\xi}})\overline{{\eta}}D_{\mu}T\psi^{\mu}\eta=-\alpha^{\prime}\frac{1}{\partial_{\tau}}(\overline{{\eta}}D_{\mu}T\psi^{\mu}\eta)(\overline{{\eta}}D_{\nu}T\psi^{\nu}\eta)=\alpha^{\prime}(\overline{{\eta}}D_{\mu}T\psi^{\mu}\eta)\frac{1}{\partial_{\tau}}(\overline{{\eta}}D_{\nu}T\psi^{\nu}\eta)\,
W^{[2,3]}(\Phi^{1},\Phi^{2})={\frac{(\Phi^{1})^{6}}{6}}-(\Phi^{1})^{4}\Phi^{2}+{\frac{2}{3}}(\Phi^{1})^{2}(\Phi^{2})^{2}-(\Phi^{2})^{3}-\mu\Phi^{1}~.
M(s,t,u)=\sum_{i=0}^{\infty}{\alpha}_{i}^{(s)}(s){\nu}_{s}^{i}+\sum_{R_{t}}^{}\frac{\rho^{(st)}(s)}{(\sigma-2M_{R}^{2})-s+{\nu}_{s}}+\sum_{R_{u}}^{}\frac{\rho^{(su)}(s)}{(\sigma-2M_{R}^{2})-s-{\nu}_{s}}\;
\left[T_{\underline{{p}}}^{(N)},\,T_{\underline{{q}}}^{(N)}\right]\=\\frac{N}{2\piM}\;\mathrm{sin}\left(\frac{2\piM}{N}\left(p_{1}q_{2}-p_{2}q_{1}\right)\right)\,T_{\underline{{p}}+\underline{{q}}\,(\mathrm{mod}\,N)}^{(N)}\.
\frac{d\phi}{dr}=\frac{L}{r^{2}}\,\frac{\;\;1-\frac{1}{(1/\beta^{\prime})+2m(E-V)}\left(mr\frac{\partialV}{\partialr}\right)\;\;}{\sqrt{2m(E-V)-\frac{L^{2}}{r^{2}}}}\;.
q_{T}^{r_{1}r_{2}\tau\tau}(T_{s},\vec{\eta})=\sum_{i=1}^{N}[\eta_{i}^{r_{1}}\eta_{i}^{r_{2}}\sqrt{m_{i}^{2}+{\vec{\kappa}}_{i}^{2}}](T_{s})+M[\eta^{r_{1}}\eta^{r_{2}}-\eta^{r_{1}}R_{+}^{r_{2}}-\eta^{r_{2}}R_{+}^{r_{1}}](T_{s}),
\langleA_{\mu}^{a}(x)A_{\nu}^{b}(y)\rangle_{\mathrm{\scriptsize~Feyn}}=\delta^{ab}\,D_{\mu\nu}^{\mathrm{\scriptsize~Feyn}}(x,y)=-{\frac{\Gamma(D/2-1)}{4\pi^{D/2}}}\,g_{\mu\nu}\,\delta^{ab}\,[(x-y)^{2}]^{1-D/2}\,.
\overline{{\langle\mathrm{ground};\mathrm{ex|}}}\hat{j}_{R}(x)\overline{{|\mathrm{ground};\mathrm{ex\rangle}}}=\tilde{f}(x;x_{0})-\tilde{f}(x;y_{0})-\frac{1}{1}{L}\hbar[\frac{eb\mathrm{L}}{2\pi}],
R_{\mu\nu}-{\frac{1}{2}}g_{\mu\nu}~^{(4)}R=-\ell_{P}^{2}\frac{\delta\stackrel{\sim}{\calL}_{F}^{\prime}}{\deltag^{\mu\nu}}-\ell_{P}^{2}e^{-1}\frac{\delta{\calL}_{g}}{\deltag^{\mu\nu}}.
|0_{M}\rangle=\left(e^{-S^{(+)}/2}e^{\sum_{\sigma}\sum_{p}b_{p}^{(\sigma)\dagger}\bar{b}_{\tilde{p}}^{(-\sigma)\dagger}}e^{S^{(+)}/2}\right)\left[\,e^{-S^{(+)}/2}|0(\epsilon)\rangle\right]\,,
M_{p}^{2}=M_{5}^{3}\int_{-\infty}^{\infty}e^{2A_{0}(y)}\,dy=\frac{4M_{5}^{3}\sqrt{\pi}}{a(k^{2}-1)}\frac{\Gamma\left((k^{2}+3)/4)\right)}{\Gamma\left((k^{2}+1)/4\right)}\;,
{\calA}_{\psi}^{\mathrm{in}}={\frac{{\calF}(0)}{{\calF}(\infty)}}={\frac{4\sqrt{A_{1}}(x_{+}^{2}-x_{-}^{2})}{\sqrt{1-\mu}}}{\frac{|C_{1}|^{2}}{|\tilde{\alpha}-i\tilde{\beta}|^{2}}}.
Z=\int[DP_{ab}^{k}][D\omega_{k}^{ab}]det(G^{ijmn}(\tilde{g}))\delta(\phi^{ij})\delta(\chi^{mn})detM_{\alpha\beta}\delta(H_{\perp})\delta(H_{i})\delta(J_{ab})\exp\frac{i}{\hbar}S
|f^{\prime}|\leq|f|\left({\frac{N}{|w_{1}|}}+{\frac{1}{|w_{1}-\beta|}}+{\frac{1/2}{|1-w_{1}|}}\right)\leq{\frac{\sqrt{2}}{1+\beta}}\left(N+{\frac{1}{1+\beta}}+{\frac{1}{4}}\right),
F^{k_{s},\alpha_{s}}=dA^{k_{s},\alpha_{s}}+\frac{1}{2}C_{i_{p},\beta_{p}\;j_{q},\gamma_{q}}^{k_{s},\alpha_{s}}\;A^{i_{p},\beta_{p}}\wedgeA^{j_{q},\gamma_{q}}\,:=\,DA^{k_{s},\alpha_{s}}\quad,
\vec{\calJ}(\tau)\approx\vec{\calS}(\tau)=\intd^{3}\sigma_{o}\,\vec{\Re}(\tau,\vec{\sigma}_{o})\times\vec{\wp}(\tau,\vec{\sigma}_{o})=\sum_{\underline{{n}}}\,\vec{r}_{\underline{{n}}}(\tau)\times\vec{p}_{\underline{{n}}}(\tau).
\left(\begin{array}{cc}{\frac{\tilde{B}_{\lambda}}{(k^{2})^{3/2-\beta_{\lambda}}}}&{i\vartheta\frac{\tilde{B}_{\lambda\sigma}}{(k^{2})^{\beta_{\lambda\sigma}}}}\\{i\vartheta\frac{\tilde{B}_{\lambda\sigma}}{(k^{2})^{\beta_{\lambda\sigma}}}}&{\frac{\tilde{B}_{\sigma}}{(k^{2})^{3/2-\beta_{\sigma}}}}\\\end{array}\right)
\tilde{H}=H_{c}-\frac{1}{m^{2}}\int\Phi^{1a}\Phi^{2a}-\frac{2g}{m^{2}}f^{abc}f^{cge}\intA_{0}^{b}A_{0}^{g}\Phi^{1a}\Phi^{1e}+O(\Phi\Phi\Phi)
\hat{\calR}=Q^{2\left(\hat{\zeta}\otimes\hat{H}+\hat{H}\otimes\hat{\zeta}\right)}\lambda^{2\left(\hat{\zeta}\otimes\hat{H}-\hat{H}\otimes\hat{\zeta}\right)}\left(1\!\mathrm{l}\otimes1\!\mathrm{l}+\sigmaQ^{\hat{\zeta}}\lambda^{\hat{\zeta}}\hat{X}_{+}\otimesQ^{-\hat{\zeta}}\lambda^{\hat{\zeta}}\hat{X}_{-}\right)
\Box^{(3)}\Omega=(1+l_{0}^{2}K_{a}^{b}K_{b}^{a}+3N+\frac{3}{2}N^{2})\Omega+\frac{3}{2}N\Omega^{2}+\frac{3}{2}\Omega^{2}+\frac{1}{2}\Omega^{3},
\left[E^{2}-P_{z}^{2}-\frac{m^{2}}{4}-4\omega\left(\frac{E}{m}+(N+\left|m\right|+\frac{1}{2})\right)\right]+i\omegaP_{z}b_{-}+i\omega\left(N+\left|m\right|+\frac{1}{2}\right)P_{z}b_{+}=0
\gamma_{2a}\equiv-\left(\left(\mathrm{D}_{i}\right)_{a}^{\;\;b}\pi_{b}^{i}-\mathrm{ig}\bar{\psi}_{\alpha}^{\;B}\left(\gamma^{0}\right)_{\;\;\beta}^{\alpha}T_{a\;\;B}^{A}\psi_{\;A}^{\beta}\right)\approx0,
\tau_{1}(t_{f})-\tau_{1}(t_{i})\equiv\int_{t_{i}}^{t_{f}}\frac{ds}{a_{1}^{2}(s)}=\frac{1}{\alpha(\alphat_{i}+\beta)}-\frac{1}{\alpha(\alphat_{f}+\beta)}=\frac{T}{a(t_{f})a(t_{i})},
\langleT_{00}\rangle=\langleT_{11}\rangle={\frac{1}{2}}\int_{0}^{\infty}d\omega\Biggl\{(\partial_{0}\phi_{\omega})(\partial_{0}\phi_{\omega}^{\ast})+(\partial_{1}\phi_{\omega})(\partial_{1}\phi_{\omega}^{\ast})\Biggr\},
\langle{\hat{\sigma}}_{1}{\hat{\sigma}}_{n_{1}}...{\hat{\sigma}}_{n_{s}}\rangle_{g}=\sum_{i=1}^{s}C({\hat{\sigma}}_{1},{\hat{\sigma}}_{n_{i}})\langle{\hat{\sigma}}_{n_{1}}...{\hat{\sigma}}_{n_{s}}\rangle_{g}=(2g-2+n)\langle{\hat{\sigma}}_{n_{1}}...{\hat{\sigma}}_{n_{s}}\rangle_{g}
(\partial^{\lambda\dot{\mu}})=\left[\begin{array}{cc}{\partial^{1\dot{1}}}&{\partial^{1\dot{2}}}\\{\partial^{2\dot{1}}}&{\partial^{2\dot{2}}}\\\end{array}\right]=\left[\begin{array}{cc}{\partial_{0}+\partial_{3}}&{\partial_{1}+i\partial_{2}}\\{\partial_{1}-i\partial_{2}}&{\partial_{0}-\partial_{3}}\\\end{array}\right],
\sum_{\mu,\nu\in{\calR}}\left((\rho\cdot\hat{H})^{2}\hat{s}_{\rho}\right)_{\mu\nu}=\sum_{\mu,\nu}(\rho\cdot\mu)^{2}(\hat{s}_{\rho})_{\mu\nu}=\sum_{\mu}(\rho\cdot\mu)^{2}=C_{\calR}|\rho|^{2},
\chi_{s}=-\frac{2l}{l_{1}\cdotsl_{5}}\left[l_{1}^{3}+\sum_{i=2}^{5}l_{i}^{2}{\hat{l}}_{i}+\sum_{i<j<k\atopi,j,k\in\{2,..,5\}}2l_{i}l_{j}l_{k}\right]
N_{\pm}=\pm\left(I_{H}^{\pm}-2I_{S}^{\pm}-I_{C}^{\pm}-I_{W}^{\pm}\right)\mpS^{\pm}2l_{W}^{\pm}\pm{\calL}_{W}^{\pm}-2\left(S^{\pm}\right)^{2}\mp2S^{\pm}\left(\displaystyle\frac{\delta}{2}-k_{W}^{\pm}+k_{\pm}\right).
S_{2}=\frac{1}{2}\alpha_{\mu}^{a}\left[{-(D^{2})^{ab}\delta_{\mu\nu}+(D_{\nu}D_{\mu})^{ab}-\frac{1}{\xi}(D_{\mu}D_{\nu})^{ab}-igF_{\mu\nu}^{ab}}\right]\alpha_{\nu}^{b}+\bar{c}^{a}\left[{(D^{2})^{ab}}\right]c^{b},
\chi_{n}\rightarrowU_{r}(\tilde{g})\;\chi_{n}\;,\quad\tilde{A}_{m}^{\,(r)}\rightarrowU_{r}(\tilde{g})\left(\tilde{A}_{m}^{\,(r)}+\partial_{m}\right)U_{r}^{-1}(\tilde{g})\;,\quad\tilde{g}(\vec{x})\inG\;,
{\frac{1}{32\omega^{2}}}\left(-{\frac{32\omegaz_{0}^{2}}{l^{2}}}+9+3\sqrt{-64\omega{\frac{z_{0}^{2}}{l^{2}}}+9-64{\frac{z_{0}^{2}}{l^{2}}}\omega^{2}}\right)<z_{0}^{2}\varepsilon^{2}<{\frac{1}{8\omega+3}}\left(-3-{\frac{5z_{0}^{2}}{l^{2}}}\right)\,.
G_{l+\nu}^{(n)}(r^{\prime\prime},r^{\prime};E)=g^{n}\,\left[G_{l+\nu}^{(0)}(a,a;E)\right]^{n-1}\,G_{l+\nu}^{(0)}(r^{\prime\prime},a;E)\,G_{l+\nu}^{(0)}(a,r^{\prime};E)\;,
0=\frac{\partialV}{\partial\mathrm{Re}F}=-2\left(\frac{N_{c}}{\gamma}-\frac{N_{f}}{\tilde{\gamma}}\right)\,\mathrm{ln}\,\left(\frac{N_{c}|F|}{\gamma\Lambda_{QCD}^{4}}\right)-2\,\mathrm{ln}\,\left(\frac{|\mathrm{det}F_{T}|}{\Lambda_{QCD}^{3N_{f}}}\right)+2\frac{N_{f}}{\tilde{\gamma}}\,\mathrm{ln}\left(\frac{N_{f}\gamma}{N_{c}\tilde{\gamma}}\right)\.
M_{F4}^{form-med.}=-\frac{1}{12}\left(\frac{\kappa}{24}\right)^{2}\left(C^{F~\alpha\beta\rho}(k_{1},k_{2})\frac{1}{s}\:C_{\alpha\beta\rho}^{F}(k_{3},k_{4})+\mathrm{perm.}\right),
H_{i}\toH_{Bi}+\delta\dot{b}_{i},\quadH\toH_{B}+\frac13\sum_{i}\delta\dot{b}_{i},\quad\sum_{i}H_{i}^{2}\to\sum_{i}H_{Bi}^{2}+2\sum_{i}H_{Bi}\,\delta\dot{b}_{i},\quadV\toV_{B}+V_{B}\sum_{i}\deltab_{i}.
\mathbf{G}^{\mu\nu}=\Phi\star\overline{{Z}}^{\mu\nu}-Z^{\mu\nu}\star\overline{{\Phi}},\quad\widehat{\mathbf{{Q}}}_{\mu\nu}\star\Phi=0,\qquad\widehat{\mathbf{{Q}}}_{\mu\nu}\starZ^{\mu\nu}=0,
\sum_{j=1}^{p-1}\frac{\partial}{\partialx_{j}}\frac{\partial}{\partialx_{2}}\frac{\partial^{2}}{\partialx_{3}^{2}}\dots\frac{\partial^{p-2}}{\partialx_{p-1}^{p-2}}f\biggr|_{x_{1}=x_{2}=\dots=x_{p-1}}=0.
\begin{array}{ccc}{j+(k3)\rightarrow(jk3),\,\,\,\,}&{(jk3)+j\rightarrow(k3),\,\,\,\,}&{(k3)+(jk3)\rightarrowj,}\\{k+(j3)\rightarrow(jk3),\,\,\,\,}&{(jk3)+k\rightarrow(j3),\,\,\,\}&{(j3)+(jk3)\rightarrowk,}\\\end{array}
2T_{ij}^{\alpha\beta}=\left(\begin{array}{cccccc}{0}&{2n_{0}}&{-n_{0}}&{0}&{n_{0}}&{-2n_{0}}\\{-2n_{1}}&{0}&{2n_{1}}&{-n_{1}}&{0}&{n_{1}}\\{n_{2}}&{-2n_{2}}&{0}&{2n_{2}}&{-n_{2}}&{0}\\{0}&{n_{3}}&{-2n_{3}}&{0}&{2n_{3}}&{-n_{3}}\\{-n_{4}}&{0}&{n_{4}}&{-2n_{4}}&{0}&{2n_{4}}\\{2n_{5}}&{-n_{5}}&{0}&{n_{5}}&{-2n_{5}}&{0}\\\end{array}\right)
\begin{matrix}{\psi_{x}(\mathrm{e}^{2\pii}z)=\mathrm{e}^{2\pii{\frac{k}{n}}}\psi_{x}(z),}&{\psi_{y}(\mathrm{e}^{2\pii}z)=\mathrm{e}^{2\pii{\frac{n-k}{n}}}\psi_{y}(z),}\\{\tilde{\psi}_{x}(\mathrm{e}^{-2\pii}\bar{z})=\mathrm{e}^{2\pii{\frac{n-k}{n}}}\tilde{\psi}_{x}(\bar{z}),}&{\tilde{\psi}_{y}(\mathrm{e}^{-2\pii}\bar{z})=\mathrm{e}^{2\pii{\frac{k}{n}}}\tilde{\psi}_{y}(\bar{z}),}\\\end{matrix}
\frac{1}{\sqrt{g}}\left(-{\frac{\deltaS}{\deltag_{\mu\nu}}}{\frac{\deltaS}{\deltag^{\mu\nu}}}-\frac{1}{2}{\frac{\deltaS}{\delta\phi^{I}}}{\frac{\deltaS}{\delta\phi^{J}}}G^{IJ}+\frac{1}{d-1}\left(g^{\mu\nu}{\frac{\deltaS}{\deltag^{\mu\nu}}}\right)^{2}\right)~=~\sqrt{g}(R^{d}+2\Lambda+V(\phi)+\frac{1}{2}G_{IJ}\nabla_{\mu}\phi^{I}\nabla^{\mu}\phi^{J})
(\sigma_{0})_{\alpha\beta}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\quad(\sigma_{1})_{\alpha\beta}=\left(\begin{array}{cc}{1}&{0}\\{0}&{1}\\\end{array}\right)\quad(\sigma_{2})_{\alpha\beta}=\left(\begin{array}{cc}{-i}&{0}\\{0}&{i}\\\end{array}\right)
f^{(2)}=\left(\begin{array}{cccc}{0}&{-\delta_{ij}}&{-a_{i}}&{-\pi_{i}}\\{\delta_{ij}}&{0}&{0}&{-a_{i}}\\{a_{i}}&{0}&{0}&{0}\\{\pi_{i}}&{a_{i}}&{0}&{0}\\\end{array}\right).
Z=\left(\begin{array}{ccc}{\frac{1}{\sqrt2}\,v^{-\frac12}}&{-\frac{1}{\sqrt2}\,v^{-\frac12}\,b^{T}\eta}&{-\frac{1}{2\sqrt2}\,v^{-\frac12}(b^{T}\eta\,b)-\frac{1}{\sqrt2}\,v^{\frac12}}\\{0}&{u}&{u\,b}\\\end{array}\right),
F_{k}(\lambda)\equivL_{k}(\lambda)L_{k-1}(\lambda)\equiv\left(\begin{array}{cc}{F_{11}^{(k,k-1)}(\lambda)}&{F_{12}^{(k,k-1)}(\lambda)}\\{F_{21}^{(k,k-1)}(\lambda)}&{F_{22}^{(k,k-1)}(\lambda)}\\\end{array}\right)\,,
L=-1/4F_{\mu\nu}F^{\mu\nu}+1/2\partial_{\mu}\phi\partial^{\mu}\phi+1/2M^{2}\phi^{2}-\frac{\lambda}{24}\phi^{4}+\sum_{j=1}^{N}i\bar{\psi_{j}}\gamma^{\mu}(\partial_{\mu}-eA_{\mu})\psi_{j}-g_{j}\bar{\psi_{j}}\psi_{j}\phi
\frac{1}{2J_{R}}\sum_{l,l^{\prime}=0}^{2J_{R}}\frac{1}{2}[\delta_{l,l^{\prime}-1}+\delta_{l,l^{\prime}+1}]\beta^{nl-n^{\prime}l^{\prime}}=\delta_{n,n^{\prime}}\frac{\beta^{n}+\beta^{-n}}{2}
\renewcommand{\arraystretch}{1}X_{L}(a,b)=\!\!\!\sum_{\begin{array}{c}{\scriptstyle\sigma_{1},\ldots,\sigma_{L-1}=0}\\{\scriptstyle|\sigma_{j+1}\!-\!\sigma_{j}|=1}\\\end{array}}^{r-2}\!\!q^{\;\;\displaystyle\sum_{k=1}^{L}k|\sigma_{k+1}-\sigma_{k-1}|/4}\quad\sigma_{0}=a-1,\;\sigma_{L}=b-1,\;\sigma_{L+1}=b,
S_{\mathrm{Sugra+YM}}^{(D)}={\textstyle\frac{1}{2}}\intd^{D}x\sqrt{-\hat{g}}\e^{-2\hat{\phi}}\left[-\hat{R}+4(\partial\hat{\phi})^{2}-{\textstyle\frac{3}{4}}\hat{H}^{2}+{\textstyle\frac{1}{4g^{2}}}\hat{F}_{\hat{\mu}\hat{\nu}}^{I}\hat{F}_{I}^{\hat{\mu}\hat{\nu}}\right]\,,