harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[\begin{array}{ccl}{\{\pi_{\underline{{\underline{{\alpha}}}}},A_{\underline{{\underline{{\beta}}}}}\}}&{}&{\;\;\;\;\;\;\;0}\\{0}&{}&{\;\;\{\chi_{\underline{{\underline{{\alpha}}}}},\chi_{\underline{{\underline{{\beta}}}}}\}}\\\end{array}\right]\;
\sum_{i=1}^{r}\,\Lambda(H_{i})\Lambda^{\prime}(H^{i})=(\lambda,\lambda^{\prime})\,,~~~~\forall\Lambda=(\lambda,\kappa,\tau)\,,\,\Lambda^{\prime}=(\lambda^{\prime},\kappa^{\prime},\tau^{\prime})\,\in\,{\calH}^{*}
A_{1}^{\left(0\right)}={\frac{{\Phi}}{{2\pir^{2}}}}x^{2},\;A_{2}^{\left(0\right)}=-{\frac{{\Phi}}{{2\pir^{2}}}}x^{1},\;A_{0}^{\left(0\right)}=A_{3}^{\left(0\right)}=0,\;r^{2}=\left(x^{1}\right)^{2}+(x^{2})^{2}.
\mathcal{G}_{1}^{\mu\nu}:=\frac{1}{2}(\tilde{\Phi}^{\ast}F^{\mu\nu}\Phi+\Phi^{\ast}F^{\mu\nu}\tilde{\Phi}),\quad\quad\mathcal{G}_{2}^{\mu\nu}:=\frac{i}{2}(\tilde{\Phi}^{\ast}F^{\mu\nu}\Phi-\Phi^{\ast}F^{\mu\nu}\tilde{\Phi})
\mathcal{B}_{\Sigma}\;=\intd^{D}x\,\left(\frac{\delta\Sigma}{\delta\varphi^{i}}\frac\delta{\delta\varphi^{i*}}\;+\;\frac{\delta\Sigma}{\delta\varphi^{i*}}\frac\delta{\delta\varphi^{i}}\right)\;\;,
G_{b}^{(n)}(\zeta)=c_{n}^{\prime}\prod_{1\leqslantj<k\leqslant2n}\zeta_{j}g(z_{j}/z_{k})g(z_{j}z_{k})\prod_{j=1}^{2n}\varphi_{1}(z_{j};r)g_{b}(z_{j})\times\overline{{G}}_{b}^{(n)}(\zeta).
Z_{LBF}=\sum_{\{J_{i}\}}\prod_{\mathrm{\scriptsize~site}}\Lambda^{-1}\prod_{\mathrm{\scriptsize~link}}\Lambda\prod_{\mathrm{\scriptsize~triangle}}(2J+1)\prod_{\mathrm{\scriptsize~tetrahedron}}(2J+1)\prod_{\mathrm{\scriptsize~4-simplex}}\{\mathrm{15-~j~}\}.
\widetilde{WZ}^{(p+1)}=(2\pi\alpha^{\prime})\left\{dc+{\textstyle\frac{1}{2\pi\alpha^{\prime}}}\left[C+{\textstyle\frac{m}{2}}\omega(e)e^{2\pi\alpha^{\prime}{\calF}(e)}\right]e^{-2\pi\alpha^{\prime}{\calF}(b)}\right\}\,.
G({\bfx}^{\prime\prime},{\bfx}^{\prime};T\vert{\hat{\bfA}})=\int[d^{N}{\bfx}]exp\biggl(i\sum_{a=1}^{N}{\frac{m}{2}}{\dot{\bfx}}_{a}^{2}\biggr)\,{\bfT}exp\biggl(ie\sum_{a=1}^{N}\int_{0}^{T}dt\,{\dot{\bfx}}_{a}\cdot{\hat{\bfA}}({\bfx}_{a})\biggr)\,,
\left(\omega^{g_{1},g_{2},g_{3}}\right)\,\left(\omega^{g_{2},g_{1},g_{3}}\right)^{-1}\,\left(\omega^{g_{3},g_{2},g_{1}}\right)^{-1}\,\left(\omega^{g_{3},g_{1},g_{2}}\right)\,\left(\omega^{g_{2},g_{3},g_{1}}\right)\,\left(\omega^{g_{1},g_{3},g_{2}}\right)^{-1}
({\gamma}^{\mu}i{\partial}_{\mu}-m_{2}){\psi}_{2}(x,t)-\frac{1}{2}e_{2}b(t){\gamma}^{1}{\psi}_{2}(x,t)+e_{2}\int_{0}^{\mathrm{L}}dyD(x,y|\mathrm{L})J^{0}(y,t){\gamma}^{0}{\psi}_{2}(x,t)=0.
\begin{array}{ll}{\begin{array}{ll}{\varphi(r)=\eta}&{r\rightarrow\infty}\\{\varphi(r)=0}&{r=0}\\\end{array}}&{\begin{array}{ll}{P(r)=0}&{r\rightarrow\infty}\\{P(r)=1}&{r=0.}\\\end{array}}\\\end{array}
\tilde{R}_{2}(\nu,\tau)=\int_{1}^{\infty}d\alpha(\sum_{\ell=1}^{7}\Phi_{\ell}^{(3)}(\alpha)\left\{\frac{1}{(\alpha+\tau)^{\ell}}-\frac{1}{\alpha^{\ell}}\right\})[\alpha^{\frac{\nu}{2}}+\alpha^{\frac{-\nu}{2}}].
Z\approxZ_{\mathrm{down}}{\frac{r}{r_{+}}}\left({\frac{r-r_{-}}{r_{+}-r_{-}}}\right)^{2}\left({\frac{r-r_{+}}{r-r_{-}}}\right)^{-i\nu}\left[1+{\frac{4(1+i\nu)}{1-2i\nu}}\left({\frac{r-r_{+}}{r-r_{-}}}\right)+{\frac{(1+i\nu)(1+2i\nu)}{(1-i\nu)(1-2i\nu)}}\left({\frac{r-r_{+}}{r-r_{-}}}\right)^{2}\right].
\delta^{2}S_{\mathrm{eff}}/\deltaA_{\mu}(x)\deltaA_{\nu}(y)=<j^{\mu}(x)j^{\nu}(y)>=-4e^{2}NK\left(\frac{\partial^{\mu}\partial^{\nu}}{\partial^{2}}-g^{\mu\nu}\right)\delta^{3}(x-y)\,\,\,.
S_{E}=\intd^{3}{\bfx}\left\{\frac{1}{2}\sum_{n}(\partial_{\mu}\phi_{n}^{a}\partial_{\mu}\phi_{-n}^{a}-\mu^{2}\phi_{n}^{a}\phi_{-n}^{a})+\frac{\lambdaT}{4}\sum_{n_{1},n_{2},n_{3},n_{4}}\delta(n_{1}+n_{2}+n_{3}+n_{4})\phi_{n_{1}}^{a}\phi_{n_{2}}^{a}\phi_{n_{3}}^{b}\phi_{n_{4}}^{b}\right\},
\hat{E}^{\mu\nu}=\bar{\varepsilon}^{\mu\nu\rho}\delta_{\rho}\varepsilon+{\frac{1}{2\sqrt{2}}}e^{-\varphi/2}\bar{\varepsilon}(V_{R}L({\calF}-i\gamma^{5}\tilde{\calF})^{\mu\nu})^{a}\Gamma^{a}\varepsilon+\cdots,
Z^{(1)}(k;l_{1},l_{2})=(-1)^{2l_{1s}+2l_{2s}}16^{2l_{1s}}\prod_{p=1}^{\infty}\frac{[1+(-1)^{2l_{2}}k^{p}k^{(2l_{1}-1)/2}]^{8}}{[1-k^{p}]^{8}}\,.
\bar{h}_{\quad|\alpha}^{\alpha\mu}=\frac{2\Omega^{2}}{1-\Omega^{2}}\delta\phi^{|\mu}-\left[\frac{4l^{2}}{3}\frac{\Omega^{2}}{1-\Omega^{2}}\left(\frac{d/l}{2(1-\Omega^{2})}-\frac{1}{4}\right)-\frac{l^{2}}{4}\right]\Box\delta\phi^{|\mu}+\frac{l^{2}}{6}\left[\frac{d/l}{2(1-\Omega^{2})}-\frac{1}{4}\right]\Box\bar{h}^{|\mu}\,
F^{0}=\left(\begin{array}{ccccc}{0}&{-E_{0}}&{0}&{0}&{\dot{T}}\\{E_{0}}&{0}&{0}&{-B_{2}}&{0}\\{0}&{0}&{0}&{B_{1}}&{0}\\{0}&{B_{2}}&{-B_{1}}&{0}&{0}\\{-\dot{T}}&{0}&{0}&{0}&{0}\\\end{array}\right).
T^{00}={\frac{2cosh^{2}(\gamma)}{r^{2}g^{2}}}\left[C^{2}csc^{2}(Cr)\Big(1-Cr\;cot(Cr)\Big)^{2}+{\frac{\Big(C^{2}r^{2}csc^{2}(Cr)-1\Big)^{2}}{2r^{2}}}\right]
\int\frac{d^{4}k}{\left(2\pi\right)^{4}}\frac{\partial}{\partialk_{\nu}}\left(\frac{k_{\mu}}{\left(k^{2}-m^{2}\right)^{\alpha}}\right)=\int\frac{d^{4}k}{\left(2\pi\right)^{4}}\frac{g_{\mu\nu}}{\left(k^{2}-m^{2}\right)^{\alpha}}-2\alpha\int\frac{d^{4}k}{\left(2\pi\right)^{4}}\frac{k_{\mu}k_{\nu}}{\left(k^{2}-m^{2}\right)^{\alpha+1}}.
(\kappa_{\tau}\frac{\partial}{\partial\kappa_{\tau}}+\beta_{\tau}\frac{\partial}{\partialu_{\tau}}-\frac{1}{2}N\gamma_{\phi(\tau)}(u_{\tau})+\gamma_{\phi^{2}(\tau)}(u_{\tau})t\frac{\partial}{\partialt})\Gamma_{R(\tau)}^{(N)}=0.
k=(v_{1}\cdot\deltav_{2})-(v_{2}\cdot\deltav_{1})-\frac{1}{|\tau|^{2}}\Big(\,\epsilon_{mnkl}v_{2}^{m}v_{1}^{n}\tau^{k}\delta\tau^{l}-(v_{2}\cdot\tau)(v_{1}\cdot\delta\tau)-\,(v_{1}\cdot\tau)(v_{2}\cdot\delta\tau)\Big)\,.
\Phi_{\rho}^{L}={\frac{1}{3}}\gamma^{\nu}[D_{\nu}\Psi_{\rho}^{R}-D_{\rho}\Psi_{\nu}^{R}]+{\frac{i}{12}}\gamma_{5}\gamma^{\lambda}\epsilon_{\lambda\rho}^{\sigma\tau}[D_{\sigma}\Psi_{\tau}^{R}-D_{\tau}\Psi_{\sigma}^{R}]\,.
{\frac{1}{2}}(h_{1}-h_{2})^{\prime}=\left({\frac{1}{2}}{\frac{d^{\prime}}{d}}+{\frac{3}{r}}+{\frac{1}{6}}{\frac{f^{\prime}}{f}}\right)h_{2}-\left({\frac{1}{2}}{\frac{d^{\prime}}{d}}-{\frac{1}{3}}{\frac{f^{\prime}}{f}}\right)h_{1},
\begin{array}{rclcrcl}{\mathrm{dim}\,\mathbb{D}_{1}}&{=}&{1}&{;}&{\mathrm{dim}\,\mathbb{D}_{2}}&{=}&{3}\\{\mathrm{dim}\,\mathbb{D}_{3}}&{=}&{6}&{;}&{\mathrm{dim}\,\mathbb{D}_{4}}&{=}&{10}\\{\mathrm{dim}\,\mathbb{D}_{5}}&{=}&{16}&{;}&{\mathrm{dim}\,\mathbb{D}_{6}}&{=}&{27}\\\end{array}
\left.\frac{1}{(x^{2})^{D/2+k+\epsilon}}\right|_{\epsilon\to0}\simeq\frac{1}{\epsilon}\frac{\pi^{D/2}4^{-k}}{\Gamma\left(\frac{D}{2}+k\right)\Gamma\left(k+1\right)}\Box^{k}\delta(x).
\langle0|q|0\rangle=\left(\begin{array}{cc}{q_{0}I_{n}}&{0}\\{0}&{0}\\\end{array}\right)\qquad\langle0|{\bar{q}}|0\rangle=\left(\begin{array}{cc}{\bar{q}_{0}I_{n}}&{0}\\{0}&{0}\\\end{array}\right)
{\frac{\partial\,\overline{{\mathrm{\Large~v~}}}_{D}(x)}{\partial(m/\Lambda)}}={\frac{2m}{\Lambda}}\left[{\frac{(4\pi)^{D/2}}{{\mathrm{tr}}{\bf1}\,g^{2}\Lambda^{D-2}}}+f_{D}(x)\right]=0\,
T_{\Omega}z=z^{\prime}=\frac{\psi_{2}^{\prime}}{\psi_{1}^{\prime}}=\frac{\omega_{2}\psi_{1}+\omega_{1}^{*}\psi_{2}}{\omega_{1}\psi_{1}-\omega_{2}^{*}\psi_{2}}=\frac{\omega_{2}+\omega_{1}^{*}z}{\omega_{1}-\omega_{2}^{*}z}
\langleE_{\perp}(r)E_{\perp}(r^{\prime})\rangle_{eff}=\frac{1}{\pir^{2}a^{2}}\int_{0}^{x_{0}}xdx\sum_{l=1}^{\infty}\frac{2l+1}{4\pi}\{B_{F}(x)e_{l}^{2}(x\tilde{r})-B_{G}(x)[e_{l}\prime(x\tilde{r})]^{2}\},
\int\!\frac{\mathrm{d}q_{0}}{\sqrt{2E-V^{2}(x_{qc})}}=\int\!\frac{\big[A-U(x_{qc}(\tau))-V(x_{qc}(\tau))V^{\prime}(x_{qc}(\tau))q_{0}(\tau)\big]}{2E-V^{2}(x_{qc}(\tau))}\mathrm{d}\tau.
det^{\prime}\Bigl(\astd_{\scriptscriptstyle1}+(\astd_{\scriptscriptstyle1})^{\scriptscriptstyle2}\Bigr)=e^{i\pi\psi}det^{\prime}(\astd_{\scriptscriptstyle1})det^{\prime}(\mathrm{1\mkern-5muI}+\astd_{\scriptscriptstyle1})
\left\{j_{M}^{0}\left(t,\sigma\right),j_{N}^{0}\left(t,\sigma^{\prime}\right)\right\}_{PB}=-\delta_{M}\phi^{i}\frac{\partial\delta_{N}\phi^{j}}{\partial\phi^{i}}\Lambda_{j}\,\delta\left(\sigma-\sigma^{\prime}\right)+\delta_{N}\phi^{j}\frac{\partial\delta_{M}\phi^{i}}{\partial\phi^{j}}\Lambda_{i}\,\delta\left(\sigma-\sigma^{\prime}\right)\quad.
Z_{\mathrmWKB}[\gamma,\beta]=e^{-\beta\,\ell[\gamma]\pm\frac{\pi}{4}i(\eta_{H}(0)-\zeta_{H}(0))}e^{\frac{1}{2}\zeta_{H}^{\prime}(0)}\,(\beta\,\ell[\gamma])^{-\frac{1}{2}\zeta_{H}(0)},
I[\phi_{s},g_{\mu\nu}]=\frac12\int_{\calM}dV\left[\left(\nabla\phi_{s}\right)^{2}+\left(m_{s}^{2}+\xi_{s}R\right)\phi_{s}^{2}\right]+\xi_{s}\int_{\partial{\calM}}dv~K\phi_{s}^{2}~~~,
(2\pi\sqrt{\lambda_{\bfg}})^{\zeta(A_{f})}\,e^{-\frac{i\pi}{4}\eta(A_{f},g)}\,k^{-\zeta(A_{f})/2}\,\tau(M,A_{f},\langle\cdot\,,\cdot\rangle_{H(A_{f})})^{1/2}
\int_{G}dg\,\overline{{\langlel_{3},m_{3}|T_{g}|l_{3},m_{3}^{\prime}\rangle}}\langlel_{1},m_{1}|T_{g}|l_{1},m_{1}^{\prime}\rangle\langlel_{2},m_{2}|T_{g}|l_{2},m_{2}^{\prime}\rangle=\hat{C}_{m_{3}^{\prime}m_{1}^{\prime}m_{2}^{\prime}}^{l_{3}l_{1}l_{2}}\overline{{\hat{C}_{m_{3}m_{1}m_{2}}^{l_{3}l_{1}l_{2}}}}.
\Psi_{\sigma}^{(n)}(w|\zeta)=\vartheta_{\sigma}^{(n)}(w|\zeta)\prod_{a\inA}\prod_{j=1}^{2n}x^{-\frac{(v_{a}-u_{j})^{2}}{2r}}\psi\Bigl(\frac{w_{a}}{z_{j}}\Bigr)\prod_{1\leqslantj<k\leqslant2n}x^{-\frac{(u_{j}-u_{k})^{2}}{4r}},
I_{11}=\frac{1}{2\kappa_{11}{}^{2}}\intd^{11}x\sqrt{-G}\left[R_{G}-\frac{1}{2\cdot4!}K_{\scriptscriptstyleMNPQ}^{2}\right]-\frac{1}{12\kappa_{11}{}^{2}}\intC_{3}\wedgeK_{4}\wedgeK_{4}\,
S_{\mu}=R_{\mu}+\frac{1}{3}\left(1-\frac{3N_{c}^{\prime}}{N_{f}}-\gamma_{q}\right)\left(K_{\mu}^{(q)}-2K_{\mu}^{(M)}\right)-\frac{1}{3}\left(2\gamma_{q}+\gamma_{M}\right)K_{\mu}^{(M)}
T^{5}=\left(\begin{matrix}{\gamma^{\mu\nu}\omega_{1\mu}^{5b}e_{\nu}^{b}}&{g^{\mu\nu}\gamma^{5}V_{\nu}(e_{\mu}^{b}l^{5b}-\partial_{\mu}(\frac{1}{\sqrt{V^{2}}}))}\\{-g^{\mu\nu}\gamma^{5}V_{\nu}(e_{\mu}^{b}\tilde{l}^{5b}-\partial_{\mu}(\frac{1}{\sqrt{V^{2}}}))}&{\gamma^{\mu\nu}\omega_{2\mu}^{5b}e_{\nu}^{b}}\\\end{matrix}\right)
\begin{array}{rl}{\left(p_{\mu}\gamma_{\mu}+p_{7}\gamma_{5}+ip_{8}\right)\psi_{1}+\gamma_{5}\left(p_{5}-ip_{6}\right)\psi_{2}}&{=0\,,}\\{\left(p_{\mu}\gamma_{\mu}-p_{7}\gamma_{5}+ip_{8}\right)\psi_{2}+\gamma_{5}\left(p_{5}+ip_{6}\right)\psi_{1}}&{=0\,.}\\\end{array}
\begin{array}{l}{\lbrack\tilde{q}^{+},\tilde{p}^{-}]=i\,,}\\{\left[\tilde{\alpha}_{n}^{+}\,,\tilde{\alpha}_{m}^{-}\right]=n\,\delta_{n+m,0}\,\,\,,}\\{\left[\tilde{s}_{n}\,,\tilde{s}_{m}\right]=\left(\frac{k}{2}-1\right)\,n\,\delta_{n+m,0}\,\,.\,}\\\end{array}
U(\varphi,\sigma)=\frac{\lambda_{\phi}}{4}(|\varphi|^{2}-\eta^{2})^{2}+f|\phi|^{2}|\sigma|^{2}+\frac{\lambda_{\sigma}}{4}|\sigma|^{4}-\frac{m_{\sigma}^{2}}{2}|\sigma|^{2},
E=+E_{n}\,,\;n\geq1\;\;\psi_{1}^{(+)}=\sqrt{\frac{E_{n}+m}{2E_{n}}}{e^{ipx_{2}-\frac{1}{2}{\zeta}^{2}}}\left(\begin{array}{c}{N_{n}H_{n}(\zeta)}\\{-\frac{\sqrt{E_{n}^{2}-m^{2}}}{E_{n}+m}N_{n-1}H_{n-1}(\zeta)}\\{0}\\{0}\\\end{array}\right)
p_{\mu_{1},\cdots,\mu_{n}}\equiv\frac{\partial(x_{1},\cdots,x_{D})}{\partial(\xi_{1},\cdots,\xi_{n},\phi_{1},\cdots,\phi_{D-n})}\frac{\partial(\phi_{1},\cdots,\phi_{D-n})}{\partial(x_{\nu_{1}},\cdots,x_{\nu_{D-n}})}\times\varepsilon_{\mu_{1},\cdots,\mu_{n},\nu_{1},\cdots,\nu_{D-n}}.
D_{\psi_{i}}=\sum_{N=0}^{+\infty}{\left(\frac{\Gamma_{i}e^{-{\frac{\hbar\omega_{i}}{kT}}}}{1-\left(1-\Gamma_{i}\right)e^{-{\frac{\hbar\omega_{i}}{kT}}}}\right)}^{N}\left|\rho_{i}^{N}\right>\left<\rho_{i}^{N}\right|\;.
V_{3}=\frac{\hat{e}}{e_{m}}\left(\frac{N_{2}^{(1)}\bar{N}_{2}^{(1)}N_{2}^{(2)}\bar{N}_{2}^{(2)}}{N_{5}^{(1)}\bar{N}_{5}^{(1)}N_{5}^{(2)}\bar{N}_{5}^{(2)}}\right)^{\frac14}
\begin{array}{l}{\displaystyle\sum_{i,j}\int\!\mathrm{d}^{4}x_{1}\dots\mathrm{d}^{4}x_{i}\,\mathrm{d}^{4}y_{1}\dots\mathrm{d}^{4}y_{j}\enspacef_{i}^{\ast}(x_{1},\dots,x_{i})\,f_{j}(y_{1},\dots,y_{j})}\\{\qquad\times{\calS}(\thetax_{1},\dots,\thetax_{i},y_{1},\dots,y_{j})\ge0\,,}\\\end{array}
\frac{1}{6}\epsilon_{\nu\sigma\tau\mu}\partial^{\mu}\varphi=\partial_{\nu}B_{\sigma\tau}+\partial_{\sigma}B_{\tau\nu}+\partial_{\tau}B_{\nu\sigma}+\frac{1}{6}\epsilon_{\nu\sigma\tau\rho}\int(dy)f^{\rho}(x-y)J(y)\.
g_{\mu\nu}^{E}dx^{\mu}dx^{\nu}=-{\frac{1}{\sqrt{H_{1}H_{2}H_{3}H_{4}}}}fdt^{2}+\sqrt{H_{1}H_{2}H_{3}H_{4}}\left[f^{-1}dr^{2}+r^{2}d\Omega_{2}^{2}\right],
D_{\mu}Q_{i}=\partial_{\mu}Q_{i}-\frac{i}{2}\left(\begin{array}{ll}{A_{\mu}^{(i)}}&{\sqrt{2}W_{\mu}^{(i)-}}\\{\sqrt{2}W_{\mu}^{(i)+}}&{-A_{\mu}^{(i)}}\\\end{array}\right)Q_{i}+\frac{i}{2}Q_{i}\left(\begin{array}{ll}{A_{\mu}^{(i+1)}}&{\sqrt{2}W_{\mu}^{(i+1)-}}\\{\sqrt{2}W_{\mu}^{(i+1)+}}&{-A_{\mu}^{(i+1)}}\\\end{array}\right),
\frac{\partial\omega_{j}}{\partiala_{i}}=\frac{\partial\omega_{i}}{\partiala_{j}},\\\frac{\partial\omega_{i}}{\partialT_{n}}=\frac{\partial\Omega_{n}}{\partiala_{i}},\\,frac{\partial\Omega_{m}}{\partialT_{n}}=\frac{\partial\Omega_{n}}{\partialT_{m}}.
|B_{0},\theta,\eta\rangle=\sum_{N=0}^{4}{\frac{1}{(2N)!}}(ip^{+})^{N-2}A_{a_{1}a_{2}\dotsa_{2N}}^{2N}\theta^{a_{1}}\theta^{a_{2}}\dots\theta^{a_{2N}}|0,\eta\rangle.
{\calG}_{\omega}(r,r^{\prime},\theta)=\sum_{n=0}^{\infty}r^{1-D/2}\left[e_{n}H_{\nu}^{(1)}(kr)+f_{n}H_{\nu}^{(2)}(kr)\right]C_{n}^{(-1+{D/2})}(z)\quad(r^{\prime}>r>a).
N_{Q}(-i\Omega_{m},\stackrel{\rightharpoonup}{p})=-\frac{d_{Q}(R)}{2}\int\frac{d^{3}l}{(2\pi)^{3}}T\sum_{n=-\infty}^{\infty}\mathrm{tr}S_{Q}(-i\omega_{n}+\mu_{Q},\stackrel{\rightharpoonup}{l})S_{Q}(-i\omega_{n}-i\Omega_{m}+\mu_{Q},\stackrel{\rightharpoonup}{l}+\stackrel{\rightharpoonup}{p}).
\left[x\sum_{k=0}^{n_{c}}(-1)^{k}s_{k}(\phi^{2})x^{n_{c}-k}+2\Lambdam_{1}\cdotsm_{n_{f}}\right]^{2}-4\Lambda^{2}\prod_{i=1}^{n_{f}}(x+m_{i}^{2})=-4\Lambda^{2}x\prod_{a=1}^{n_{c}}(x-\alpha_{a})^{2}.
\bar{N}_{nm}^{rs}=\frac{1}{nm}\oint_{z^{r}}\frac{dz}{2\pii}\oint_{z^{s}}\frac{dw}{2\pii}\frac{1}{(z-w)^{2}}(-1)^{n(r-1)+m(s-1)}(f(z))^{(-1)^{r}n}(f(w))^{(-1)^{s}m}
\psi_{\pm}^{(+)}={\frac{C}{r^{(d-2)/2}}}\left(\begin{matrix}{iJ_{n+d/2}(kr)Z_{+}^{(n)}(\Omega)}\\{\varepsilonJ_{n+(d-2)/2}(kr)Z_{+}^{(n)}(\Omega)}\\\end{matrix}\right),
C_{IJK}\,h^{I}h^{J}h^{K}=3\left\{h^{1}\,\big(h^{2}\big)^{2}-h^{1}\,\big(h^{\mu}\big)^{2}-h^{2}\,\big(h^{i}\big)^{2}+\gamma_{\muij}\,h^{\mu}\,h^{i}\,h^{j}\right\}.
\Gamma^{A}=\left(\begin{matrix}{0}&{\gamma_{I\dot{B}}^{A}}\\{\gamma_{\dot{A}J}^{A}}&{0}\\\end{matrix}\right),\qquadA=1,\ldots,7\quad\mathrm{and}\qquad\Gamma^{8}=\left(\begin{matrix}{0}&{\delta_{I\dot{B}}}\\{-\delta_{\dot{A}J}}&{0}\\\end{matrix}\right).
\psi_{\frac{1}{2}\frac{1}{2}}=\psi_{-\frac{1}{2}-\frac{1}{2}}^{*}=\frac{i}{\sqrt{2}}f_{2}e^{-i\phi},\quad\psi_{\frac{1}{2}-\frac{1}{2}}=-\psi_{\frac{1}{2}-\frac{1}{2}}=-\frac{i}{\sqrt{2}}f_{1}.
F\left(\eta,\dot{\eta}\right)=\dot{\eta}-\left\{\eta\,,H^{\left(1\right)}\right\}=0\,,\,\,\Phi^{\left(1\right)}\left(\eta\right)=0\,,\;H^{\left(1\right)}=H\left(\eta\right)+\lambda\Phi^{\left(1\right)}\left(\eta\right)\,.
S[B_{(p+2)}]=\intd^{d}x\\sqrt{|g|}\\left[{\textstyle\frac{(-1)^{(p+2)}}{2\cdot(p+3)!}\frac{1}{e_{(p+1)}^{2}}}H_{(p+3)}^{2}+{\textstyle\frac{(-1)^{(p+1)}}{2\cdot(p+2)!}\frac{1}{e_{(p)}^{2}}}B_{(p+2)}^{2}\right]\,,
\epsilon_{eacd}\,t_{L}^{cd}\,h_{b}^{\,\,\bar{a}}\,F^{++ab}=0\,,\qquad\epsilon_{\bar{e}\bar{b}\bar{c}\bar{d}}\,t_{R}^{\bar{c}\bar{d}}\,h_{\,\,\bar{a}}^{a}\,F^{++\bar{a}\bar{b}}=0
x^{\mu}(\tau)=(1-v^{2})^{-1/2}\left(\begin{array}{c}{\tau}\\{v\tau}\\\end{array}\right)\;\mathrm{~and~}\;u^{\mu}(\tau)=(1-v^{2})^{-1/2}\left(\begin{array}{c}{1}\\{v}\\\end{array}\right),
\delta_{Q}x^{\mu}=-2i[\theta_{2}\sigma^{\mu}\bar{\epsilon}^{i}1_{i}^{2}+\theta_{3}\sigma^{\mu}\bar{\epsilon}^{i}1_{i}^{3}+\theta_{4}\sigma^{\mu}\bar{\epsilon}^{i}1_{i}^{4}-1_{1}^{i}\epsilon_{i}\sigma^{\mu}\bar{\theta}^{1}]\,.
A_{n}={{\frac{1}{4}}\sum_{m}}\Biggl[{\frac{1}{(n-m-z)^{2}}}+{\frac{1}{(n-m+z)^{2}}}-{\frac{1}{(n-m+z)^{2}}}-{\frac{1}{(n+m-z)^{2}}}\Biggr]
\delta\,S_{1}\,[\,\phi,\chi]_{\vert_{\epsilon=0}}={\frac{\deltaS_{0}[\overline{\psi},A]}{\delta\overline{\psi}^{r}}}\,\,\delta{\overline{\psi}}^{r}\,+{\frac{\deltaS_{0}[\overline{\psi},A]}{\deltaA^{m}}}\,\,\deltaA^{m}\,,
m_{ab}^{2}=-\frac{m_{\mathrm{string}}^{2}}{2\pi}\left|\frac{\epsilonI_{ab}R_{2}/R_{1}}{|n_{a}-m_{a}R_{2}/R_{1}||n_{b}-m_{b}R_{2}/R_{1}|}-\frac{\tilde{\epsilon}\tilde{I}_{ab}\tilde{R}_{2}/\tilde{R}_{1}}{|\tilde{n}_{a}-\tilde{m}_{a}\tilde{R}_{2}/\tilde{R}_{1}||\tilde{n}_{b}-\tilde{m}_{b}\tilde{R}_{2}/\tilde{R}_{1}|}\right|
\Xi_{g\geq2}(\kappa)={\frac{1}{4\sqrt{\pi}}}\int_{0}^{\infty}{\frac{ds}{s^{3/2}}}\left(k_{H}(s)\Theta_{\Sigma_{g}}({\frac{s}{\kappa}})-(g-1)\sqrt{\frac{\pi}{s}}(({\frac{1}{3}}-{\frac{s}{20}})\kappa-{\frac{s}{9}})\right)e^{-{s/4}}~~.
\begin{array}{c}{4=(1,0,0)\,,~~~~~\overline{{4}}=(0,0,1)\,,}\\{{}}\\{6=(0,1,0)\,,}\\{{}}\\{10=(2,0,0)\,,~~~~~\overline{{10}}=(0,0,2)}\\{{}}\\{15=(1,0,1)\,,}\\{{}}\\{20=(1,1,0)\,,~~~~~20^{\prime}=(0,2,0)\,,~~~~~\overline{{20}}=(0,1,1)\,,}\\{{}}\\{36=(2,0,1)\,,~~~~~\overline{{36}}=(1,0,2)\,.}\\\end{array}
\tilde{\Gamma}_{_C}^{^\Lambda\mu}(p,q)=e\;\frac{\left(q^{\mu}+{\tilde{q}}^{\mu}\right)}{q^{2}}\left[P_{-}\,\tilde{\Gamma}_{_C}^{^\Lambda}(p+q)-\tilde{\Gamma}_{_C}^{^\Lambda}(p)P_{+}\right]
\Phi^{\prime\prime}-\partial_{\alpha}\partial^{\alpha}\Phi+\Phi\biggl[\frac{5}{2}{\calH}^{\prime}+\frac{\varphi^{\prime\prime\prime}}{\varphi^{\prime}}-\frac{9}{4}{\calH}^{2}-{\calH}\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}-2\biggl(\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}\biggr)^{2}\biggr]=0.
(-1)^{\sum_{i=1}^{N}(N-i)|\alpha_{i}|}\tilde{\phi}_{1}\circ\tilde{\phi}_{2}(\tilde{\alpha}_{1},\ldots,\tilde{\alpha}_{N})=(-1)^{(n_{1}-1)|\phi_{2}|}\Pi\Bigl(\phi_{1}\circ\phi_{2}(\alpha_{1},\ldots,\alpha_{N})\Bigr).
{\calL}=\frac{1}{2g}[\sum_{i=1}^{9}\mathrm{tr}\dot{X}_{n}^{i}\dot{X}_{-n}^{i}-\sum_{j=2}^{9}\mathrm{tr}S_{n}^{j}(S_{n}^{j})^{+}-\frac{1}{2}\sum_{j,k=2}^{9}\mathrm{tr}T_{n}^{jk}(T_{n}^{jk})^{+}],
_{\begin{array}{l}{\left[:X^{-}\,P^{+}X^{-}:\,-ik\partial_{z}X^{-}+2X^{-}S\right](z)\times}\\{\quad\times\,\left[\,:X^{-}\,P^{+}X^{-}:\,-ik\partial_{w}X^{-}+2X^{-}S\right](w)}\\{\quad\quad\quad\quad\rightarrow\,\,\,0+\cdots}\\\end{array}}
S=\sum_{n=0}^{\infty}\intd^{6}x\\frac{(-1)^{n}}{3}\left(-\frac{1}{4}F_{{\alpha}{\beta}{\gamma}}^{(n)}F_{(n)}^{{\alpha}{\beta}{\gamma}}-{\Lambda}_{(n+1)}^{{\alpha}{\beta}{\gamma}}(F_{{\alpha}{\beta}{\gamma}}^{(n)}-F_{{\alpha}{\beta}{\gamma}}^{(n+1)})+{\Lambda}_{(n+1)}^{{\alpha}{\beta}{\gamma}}{\Lambda}_{{\alpha}{\beta}{\gamma}}^{(n+2)}\right)\.
m_{I}^{2}=\Bigl[{\frac{2\pi^{3}}{3}}-{\frac{\pi}{3}}\Big(\mathrm{ln}\Big({\frac{2\pi^{2}}{3}}\Big)\Big)^{2}\Big]T^{2}{\hat{g}}^{4}+{\frac{4\pi}{3}}T^{2}\mathrm{ln}\Big({\frac{1}{\hat{g}}}\Big)-{\frac{4\pi}{3}}T^{2}\Big(\mathrm{ln}\Big({\frac{1}{\hat{g}}}\Big)\Big)^{2}+0({\hat{g}}^{5})
\frac{\delta}{\delta\Phi}\left(\sqrt{-g_{(i)}}L_{(i)}^{\mathrm{matter}}\right)=\frac{\partial\tilde{g}_{(i)}^{~\mu\nu}}{\partial\Phi}\frac{\delta}{\delta\tilde{g}_{(i)}^{~\mu\nu}}\left(\sqrt{-g_{(i)}}L_{(i)}^{\mathrm{matter}}\right)=\sqrt{-g_{(i)}}\frac{1}{\Omega_{(i)}}\frac{d\Omega_{(i)}}{d\Phi}\,\tau_{(i)}
\bar{u}_{\sigma_{1}}^{LF}(k_{1})=\sum_{\sigma_{1}^{\prime}=\pm\frac{1}{2}}\beta_{\sigma_{1}\sigma_{1}^{\prime}}^{*}(k_{1})\bar{u}_{\sigma_{1}^{\prime}}(k_{1}),\quad\bar{u}_{\sigma_{2}}^{LF}(k_{2})=\sum_{\sigma_{2}^{\prime}=\pm\frac{1}{2}}\beta_{\sigma_{2}\sigma_{2}^{\prime}}^{*}(k_{2})\bar{u}_{\sigma_{2}^{\prime}}(k_{2}),
\left(\begin{matrix}{{}~1}&{~0}&{~0}&{~0}\\{{}~0}&{~1}&{~0}&{~0}\\{{}~0}&{~0}&{~1}&{~0}\\{{}~0}&{~0}&{~0}&{~1}\\{-1}&{-1}&{-1}&{-1}\\\end{matrix}\right)
\begin{array}{c}{dX^{a}\Gamma_{a}\wedge\nabla\left((i_{*}\xi)^{b}\Gamma_{b})S\right)=-\nabla\left(dX^{a}\Gamma_{a}(i_{*}\xi)^{b}\Gamma_{b}S\right)=}\\{=-{\frac{1}{2}}\nabla\left(dX^{a}g_{ab}(i_{*}\xi)^{b}S+\sum_{n}dX^{a}\omega_{ab}^{n}(i_{*}\xi)^{b}\hat{I}_{n}S\right)}\\\end{array}
\begin{array}{l}{\Psi(z,\bar{z})=S_{-1/2}\phi(z,\bar{z});}\\{\bar{\Psi}(z,\bar{z})=\bar{S}_{-1/2}\phi(z,\bar{z});}\\{F(z,\bar{z})=S_{-1/2}\bar{S}_{-1/2}\phi(z,\bar{z})}\\\end{array}
{\delta^{(0)}(\gamma^{\mu}\psi_{\mu}^{*})_{a}=-{\frac{3}{4}}i\lambda_{a}^{*}(\bar{\epsilon}\lambda)+{\frac{1}{1920}}i(\gamma^{\rho_{1}\dots\rho_{5}}\epsilon^{*})_{a}\left(\bar{\lambda}\gamma_{\rho_{1}\dots\rho_{5}}\lambda\right)+\dots,}
S_{PV}(\Phi,\Phi^{*},\underline{\Phi}_{i})=\frac{1}{2}\sum_{i=1}^{N}{\underline{\Phi}}_{i}^{A}S_{AB}{\underline{\Phi}}_{i}^{B}-\frac{1}{2}M_{i}^{2}{\underline{\Phi}}_{i}^{A}T_{AB}{\underline{\Phi}}_{i}^{B},
\left.\left[F,G\right]^{*}\right|_{z,y}=\left[F,G\right]-\left[F,\chi_{\alpha_{0}}\right]\mu^{\alpha_{0}\beta_{0}}\left[\chi_{\beta_{0}},G\right]-\left[F,y_{\alpha_{1}}\right]\omega^{\alpha_{1}\beta_{1}}\left[y_{\beta_{1}},G\right],
\intd^{2}k\frac{-(q_{2}+q_{3}+q_{4})\cdotq_{2}\;q_{3}^{2}\;q_{4}^{2}}{(k^{2}\!+\!m^{2})((k\!+\!q_{2})^{2}\!+\!m^{2})((k\!+\!q_{2}\!+\!q_{3})^{2}\!+\!m^{2})((k\!+\!q_{2}\!+\!q_{3}\!+\!q_{4})^{2}\!+\!m^{2})}\;.
{\mathrm{det}}_{sp}(A)=\frac{1}{4!}\lbrack(\mathrm{tr}A)^{4}-6(\mathrm{tr}A)^{2}(\mathrm{tr}A^{2})+8\mathrm{tr}A\;\mathrm{tr}A^{3}+3(\mathrm{tr}A^{2})^{2}-6\mathrm{tr}A^{4}\rbrack,
\begin{array}{l}{D_{\mu}^{ij}[A](x)\langleT[j^{\mu,j}(x)\frac{\deltaj^{\beta,l}}{\deltaA_{\lambda}^{k}}(y)]\rangle}\\{+ig\delta^{4}(x-y)(f^{bil}\langle\frac{\deltaj^{\beta,k}}{\deltaA_{\lambda}^{b}}(x)\rangle+f^{kia}\langle\frac{\deltaj^{\beta,a}}{\deltaA_{\lambda}^{l}}(x)\rangle)=0.}\\\end{array}
\left[\frac{1}{N^{2}}\frac{1}{A(\lambda)}\left(\frac{d}{d\lambda}+\frac{(-)^{n}\alpha}{2\lambda}\right)\frac{1}{A(\lambda)}\left(\frac{d}{d\lambda}-\frac{(-)^{n}\alpha}{2\lambda}\right)+1-\left(\frac{\lambda}{2q}\right)^{2}\right]\psi_{n}=0.
\mathrm{Det}^{r}(d)\rightarrowe^{{\frac{-i}{8\pi^{2}}}\int\left(fe^{2}\psi\tilde{F}^{\mu\nu}F_{\mu\nu}+{\frac{1}{3}}\;f^{3}\psi\tilde{G}^{\mu\nu}G_{\mu\nu}\right)d^{4}x}\;\mathrm{Det}^{r}(d).
\Phi=\left[G_{1}^{0}G_{2}^{0}K_{12}+G_{2}^{0}G_{3}^{0}K_{23}+G_{3}^{0}G_{1}^{0}K_{31}+G_{1}^{0}G_{2}^{0}G_{3}^{0}K_{123}\right]\Phi
r_{0}^{\ast}=\frac{\delta}{\varepsilon\varepsilon^{\ast}}\varepsilon^{\ast}=\frac{\sqrt{2}\delta_{0}}{m_{0}}\frac{4\pim_{0}^{2}}{g^{2}N}\sqrt{1-\sqrt{1-\left(\frac{g^{2}N}{4\pim_{0}^{2}\alpha_{0}}\right)^{2}}}\,,
\left[2p^{+}P^{-}+(p^{+})^{2}\left(\sum_{i=1}^{4}\frac{\mu^{2}}{9}\partial_{p^{i}}^{2}+\sum_{i^{\prime}=5}^{8}\frac{\mu^{2}}{36}\partial_{p^{i^{\prime}}}^{2}\right)-p_{I}^{2}+\alphap^{+}\right]\varphi=0.
(m^{-1})_{a}^{~b}={\frac{1}{1-{{\frac{2}{3}}k^{2}}}}(2\delta_{a}^{~b}-m_{a}^{~b}),\quadk_{a}^{~b}\equivh_{acd}h^{bcd},\quadk^{2}\equiv{\frac{1}{6}}\mathrm{tr}({kk}).
a_{0}\left(x\right)=\chi\frac{\xi_{1}^{2}\kappa+\eta}{\xi_{1}}\left(C_{1}e^{-x\xi_{1}}-C_{2}e^{x\xi_{1}}\right)+\chi\frac{\xi_{2}^{2}\kappa+\eta}{\xi_{2}}\left(C_{3}e^{-x\xi_{2}}-C_{4}e^{x\xi_{2}}\right)+C_{6}
S_{G,A}=\frac12\int_{0}^{T}d\tau\left[\begin{array}{c}{\left(D_{\tau}X_{i}^{M}\right)\varepsilon^{ij}X_{j}^{N}\,\,G_{MN}\left(X_{1},X_{2}\right)}\\{+\left(D_{\tau}X_{i}^{M}\right)\varepsilon^{ij}A_{jN}\left(X_{1},X_{2}\right)\,.\,}\\\end{array}\right]
\begin{array}{rclrcl}{B_{\alpha\beta}}&{=}&{0\,,}&{H_{\alpha\beta\gamma}}&{=}&{0\,,}\\{}&{}&{}&{}&{}&{}\\{G_{mn}}&{=}&{-e^{2\rho_{m}}\delta_{mn}\,,}&{B_{mn}}&{=}&{0\,.}\\\end{array}
\begin{array}{c}{b_{1}=q^{-N_{2}-\frac{1}{2}N_{1}}\t_{1}=q^{\frac{1}{2}N_{2}}a_{1}}\\{b_{2}=q^{-\frac{1}{2}N_{2}}t_{2}=q^{-\frac{1}{2}N_{1}}a_{2}}\\{b_{1}^{+}=t_{1}^{+}q^{-N_{2}-\frac{1}{2}N_{1}}=a_{1}^{+}q^{\frac{1}{2}N_{2}}}\\{b_{2}^{+}=t_{2}^{+}q^{-\frac{1}{2}N_{2}}=a_{2}^{+}q^{-\frac{1}{2}N_{1}}}\\\end{array}
H(\alpha_{i};1-\eta)=\eta^{\frac{b\alpha_{1}}{2}+\frac{3}{8}}(1-\eta)^{\frac{b\alpha_{2}}{2}}F(\frac{ib}{2}p_{+1+2+3}+\frac{3}{4},\,\frac{ib}{2}p_{+1+2-3}+\frac{3}{4};\,ibp_{2}+1;\,1-\eta),
{\bfC}^{N}=(F_{\Lambda_{1}}\oplus\dots\oplusF_{\Lambda_{r}})\bigoplus_{\nu}(G_{\nu}^{\zeta_{\nu}})^{s(\nu)}\bigoplus_{\epsilon}\bigl((F_{\epsilon}^{\eta_{\epsilon}^{1}})^{\tau^{1}(\epsilon)}\oplus(F_{\epsilon}^{\eta_{\epsilon}^{2}})^{\tau^{2}(\epsilon)}\oplus(F_{\epsilon}^{\eta_{\epsilon}^{3}})^{\tau^{3}(\epsilon)}\bigr)
\sum_{a,\ldots,f=3}^{n}\xi_{12a}\xi_{12b}\xi_{12c}\xi_{12d}\xi_{12e}\bar{\theta}_{12f}F_{abcdef}^{p_{1}-6\vertp_{2}-6\vert\ldotsp_{a}-1\ldotsp_{f}-1\ldots\vertp_{n}}\;.
\hat{q}(t)=\sqrt{\frac{\hbar}{2m\omega}}\left[a(t)+a^{\dagger}(t)\right]=\sqrt{\frac{\hbar}{2m\omega}}\left[a(t_{0})\,e^{-i\omega(t-t_{0})}+a^{\dagger}(t_{0})\,e^{i\omega(t-t_{0})}\right]\,
D_{m}(p)\equiv\sum_{n=0}^{\infty}D_{n,m}\frac{1}{p^{n+1}}=\sum_{i=1}^{N}\sum_{n=0}^{\infty}\frac{\lambda_{i}^{n}}{P^{n+1}}\frac{\partial^{m}}{\partial\lambda_{i}^{m}}=\sum_{i=1}^{N}\frac{1}{P-{\lambda}_{i}}\frac{\partial^{m}}{\partial\lambda_{i}^{m}}