harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
B(t)=\frac{\Gamma(\tilde{c})\Gamma(\tilde{a}+\tilde{b}-\tilde{c})}{\Gamma(\tilde{a})\Gamma(\tilde{b})}\frac{{e}^{i\omega_{0}t}}{\sqrt{2\omega_{0}}}+\frac{\Gamma(\tilde{c})\Gamma(\tilde{c}-\tilde{a}-\tilde{b})}{\Gamma(\tilde{c}-\tilde{a})\Gamma(\tilde{c}-\tilde{b})}\frac{{e}^{-i\omega_{0}t}}{\sqrt{2\omega_{0}}}\;.
\left(\begin{matrix}{A}&{B}\\{C}&{D}\\\end{matrix}\right)=<\binom{\tilde{\Phi}(x_{1})\tilde{\Phi}(x_{3})}{\tilde{C}(x_{1})\tilde{C}(x_{3})}(\tilde{\Phi}(x_{2})\tilde{\Phi}(x_{4}),\tilde{C}(x_{2})\tilde{C}(x_{4}))>
\begin{array}{rcl}{(\mathrm{NC~vertices})}&{=}&{\displaystyle\left[e^{-i(q_{\lambda},-p_{1\lambda_{1}},p_{\lambda})}+e^{-i(p_{\lambda},-p_{1\lambda_{1}},q_{\lambda})}\right]}\\{}&{\times}&{\displaystyle\left[e^{-i(q_{\lambda},+p_{2\lambda_{2}},p_{\lambda})}+e^{-i(p_{\lambda},+p_{2\lambda_{2}},q_{\lambda})}\right],}\\\end{array}
L_{B}=-{\frac{a}{4}}C_{\mu\nu}^{a}C^{\mu\nua}-{\frac{1}{2g}}B_{\mu}^{a}B^{\mua}+\alpha\epsilon^{\mu\nu\lambda}C_{\mu}^{a}(\partial_{\nu}C_{\lambda}^{a}+{\frac{1}{3}}f^{abc}C_{\nu}^{b}C_{\lambda}^{c}).
\Psi^{\prime}(t,\vec{x}\,)\,=\,\sum_{\sigma=0,\pm1}\int{\frac{d^{3}p}{(2\pi)^{3}}}{\frac{1}{2\,\omega_{\vec{p}}}}\Big[S(T)\,u_{\sigma}^{\ast}(\vec{p}\,)\,a_{\sigma}^{\dagger}(\vec{p}\,)\,e^{ip\cdotx^{\prime}}+S(T)\,v_{\sigma}^{\ast}(\vec{p}\,)\,b_{\sigma}(\vec{p}\,)\,e^{-ip\cdotx^{\prime}}\Bigr]\quad,
G(p)_{\mu\nu}=-i\left[\int_{0}^{\infty}ds\frac{\rho_{1}(s)}{p^{2}-s^{2}+i\epsilon}{\calP}_{\mu\nu}(p)-iM\epsilon_{\mu\nu\rho}\frac{p^{\rho}}{p^{2}}\int_{0}^{\infty}ds\frac{\rho_{2}(s)}{p^{2}-s^{2}+i\epsilon}\right]
Z^{D}\!=\!\!\int{\calD}g_{++}{\calD}c{\calD}b^{--}{\calD}\chi_{+}{\calD}\chi_{-}~\mathrm{exp}\!\left[i\Gamma^{D}(g)+i\!\int\!d^{2}\xi~(b^{--}\nabla_{-}\chi_{-}+c\nabla_{+}\chi_{-}+c\nabla_{-}\chi_{+})\right]
C(n,\tau)=\left(\theta_{3}^{4}(0,\tau)-\theta_{4}^{4}(0,\tau)-e^{i\pin}\theta_{2}^{4}(0,\tau)\right)\left(\bar{\theta}_{3}^{4}(0,\tau)-\bar{\theta}_{4}^{4}(0,\tau)-e^{i\pin}\bar{\theta}_{2}^{4}(0,\tau)\right)
g_{y\bar{y}}=\partial_{y}\partial_{\bar{y}}K=\fracn{n+1}+{\frac{|y|^{2(n-1)}}{(n-1)!}}{\frac{e^{-|y|^{2}}}{Q(|y|^{2})}}\left(n-|y|^{2}-{\frac{|y|^{2n}}{(n-1)!}}{\frac{e^{-|y|^{2}}}{Q(|y|^{2})}}\right),
\frac{U^{4}/2}{\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}}}
I={\frac{1}{16\piG}}\left[\int_{\calM}\sqrt{-g}d^{4}xR+2\int_{\partial{\calM}}K\sqrt{-h}d^{3}x\right]-{\frac{1}{4\pi}}\sum_{s}\mu_{s}\int\sqrt{\sigma}_{s}d^{2}\zeta_{s}~~~.
\begin{array}{c}{\displaystyle{z_{0}^{-1}\delta\left(\frac{z_{1}-z_{2}}{z_{0}}\right)Y_{M}(u,z_{1})Y_{M}(v,z_{2})-z_{0}^{-1}\delta\left(\frac{z_{2}-z_{1}}{-z_{0}}\right)Y_{M}(v,z_{2})Y_{M}(u,z_{1})}}\\{\displaystyle{=z_{2}^{-1}\left(\frac{z_{1}-z_{0}}{z_{2}}\right)^{-r/K}\delta\left(\frac{z_{1}-z_{0}}{z_{2}}\right)Y_{M}(Y(u,z_{0})v,z_{2})}}\\\end{array}
p\cdot\varepsilon^{+}(k;q)=\frac{\langleqp\rangle[pk]}{\sqrt{2}\;\langleqk\rangle}\;\mathrm{sign}(k_{0})\,,\quadp\cdot\varepsilon^{-}(k;q)=\frac{\langlekp\rangle[pq]}{\sqrt{2}\;[kq]}\;\mathrm{sign}(k_{0})\,.
f_{e+f+g+h}^{(2)}=\frac{g_{1}^{2}}{8\pi^{2}}\frac{g_{2}^{2}}{8\pi^{2}}\int_{\lambda_{1}^{2}}^{1}\frac{d\alpha_{1}}{\alpha_{1}}\int_{\lambda_{1}^{2}/\alpha_{1}}^{1}\frac{d\beta_{1}}{\beta_{1}}\int_{\lambda_{2}^{2}}^{1}\frac{d\alpha_{2}}{\alpha_{2}}\int_{\lambda_{2}^{2}/\alpha_{2}}^{1}\frac{d\beta_{2}}{\beta_{2}}.
N_{z}=p_{z}\frac{\partial}{\partialE}+\left(\frac{{\tilde{L}}_{p}}{2}{\vec{p}}^{2}+\frac{1-e^{-2{\tilde{L}}_{p}E}}{2{\tilde{L}}_{p}}\right)\frac{\partial}{\partialp_{z}}-{\tilde{L}}_{p}p_{z}\left(p_{j}\frac{\partial}{\partialp_{j}}\right)~.
F_{12}^{\mathrm{vec}}=\frac{(q_{\mathrm{vec}}^{I})_{1}(q_{\mathrm{vec}}^{I})_{2}}{r_{12}^{2}}-\frac{(\sigma_{\mathrm{vec}}^{I})_{1}(\sigma_{\mathrm{vec}}^{I})_{2}}{r_{12}^{2}}=0\.
UDU^{*}\;=\;\left(\begin{array}{cc}{\nabla\!\!\!\!/}&{k\,\gamma^{5}\,e^{-Re\,\sigma}{\binom{0}{1}}}\\{k^{*}\gamma^{5}\,e^{-Re\,\sigma}\,(0,1)\quad}&{\nabla\!\!\!\!/}\\\end{array}\right)\;+\;\left(\begin{array}{cc}{0}&{0}\\{0}&{i\,\partial\!\!\!/\,Im\,\sigma_{2}}\\\end{array}\right)
{\widetildeT}_{\mu\nu}=T_{\mu\nu}-{\widehatM}_{P}^{D-4}\biggl[-\partial^{\lambda}\partial_{\lambda}H_{\mu\nu}+\partial_{\mu}\partial_{\nu}(\chi+g)-{\frac{1}{2}}\eta_{\mu\nu}\partial^{\lambda}\partial_{\lambda}\chi\biggr]~.
\begin{array}{cc}{\sigma_{z}=\left(\begin{matrix}{1}&{0}\\{0}&{0}\\\end{matrix}\right)}&{\sigma_{\bar{z}}=\left(\begin{matrix}{0}&{0}\\{0}&{1}\\\end{matrix}\right)}\\{\sigma_{y}=\left(\begin{matrix}{0}&{0}\\{i}&{0}\\\end{matrix}\right)}&{\sigma_{\bar{y}}=\left(\begin{matrix}{0}&{i}\\{0}&{0}\\\end{matrix}\right)}\\\end{array}
{\calW}=\epsilon_{\alpha\beta\gamma}\Phi_{\alpha}\chi_{\beta}\chi_{\gamma}+\chi_{1}P^{1}{\widehatR}^{1}+{\widetilde\chi}_{1}P^{1}R^{1}+{\widehat\chi}_{1}{\widehatP}^{1}R^{1}+\dots~.
S^{\prime}=\intd^{2}\sigma\left(-e^{-\phi}\sqrt{-\mathrm{det}\,(G_{ij}+F_{ij}-b_{ij})}+\frac{1}{2}\tilde{H}^{ij}(F_{ij}-2\partial_{i}A_{j})+\frac{1}{2}e^{-\phi}\epsilon^{ij}C_{ij}-\frac{1}{2}C_{0}\epsilon^{ij}F_{ij}\right)
1\simeq\frac{2\alpha}{\pi}T|eH|\int_{-\infty}^{\infty}dq_{3}\int_{0}^{\infty}d\hat{q}_{\perp}^{2}\mathrm{e}^{-\hat{q}_{\perp}^{2}}\frac{1}{Q_{2}}\frac{1}{Q_{1}+\pi^{2}T^{2}}.
\tilde{S}_{Q2f}^{pot}(B_{n,i})=\frac{2g}{N^{2}}\sum_{j,k}\mathrm{Tr}\,B_{1,j}^{2}\mathrm{Tr}\,B_{2,k}^{2}\sum_{i}U_{ij}U_{ij}U_{ik}U_{ik}=\frac{2g}{N^{2}L}\sum_{j,k}\mathrm{Tr}\,B_{1,j}^{2}\mathrm{Tr}\,B_{2,k}^{2}
M^{\mu_{n}(i)}{}_{(j)}\left(p\right)f_{\left[k\right]}^{\mu_{n}(j)}\left(p\right)=\lambda_{\left[k\right]}^{\mu_{n}}\left(p\right)f_{\left[k\right]}^{\mu_{n}(i)}\left(p\right)\;\,,\;\,p\inV_{\mu_{n}}^{+}\;\,,
S=\left(\begin{array}{llll}{\partial_{1}X^{0}}&{\partial_{1}X^{1}}&{...}&{\partial_{1}X^{p}}\\{\partial_{2}X^{0}}&{\partial_{2}X^{1}}&{...}&{\partial_{2}X^{p}}\\{...}&{...}&{...}&{...}\\{G^{0\nu}n_{\nu}}&{G^{1\nu}n_{\nu}}&{...}&{G^{p\nu}n_{\nu}}\\\end{array}\right)\.
Z(t,t^{*})=c\,\sum_{\{h^{e},h^{o}\}}{\frac{\prod_{i}(h_{i}^{e}-1)!!h_{i}^{o}!!}{\prod_{i,j}(h_{i}^{e}-h_{j}^{o})}}~\chi_{\{h\}}(A)~\chi_{\{h\}}(B).
S_{\mathrm{\tiny\it~brane}}=\intd^{7}x\sqrt{-g}\,\mathcal{L}_{m}\quad\mathrm{with}\quad\mathcal{L}_{m}=-\frac{1}{4}G_{MN}^{\tilde{a}}G^{\tilde{a}MN}-\frac{1}{2}\mathcal{D}_{M}\Phi^{\tilde{a}}\mathcal{D}^{M}\Phi^{\tilde{a}}-\frac{\lambda}{4}\left(\Phi^{\tilde{a}}\Phi^{\tilde{a}}-\eta^{2}\right)^{2}~,
\mathcal{M=}\left(\pi\right)\frac{\sqrt{E^{2}-s^{2}m^{2}}}{2E}\left(2s\right)\delta_{l,l^{\prime}}\delta_{s,s^{\prime}}\intJ_{l+1}\left(p^{\bot}\rho\right)\,J_{l}\left(p^{\bot}\rho\right)d\rho.
A_{i}^{(1)}=\frac{h^{2}k_{i}-(k.h)h_{i}}{k^{2}h^{2}-(k.h)^{2}}\,,\,\,\,\,A_{i}^{(2)}=\frac{k^{2}h_{i}-(k.h)k_{i}}{k^{2}h^{2}-(k.h)^{2}}\,.
z_{0}^{2}\varepsilon^{2}>{\frac{1}{32}}\left(-32\omegaz_{0}^{2}+9l^{2}+3\sqrt{-64\omegaz_{0}^{2}l^{2}+9l^{4}-64l^{2}z_{0}^{2}\omega^{2}}\right)/(\omega^{2}l^{2})\,,
\begin{array}{cl}{(2,1)\times2^{2}~=(3,1)+(1,1)+(2,2)~~~~~~~}&{\mathrm{:\,massless~tensor,~Maxwell,}}\\{{}}&{{}}\\{(2,1,1)\times2^{2}=(3,1,1)+(1,1,1)+(2,1,2)~}&{\mathrm{:\,massive~tensor}\,,}\\\end{array}
\left(m+\frac{1}{2r_{1}}+\frac{1}{2r_{2}}\right)dr^{2}+\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)drdR+\left(2M-\frac{2}{R}+\frac{1}{2r_{1}}+\frac{1}{2r_{2}}\right)dR^{2}
\begin{array}{l}{\tau^{0}=s!w(w-1)\cdots(w-s+1)f,}\\{\tau^{1}=s!(w-1)\cdots(w-s+1)\nablaf,\quad\mathrm{and}}\\{\tau^{k}=\frac{1}{k(w-k+1)}\left(\nabla\circ\tau^{k-1}+\Phi\circ\tau^{k-2}\right)\quad\mathrm{for~}2\leqk\leqs.}\\\end{array}
{\calL}_{\{n_{q}\},\{m_{q}\}}^{\{a_{i_{l}j_{l^{\prime}}},a_{i_{l}l^{\prime}},a_{l^{\prime}j_{l^{\prime}}},a_{ll^{\prime}}-r_{l}\cdotr_{l^{\prime}}\},\{n_{v_{q}}\}}(z_{q})\vec{\calS}_{\{n_{v_{q}}\}}^{\{a_{i_{l}j_{l^{\prime}}},a_{i_{l}l^{\prime}},a_{lj_{l^{\prime}}},a_{ll^{\prime}}-r_{l}\cdotr_{l^{\prime}}\}}(z_{v_{q}})=0~~~for~~~\{n_{q}\neq0\}
M_{string}^{2}=(Q_{1}^{(1)}+Q_{1}^{(2)})^{2}+{\frac{4}{\alpha^{\prime}}}(N_{R}-{\frac{1}{2}})=(Q_{1}^{(1)}-Q_{1}^{(2)})^{2}+{\frac{4}{\alpha^{\prime}}}(N_{L}-1),
L_{\alpha\beta}=\frac{1}{2}\{\hat{y}_{\alpha},\,\hat{y}_{\beta}\},\qquad\bar{L}_{\dot{\alpha}\dot{\beta}}=\frac{1}{2}\{\hat{\bar{y}}_{\dot{\alpha}},\,\hat{\bar{y}}_{\dot{\beta}}\},\qquadP_{\alpha\dot{\beta}}=\hat{y}_{\alpha}\,\hat{\bar{y}}_{\dot{\beta}}
P(A)=\frac{D^{2}(k^{2}-1)}{2(1+q)Q}\left[1-\frac{2}{\Omega^{1+q}}\frac{k}{\sqrt{k^{2}-1}}Q(-\mu/2)^{-1/2}+\frac{Q^{2}(-\mu/2)}{\Omega^{2(1+q)}}\right].
I_{\rho}^{\prime}(\rhoz)\sim{\frac{1}{\sqrt{2\pi\rho}}}e^{\rho\eta}{\frac{(1+z^{2})^{\frac{1}{4}}}{z}}\left[1+\sum_{k=1}^{\infty}{\frac{v_{k}(t)}{\rho^{k}}}\right],
{\mathcal{L}}=-\frac{1}{8}\int{d^{2}\Theta2{\mathcal{E}}\left(\bar{\mathcal{D}}^{2}-8R\right)f}-\frac{1}{8}\int{d^{2}\bar{\Theta}2\bar{\mathcal{E}}\left({\mathcal{D}}^{2}-8R^{\dagger}\right)f}.
2\mathrm{Im}\;S_{\mathrm{ABCD}}^{(IA)}=S_{\mathrm{ABCD}}^{(IA)}-(S_{\mathrm{ABCD}}^{(IA)})^{*}=S_{\mathrm{ABCD}}^{(IA)}+S_{\mathrm{D^{\prime}C^{\prime}B^{\prime}A^{\prime}}}^{(IA)}=S_{E}^{(IA)}\;,
{\calL}\,=\,\pi\dot{\phi}-\frac12\pi^{2}-\frac12\left(\phi^{\prime}\right)^{2}-\left(e_{-}A_{0}+e_{+}A_{1}\right)\pi\,+\,\left(e_{-}A_{1}+e_{+}A_{0}\right)\phi^{\prime}\,+\,\frac12A_{\mu}M^{\mu\nu}A_{\nu}
\mathrm{Det}^{g}(d)=\mathrm{Det}^{r}(d)\cdote^{{\frac{i}{8\pi^{2}}}\intd^{4}x\intd^{4}yB^{\mu}(x){\frac{\partial}{\partialx^{\mu}}}G(x,y)\left[fe^{2}\tilde{F}^{\mu\nu}(y)F_{\mu\nu}(y)+{\frac{1}{3}}f^{3}\tilde{G}^{\mu\nu}(y)G_{\mu\nu}(y)\right]}.
\left(\begin{array}{l}{s}\\{v}\\{t}\\\end{array}\right)(4,2;3,1)=-\frac{1}{4}\left[\begin{array}{ccc}{1}&{1}&{1}\\{5}&{-3}&{1}\\{10}&{2}&{-2}\\\end{array}\right]\left(\begin{array}{l}{s}\\{v}\\{t}\\\end{array}\right)(4,1;3,2)
\mathrm{either}\quad{\calS}^{ab}=\tilde{S}^{ab}=-\frac{i}{4}[\tilde{a}^{a},\tilde{a}^{b}]=-\frac{i}{4}[\tilde{\gamma}^{a},\tilde{\gamma}^{b}],\quad\mathrm{or}\qquad{\calS}^{ab}=\tilde{\tilde{S}}{}^{ab}=-\frac{i}{4}[\tilde{\tilde{a}}{}^{a},\tilde{\tilde{a}}{}^{b}].
g_{\mu\nu}=\left(\begin{array}{cc}{h_{ij}-Ba_{i}a_{j}}&{-Ba_{i}}\\{-Ba_{j}}&{-B}\\\end{array}\right)~~,~~g^{\mu\nu}=\left(\begin{array}{cc}{h^{ij}}&{-a^{i}}\\{-a^{j}}&{a^{2}-B^{-1}}\\\end{array}\right)~~~,
S=-\frac{1}{2\pi}\intd^{2}\sigma\left[\partial_{\alpha}X^{\mu}\partial^{\alpha}X_{\mu}-i\bar{\psi}^{\mu,j}\rho^{\alpha}\partial_{\alpha}\psi_{\mu,j}+i\bar{\phi}^{\mu,k}\rho^{\alpha}\partial_{\alpha}\phi_{\mu,k}\right].
G_{1}^{-1}\left(p_{1}\right)G_{2}^{-1}\left(p_{2}\right)\psi^{12}\left(p,P\right)=g^{2}i\int\frac{d^{4}q}{\left(2\pi\right)^{4}}\,G_{3}\left(p-q\right)\psi^{12}\left(q,P\right)
Z^{m}(\sigma,\tau)=\left(z_{0}^{m}+\frac{p_{0}^{m}}{l}\tau\right)\;+\;\sum_{k\neq0}\left(z_{k}^{m}+\frac{p_{k}^{m}}{l}\tau\right)e^{\textstyle{\frac{2\piik\sigma}{l}}}\,
ds_{11}^{2}=dx_{1,2}^{2}+\frac{d\rho^{2}}{\left(1-\frac{l^{10/3}}{\rho^{10/3}}\right)}+\frac{9}{100}\rho^{2}\left(1-\frac{l^{10/3}}{\rho^{10/3}}\right)(\tilde{\omega}_{i}-A^{i})^{2}+\frac{9}{20}\rho^{2}ds_{4}^{2},
\left(\frac{i\epsilon_{\mu\lambda\alpha}p_{\lambda}}{p^{2}}\right)\left(\frac{i\alpha\epsilon_{\alpha\sigma\beta}p_{\sigma}}{\sqrt{p^{2}}}\right)\left(\frac{i\epsilon_{\beta\kappa\nu}p_{\kappa}}{p^{2}}\right)=\frac{i\alpha\epsilon_{\mu\lambda\nu}p_{\lambda}}{\left(p^{2}\right)^{3/2}}\,.
F_{\left\{n_{1},\cdots,n_{g}\right\}}\left({\bfr}_{1}\cdots{\bfr}_{\alpha}^{\prime}\cdots{\bfr}_{N}\right)=e^{-i\frac{\theta_{1k}}{\kappa}}e^{-i\frac{2\pi}{\kappa}n_{k}}F_{\left\{n_{1},\cdots,n_{g}\right\}}\left({\bfr}_{1}\cdots{\bfr}_{\alpha}\cdots{\bfr}_{N}\right),
S=T_{2}\intd^{3}\zeta(\frac{1}{2}v^{2}+\frac{1}{8}h(v^{2})^{2}+\frac{i}{16}v^{2}v^{\hat{I}}\partial_{\hat{J}}h\psi_{a}^{\alpha}(\gamma^{\tilde{I}\tilde{J}})_{ab}\psi_{\alphab}).
\begin{array}{lll}{\mathrm{A}_{2n-1}^{(1)}:}&{r=2}&{u=n-2\quad(n\geq2)}\\{\mathrm{A}_{2n}^{(1)}:}&{r=1}&{u=n-1}\\{\mathrm{C}_{n}^{(1)}:}&{r=2}&{u=n-2\quad(n\geq2)}\\{\mathrm{A}_{2n-1}^{(2)}:}&{r=1}&{u=n-1}\\{\mathrm{A}_{2n}^{(2)}:}&{r=1}&{u=n-1.}\\\end{array}
\sum_{m=0}^{\infty}\sum_{j=1}^{23}{\frac{-(\Gamma({\frac{j}{24}}+m))^{2}\pi\Gamma({\frac{1}{2}}+{\frac{j}{24}}+m)(-)^{m}\prod_{j\neqk=1}^{23}\Gamma({\frac{k-j}{24}}-m)}{({\frac{j}{24}}+m)sin[\pi({\frac{j}{24}}+m)]m!(m+{\frac{j}{24}})\prod_{k=-2}^{8}\Gamma({\frac{k}{8}}-m-{\frac{j}{24}})\prod_{l=10}^{1}\Gamma({\frac{l}{12}}-m-{\frac{j}{24}})}}z^{-m-{\frac{j}{24}}}.
\begin{array}{llll}{{{R}^{0}}_{101}({\Gamma})}&{=}&{cJ^{0}\frac{H^{\prime}}{2}={{R}^{0}}_{202}({\Gamma})\nonumber}\\{{{R}^{0}}_{330}({\Gamma})}&{=}&{J^{0^{\prime}}\nonumber}\\{{{R}^{1}}_{212}({\Gamma})}&{=}&{{c}^{2}\frac{H^{2}}{4}\nonumber}\\{{{R}^{2}}_{332}({\Gamma})}&{=}&{-{{R}^{1}}_{331}({\Gamma})=-\frac{1}{2}(H^{\prime\prime}+\frac{1}{2}H^{2})\nonumber}\\\end{array}
\nu_{0}^{TM}\simeq\frac{{n_{2}}\,R\,\Omega}{c}\,\left\{1-k_{1}^{TM}\,\left(\frac{{n_{2}}\,R\,\Omega}{c}\right)^{-2/3}+k_{2}^{TM}\,\left(\frac{{n_{2}}\,R\,\Omega}{c}\right)^{-4/3}+{\calO}\left(\frac{R\,\Omega}{c}\right)^{-2}\right\},
32{\delta_{\alpha}}^{\delta}{\delta_{\gamma}}^{\beta}=2{\delta_{\alpha}}^{\beta}{\delta_{\gamma}}^{\delta}-\Gamma^{ab}{}_{\alpha}{}^{\beta}\Gamma_{ab}{}_{\gamma}{}^{\delta}+\frac2{4!}\Gamma^{abcd}{}_{\alpha}{}^{\beta}\Gamma_{abcd}{}_{\gamma}{}^{\delta}
F_{\mu\nu}=\left(\partial_{\mu}A_{\nu}^{D}-\partial_{\nu}A_{\mu}^{D}+{\frac{1}{2}}f^{BCD}\{A_{\mu}^{B},A_{\nu}^{C}\}-{\frac{i}{2}}d^{BCD}[A_{\mu}^{B},A_{\nu}^{C}]\right)T^{D}
S_{F}=-\intd^{2}\theta\Bigl\{\frac{\deltaF}{\delta\Phi^{A}(\theta)}\frac{\partial^{2}\Phi^{A}(\theta)}{\partial\theta^{2}}+\mathrm{~\frac{1}{2}~}\epsilon_{ab}\intd^{2}\bar{\theta}\,\frac{\partial\Phi^{A}(\theta)}{\partial\theta_{a}}\frac{\delta^{2}F}{\delta\Phi^{A}(\theta)\delta\Phi^{B}(\bar{\theta)}}\frac{\partial\Phi^{B}(\bar{\theta})}{\partial\bar{\theta}_{b}}\Bigr\}.
W_{\infty}^{3,D}={\frac{1}{16}}(1-\gamma)^{-1}\sum_{k=0}^{\infty}(-1)^{k}\beta^{k}(k+1)(k+3)(k+5)\sum_{m=0}^{\infty}(m^{2+k}-m^{1+k})\alpha_{m}^{-k-7}.
{\bfJ}\cdot\widehat{\bfp}\,\,\Theta_{[j]}\,\left[\phi_{_{L,R}}^{h}({p}^{\mu})\right]^{\ast}\,=\,-\,h\,\Theta_{[j]}\,\left[\phi_{_{L,R}}^{h}({p}^{\mu})\right]^{\ast}\quad.
\sum_{n=\frac{1}{2}}^{R/2}~\frac{\trianglek_{n}^{+}}{k_{n}^{+}}~f(k_{n}^{+})=\sum_{n=\frac{1}{2}}^{R/2}~\frac{\trianglek_{n}^{+}}{k_{n}^{+}}~e^{-\sigma^{2}/(\Lambda^{\prime}k_{n}^{+})}
\begin{array}{cc}{\vec{x}_{1}=\vec{x}_{CM}+\hat{n}\,\displaystyle{\sqrt{(vt)^{2}+({\scriptstyle{R}}\kappa/2v)^{2}}}\,,~}&{~\vec{x}_{2}=2\vec{x}_{CM}-\vec{x}_{1}\,,}\\\end{array}
S_{p}=\frac{2V_{3}N\sqrt{2m\lambda}l^{4}}{\pi\lambda^{2}(2r_{+}^{4}+3l^{2}r_{+}^{2}+l^{4})}\frac{l^{2}+2r_{+}^{2}}{l^{2}-2r_{+}^{2}}\frac{1}{\triangle_{*}}\left[\frac{m(l^{2}-2r_{+}^{2})}{l^{4}}-\frac{u^{2}(u^{2}+l^{2})-m}{u^{2}+l^{2}}-u^{2}\triangle_{*}\right],
F(p)=\sum_{|\alpha|\leq\sigma}\frac{p^{\alpha}}{\alpha!}\frac{\partial^{|\alpha|}F}{\partialp^{\alpha}}(0)+\sum_{|\alpha|=\sigma+1}\frac{p^{\alpha}}{\alpha!}\int_{0}^{1}{\mathrmd}t\;(1-t)^{|\alpha|-1}\frac{\partial^{|\alpha|}F}{\partialp^{\alpha}}(tp),
\left(A_{k}^{\mu}+S^{\mu}\,\!\!_{\nu}B_{k}^{\nu\dagger}\right)\left|B_{X}\right\rangle=0,\qquad\left(A_{k}^{\mu\dagger}+S^{\mu}\,\!\!_{\nu}B_{k}^{\nu}\right)\left|B_{X}\right\rangle=0,
[det(\eta_{\mu\nu}+M_{\mu\nu})]^{\frac{1}{2}}=\frac{[1+(H_{i5})^{2}-\frac{1}{4!}\epsilon^{ijkl}\epsilon^{mnpq}H_{mi}H_{nj}H_{pk}H_{ql}]}{[-det(\eta_{ab}+i{}^{*}H_{ab})]^{\frac{1}{2}}}.
\Gamma^{n}(b)=\intd^{3}y_{1}....d^{3}y_{n}F_{\mu_{1}\nu_{1}}(y_{1})....F_{\mu_{n}\nu_{n}}(y_{n})\Omega^{\mu_{1}\nu_{1}...\mu_{n}\nu_{n}}(y_{1},...,y_{n})\;,
ds^{2}=\frac{x^{2}}{\sqrt{c_{1}c_{5}Q_{1}Q_{5}}}(-dt^{2}+dx_{9}^{2})+\frac{\sqrt{c_{1}c_{5}Q_{1}Q_{5}}}{x^{2}}dx^{2}+\sqrt{c_{1}c_{5}Q_{1}Q_{5}}d\Omega_{3}^{2}+\cdots
\frac{1}{(k\!\cdot\!n)\:(p-k)\!\cdot\!n}\rightarrow\frac{1}{p\!\cdot\!n}\left[\frac{1}{k\!\cdot\!n}+\frac{1}{(p-k)\!\cdot\!n}\right]\:,\:\:\:\:\:\:\:p\!\cdot\!n\neq0
{\widehat\Sigma}=\left(\begin{array}{cc}{\Sigma_{RR},}&{\Sigma_{RA}}\\{\Sigma_{AR},}&{\Sigma_{AA}}\\\end{array}\right)\,,\qquad{\bar{\Sigma}}=\left(\begin{array}{cc}{\Gamma_{aa}^{(2)},}&{\Gamma_{ar}^{(2)}}\\{\Gamma_{ra}^{(2)},}&{\Gamma_{rr}^{(2)}}\\\end{array}\right)\,.
{\calO}\equiv\intd^{4}x\,\mathrm{tr}\,\phi^{2},\qquad\bar{Q}_{\mu}{\calO},\qquad\bar{Q}_{\mu}\bar{Q}_{\nu}{\calO},\qquad\bar{Q}_{\mu}\bar{Q}_{\nu}\bar{Q}_{\rho}{\calO},\qquad\bar{Q}_{\mu}\bar{Q}_{\nu}\bar{Q}_{\rho}\bar{Q}_{\sigma}{\calO}.
\eta=(n-m)(t_{1}\tilde{\epsilon}+t_{2}\tilde{\epsilon}^{2})+(m-1)\left[{\frac{t_{1}}{X_{0}}}\tilde{\epsilon}+\left({\frac{t_{2}}{X_{0}}}-{\frac{t_{1}X_{1}}{X_{0}^{2}}}\right)\tilde{\epsilon}^{2}\right]-\tilde{\epsilon}+O(\tilde{\epsilon}^{3}).
\begin{array}{l}{R_{12}X_{1}^{\prime}X_{2}^{\prime}=R_{12}(T_{1}X_{1})(T_{2}X_{2})=R_{12}T_{1}T_{2}X_{1}X_{2}}\\{\,}\\{=T_{2}T_{1}R_{12}X_{1}X_{2}=qT_{2}T_{1}X_{2}X_{1}=qX_{2}^{\prime}X_{1}^{\prime}\quad}\\\end{array}
\tilde{S}_{ij}^{-1}(\vec{x},\vec{y})=\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)\,\Bigl(\eta_{ij}-\frac{1}{m^{2}}\,\partial_{i}\partial_{j}\Bigr)\,\frac{1}{\nabla^{2}+m^{2}}\,\delta(\vec{x}-\vec{y})
\Sigma_{-;1;3}^{-;2}+\Sigma_{-;2;3}^{-;1},\quad\Sigma_{-;1;2}^{-;3}+2\Sigma_{-;3;2}^{-;1}+3\Sigma_{-;1,2;2}^{-;1,1},\quad\Sigma_{1;1;2}^{1;2}+\Sigma_{1;2;2}^{1;1}.
\Gamma_{4,tree}^{\hat{\varepsilon}^{0}a_{0}...\hat{\varepsilon}^{3}a_{3}}\propto\left[(\hat{\varepsilon}^{0}\cdot\hat{\varepsilon}^{1})(\hat{\varepsilon}^{2}\cdot\hat{\varepsilon}^{3})f^{a_{0}a_{3}e}f^{a_{1}a_{2}e}+perm.\right]\neq0.
\hat{C}^{\hat{a}\hat{b}}={\frac{1}{K}}\,\left[(1+{\frac{1}{12}}\hat{{\calH}}^{2})\,\hat{g}^{\hat{a}\hat{b}}-\,{\frac{1}{4}}(\hat{{\calH}}^{2})^{\hat{a}\hat{b}}\right]\,,
\int_{\zeta(1)}^{1}\epsilon^{-1/\gamma}\left(1-\epsilon^{2}\right)^{-1/2}d\epsilon=\int_{0}^{1}\epsilon^{-1/\gamma}\left(1-\epsilon^{2}\right)^{-1/2}d\epsilon-\int_{0}^{\zeta(1)}\epsilon^{-1/\gamma}\left(1-\epsilon^{2}\right)^{-1/2}d\epsilon.
S_{0}^{+Q}L=\{\left(\begin{matrix}{0}&{-i}\\{-i}&{0}\\\end{matrix}\right),\left(\begin{matrix}{i}&{0}\\{0}&{i}\\\end{matrix}\right),\left(\begin{matrix}{-1}&{0}\\{0}&{1}\\\end{matrix}\right),\left(\begin{matrix}{0}&{-i}\\{i}&{0}\\\end{matrix}\right)\}
\times\left(e^{t_{m}\partial^{2}}(\partialA+A\partial+AA)e^{t_{m-1}\partial^{2}}(\partialA+A\partial+AA)\ldots(\partialA+A\partial+AA)e^{t_{1}\partial^{2}}\right)_{C_{i}z_{i},C_{i+1}z_{i+1}}
\left(S_{I}^{64}\right)_{\alpha\beta}=\Gamma_{\alpha\beta}^{\bar{M}_{1}\bar{M}_{2}\bar{M}_{3}}P_{\bar{M}_{1}}F_{\bar{M}_{2}\bar{M}_{3}}^{42}+\Gamma_{\alpha\beta}^{\bar{M}_{1}\bar{M}_{2}\cdots\bar{M}_{7}}\,\,Z_{\bar{M}_{1}\cdots\bar{M}_{5}}^{+}F_{\bar{M}_{6}\bar{M}_{7}}^{42}
\begin{array}{rcl}{\int_{11}-\textstyle{\frac{1}{2}}\hat{\hat{G}}{}^{\star}\hat{\hat{G}}}&{=}&{\int_{9}\left\{-\textstyle{\frac{1}{2}}K^{6/7}F_{(4)}{}^{\star}F_{(4)}\,+\,\textstyle{\frac{1}{2}}K^{-3/7}\vec{F}_{(3)}^{\T}{\calM}^{-1}{}^{\star}\vec{F}_{(3)}\right.}\\{}&{}&{}\\{}&{}&{\left.-\textstyle{\frac{1}{2}}K^{-12/7}F_{(2)}{}^{\star}F_{(2)}\right\}\;.}\\\end{array}
\sqrt{-G}(R_{AB}-\frac{1}{2}G_{AB}R_{5})=-k_{5}^{2}\sum_{\alpha}\sqrt{-g_{\alpha}}g_{mn}^{\alpha}\delta_{A}^{m}\delta_{B}^{n}{\calL}_{\alpha}\delta(y-y_{\alpha})+2k_{5}^{2}\sqrt{-G}\LambdaG_{AB}.
A=\frac{\partial^{2}F}{(\partialv^{1})^{2}}\mid_{(t^{*},p_{i})},\quad\quadB=\frac{\partial^{2}F}{\partialv^{1}\partialt}\mid_{(t^{*},p_{i})},\,quad\quadC=\frac{\partial^{2}F}{\partialt^{2}}\mid_{(t^{*},p_{i})}.
N^{\mu\nu}=\overline{{\epsilon^{\prime}}}\gamma^{\mu\nu\lambda}\delta_{\epsilon}\psi_{\lambda}=\overline{{\epsilon^{\prime}}}\gamma^{\mu\nu\lambda}[\nabla_{\lambda}+{\textstyle{\frac{1}{4}}}H_{\lambda\rho\sigma}\gamma^{\rho\sigma}]\epsilon
L_{KdV}={\frac{1}{2}}\int_{-\infty}^{+\infty}dxdyu(x)\epsilon(x-y){\frac{\partialu(y)}{\partialt}}-\intdx[{\frac{u^{3}}{6}}-{\frac{1}{2}}({\frac{\partialu}{\partialx}})^{2}]
S=\frac{1}{8\pi}\int_{\Sigma}d^{2}x\left\{\partial\varphi\bar{\partial}\varphi+e^{\sqrt{\frac{2}{k}}\varphi}\left((\partial_{t}\gamma^{1})^{2}+(\partial_{x}\gamma^{1})^{2}+2\partial_{x}\gamma^{1}\right)\right\},
H_{c}=\dot{q}{\frac{\partialL}{\partial\dot{q}}}+{\frac{\partialL}{\partial(\partial_{t}\psi)}}\dot{\psi}+{\frac{\partialL}{\partial(\partial_{t}{\bar{\psi}})}}\dot{\bar{\psi}}-L={\frac{1}{2}}\left\{p^{2}+\right(U^{\prime}(q)\left)^{2}+U^{\prime\prime}(q)[\bar{\psi},\psi]_{-}\right\},
f^{(2)}=\left(\begin{array}{cccc}{0}&{-\delta_{ab}}&{\sigma_{a}}&{\pi_{a}}\\{\delta_{ba}}&{0}&{0}&{\sigma_{a}}\\{-\sigma_{b}}&{0}&{0}&{0}\\{-\pi_{b}}&{-\sigma_{b}}&{0}&{0}\\\end{array}\right)\,\delta(x-y),
\sum_{\lambda}\epsilon_{i}^{P(\lambda)}(\vec{q})\epsilon_{j}^{P(\lambda)}(\vec{q})=R_{ik}(q)\left(\delta_{kl}+{\frac{q_{k}q_{l}}{M^{2}}}\right)R_{lj}(k)=R_{ij}(q)=\delta_{kl}-{\frac{q_{k}q_{l}}{\vec{q}^{2}+M^{2}}}~.
\tilde{\theta}=\theta+\alpha\phi_{2}~,~\tilde{\Pi}_{\theta}=\Pi_{\theta}~,~\tilde{\alpha}=\alpha+{\frac{\phi_{1}}{\Pi_{\theta}}}~,~\tilde{\Pi}_{\alpha}=\Pi_{\alpha}-\Pi_{\theta}\phi_{2}~,~\tilde{\beta}=\beta-\phi_{2}~,~\tilde{\Pi}_{\beta}=\Pi_{\beta}~,~\tilde{\phi}_{i}=0.
\begin{array}{ll}{B_{1}={{\hat{A}}^{*\mu}}A_{\mu}}&{d_{1}\le0\,,}\\{B_{2}=\bar{\Psi}{}^{*}\Psi+\bar{\Psi}\Psi^{*}\qquad}&{d_{2}\le0\,,}\\{B_{3}=\varphi^{*}\varphi}&{d_{3}\le0\,.}\\\end{array}
v_{11}/v_{12}+\zeta(\mu_{jk})-\zeta(x)-\zeta(\mu_{jk}-x)={\frac{\sigma(\mu_{jk})\sigma(x-\mu_{jm})\sigma(x-\mu_{jk}+\mu_{jm})}{\sigma(\mu_{jm})\sigma(\mu_{jk}-\mu_{jm})\sigma(x-\mu_{jk})\sigma(x)}}.
\left\{\left(\begin{array}{c}{u}\\{0}\\\end{array}\right),\left(\begin{array}{c}{v}\\{0}\\\end{array}\right)\right\}=\left\{\left(\begin{array}{c}{0}\\{u}\\\end{array}\right),\left(\begin{array}{c}{0}\\{v}\\\end{array}\right)\right\}=0
D_{ij}(t-t^{\prime})=T_{ij}f_{i}(t)f_{j}(t^{\prime})=\hbar\left[A_{i}B_{j}\theta(t-t^{\prime})e^{-i\omega(t-t^{\prime})}+A_{j}B_{i}\theta(t^{\prime}-t)e^{i\omega(t-t^{\prime})}\right],
F_{2}(D,\beta)=-{\frac{1}{\pi^{D/2}\beta^{D}}}\sum_{n=0}\gamma_{D-1,n}\Gamma\left({\frac{D-2n-1}{2}}\right)\zeta(D-2n-1)d_{n}\left({\frac{\beta}{2}}\right)^{2n+1}~~~.
\langle{\calD}_{\alpha}v,{\calD}_{\beta}v\rangle=\int_{c_{\Lambda}}\Omega_{(\alpha)}^{(2,1)}\cdotq^{\Lambda\Sigma}\cdot\int_{c_{\Sigma}}\Omega_{(\beta)}^{(2,1)}=-\int_{CY}\Omega_{(\alpha)}^{(2,1)}\wedge\Omega_{(\beta)}^{(2,1)}=0\.
{\calH}_{W}=\sqrt{W_{1}\left(\Pi+\sqrt{1+\frac{W_{2}}{B_{0}^{2}}}B\right)^{2}+W_{1}\left(\frac{\sqrt{1+\frac{W_{2}}{B_{0}^{2}}}B}{R}\Pi-R\right)^{2}}\geq\sqrt{W_{1}}\left(\Pi+\sqrt{1+\frac{W_{2}}{B_{0}^{2}}}B\right).
\delta\varepsilon=\{\delta\varepsilon^{ij}\}=\left(\begin{array}{ccccc}{0}&{\alpha_{1}}&{\alpha_{2}}&{\alpha_{3}}&{0}\\{-\alpha_{1}}&{0}&{0}&{0}&{-\alpha_{1}}\\{-\alpha_{2}}&{0}&{0}&{0}&{-\alpha_{2}}\\{-\alpha_{3}}&{0}&{0}&{0}&{-\alpha_{3}}\\{0}&{\alpha_{1}}&{\alpha_{2}}&{\alpha_{3}}&{0}\\\end{array}\right)
CPT:\Psi(\vec{x},t)\rightarrow-iL^{0}e^{i(\alpha_{P}+\alpha_{C}+\alpha_{T})}\gamma^{0}\gamma_{5}\bar{\Psi}^{T}(-\vec{x},-t)~,~~~L^{0}=\left(\begin{array}{cc}{0}&{e^{i\eta}I_{4}}\\{e^{-i\eta}I_{4}}&{0}\\\end{array}\right)
B_{\alpha\beta}=\frac{-2\chi_{[\alpha}^{L}\chi_{\beta]}^{R}}{|\vec{\chi}^{R}|^{2}+(\vec{\chi}^{R}\vec{\chi}^{L})}\quad,\quadB_{\alpha\mu}=A_{\mu\alpha}^{(2)}+B_{\alpha\beta}A_{\mu}^{(1)\beta}\,
W=\left(\begin{matrix}{2}&{-1}&{0}&{0}&{\dots}\\{-1}&{2}&{-1}&{0}&{\dots}\\{0}&{-1}&{2}&{-1}&{\dots}\\{\vdots}&{\ddots}&{\ddots}&{\ddots}&{\vdots}\\{0}&{\dots}&{0}&{-1}&{2}\\\end{matrix}\right)
{\calH}=\mathrm{tr}\left[\vec{E}_{\bot}^{2}+\vec{B}^{\,2}\right]+\frac{1}{2}{\calE}_{3}^{2}+\frac{1}{2L^{2}}e_{3}^{\dagger}\left(\vec{x}_{\perp}\right)e_{3}\left(\vec{x}_{\perp}\right)+\frac{1}{2}\vec{\eta}\,^{2}\left(\vec{x}_{\perp}\right)\.\nonumber
\sum_{m_{1},\cdots,m_{l}\neq0}\epsilon^{m_{1}+\cdots+m_{l}}\oint\prod_{i=1}^{l}\frac{dt_{i}}{2\pii}\mathrm{e}^{-N\widetilde{\Phi}_{m_{i}}(t_{i})}f(m_{1},t_{1})\mathrm{det}^{(l)}B.
\frac{\dot{\alpha}^{2}}{\alpha^{2}}+\frac{\dot{\alpha}}{\alpha}\frac{\dot{\beta}}{\beta}-\frac{1}{12}\frac{\dot{\gamma}^{2}}{\gamma^{2}}-\frac{\alpha_{0}^{2}}{18}\frac{1}{\gamma^{2}}=-\frac{\sqrt{2}\alpha_{0}}{3}\frac{1}{\beta\gamma}\left(\delta(y)-\delta(y-\pi\rho)\right)\;,
V_{x}=3\,{x^{8}}+8\,{x^{7}}\,{y^{2}}-7\,{x^{6}}\,{y^{4}}+6\,{x^{5}}\,{y^{6}}-4\,{x^{3}}\,{y^{7}}+2\,x\,{y^{8}}=0
A_{1\,9}=\frac{i}{24}\times\frac{1}{3}\times(-\sqrt{3})\times2\times[3\times2i\sqrt{3}\mathrm{Tr}\tilde{B}_{2}^{9}(-2\mathrm{Tr}\tilde{(}{B}_{2}^{5_{3}})^{2})]=-2\;.