harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\prod_{a=1}^{M}a_{-m_{a}}^{i_{a}}|0\rangle*\prod_{b=1}^{N}a_{-n_{b}}^{j_{b}}|0\rangle=\prod_{a=1}^{M}a_{-m_{a}}^{i_{a}}\prod_{b=1}^{N}a_{-n_{b}}^{j_{b}}|0\rangle+\ldots\,,
R(\theta)=\frac{1}{\lambda-\lambda^{-1}}\left(\begin{array}{cccc}{e^{\theta}\lambda-e^{-\theta}\lambda^{-1}}&{}&{}&{}\\{}&{e^{\theta}-e^{-\theta}}&{\lambda-\lambda^{-1}}&{}\\{}&{\lambda-\lambda^{-1}}&{e^{\theta}-e^{-\theta}}&{}\\{}&{}&{}&{e^{-\theta}\lambda-e^{\theta}\lambda^{-1}}\\\end{array}\right)
\partial^{\mu}{\cal{G}}_{\mu\mu_{2n}...\mu_{s}}^{\{n-1\}}\;\;=\;\;{\frac{(s+1-2n)}{(s+2-2n)}}\,\partial_{(\mu_{2n}}\,{\cal{G}}_{\mu_{2n+1}...\mu_{s})}^{\{n\}}
S=\frac{1}{4\pi\alpha^{\prime}}\int_{\calM}d^{2}z\left(G_{ij}\partial_{\alpha}X^{i}\partial^{\alpha}X^{j}-2\pii\alpha^{\prime}B_{ij}\epsilon^{\alpha\beta}\partial_{\alpha}X^{i}\partial_{\beta}X^{j}\right).
{\bfB}=U_{3}^{-1}({\bfr})\left(\begin{matrix}{{\frac{\hat{\bfy}}{2y^{2}}}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{-{\frac{\hat{\bfy}}{2y^{2}}}}\\\end{matrix}\right)U_{3}({\bfr})+O(b/y^{3})\,.
ds^{2}=-\Big((1+{\frac{q_{0}}{r^{2}}})(1+{\frac{q_{1}}{r^{2}}})(1+{\frac{q_{2}}{r^{2}}})\Big)^{\frac{-2}{3}}dt^{2}+\Big((1+{\frac{q_{0}}{r^{2}}})(1+{\frac{q_{1}}{r^{2}}})(1+{\frac{q_{2}}{r^{2}}})\Big)^{\frac{1}{3}}
K\rightarrowK-\frac{\mu^{2s}}{(4\pi)^{3/2}}[\frac{-\pi^{3/2}}{6}\int_{S^{2}}ds(\frac{\lambda}{2}\hat{\phi}^{2})^{-s}+\frac{\sqrt{\pi}}{2}\int_{s^{2}}(\frac{\lambda}{2}\hat{\phi}^{2})^{1-s}ds]
{\bfD}_{\mu_{2}}(\sigma_{2}){\bfV}^{\mu_{1}}(\sigma_{1})={\frac{\delta{\bfV}^{\mu_{1}}(\sigma_{1})}{\deltaz^{\mu_{2}}(\sigma_{2})}}+\intd\sigma_{3}\,{\bf\Gamma}_{\mu_{2}\mu_{3}}^{\mu_{1}}(\sigma_{1},\sigma_{2},\sigma_{3}){\bfV}^{\mu_{3}}(\sigma_{3})=(D_{\mu_{2}}V^{\mu_{1}})|_{z(\sigma_{1})}\,\delta(\sigma_{1}-\sigma_{2})
\sum_{s_{i}}\sum_{s_{j}}\sum_{s_{k}}\,_{(N_{j},N_{k},0)}^{((N_{i}-N_{j}-N_{k})/2,0,0)}\phi_{(t)}^{(s)}\frac{N_{i}!N_{j}!N_{k}!}{2^{\Sigmas}s_{i}!s_{j}!s_{k}!t_{ij}!t_{ik}!t_{jk}!}=\frac{1}{2^{N}(d/2)_{N}}
h(t)=2\,{\mu_{+}}\,q_{1}(x_{0}^{+})\,\left(\begin{array}{rr}{\ell{\calK}_{0}^{\prime}}&{-\ell{\calF}_{0}^{\prime}}\\{-{\calK}_{0}}&{{\calF}_{0}}\\\end{array}\right)J_{0}\,\left(\begin{array}{rr}{{\calF}_{0}}&{\ell{\calF}_{0}^{\prime}}\\{{\calK}_{0}}&{\ell{\calK}_{0}^{\prime}}\\\end{array}\right)\,q_{1}^{-1}(x_{0}^{+})\qquad.
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}\mathrm{d}t(1-t)^{\sigma}\frac{\partial^{|\alpha|}F}{\partialp^{\alpha}}(tp).
{\mathrm{Res}}_{z=0}\frac{H_{\mu}^{(1)}(\lambdaz)}{H_{\nu}^{(1)}(z)}F(z)={\mathrm{Res}}_{z=0}\frac{J_{\nu}(z)J_{\mu}(\lambdaz)+Y_{\nu}(z)Y_{\mu}(\lambdaz)}{J_{\nu}^{2}(z)+Y_{\nu}^{2}(z)}F(z)
z(1-z)\partial_{z}^{2}g(z)+(1+2\kappa)(1-z)\partial_{z}g(z)+\left[\frac{1}{z}\left(\kappa^{2}+\frac{\omega^{2}\ell^{4}}{4r_{H}^{2}}\right)-\kappa^{2}\right]g(z)=0,
\frac{\partial}{\partialx^{\mu}}\Bigl(Tj^{\mu}(x){\calO}(0)\Bigr)=[j^{0}(x),{\calO}(0)]\delta(x^{0})+T\biggl(\frac{\partial}{\partialx^{\mu}}j^{\mu}(x){\calO}(0)\biggr)
\left(\left(D^{i}\right)_{\;\;b}^{a}+f_{\;\;bc}^{a}\pi^{0ic}\right)\Phi_{i}^{(2)b}=-\epsilon^{0ijk}f_{\;\;bc}^{a}\chi_{i}^{(1)b}\left(D_{j}\right)_{\;\;d}^{c}\chi_{k}^{(1)d}\approx0.
S_{b}=-\frac{1}{4g^{2}}Tr\left([A_{\mu},A_{\nu}]^{2}+2\sum_{i=1}^{3}[A_{\mu},B_{i}][A^{\mu},B_{i}^{\dagger}]+\frac{1}{2}\sum_{i,j=1}^{3}([B_{i},B_{j}^{\dagger}][B_{i}^{\dagger},B_{j}]+[B_{i},B_{j}][B_{i}^{\dagger},B_{j}^{\dagger}])\right).
\begin{array}{l}{<0|T\left(\varphi_{1}\varphi_{2}\varphi_{3}(\varphi\star\varphi\star\varphi)(x_{4})\right)|0>}\\{=\tau_{34}\{D_{14},D_{24},D_{34}\}_{\star}+\tau_{43}\{D_{14},D_{24},D_{43}\}_{\star},\mathrm{~for~}T_{12}T_{34},}\\\end{array}
\begin{array}{lcl}{I}&{\equiv}&{\{(\sigma,\tau):~0\leq\sigma<2\pi,~\tau_{i}\leq\tau<\tau_{0}\}}\\{}&{}&{}\\{II}&{\equiv}&{\{(\sigma,\tau):~0\leq\sigma<\sigma_{0},~\tau_{0}<\tau\leq\tau_{f}\}}\\{}&{}&{}\\{III}&{\equiv}&{\{(\sigma,\tau):~\sigma_{0}\leq\sigma<2\pi,~\tau_{0}<\tau\leq\tau_{f}\}}\\\end{array}
(A_{\xi},\phi_{\xi}):\qquad\begin{array}{rlrcl}{x}&{\to}&{x^{\prime}}&{=}&{\phi_{\xi}(x)}\\{e(x)}&{\to}&{e^{\prime}(x^{\prime})}&{=}&{e[\phi_{\xi}(x)]}\\{\hat{e}(x)}&{\to}&{\hat{e}^{\prime}(x^{\prime})}&{=}&{\hat{e}[\phi_{\xi}(x)]}\\{\psi(x)}&{\to}&{\psi^{\prime}(x^{\prime})}&{=}&{\rho[A_{\xi}(x)]\psi(x)\,.}\\\end{array}\qquad
\int_{0}^{1}dxh_{strong}^{4}(x)=\frac{4\pi^{6}}{5A^{5}}(\theta_{3}^{12}-\theta_{2}^{4}\theta_{3}^{8}-\theta_{2}^{8}\theta_{3}^{4}+\theta_{2}^{12})+\frac{6\pi^{4}}{5A^{4}}(\theta_{3}^{8}+\frac{2}{3}\theta_{3}^{4}\theta_{2}^{4}+\theta_{2}^{8})
\begin{array}{lll}{X_{37_{1}}=\gamma_{\omega_{2}}^{\frac{1}{2n}},}&{X_{37_{2}}=\gamma_{\omega_{1}}^{-\frac{1}{2n}}\gamma_{\omega_{2}}^{-\frac{1}{2n}},}&{X_{37_{3}}=\gamma_{\omega_{1}}^{\frac{1}{2n}}.}\\\end{array}
S^{n}=\frac{_1}{^2}T\alpha^{2}D_{n}^{+}\varphi_{n}^{\alpha}D_{n}^{+}\varphi_{n}^{\alpha}+\frac{_1}{^4}\mu_{n}T\left(\varphi_{n}^{\alpha}\varphi_{n}^{\alpha}-1\right)+
p^{2}\xi_{11}(p,q)-p.q\xi_{02}(p,q)={\frac{1}{2}}\left\{-{\frac{1}{2}}Z_{0}((p+q)^{2};m^{2})+{\frac{1}{2}}Z_{0}(q^{2};m^{2})+p^{2}\xi_{01}(p,q)\right\}\,.
\sum_{Q}m_{Q}^{4}K_{Qr}(p)=\tilde{k}_{r},\sum_{Q}m_{Q}^{4}K_{Qi}(p)=\tilde{k}_{i},\\\sum_{Q}m_{Q}^{4}H_{Q}(p)=\tilde{h},\\,sum_{Q}m_{Q}^{4}S_{Q}^{I}(p)=\tilde{s}^{I}.
\begin{array}{cl}{}&{\left.\displaystyle\sum_{s=\pm1}t_{k+s}^{k}(u_{2n-1}-u_{0})^{\varepsilon_{2n-1}}t_{k}^{k+s}(u_{2n}-u_{0})^{\varepsilon_{2n}}\right|_{u_{2n}=u_{2n-1}-1}}\\{=}&{\sqrt{-1}\{k\}\delta_{\varepsilon_{2n-1}+\varepsilon_{2n},0}.}\\\end{array}
\int_{B_{2}}\left(\kappa_{2}\,u_{\mu\nu}^{0}(x)T^{\mu\nu}+\kappa_{2}\sum_{n=1}^{\infty}\frac{\psi_{n}(R)e^{2kR}}{N_{0}}\,u_{\mu\nu}^{n}(x)T^{\mu\nu}-\frac{\kappa_{1}}{\sqrt{3}}\varphi\,T_{\mu}^{\mu}\right)dz.
\sum_{n\neq0}n\left[N_{n}+\widetilde{N}_{n}+\sum_{I=3}^{8}N_{n}^{(I)}+\sum_{b=1}^{8}F_{n}^{(b)}\right]=\frac{\sqrt{f}}{l_{s}}mR_{T}(\tilde{a}_{0}^{\dagger}+\tilde{a}_{0})~.
\left|\frac{\mathrm{det}^{\prime}L_{-}}{\left(\mathrm{det}^{\prime}(-{\triangle}_{0})\right)^{\frac{1}{2}}\left(\mathrm{det}^{\prime}(-{\triangle}_{1})\right)^{\frac{1}{2}}}\right|={\tau}_{R}^{-2}({\calM}).
Q^{a}=-i\frac{2J}{1+\epsilon\vert\xi\vert^{2}}\left(T_{00}^{a}+T_{0i}^{a}\xi_{i}+\epsilonT_{i0}^{a}\bar{\xi}_{i}+\epsilonT_{ij}^{a}\bar{\xi}_{i}\xi_{j}\right).
[D,Q^{\pmi}]=-\frac{1}{2}Q^{\pmi}\,,\quad[D,Q_{i}^{\pm}]=-\frac{1}{2}Q_{i}^{\pm}\,,\qquad[D,S^{\pmi}]=\frac{1}{2}S^{\pmi}\,,\quad[D,S_{i}^{\pm}]=\frac{1}{2}S_{i}^{\pm}\,,
\{Q_{(\alpha)},Q_{(\beta)}\}=g[\mathcal{C}\Gamma_{s}(\hat{M}^{\hat{a}\hat{b}})]_{\alpha\beta}\hat{M}_{\hat{a}\hat{b}}=-i(\mathcal{C}\gamma^{a})_{\alpha\beta}P_{a}-{\textstyle\frac{g}{2}}(\mathcal{C}\gamma^{ab})_{\alpha\beta}M_{ab}\,,
\begin{array}{rcccl}{d\psi_{+}\wedgeJ}&{=}&{\psi_{+}\wedgedJ}&{=}&{{\calW}_{1}^{+}\,J\wedgeJ\wedgeJ\,,}\\{d\psi_{-}\wedgeJ}&{=}&{\psi_{-}\wedgedJ}&{=}&{{\calW}_{1}^{-}\,J\wedgeJ\wedgeJ\,,}\\\end{array}
{\Psi}_{AR}^{(w)}\simeqN_{w}[\Psi_{R}^{(w)}+e^{-\piw/\kappa}\Psi_{L}^{(-w)}],\qquad{\Psi}_{AL}^{(w)}\simeqN_{w}[\Psi_{L}^{(w)}+e^{-\piw/\kappa}\Psi_{R}^{(-w)}].
\phi=\left(\nabla^{\dot{A}a}\nabla_{\dot{A}}^{b}\left(\nabla_{a}^{\dot{B}}\nabla_{\dot{B}b}+16X_{ab}\right)-\nabla^{\dot{A}a}\nabla_{a}^{\dot{B}}\left(\nabla_{\dot{A}}^{b}\nabla_{\dot{B}b}-16iY_{\dot{A}\dot{B}}\right)\right)W^{2}\overline{{W}}^{2}
Im\epsilon_{\pm}=\mp\left[\frac12\left\{\left[\left(R^{2}-\frac14\Gamma^{2}\right)^{2}+\left(\vec{R}\cdot\vec{\Gamma}\right)^{2}\right]^{\frac12}-\left(R^{2}-\frac14\Gamma^{2}\right)\right\}\right]^{\frac12}.
S~=~\intd^{2}x~\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+eA_{\nu}(\eta^{\mu\nu}-\epsilon^{\mu\nu})\partial_{\mu}\phi+\frac{1}{2}e^{2}A_{\mu}A^{\mu}~\right].
\ddot{\Phi}+\left(H-2\frac{\ddot{\sigma}}{\dot{\sigma}}\right)\dot{\Phi}+\left(2\dot{H}-2H\frac{\ddot{\sigma}}{\dot{\sigma}}\right)\Phi+\frac{k^{2}}{a^{2}}\Phi=-\frac{8\pi}{\mp^{2}}\,V_{s}\,\deltas\,.
\begin{array}{lcl}{{\mathcalL}_{w}}&{=}&{w_{-1}\starp+w_{0}}\\{}&{=}&{\{u_{-1}v_{-1}^{\prime}-u_{-1}^{\prime}v_{-1}\}\starp+\{u_{-1}v_{0}^{\prime}-u_{0}^{\prime}v_{-1}\}.}\\\end{array}
\Omega_{u}\left(1+r\,e^{2\pii}\right)\,=\,\left(\begin{matrix}{1}&{-2}\\{0}&{1}\\\end{matrix}\right)\,\left(\begin{matrix}{f^{(1)}(1+r)}\\{f^{(2)}(1+r)}\\\end{matrix}\right)=T_{1}\,\Omega_{u}(1+r)\.
({\widehat\Lambda}^{\alphaA}(x))_{~v}^{u}=(\sigma_{\nu})_{~\dot{\beta}}^{\alpha}\left(w^{\dot{\beta}u}\,{\bar{\mu}}_{~v}^{A}+\mu^{Au}\,{\bar{w}}_{~v}^{\dot{\beta}}\right)\frac{(x-x_{0})^{\nu}}{\sqrt{(x-x_{0})^{2}\Big[(x-x_{0})^{2}+\rho^{2}\Big]^{3}}}~~.
\left(I_{3};I_{3}^{\mu};I_{3}^{\mu\nu}\right)=\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\left(1;k^{\mu};k^{\mu}k^{\nu}\right)}{\left[(k+k_{1})^{2}-m^{2}\right]\left[(k+k_{2})^{2}-m^{2}\right]\left[(k+k_{3})^{2}-m^{2}\right]}.
L_{5}^{(A)dS}=\kappa\cdot\epsilon_{abcde}\left[\frac{1}{l}e^{a}R^{bc}R^{de}\pm\frac{2}{3l^{3}}e^{a}e^{b}e^{c}R^{de}+\frac{1}{5l^{5}}e^{a}e^{b}e^{c}e^{d}e^{e}\right].
\rho_{m_{0}\rightarrowm_{1}\!+m_{2}\!+m_{3}}^{D}\!=\!\!\left(\!\prod_{i=1}^{3}\!\int_{0}^{\infty}\!\frac{\mu_{i}d\mu_{i}}{(2\pi)^{n/2}}\!\right)\!\!\frac{2^{3-n/2}\theta(\lambda_{E})\pi^{n/2-1}}{\lambda_{E}^{3-n}(\mu^{2},\mu_{2}^{2},\mu_{3}^{2})\Gamma(n\!-\!1)}\rho_{m_{0}\rightarrowm_{1\mu_{1}}\!+m_{2\mu_{2}}\!+m_{3\mu_{3}}}^{D-n}.
\int\frac{d^{d}q}{{(2\pi)^{d}}}\int\frac{dq_{5}}{2\pi}g(q_{\mu},q_{5})+\int\frac{d^{d}q}{{(2\pi)^{d}}}\sum_{k\neq0}\int\frac{dq_{5}}{2\pi}e^{i(2\pikR)q_{5}}g(q_{\mu},q_{5}).
M_{k(k_{1}k_{2}...k_{r})}^{J}(x)=\underset{\substack{x_{i}\rightarrowx\,(\mathrm{all}\,i)}}{\mathrm{lim}}\{:\sum_{\{I_{i}\}}\gamma_{k,k_{1}k_{2}...k_{r}}^{J,I_{1}I_{2}...I_{r}}\prod_{i=1}^{r}O_{k_{i}}^{I_{i}}(x_{i}):\}
\frac{\partial}{\partialy}\intdx\,x\,a(x,y,t)=\frac{\partial}{\partialy}(y\,p_{0}(y,t))+{\frac{1}{2}}\frac{\partial^{2}}{\partialy^{2}}\intdx\,x^{2}\varphi(x,y,t)-\frac{\partialp_{0}}{\partialt}\,.
R_{\mu\nu}-\frac{1}{4~\cdot4!}(\tilde{F}_{5})_{\mu\rho\sigma\tau\delta}(\tilde{F}_{5})_{\nu}^{\,\,\,\rho\sigma\tau\delta}+2\kappa^{2}\frac{\delta\cal{L}}{\deltaG^{\mu\nu}}=\frac{1}{2}\left[\partial_{\mu}\phi\partial_{\nu}\phi+\mathrm{e}^{2\phi}\partial_{\mu}C_{0}\partial_{\nu}C_{0}\right]+T_{\mu\nu}
\epsilon\;=\;\left(\begin{array}{c}{\epsilon_{1}}\\{-\epsilon_{2}}\\\end{array}\right)\;,\;\Psi_{\mu}\;=\;\left(\begin{array}{c}{\Psi_{1\mu}}\\{-\Psi_{2\mu}}\\\end{array}\right)\;,\;\tilde{\chi}\;=\;\left(\begin{array}{c}{\tilde{\chi}_{1}}\\{-\tilde{\chi}_{2}}\\\end{array}\right)\;,\;\chi\;=\;\left(\begin{array}{c}{\chi_{1}}\\{\chi_{2}}\\\end{array}\right)\;,
{\calL}_{g,4A}=-\frac{1}{4}\sum_{n,m,l,k\neq(0,0,0)}(\sum_{j=0}^{N}a_{jn}a_{jm}a_{jl}a_{jk})g_{L}f^{abc}g_{L}f^{ade}\tilde{A}_{\mu}^{n}\tilde{A}_{\nu}^{m}\tilde{A}^{l~\mu}\tilde{A}^{k~\nu},
\omega_{0}^{\prime}=1,\\\\omega_{1}^{\prime}={\frac{1}{\sqrt{P^{2}}}}\notP,\\\\omega_{2}^{\prime}={\frac{1}{\sqrt{k^{2}}}}\notk,\\\\omega_{3}^{\prime}={\frac{-i}{2\sqrt{P^{2}k^{2}}}}[\notP,\notk]
\begin{array}{l}{{\frac{\delta}{\deltaC_{\mu}(\sigma)}}W[C]=p^{\mu}(\sigma)=(p^{0},p^{1},p^{2},p^{3})=c(\sigma)\lambda^{\mu}(x(\sigma))=}\\{=c(\sigma)\left({\frac{1+x_{-}^{2}+x_{1}^{2}+x_{2}^{2}}{2}},\;\;-x_{2}+x_{-}x_{1},\;\;x_{1}+x_{-}x_{2},\;\;-{\frac{1+x_{-}^{2}-x_{1}^{2}-x_{2}^{2}}{2}}\right)}\\\end{array}
S_{\mathrm{ef}}(\lambda)=S_{\mathrm{ef}}(\lambda=0)+\frac{i}{2}\lambda\left[\mathrm{Tr}V_{1\nu}^{~\mu}(\nabla)-\mathrm{Tr}V_{2\nu}^{~\mu}(\nabla)\right]-\frac{i}{4}\lambda^{2}\mathrm{Tr}[V_{1\nu}^{~\mu}(\nabla)]^{2}+{\calO}(({\calE}^{\mu\nu})^{2}),
\dot{r}^{2}=\left[{\frac{RR^{\prime}}{2{\calM}}}(F_{i}-F_{o})+{\frac{\calM}{2RR^{\prime}}}\right]^{2}-F_{i}=\left[{\frac{m}{\calM}}\left(1-{\frac{1}{(1+a^{2})}}{\frac{r_{-}}{r}}\right)+{\frac{\calM}{2RR^{\prime}}}\right]^{2}-F_{i}.
ds^{2}=\frac{1-\frac{2M}{\bar{r}}}{1-\frac{Q^{2}}{M\bar{r}}}dt^{2}+\frac{d\bar{r}^{2}}{\left(1-\frac{2M}{\bar{r}}\right)\left(1-\frac{Q^{2}}{M\bar{r}}\right)}+r^{2}d\Omega^{2}
\mathbf{M}\,[\phi(\alpha,s)]\,=\,\sum_{k}\frac{-b_{k}/\alpha}{s-h+\lambda_{k}}\,+\,\int_{1}^{\infty}\phi(\tau)\tau^{s-1}\,d\tau\,+\,\frac{1}{\alpha}\,\int_{1}^{\infty}\phi(\tau)\tau^{h-s-1}\,d\tau.
\frac{d\rho_{0}}{d\Lambda}=-\frac{1}{2}\frac{\lambda_{\chi}\,\frac{\partial}{\partial\rho}\left.\left(\frac{\partialU_{\Lambda}}{\partial\Lambda}\right)\right|_{(\rho_{0},\zeta_{0})}+\lambda_{\varphi\chi}\,\frac{\partial}{\partial\zeta}\left.\left(\frac{\partialU_{\Lambda}}{\partial\Lambda}\right)\right|_{(\rho_{0},\zeta_{0})}}{\lambda_{\varphi}\lambda_{\chi}-\lambda_{\varphi\chi}^{2}}\;,
{\calG}_{\mathrm{rel}}^{(2)}=\frac{\mu}{2}\left((1+r_{0}/r)\,d{{\bfr}}^{2}+r_{0}^{2}(1+r_{0}/r)^{-1}(d{\psi}+{\bfw}({\bfr})\cdotd{{\bfr}})^{2}\right).
\left(\zeta+a_{0}\frac{x_{1}^{4-n}z^{12-2n}}{\zeta}\right)\to\zeta+a_{0}\frac{1}{\zeta}z^{2}\prod_{i=1}^{N_{s}}(x_{1}-m_{i}^{4})\prod_{j=1}^{N_{f}}(z^{2}-m_{j}^{2})
W[A]=-\int_{0}^{\infty}\,\frac{dT}{T}e^{-Tm}\int_{x(0)=x(T)}{\calD}x(\tau)\delta[\dot{x}(\tau)^{2}-1]tr\Phi(C)Tr_{c}Pexp\left[ig\int_{0}^{T}\dot{x}\cdotA(x(\tau))\right],
H=\frac{1}{2}p_{i}^{A}p_{i}^{A}+\frac{1}{4}G_{ij}^{A}G^{ij\,A}-A^{0\,A}\nabla^{i}p_{i}^{A}-\frac{m^{2}}{2}A^{0\,A}A^{0\,A}+\frac{m^{2}}{2}A^{i\,A}A^{i\,A}.
W_{\vec{k}}^{(c)}=\sum_{n\ge1}{\frac{(-1)^{n-1}}{n}}\sum_{\vec{k}_{1},\cdots,\vec{k}_{n}}\delta_{\sum_{i=1}^{n}\vec{k}_{i},\vec{k}}\prod_{i=1}^{n}{\frac{W_{\vec{k}_{i}}}{z_{\vec{k}_{i}}}}.
x=\left(\begin{array}{cc}{v_{1}}&{-v_{2}^{\ast}}\\{v_{2}}&{v_{1}^{\ast}}\\\end{array}\right)\left(\begin{array}{cc}{u_{1}^{\ast}}&{u_{2}^{\ast}}\\{-u_{2}}&{u_{1}}\\\end{array}\right)~\frac{1}{\midu_{1}\mid^{2}+\midu_{2}\mid^{2}}
\left(\frac{\delta\rho}{\rho}\right)_{\Lambda}\sim\left(\frac{R_{\Lambda}}{R_{EQ}}\right)\left(\frac{\delta\rho}{\rho}\right)_{0}=(1+z)_{EQ}\left(\frac{\delta\rho}{\rho}\right)_{0}\sim\left(\frac{T_{EQ}}{T_{\Lambda}}\right)\left(\frac{\delta\rho}{\rho}\right)_{0}\sim(3\times10^{4})\,\left(\frac{\delta\rho}{\rho}\right)_{0}\.
d\mu_{\Lambda}=Z^{-1}e^{-\int_{\Lambda}d^{2}x:\lambda\phi(x)^{4}+\sigma\phi(x)^{2}:_{m_{0}}+m_{0}^{2}/2\int_{R^{2}}dx^{2}:\left(\phi\left(x\right)-\xi\right)^{2}:_{m_{0}}}d\mu_{m_{0},\xi},
\pi_{i}=g\bar{\Gamma}\psi_{i}^{\prime}\sqrt{2V_{eff}-\rho_{f}-p_{f}}e^{3T},\\\\pi_{f}=g\bar{\Gamma}({\frac{n_{f}^{2}\theta_{f}^{\prime}}{\rho_{f}+p_{f}}})\sqrt{2V_{eff}-\rho_{f}-p_{f})}e^{3T},
(\nabla_{1}-i\sigma_{p}\nabla_{2})\sum_{I}q_{p}^{I}(\sigma_{p}a_{1}^{I}+i\sigma_{p}\deltaa_{2}^{I})+2i\sigma_{p}\sum_{p^{\prime}}\beta_{pp^{\prime}}\Psi_{p^{\prime}}^{*}\delta\Psi_{p^{\prime}}=0\,.
G^{strip}(t_{1},\theta_{1},t_{2},\theta_{2})\;=\;\frac{1}{2\pi}\,\left(\begin{array}{cc}{\frac{1}{\bar{w}_{1}-\bar{w}_{2}}}&{\frac{1}{w_{1}-\bar{w}_{2}}}\\{}&{}\\{\frac{1}{\bar{w}_{1}-w_{2}}}&{\frac{1}{w_{1}-w_{2}}}\\\end{array}\right)\,.
\begin{aligned}{\Psi^{m_{1}m_{2}m_{3}p}\Psi_{n_{1}n_{2}n_{3}p}}&{{}=6\delta_{n_{1}n_{2}n_{3}}^{m_{1}m_{2}m_{3}}+9\Psi^{[m_{1}m_{2}}{}_{[n_{1}n_{2}}\delta_{n_{3}]}^{m_{3}]},}\\{\Psi^{m_{1}m_{2}p_{1}p_{2}}\Psi_{n_{1}n_{2}p_{1}p_{2}}}&{{}=12\delta_{n_{1}n_{2}}^{m_{1}m_{2}}+4\Psi^{m_{1}m_{2}}{}_{n_{1}n_{2}},}\\{\Psi^{mp_{1}p_{2}p_{3}}\Psi_{np_{1}p_{2}p_{3}}}&{{}=42\delta_{n}^{m}.}\\\end{aligned}
Q_{1}=\frac{1}{\sqrt{2m}}\sigma_{a}e_{a}^{\mu}(x)[-i\partial_{\mu}-A_{\mu}+\frac{1}{2}\sigma_{3}\omega_{\mu}(x)]\equiv\frac{1}{\sqrt{2m}}\sigma_{a}e_{a}^{\mu}(x)(-iD_{\mu}),
\Phi^{<}(\eta)=\Phi_{\mathrm{i}}+\frac{1}{3}\eta_{\mathrm{i}}\Phi_{\mathrm{i}}^{\prime}-\frac{1}{3}\eta_{\mathrm{i}}\Phi_{\mathrm{i}}^{\prime}\frac{\eta_{\mathrm{i}}^{3}}{\eta^{3}}.
[\hat{x}^{\mu},\hat{\pi}_{\nu}]=\imath\{x^{\mu},\pi_{\nu}\}_{D(\Phi_{4}^{(1)})}=\imath\delta_{\nu}^{\mu}\;,\;\;[\hat{\psi}^{a},\hat{\psi}^{b}]_{+}=\imath\{\psi^{a},\psi^{b}\}_{D(\Phi_{4}^{(1)})}=-\frac{1}{2}\eta^{ab}\;.
\sum_{n=1}^{\infty}(C^{\prime}U^{\prime})_{mn}v_{n}^{(\kappa)}=\nu(\kappa)v_{m}^{(\kappa)}\quad\mathrm{and}\quad\sum_{n=1}^{\infty}(U^{\prime}C^{\prime})_{mn}v_{n}^{(\kappa)}=\overline{{\nu}}(\kappa)v_{m}^{(\kappa)},
d\ell^{2}=\varepsilon\frac{d\eta^{2}}{\prod_{i\neqj,j+1}^{4}(z_{j}-z_{i})(\hat{\eta}^{2}-\eta^{2})}+\prod_{i\neqj,j+1}^{4}(z_{j}-z_{i})(\hat{\eta}^{2}-\eta^{2})d\psi^{2}
S=T\intd^{3}x\sqrt{-\gamma}[-\frac{1}{2}\gamma^{ab}L_{a}^{\mu}L_{b\mu}+\frac{1}{2}-\frac{\varepsilon^{abc}}{\sqrt{-\gamma}}(L_{a}^{\mu}L_{b}^{\nu}L_{c\mu\nu}+\frac{9}{10}L_{a}^{\mu}L_{b}^{\alpha}L_{c\mu\alpha}-\frac{1}{5}L_{a}^{\alpha}L_{b}^{\beta}L_{c\alpha\beta})]
S_{m}=\intd^{4}x\sqrt{{g}}[-\frac{1}{2}D_{\mu}\phi(D^{\mu}\phi)^{*}-\frac{1}{2}D_{\mu}\Sigma(D^{\mu}\Sigma)^{*}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}H_{\mu\nu}H^{\mu\nu}-V(|\phi|,|\Sigma|,\Lambda)],
\tilde{\calL}=\frac{1}{8\pi}\epsilon^{\mu\nu\lambda}(B_{\mu}^{a}\partial_{\nu}B_{\lambda}^{a}+\frac{1}{3}\epsilon^{abc}B_{\mu}^{a}B_{\nu}^{b}B_{\lambda}^{c})+\frac{1}{2g^{2}}(B_{\mu}^{a}+\omega_{\mu}^{a})(B^{\mu,a}+\omega^{\mu,a})
\psi=\left(\begin{array}{c}{1}\\{0}\\\end{array}\right)\otimes\left(\begin{array}{c}{\lambda^{\alpha,A}}\\{0}\\\end{array}\right)+\left(\begin{array}{c}{0}\\{1}\\\end{array}\right)\otimes\left(\begin{array}{c}{0}\\{\bar{\lambda}_{\bar{\alpha},A}}\\\end{array}\right)
P\psi=\left(\begin{array}{c}{0_{4\times1}}\\{\sqrt{m}\varphi}\\\end{array}\right),\P^{\mu}\psi=\frac{i}{\sqrt{m}}\left(\begin{array}{c}{0_{4\times1}}\\{\partial^{\mu}\varphi}\\\end{array}\right),\\square\varphi+m^{2}\varphi=0.
R_{\mu\nu}(W)=W_{\mu\nu,\beta}^{\beta}-{\frac{1}{2}}(W_{\mu\beta,\nu}^{\beta}+W_{\nu\beta,\mu}^{\beta})-W_{\alpha\nu}^{\beta}W_{\mu\beta}^{\alpha}+W_{\alpha\beta}^{\beta}W_{\mu\nu}^{\alpha},
B_{0}(M_{*}^{2})=\frac{\Gamma(\frac{d}{2})\,\Gamma(d-2+2k)\,\Gamma(k+\frac{1}{2})}{\sqrt{\pi}\,\Gamma(d-2)\,\Gamma(2k+1)\,\Gamma(k+\frac{d}{2})}M_{\ast}^{2k}\\cdot
\frac{\partialS_{2}}{\partial\tau^{A}}\delta_{1}\tau^{A}=-\mathscr{F}_{At}\mathscr{J}^{A}{}_{t}t^{2}-(\mathscr{F}_{tA}\mathscr{J}^{A}{}_{B}+\mathscr{F}_{AB}\mathscr{J}^{A}{}_{t})t\tau^{B}-\mathscr{F}_{AB}\mathscr{J}^{B}{}_{C}\tau^{A}\tau^{C}.
S_{GF+FP}=-\intd^{4}xi\delta_{B}\left[\bar{C}^{a}\left\{D_{\mu}[a]A^{\mu}+{\frac{\alpha}{2}}B\right\}^{a}-i{\frac{\zeta}{2}}gf^{abi}\bar{C}^{a}\bar{C}^{b}C^{i}-i{\frac{\zeta}{4}}gf^{abc}C^{a}\bar{C}^{b}\bar{C}^{c}\right],
\left(\begin{array}{cc}{0}&{-B}\\{B}&{0}\\\end{array}\right)\left(\begin{array}{c}{\dot{V}_{\mathrm{even}}}\\{\dot{V}_{\mathrm{odd}}}\\\end{array}\right)=\lambda\left(\begin{array}{cc}{-{\frac{1}{2}}B^{2}}&{0}\\{0}&{-{\frac{1}{2}}B^{2}}\\\end{array}\right)\left(\begin{array}{c}{\ddot{V}_{\mathrm{even}}}\\{\ddot{V}_{\mathrm{odd}}}\\\end{array}\right),
S_{self}=\frac{e^{2}{\hbar}^{2}}{2}\overline{{\sum}}_{n,m,r,s}\delta({\omega}_{nm}+{\omega}_{rs})\int_{-\infty}^{\infty}\frac{dq_{1}}{\frac{1}{c}{\omega}_{nm}^{2}-q_{1}^{2}+i\epsilon}\cdot(T_{nm}^{\mu}(q_{1})T_{rs,{\mu}}(-q_{1})-T_{nm}^{0}(0)T_{rs}^{0}(0)),
{\calH}_{W}=\frac{1}{2}\Pi_{-}^{a}\Pi_{-}^{a}+\frac{1}{2}F_{12}^{a}F_{12}^{a}+\frac{1}{2\eta^{2}}\sum_{i=1,2}\left(\Pi_{i}^{a}-F_{-i}^{a}\right)^{2}.
L_{rel}=\frac{1}{2}\sum_{k=1}^{N-1}({\dot{\rho}}_{k}^{2}+{\rho}_{k}^{2}{\dot{\phi}}_{k}^{2})-\frac{1}{2}\sum_{k=1}^{N-1}{\omega}^{2}{\rho}_{k}^{2}+\alpha\hbar\sum_{k=1}^{N-1}k{\dot{\phi}}_{k},
Q_{ij}=N_{00}^{11}\left(\begin{array}{cccc}{2}&{1}&{0}&{0}\\{1}&{2}&{0}&{0}\\{0}&{0}&{2}&{1}\\{0}&{0}&{1}&{2}\\\end{array}\right)+T\left(\begin{array}{cccc}{1}&{1}&{0}&{0}\\{1}&{1}&{0}&{0}\\{0}&{0}&{1}&{1}\\{0}&{0}&{1}&{1}\\\end{array}\right)+D\frac{1}{1-S\tilde{N}}SD^{T}\,.
N^{AB}=\bar{\epsilon}_{1}\Gamma^{ABC}D_{C}\epsilon_{2}+\textstyle{\frac{1}{8}}\bar{\epsilon_{1}}\Gamma^{C_{1}C_{2}}\epsilon_{2}F_{\\C_{1}C_{2}}^{AB}+\textstyle{\frac{1}{96}}\bar{\epsilon}_{1}\Gamma^{ABC_{1}\cdotsC_{4}}\epsilon_{2}F_{C_{1}\cdotsC_{4}}.
S_{n}^{j}=\sum_{q}\left(X_{q}^{1}X_{n-q}^{j}-X_{q}^{j}X_{n-q}^{1}\right)-2\piRnX_{n}^{j}=\sum_{q}\left([X_{q}^{1},X_{n-q}^{j}]\right)-2\piRnX_{n}^{j}
N_{n}={\frac{[(2n+a+b+c+d-1)\Gamma(n+a+b+c+d-1)\Gamma(n+a+c)\Gamma(n+a+d)]^{1/2}}{\Gamma(a+c)\Gamma(a+d)n!\,[\Gamma(n+b+c)\Gamma(n+b+d)]^{1/2}}}.
\rho^{(k_{1},k_{2},\omega,\pm)}=\frac{1}{2}\Bigl(5\omega^{2}+\frac{1}{V}(\tau\kappa_{1}^{2}+\frac{1}{\tau}\kappa_{2}^{2})\Bigr)\pm\frac{1}{2}\sqrt{\Bigl(5\omega^{2}+\frac{1}{V}(\tau\kappa_{1}^{2}+\frac{1}{\tau}\kappa_{2}^{2})\Bigr)^{2}-16\omega^{4}}.
+\frac{1}{2}\left([\Omega,U_{\alpha\beta}^{\gamma}]\bar{\calP}_{\gamma}(-1)^{\varepsilon_{\alpha}+\varepsilon_{\beta}+\varepsilon_{\gamma}}-\bar{\calP}_{\gamma}[\Omega,U_{\alpha\beta}^{\gamma}](-1)^{(\varepsilon_{\alpha}+\varepsilon_{\beta})\varepsilon_{\gamma}}\right),
L=\left(\begin{matrix}{p_{1}}&{ime^{(q_{1}-q_{2})/2}}&{0}&{-ime^{-q_{1}}}\\{-i\,me^{(q_{1}-q_{2})/2}}&{p_{2}}&{i\,me^{q_{2}}}&{0}\\{0}&{-i\,me^{q_{2}}}&{-p_{2}}&{i\,me^{(q_{1}-q_{2})/2}}\\{ime^{-q_{1}}}&{0}&{-i\,me^{(q_{1}-q_{2})/2}}&{-p_{1}}\\\end{matrix}\right),
D(x-x^{\prime},t-t^{\prime})=\frac{2a}{(2\pi)^{5}}\intd^{4}k\intd\omega\frac{e^{-ik(x-x^{\prime})-i\omega(t-t^{\prime})}}{\omega^{2}+a^{2}(k^{2}-m^{2}+i\epsilon)^{2}}.
\Omega(p)=\frac{p_{3}+ip_{1}}{p^{0}+p_{2}},\qquad\qquad\tilde{\Omega}(p)=\frac{p_{3}-ip_{1}}{p^{0}-p_{2}},\qquad\qquadC_{n}=\frac{\sqrt{|[2n]_{\nu}!|}}{2^{n}n!},
\tilde{H}=\frac{\sqrt{\alpha^{\prime}}}{2}g^{\mu\nu}(P_{\mu}-\tilde{Q}_{_{E}}{\calA}_{\mu}-\tilde{Q}_{_{M}}\tilde{\calA}_{\mu})(P_{\nu}-\tilde{Q}_{_{E}}{\calA}_{\nu}-\tilde{Q}_{_{M}}\tilde{\calA}_{\nu})+\frac{\sqrt{\alpha^{\prime}}}{2k^{2}}({\tilde{Q}_{_{E}}}^{2}+{\tilde{Q}_{_{M}}}^{2}),
\bigl[\bigl(\omega^{ij}\bigr)^{ab}\bigr]=\left(\begin{array}{cccc}{0}&{-\,i}&{0}&{0}\\{-\,i}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{0}\\\end{array}\right)\,\delta^{ij}\,,
S_{ph}+S_{int}=\frac{1}{2}\int_{0}^{t}ds\sum_{k}\sum_{r}\sum_{l}(\dot{q}_{rl}^{2}({\bfk})+k^{2}q_{rl}^{2}({\bfk})+2q_{rl}({\bfk})J_{rl}({\bfk})
\pi_{1}={\frac{\dot{\mu}_{1}}{2}}{\frac{\mu_{1}-\mu_{2}}{(\mu_{1}-z_{1})(\mu_{1}-z_{2})}},\quad\pi_{2}={\frac{\dot{\mu}_{2}}{2}}{\frac{\mu_{2}-\mu_{1}}{(\mu_{2}-z_{1})(\mu_{2}-z_{2})}}
Q_{2}^{13}\left(Q_{2}^{\dagger}\right)^{31}u_{B}({r})={\frac{1}{m}}\left[-{\frac{\partial^{2}}{\partialr^{2}}}-{\frac{1}{r}}{\frac{\partial}{\partialr}}+{\frac{\left(2\ell-\lambda\right)^{2}}{r^{2}}}-\lambda{\frac{\delta(r)}{r}}\right]u_{B}=Eu_{B}({r})
\psi_{a_{1}\cdotsa_{2k}}\;=\;-(2k)(2k-1)\:F_{\beta}\:N_{[a_{1}}\:(N^{\dagger}N)^{-1}\:D_{a_{2}}^{\dagger}\:F_{\beta}^{-1}\:\psi_{a_{3}\cdotsa_{2k}]}\:+\:\psi_{a_{1}\cdotsa_{2k}}^{(h)}\;\;\;,
\mathrm{dim}_{q}{\tilde{\rho}}_{\ell}={\frac{([N][N+1]\ldots[N+2\ell-1])\prod_{a=1}^{N-2}([N-a][N-a+1]\ldots[N-a+\ell-1])}{[\ell]!([N+2\ell-2][N+2\ell-3]\ldots[N+\ell-1])\prod_{a=1}^{N-2}([a][a+1]\ldots[a+\ell-1])}}
<0|T\phi_{+}(x,x_{5})\phi_{+}^{*}(y,y_{5})|0>={\frac{1}{2\pi}}\int{\frac{d^{4}p}{(2\pi)^{4}}}{\frac{e^{-ip(x-y)}}{p^{2}+i\epsilon}}J(x_{5},y_{5},pR_{5})\,
M_{U}(R_{1},R_{2})=\frac{M_{p}\,g_{U}}{{2\pi}\left|\eta\left(i\frac{R_{2}}{R_{1}}\right)\right|^{2}\left|\eta\left(iR_{1}R_{2}\right)\right|^{2}R_{2}}\times\frac{1}{\sqrt{1+g_{U}^{2}\frac{Y(R_{1},R_{2})}{16\pi^{2}}}}\;
I(\epsilon)=\frac{d}l\int_{\epsilon}d\rho\,d^{d}x\,\sqrt{\hat{g}}\,\rho^{-1-\frac{d}2}+\frac2l\intd^{d}x\sqrt{\hat{g}}\,\epsilon^{-\frac{d}2}(\epsilonh-1).
\sqrt{2}M_{j}=\pm\sum_{i}\frac{\nu_{i}\phi(a_{i})}{a_{i}^{2}-m_{j}^{2}}\mathrm{det}(a_{i}^{2}-m^{2})^{1/r_{i}}\prod_{j\nei}(a_{j}^{2}-a_{i}^{2})^{1-2r_{j}/r_{i}}\Lambda_{N=2}^{\frac{4(N_{c}-1)-2N_{f}}{r_{i}}}.
\langlex_{0};x_{1}|e^{\etaa_{0}^{\dagger}a_{1}^{\dagger}}|M-1;0\rangle={\frac{e^{-{\frac{1}{2}}(x_{0}^{2}+x_{1}^{2})}}{\pi^{1/2}\sqrt{(M-1)!}}}\sum_{n=0}^{\infty}{\frac{(\eta/2)^{n}}{n!}}H_{n+M-1}(x_{0})H_{n}(x_{1}),
{\calZ}_{_I}={\calZ}\,\times\,\Big(\int\,D\overline{\psi}^{\prime\prime}D\psi^{\prime\prime}\,e^{\,i\intd^{2}z\big[\,\frac{\delta^{2}}{m}\,\overline{\psi}^{\prime\prime}\psi^{\prime\prime}\,\big]}\,\Big)\,.
{\calE}=\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)\;\;\;{\calJ}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\;\;\;{\calK}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\mathrm{\act\on\}Q=\left(\begin{array}{c}{Q^{1}}\\{Q^{2}}\\\end{array}\right).
\begin{aligned}{\delta\chi}&{{}=\frac{1}{8}\Gamma^{\mu\nu}\epsilon\,F_{\mu\nu}\,,}&{\qquad\qquad\delta\psi_{rs}}&{{}=\frac{1}{8}\Gamma^{\mu\nu}\epsilon\,R_{\mu\nurs}+\cdots\,,}\\{\deltaF_{\mu\nu}}&{{}=-8D_{[\mu}(\bar{\epsilon}\Gamma_{\nu]}\chi)\,,}&{\deltaR_{\mu\nu}{}^{rs}}&{{}=-4D_{[\mu}(\bar{\epsilon}\Gamma_{\nu]}\psi^{rs}-2\,\bar{\epsilon}\Gamma^{[r}\psi^{s]}{}_{\nu]})+\cdots\,.}\\\end{aligned}
E_{C}(a,T)=\frac{5\,\xi^{2}}{32a\pi}+\frac{8a^{3}\pi^{3}\xi^{2}{T^{4}}}{45}+\frac{512a^{5}\pi^{5}\,\xi^{2}\,{T^{6}}}{945}+{\calO}(T^{7})