harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e^{2i\piA(0}=\left(\begin{array}{ccccc}{1}&{i\piX_{1}}&{-i\piX_{2}}&{...}&{i\piX_{23}}\\{0}&{1+2i\piX_{1}}&{-2i\piX_{2}}&{...}&{2i\piX_{23}}\\{0}&{2^{2}i\piX_{1}}&{1-2^{2}i\piX_{2}}&{...}&{2^{2}i\piX_{23}}\\{...}&{...}&{...}&{...}&{...}\\{0}&{2^{23}i\piX_{1}}&{2^{23}i\piX_{2}}&{...}&{1+2^{23}i\piX_{23}}\\\end{array}\right)
y\tilde{p}_{n,l}=\frac{h_{n,l}}{h_{n-1,l}}\tilde{p}_{n-1,l}+\sum_{j=1}^{m_{1}}g_{3j-1}(n,l)\frac{h_{n,l}}{h_{n-1+3j.l}}\tilde{p}_{n-1+3j,l}.
\lambda_{1,2}=1+{{{{\rho}^{\epsilon-2}}}/2}-{{3\,{{\rho}^{\epsilon}}}/2}+{{\rho}^{2\,\epsilon}}\pm{\sqrt{{{\left(1+{{{{\rho}^{\epsilon-2}}}/2}-{{3\,{{\rho}^{\epsilon}}}/2}+{{\rho}^{2\,\epsilon}}\right)^{2}}}-{{\rho}^{2\,\epsilon-2}}}},
|Dp,f,\eta\rangle_{\psi,T,R}^{(0)}=\Bigl[C\gamma^{0}\gamma^{3}\cdots\gamma^{p}\frac{1+(1/f)\gamma^{1}\gamma^{2}}{\sqrt{1+1/f^{2}}}\frac{1+i\eta\gamma}{1+i\eta}\Bigr]_{ab}|a,\tilde{b}\rangle
{\calA}_{\nu}=-\frac1{72\pi^{2}}\mathrm{Tr}_{C}\left(5[D_{\alpha},F_{\alpha\beta}]F_{\beta\nu}+[D_{\nu},F_{\alpha\beta}]F_{\alpha\beta}+[D_{\alpha},F_{\beta\nu}]F_{\beta\alpha}\right).
{\calK}^{T}M{\calK}=\left(\begin{array}{cc}{{\calK}^{(1)}}&{{\tilde{\calK}}^{(1)}}\\\end{array}\right)e^{\varphi}\left(\begin{array}{cc}{e^{-2\varphi}+C^{(0)\,2}}&{C^{(0)}}\\{C^{(0)}}&{1}\\\end{array}\right)\left(\begin{array}{c}{{\calK}^{(1)}}\\{{\tilde{\calK}}^{(1)}}\\\end{array}\right)\,,
\left(\begin{array}{c}{A_{10}^{\prime}}\\{A_{20}^{\prime}}\\\end{array}\right)\\equiv\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{1}&{-1}\\{1}&{1}\\\end{array}\right)\\left(\begin{array}{c}{A_{10}}\\{A_{20}}\\\end{array}\right)\
g^{-1}=\left(\begin{array}{cccc}{0}&{0}&{\delta^{\,2}\,\left(x-y\right)}&{0}\\{0}&{0}&{0}&{\nabla^{-2}}\\{-\,\delta^{\,2}\,\left(x-y\right)}&{0}&{0}&{0}\\{0}&{-\,\nabla^{-2}}&{0}&{0}\\\end{array}\right).\tag{8}
\langle{\calO}(u_{1}){\calO}(u_{2})\cdots{\calO}(u_{m})\rangle_{\mathrm{CFT}}=\int\prod_{i=1}^{m}[dx_{i}G_{B\partial}^{E}(x_{i},u_{i})]\langle\Phi(x_{1})\Phi(x_{2})\cdots\Phi(x_{m})\rangle_{T}~.
\tilde{L}_{+}^{**}=+2\epsilon^{\mu\nu\rho\sigma}\partial_{\nu}G_{\rho\sigmaab}^{+}\big(M^{+}\big)_{\mu\lambda}^{-1\abcd}\epsilon^{\lambda\theta\alpha\beta}\partial_{\theta}G_{\alpha\betacd}^{+}+\dots\\\.
\begin{array}{l}{\hat{h}^{abc}=e^{\hat{K}/2}h_{lmn}(t_{33}^{1/2})^{la}(t_{33}^{1/2})^{mb}(t_{33}^{1/2})^{nc}}\\{=e^{\hat{K}/2}(\varepsilon_{lmn}+(1-e^{-2\pii/M})s_{lmn})(t_{33}^{1/2})^{la}(t_{33}^{1/2})^{mb}(t_{33}^{1/2})^{nc}.}\\\end{array}
\mathrm{~H_{nm}^{\{{\bf~R}\}}(y_1,y_2)~}=R_{12}^{m}\sum_{k=o}^{\mu_{mn}}\left(\frac{R_{11}}2\right)^{\frac{n-k}2}\frac{(-1)^{k}n!m!y_{1}^{m-k}}{(n-k)!(m-k)!k!}H_{n-k}\left(\frac{R_{11}y_{1}+R_{12}y_{2}}{\sqrt{2R_{11}}}\right).
A*B=\left.e^{\frac{\theta}{2}(\partial_{y}\partial_{\bar{y}^{\prime}}-\partial_{y^{\prime}}\partial_{\bar{y}})}A(x:y,\bar{y})B(x:y^{\prime},\bar{y}^{\prime})\right|_{y=y^{\prime},\bar{y}=\bar{y}^{\prime}}
\left\langle\prod_{i=1}^{r_{k}}\overline{{\psi}}_{k}(x_{k,i})\Gamma_{+}\psi_{k}(x_{k,i})\overline{{\psi}}_{k}(y_{k,i})\Gamma_{-}\psi_{k}(y_{k,i})\right\rangle=\frac{1}{(2\pi)^{2r_{k}}}\frac{\prod_{i>j}^{r_{k}}(x_{k,i}-x_{k,j})^{2}(y_{k,i}-y_{k,j})^{2}}{\prod_{i,j}^{r_{k}}(x_{k,i}-y_{k,j})^{2}},
\deltaL_{TG}=\varepsilon^{\mu\nu\rho}(R_{\beta\mu\nu}^{\alpha}\delta\Gamma_{\alpha\rho}^{\beta}-\bar{R}_{\beta\mu\nu}^{\alpha}\delta\bar{\Gamma}_{\alpha\rho}^{\beta})-\partial_{\mu}\,[\varepsilon^{\mu\nu\rho}N_{\beta\nu}^{\alpha}(\delta\Gamma_{\alpha\rho}^{\beta}+\delta\bar{\Gamma}_{\alpha\rho}^{\beta})]
{\cal{S}}^{{[p_{1},...,p_{N}]}_{S}}\;\;=\;\;-\left(\prod_{i=1}^{N}\frac{1}{p_{i}!}\right)\intd^{D}x\;{\bar{\psi}}^{\;{\mu}_{1}^{1}...{\mu}_{p_{1}}^{1}...{\mu}_{1}^{i}...{\mu}_{p_{i}}^{i}...{\mu}_{p_{N}}^{N}}E_{{\mu}_{1}^{1}...{\mu}_{p_{1}}^{1}...{\mu}_{1}^{i}...{\mu}_{p_{i}}^{i}...{\mu}_{p_{N}}^{N}}(\psi)
L_{B}^{r}=q^{1/2}p_{\mu}\dot{x}^{\mu}-\;\frac{e}{1+q^{2}}(p^{2}-m^{2})+{\calB}\;\dot{e}+\frac{{\calB}^{2}}{1+q^{2}}+\dot{\bar{\lambda}}\frac{d}{d\tau}(\lambda\;e),
N^{Ii}N^{Ii}=-\frac{\gamma^{\tau\tau}\Pi_{\tau}^{-}+(\gamma^{\tau\sigma}\mp1/\sqrt{-\gamma})\Pi_{\sigma}^{-}}{\gamma^{\tau\tau}\Pi_{\tau}^{+}+(\gamma^{\tau\sigma}\mp1/\sqrt{-\gamma})\Pi_{\sigma}^{+}}~.
\mathrm{Tr}\left[\Gamma_{\mu_{v}}(k_{v^{\prime}},k_{v})P_{f}(k_{v})\Gamma_{\mu_{v+1}}(k_{v},k_{v+1})\cdots\Gamma_{\mu_{v^{\prime}}}(k_{v^{\prime}-1},k_{v^{\prime}})P_{f}(k_{v^{\prime}})\right]\,;
L=\intdx\Bigg[i\hbar\Psi^{*}\partial_{t}\Psi-{\frac{\hbar^{2}}{2m}}|(\partial_{x}\mpi\kappa^{2}\rho)\Psi|^{2}+b{\frac{\hbar^{2}\kappa^{4}}{2m}}\rho^{3}\Bigg]
{\bfS}_{BH}={\frac{\bfA}{4G_{N}}};\qquad{\bfA}=2\pi^{2}(r^{2}e^{2U})_{r=0}^{\frac{3}{2}}=2\pi^{2}\Big({\frac{Z_{hor}}{3}}\Big)^{\frac{3}{2}}
{N}_{[{j_{1}},\psi_{j_{1}}][{j_{2}},\psi_{j_{2}}]}^{\quad\quad\quad\quad[{j}_{3},\psi_{j_{3}}]}=\sum_{[m]}\sum_{\psi_{m}}{\frac{{S}_{[m,\psi_{m}][{j_{1}},\psi_{j_{1}}]}{S}_{[m,\psi_{m}][{j_{2}},\psi_{j_{2}}]}{S}_{[m,\psi_{m}][{j_{3}},\psi_{j_{3}}]}^{*}}{{S}_{[m,\psi_{m}][0]}}}\;\;\;\;.
\begin{array}{rcl}{A_{P}(Q)}&{=}&{\displaystyle\frac{i(k_{p_{i+1}}-k_{p_{i}})T_{i}+c}{i(k_{p_{i+1}}-k_{p_{i}})-c}A_{P^{\prime}}(Q)}\\{}&{\equiv}&{Y_{i}(k_{p_{i+1}}-k_{p_{i}})A_{P^{\prime}}(Q)~,}\\\end{array}
\psi_{1a}(x^{\mu})=\left(\begin{matrix}{\nu_{e}}\\{e^{-}}\\{u}\\{d}\\\end{matrix}\right)~,~~\psi_{2a}(x^{\mu})=\left(\begin{matrix}{\nu_{\mu}}\\{\mu^{-}}\\{c}\\{s}\\\end{matrix}\right)\,,
I=\intdx[\Omega^{\prime}\rho^{\prime}-\frac{\mu}{8}e^{2\rho}\Omega^{1-\lambda-\gamma/4}-\frac{1}{4}\Omegaf^{\prime2}+\frac{1}{4}e^{\chif-2\rho}\Omega^{1+\gamma/4-\epsilon/2}A^{\prime2}].
=\;\bigg[\prod_{a=1}^{N}(2n_{a})!\bigg]^{\frac{1}{p}}\;\bigg[\prod_{b<c}^{N}(2n_{b}+2n_{c})!\bigg]^{\frac{1}{q(N-1)}}\;\prod_{d=1}^{N}\Big(\xi\Big)^{2n_{d}}\;,
\sum_{m=-\infty}^{\infty}\mathrm{e}^{i\delta_{n}}\mathrm{e}^{-ip_{1}^{0}\left(n\beta\right)}=\sum_{m=-\infty}^{\infty}\frac{2\pi}{\beta}\delta\left(p_{1}^{0}-\frac{2\pi\left(n+\epsilon/2\right)}{\beta}\right).
\frac12\Omega_{+q}^{-2i}+2i\gamma_{q\dot{q}}^{i}\psi_{+2\dot{q}}^{1-}=0,\-\frac12\Omega_{-q}^{-2i}+2i\gamma_{q\dot{q}}^{i}\psi_{+2\dot{q}}^{2+}=0,
{\calL}_{cc}=-2N+e^{u}V_{\dot{5}}^{1}-\bar{\rho}\tau^{3}\psi_{\dot{5}}+i\bar{\psi}_{m}\gamma^{m}\rho+\frac{1}{2}e^{u}\bar{\psi}_{a}\tau^{2}\gamma^{ab}\psi_{b}-12e^{u}t^{2}.
\left\langle\begin{array}{cc}{r}\\{t}\\\end{array}\right\rangle_{n-1}\left\langle\begin{array}{cc}{s}\\{q}\\\end{array}\right\rangle_{n-1}=\left\langle\begin{array}{cc}{r}\\{q}\\\end{array}\right\rangle_{n-1}\left\langle\begin{array}{cc}{s}\\{t}\\\end{array}\right\rangle_{n-1}\frac{1}{p_{r,n-1}-p_{s,n-1}}+
\sum_{i=1}^{N}a_{0}^{i}=0\,,\sum_{i=1}^{N}a_{k}^{i}=0,\,\sum_{i=1}^{N}\sum_{j=1}^{k}a_{j}^{i}(-1)^{k-j}=0,\,\sum_{i=1}^{N}\sum_{j=1}^{k}a_{j}^{i}(c_{m})^{j}(-1)^{k-j}
M(A,\Phi,A^{\prime},\Phi^{\prime},T)=\left(\begin{array}{cc}{-iA_{\mu}({\bfX^{\mu}})D{\bfX}^{\mu}-i\Phi_{i}({\bfX}^{\mu})\tilde{D}{\bfX}^{i}}&{\overline{{T}}({\bfX}^{\mu})}\\{T({\bfX}^{\mu})}&{-iA_{\mu}({\bfX}^{\mu})D{\bfX}^{\mu}-i\Phi_{i}^{\prime}({\bfX}^{\mu})\tilde{D}{\bfX}^{i}}\\\end{array}\right)\,
{\frac{1}{2}}\{\Gamma\,,\,\Gamma\}\,=\,{\frac{\delta^{L}\Gamma}{\delta\phi_{cl\,\,}^{A}}}{\frac{\delta^{R}\Gamma}{\delta\phi^{\ast\,A}}}\,=\,-\,i\hbar\,(\,{\calA}.\Gamma\,)
V_{abc_{n}d_{n}}^{n+1}(p,T,M(T))=\frac{1}{2}\rho^{n+1}\Bigl(\frac{\Sigma_{ren.}^{1}(p,M(T))+\Sigma_{T}^{1}(p,M(T),T)}{2}\Bigr)^{n}v_{abc_{n}d_{n}}^{(n+1)}.
\theta_{2}(\tau)=2{\frac{\eta(2\tau)^{2}}{\eta(\tau)}}\quad,\theta_{3}(\tau)={\frac{\eta(\tau)^{5}}{\eta(2\tau)^{2}\eta({\frac{\tau}{2}})^{2}}}\quad,\theta_{4}(\tau)={\frac{\eta({\frac{\tau}{2}})^{2}}{\eta(\tau)}}\,.
d{\calL}_{2}\vert_{\Psi_{-}^{I}=0}=-2iE_{\dot{q}}^{-}\wedgeE_{\dot{q}}^{-}\wedgeE^{++}+{\frac{1}{2}}E^{i}\wedge(E^{\mp\mp}\wedge\Omega^{\pm\pmi}-4iE_{q}^{+}\wedgeE_{\dot{q}}^{-}\gamma_{q\dot{q}}^{i})
\prod_{i=1}^{n-1}\left[\intd^{D}x|f_{i}(x)|^{qq_{1}/q}\right]^{\frac{1}{qq_{1}/q}}\;\left[\intd^{D}x|f_{n}(x)|^{p}\right]^{\frac{1}{p}}\;=\;\prod_{i=1}^{n}\|f_{i}\|_{q_{i}}\;.
{\calL}_{\mathrm{I}}=(\partial_{\mu}\partial^{\mu}\phi-m^{2}\phi-\frac{\lambda}{6}\phi^{3})\psi-\frac{\lambda}{4}\psi^{2}\phi^{2}-\frac{\lambda}{6}\psi^{3}\phi-\frac{\lambda}{4!}\psi^{4}\;.
c_{1}=\frac{1}{f_{1}}\sqrt{\frac{X(g_{1}+g_{2})}{p_{3}}},\qquadc_{2}=\frac{\sqrt{Xg_{1}p_{1}}}{f_{2}g_{1}},\qquadc_{3}=\frac{\sqrt{Xg_{2}p_{2}}}{f_{3}g_{2}}.
{\calL}=\frac{1}{2}\cdot\frac{\left(\partial_{\mu}\vec{\varphi}\right)^{2}}{\left(1+\frac{g^{2}}{4}\vec{\varphi}^{2}\right)^{2}}\qquad\left(\vec{\varphi}=(\varphi^{1},\cdots,\varphi^{N})\right),
\lambda_{1}=\left(\frac{\nu+1}{\nu-1}\right)^{\nu-1}\,\R_{1}=x_{1}^{-1/2}=\left(\frac{\nu+1}{\nu-1}\right)^{1/2}>1\,\s_{1}=1-\left(\frac{\nu-1}{\nu+1}\right)^{2}\,,
\left(\phi_{W}\star\psi_{W}\right)\left(\hat{x}\right)\equiv\phi\left(\hat{x}\right)\psi\left(\hat{x}\right)=\frac{1}{\left(2\pi\right)^{n}}\intd^{n}kd^{n}q\,e^{-i(k+q)\hat{x}}e^{ik\hat{x}}\tilde{\phi}\left(q+k\right)\tilde{\psi}\left(-k\right).
\theta_{(o)r}G_{(\sigma)}(z_{(o)r},z^{\prime})=(c_{r}z+d_{r})^{-2}\left(\thetaG_{(\sigma)}(z,z^{\prime})+\sum_{N_{r}=\mu_{r},\nu_{r}}\tilde{Y}_{\sigma,N_{r}}(t)\Phi_{\sigma,{N_{r}}}^{(0)}(z^{\prime})\right)
\begin{array}{lll}{}&{}&{0<(\psi-\sqrt{\phi^{2}-1})^{1/2}\leqx_{j}\leq(\psi+\sqrt{\phi^{2}-1})^{1/2}\;}\\{}&{\mathrm{~or~}}&{\phantom{0}-(\psi+\sqrt{\phi^{2}-1})^{1/2}\leqx_{j}\leq-(\psi-\sqrt{\phi^{2}-1})^{1/2}<0\;,}\\\end{array}
R_{lg}={\frac{-k_{1}-\sqrt{\left({\frac{n_{I}}{n_{T}}}\right)^{2}\omega^{2}-\left(k_{2}^{2}+\cdots+k_{m}^{2}\right)}}{-k_{1}+\sqrt{\left({\frac{n_{I}}{n_{T}}}\right)^{2}\omega^{2}-\left(k_{2}^{2}+\cdots+k_{m}^{2}\right)}}},
f_{1}(y)=\pi(\omega)^{2}\int^{y}dx\lbrace\frac{Q^{2}}{2x}-\frac{\omega^{2}R^{4}}{2x^{3}}\rbracef_{0}(x)\left[J_{1}(x)Y_{1}(y)-J_{1}(y)Y_{1}(x)\right].
\sigma_{\mathrm{abs}}^{(l+1,l)}=\frac{\pi(l+1)(l+2)}{\omega^{3}}\frac{f_{\mathrm{abs}}}{f_{\mathrm{in}}}=\frac{\pi^{2}r_{0}^{3}(l+1)(l+2)}{r_{0}\omega}|A|^{-2},
\mathrm{[}\hat{\Lambda}(k^{1}),\hat{\Lambda}(k^{\prime\,1})]=[\hat{\Lambda}^{\dagger}(k^{1}),\hat{\Lambda}^{\dagger}(k^{\prime\,1})]=0;\,\,\,[\hat{\Lambda}(k^{1}),\hat{\Lambda}^{\dagger}(k^{\prime\,1})]=\delta(k^{1}-k^{\prime\,1}),
\Psi_{k_{1},k_{1z};k_{2},k_{2z};I,I_{z}}^{r_{1},r_{2}}(\theta,\alpha,\beta,\gamma,\alpha_{I},\beta_{I},\gamma_{I})=\sqrt{\frac{p!}{m_{1}!m_{2}!m_{3}!m_{4}!}}\Psi_{1}^{m_{1}}\Psi_{2}^{m_{2}}\Psi_{3}^{m_{3}}\Psi_{4}^{m_{4}},
\int_{0}^{\pi}C_{m_{n-2}-m_{n-1}}^{m_{n-1}+\frac{1}{2}}(cos{\theta_{2}})C_{l_{n-2}-l_{n-1}}^{l_{n-1}+\frac{1}{2}}(cos{\theta_{2}})[sin{\theta_{2}}]^{m_{n-1}+l_{n-1}}d{\theta_{2}}\int_{0}^{2\pi}\frac{e^{i(\pmm_{n-1}\mpl_{n-1})\theta_{1}}}{sin{\theta_{1}}}
{\frac{1}{[(p+k)^{2}+m_{n}^{2}](p^{2}+m_{n}^{2})}}=\int_{0}^{1}{\frac{dz}{[p^{2}+2kp(1-z)+k^{2}(1-z)+m_{n}^{2}]^{2}}}=\int_{0}^{1}{\frac{dz}{[(p^{\prime})^{2}+k^{2}z(1-z)+m_{n}^{2}]^{2}}},
{\calL_{BCS}}=\psi_{\uparrow}^{\dagger}(i\partial_{t}-\epsilon(p))\psi_{\uparrow}+\psi_{\downarrow}^{\dagger}(i\partial_{t}-\epsilon(p))\psi_{\downarrow}-\lambda\psi_{\uparrow}^{\dagger}\psi_{\downarrow}^{\dagger}\psi_{\downarrow}\psi_{\uparrow}\,\,\,\,,
\psi_{\nu}^{(1)}(\zeta)=\left({\frac{\pi}{2\zeta}}\right)^{1/2}J_{\nu+{\frac{1}{2}}}(\zeta),\quad\psi_{\nu}^{(2)}(\zeta)=\left({\frac{\pi}{2\zeta}}\right)^{1/2}Y_{\nu+{\frac{1}{2}}}(\zeta),\quad
\bar{\psi}(p_{0},\vec{p})=-\frac{1}{2\pii}\int_{w_{+}(\vec{p})}^{\infty}dq_{0}\frac{f^{*}(q_{0},\vec{p},W)}{p_{0}-q_{0}+io}+\frac{1}{2\pii}\int_{-\infty}^{w_{-}(\vec{p})}dq_{0}\frac{g^{*}(q_{0},\vec{p},W)}{p_{0}-q_{0}-io}
Z_{left}(\tau)=\prod_{I=1}^{16}Z_{0,2nRA_{y}^{I}}^{0,2mRA_{y}^{I}}(\tau)+\prod_{I=1}^{16}Z_{1,2nRA_{y}^{I}}^{0,2mRA_{y}^{I}}(\tau)+\prod_{I=1}^{16}Z_{0,2nRA_{y}^{I}}^{1,2mRA_{y}^{I}}(\tau)+\prod_{I=1}^{16}Z_{1,2nRA_{y}^{I}}^{1,2mRA_{y}^{I}}(\tau)
\hat{Q}(x,l,\alpha,\bar{\beta})=\intd\mu(\gamma,\bar{\gamma})\frac{(\alpha\bar{\gamma})^{2l+ix}}{\Gamma(2l+ix+1)}\prod_{m=1}^{N}K_{m}(x)\frac{(\gamma\bar{\beta})^{2l+ix}}{\Gamma(2l+ix+1)}.
F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\;,\;D_{\mu}\varphi_{1}\equiv\partial_{\mu}\varphi_{1}+eA_{\mu}\varphi_{2}\;,\;D_{\mu}\varphi_{2}\equiv\partial_{\mu}\varphi_{2}-eA_{\mu}(\varphi_{1}+\fracme)
H\left(K;Z,Z^{\prime}\right)=B\left(K;Z,Z^{\prime}\right)+\int_{\Gamma\left(K\right)}B\left(K;Z,Z^{\prime\prime}\right)H\left(K;Z^{\prime\prime},Z^{\prime}\right)G\left(K;Z^{\prime\prime}\right)\dZ^{\prime\prime}\tag{2.2}
{\calG}_{n}=\sum_{m=0}^{\infty}g^{m}\Omega_{m}[\sum_{l=1}^{k-1}{\calG}_{k-l-1}^{(0)}{\calG}_{l+n+4m-k-1}^{(0)}+\sum_{l=k+1}^{n+4m}{\calG}_{l-k-1}^{(0)}{\calG}_{n+4m+k-l-1}^{(0)}].
\Bigr[({\varepsilon}_{abc}Q_{b}^{*}Q_{c}^{*}Q_{m}^{*}),({\varepsilon}_{abc}Q_{b}^{*}Q_{c}^{*}Q_{l}{\varepsilon}_{mn}Q_{m}^{*}Q_{n}^{*}),({\varepsilon}_{abc}Q_{a}^{*}Q_{b}^{*}Q_{c}^{*}Q_{d}Q_{m}^{*})\Bigl]
H={\frac{\hbar^{2}}{2m}}\Bigl(\int\mathrm{d}x\,\Psi^{*\prime}\Psi^{\prime}+\nu(\nu-1)\int\mathrm{d}x\,\mathrm{d}y{\frac{:\rho(t,x)\rho(t,y):}{(x-y)^{2}}}\Bigr).
n_{\mathrm{eff}}(n,\eta)=n{\frac{\lambda(\eta)}{\ell_{_\mathrm{C}}}}\mathrm{tanh}^{1/p}\biggl[\biggl({\frac{\ell_{_\mathrm{C}}}{\lambda(\eta)}}\biggr)^{p}\biggr],\quadn_{\mathrm{eff}}^{2}(n,\eta)=n^{2}+n^{2}b_{m}\biggl[{\frac{\ell_{_\mathrm{C}}}{\lambda(\eta)}}\biggr]^{2m},
\Psi_{0}^{(1)}=Ne^{-\pix_{4}^{2}/V}\theta_{2}(\tau|\bar{z})\quad;\quad\Psi_{0}^{(2)}=Ne^{-\pix_{4}^{2}/V}\theta_{3}(\tau|\bar{z})\quad;\quad\tau\equiv2i{\frac{R_{4}}{R_{5}}}\,
\int_{0}^{\infty}\frac{dt}{t^{1+s}}\left[\alpha\frac{C^{2}}{t^{2}}+\beta\frac{C}{t}\right]e^{-C/t}=s(\alphas+\alpha+\beta)\int_{0}^{\infty}\frac{dt}{t^{1+s}}e^{-C/t}
\begin{array}{lcr}{\delta\Phi^{A}(\theta)}&{=}&{\muU\Phi^{A}(\theta)+(\Phi^{A}(\theta),X-S)\mu,}\\{\delta\Phi_{A}^{*}(\theta)}&{=}&{\muV\Phi_{A}^{*}(\theta)+(\Phi_{A}^{*}(\theta),X-S)\mu.}\\\end{array}
T_{em~\mu}^{~~\mu}\equivT_{~\mu}^{\mu}-T_{~\mu}^{\mu}|_{F_{\mu\nu}=0}\approx-\frac{T_{3}}2\frac{V(T)}{\sqrt{1-\dot{T}^{2}}}\dot{T}^{2}\left(\frac{{\bfE}^{2}}{1-\dot{T}^{2}}+{\bfB}^{2}\right).
S_{g}=\intd^{D}x\sqrt{|g_{D}|}e^{-\varphi}\left[R_{D}-\omega\left(\nabla\varphi\right)^{2}+\frac{1}{4}\nablaf^{ab}\nablaf_{ab}-\frac{1}{4}e^{-2\varphi/d}f_{ab}{F_{\mu\nu}}^{a}F^{\mu\nub}\right]
\times\left\{e^{2i\omega_{12}}|z_{+}^{+}z_{+}^{-}|^{2}\,+\,\left|\frac{1}{\sqrt{2}}\left(z_{+}^{+}z_{-}^{-}+z_{-}^{+}z_{+}^{-}\right)\right|^{2}\,+\,\left|\frac{1}{\sqrt{2}}\left(z_{+}^{+}z_{-}^{-}-z_{-}^{+}z_{+}^{-}\right)\right|^{2}\,+\,e^{-2i\omega_{12}}|z_{-}^{+}z_{-}^{-}|^{2}\right\}^{n}\\\,
{\bfL}=-\intR^{2}d\Omega\,\biggl\{\dot{f}\,({\bfr}\times\nabla)f+f^{2}\,(\dot{\theta}+A_{0})\,{\bfr}\times(\nabla\theta+{\bfA})-\rho_{e}\,{\bfr}\,({\bfr}\cdot\nabla\times{\bfA})\biggr\}.
<T_{i}^{k}[\tilde{g}_{lm}]>=(\frac{\eta}{\tilde{g}})^{\frac{1}{2}}<T_{i}^{k(M)}[\eta_{lm}]>-{\frac{1}{2880\pi^{2}}}\left[{\frac{1}{6}}^{(1)}\tilde{H}_{i}^{k}-^{(3)}\tilde{H}_{i}^{k}\right]
\intd^{2}x|x|^{-a-s/4}|1-x|^{-b-t/4}=\pi\frac{\Gamma\left(1-\frac{a}{2}-\frac{s}{8}\right)\Gamma\left(1-\frac{b}{2}-\frac{t}{8}\right)\Gamma\left(\frac{a+b}{2}-1-\frac{u}{8}\right)}{\Gamma\left(2-\frac{a+b}{2}+\frac{u}{8}\right)\Gamma\left(\frac{a}{2}+\frac{s}{8}\right)\Gamma\left(\frac{b}{2}+\frac{t}{8}\right)}
E=\intd^{2}x\biggl\{\frac{1}{2}\dot{f}^{2}+\frac{1}{2}(\partial_{i}f)^{2}+\frac{1}{2}f^{2}(\dot{\theta}+A_{0})^{2}+\frac{1}{2}f^{2}(\partial_{i}\theta+A_{i})^{2}+U\biggr\}.
\left|\begin{array}{lccr}{L_{11}-\lambda}&{L_{12}}&{L_{13}}&{L_{14}}\\{L_{21}}&{L_{22}-\lambda}&{L_{23}}&{L_{24}}\\{L_{31}}&{L_{32}}&{L_{33}-2k^{2}-\lambda}&{L_{34}}\\{L_{41}}&{L_{42}}&{L_{43}}&{\Pi_{C}+k^{2}-\lambda}\\\end{array}\right|=0
W_{ik~ps}^{~jl~nm}-W_{ik~sp}^{~jl~mn}=(r_{A})_{ki}^{~nm}\delta_{p}^{~l}\delta_{s}^{~j}+(r_{A})_{ik}^{~nm}\delta_{p}^{~j}\delta_{s}^{~l}-\delta_{k}^{~n}(r_{B})_{pi}^{~lm}\delta_{s}^{~j}-
\Pi_{_h\,\rho,}^{\rho}\,^{\mu\nu}\left(k\right)=-\kappa\;\Gamma_{_h\,}^{\mu\nu}=-\kappa^{2}\omega\frac{\pi^{2}T^{4}}{180}\left(4\,\eta^{\mu0}\eta^{\nu0}-\eta^{\mu\nu}\right),
E_{l}\;=\;\sum_{a=1}^{b-N}\sum_{i_{a}=1}^{k_{a}}\;e_{a}(p_{a}^{i_{a}})\qquad,\qquadp_{l}\;=\;\sum_{a=1}^{b-N}\sum_{i_{a}=1}^{k_{a}}\;p_{a}^{i_{a}}.
F^{R}(\tau,\sigma,\nu)=\prod_{n=1}^{\infty}\frac{(1+z^{\frac{1}{2}}\zeta^{\frac{1}{2}}q^{n})(1+z^{-\frac{1}{2}}\zeta^{\frac{1}{2}}q^{n-1})(1+z^{\frac{1}{2}}\zeta^{-\frac{1}{2}}q^{n})(1+z^{-\frac{1}{2}}\zeta^{-\frac{1}{2}}q^{n-1})}{(1-q^{n})^{2}(1-zq^{n})(1-z^{-1}q^{n-1})}.
I_{YM}^{0}=\int_{M}\mathrm{Tr}F_{t}^{*}F_{t}=\frac{1}{2}\int_{M}d^{4}x\\mathrm{Tr}\left[\left(F_{\mu\nu}+{\calA}_{\mu\nu}\right)\left(F^{\mu\nu}+{\calA}^{\mu\nu}\right)\right]
S_{eff}=-8\frac{g^{2}}{(4\pi)^{2}}\,\intd^{4}x\epsilon^{\mu\nu\rho\alpha}a_{\mu}\partial_{\nu}b_{\rho\alpha}=-8\frac{g^{2}}{(4\pi)^{2}}\,{\calS}_{BF}({\calA}).
\wp=-{\frac{p-1}{256\piG}}\left[(p-3)(p^{2}-p+2)\bar{\calE}^{2}\bar{\calP}_{\phi}^{-2{\frac{p-1}{p-3}}}a^{2{\frac{p+1}{p-3}}}-(p-2)(p-7)(p+1){\frac{1}{a^{2}}}\right].
\mathrm{ind}_{1}^{D=4,6}(\mathrm{SU}(N))=-\frac{1}{N^{2}}\qquad\mathrm{and}\qquad\mathrm{ind}_{1}^{D=10}(\mathrm{SU}(N))=-\sum_{m|N\atopm>1}\frac{1}{m^{2}}.
e^{B}\left(\frac{1}{2}A^{\prime\prime}-\frac{1}{2}B^{\prime\prime}+\frac{D_{1}}{4}{A^{\prime}}^{2}-\frac{1}{4}{B^{\prime}}^{2}-\frac{1}{4}(D_{1}-1)A^{\prime}B^{\prime}\right)=\frac{\kappa^{2}}{2a^{2}}(\frac{1}{e^{2}}{P^{\prime}}^{2}+2v^{2}f^{2}P^{2})~.
D_{++}^{ab}(x)={\frac{i\delta^{ab}}{\pi^{2}}}\intd^{2}k\,e^{ikx}{\frac{k_{+}^{2}}{(k^{2}+i\epsilon)^{2}}}={\frac{\delta^{ab}(x^{-})^{2}}{\pi(-x^{2}+i\epsilon)}}\.
\nabla^{2}\equiv-{\frac{1}{\sqrtg}}\partial_{i}\sqrtgg^{ij}\partial_{j}={\frac{1}{\sqrtg}}\delta_{ij}\partial_{i}\partial_{j}=({\frac{1+\vec{x}^{2}}{2}})^{2}(\partial_{1}^{2}+\partial_{2}^{2})\,.
G_{\mathrm{scal}}(z,D)=-{(4\pi)}^{-D}{\frac{e^{2}}{2}}\int_{0}^{\infty}{\frac{dT}{T}}e^{-m^{2}T}T^{2-D}\int_{0}^{1}du_{a}\,f(z,D)G_{Bab}^{1-{\frac{D}{2}}}\nonumber\,
\tilde{D}_{1}B_{1}(w_{\pm})+\tilde{D}_{2}B_{2}(w_{\pm})+\tilde{D}_{3}=\pm\kappa_{5}^{2}l^{2}e^{-2A_{\pm}-\alpha_{\pm}^{(0)}}\bar{\tau}_{\pm(T)}^{[1]}.
=e^{2\rho}\left\{\left(1-\frac{wN}{6}e^{2\phi}\right)e^{2\phi}-\left[1-(w-2)\frac{N}{6}e^{2\phi}\right]Q^{2}e^{4\phi}-\left[1-(w+2)\frac{N}{6}e^{2\phi}\right]\Lambda\right\}.
{\calL}_{F}=\overline{{\Psi}}_{L}i\gamma^{\mu}\partial_{\mu}\Psi_{L}+\overline{{\Psi}}_{R}i\gamma^{\mu}(\partial_{\mu}+U^{\dagger}\partial_{\mu}U)\Psi_{R}-m[\overline{{\Psi}}_{R}\Psi_{L}+\mathrm{h.c.}].
{\calH}=\intd^{d}x\Bigl[{\frac{1}{2}}\sum_{i}(\partial_{\mu}\Phi_{i})^{2}+{\frac{1}{2}}\sum_{i}r_{i}\Phi_{i}^{2}+{\frac{1}{4!}}\sum_{ijkl}u_{ijkl}\;\Phi_{i}\Phi_{j}\Phi_{k}\Phi_{l}\Bigr],
\bar{\Psi}_{\mathrm{CT}}=-\frac{i}{2e^{2}}\intd^{4}x\,\sqrt{g}\,\Bigr\{\xi_{-}^{\nu}(\nabla^{\mu}T_{\mu\nu}^{+}-iV^{\mu}T_{\mu\nu}^{+}-Y_{\nu}^{+})-(\nabla^{\mu}T_{\mu\nu}^{-}-iT_{\mu\nu}^{-}V^{\mu}-Y_{\nu}^{-})\xi_{+}^{\nu}\Bigr\}.
\m^{2}(\beta)=m_{0}^{2}+\frac{\lambda_{0}}{(2\pi)^{D/2}}\sum_{n=1}^{\infty}\biggl(\frac{m_{0}}{\betan}\biggr)^{\frac{D}{2}-1}K_{\frac{D}{2}-1}(m_{0}n\beta)
X=\int\mathrm{d}x\,a\\Rightarrow\DX=(X,{\calS})+\sum_{r\geq2}\frac1{r!}\,\frac{\partial^{R}X}{\partial\xi^{B}}\,f_{A_{1}\cdotsA_{r}}^{B}\xi^{A_{r}}\cdots\xi^{A_{1}}\;.
\left\langle\Theta_{11}(\vec{x})\right\rangle=-\frac{1}{16\pi^{2}}D_{1}(\vec{x})+\frac{1}{48\pi^{2}}D_{2}(\vec{x})-\frac{1}{24\pi^{2}}D_{11}(\vec{x})+\frac{1}{64\pi^{2}}F_{1}(\vec{x})-\frac{1}{192\pi^{2}}F_{2}(\vec{x})
\partial_{t}\hat{\rho}(z)={\frac{1}{i}}\sum_{n=1}^{\infty}{\frac{1}{n!}}\left\{\partial_{\bar{z}}^{n}\widetildeV(z,\bar{z})\partial_{z}^{n}\hat{\rho}(z)-\partial_{z}^{n}\widetildeV(z,\bar{z})\partial_{\bar{z}}^{n}\hat{\rho}(z)\right\}.
c_{IR}=\frac{1}{32}(9\,U(1)_{R}^{3}-5U(1)_{R})=\frac{1}{32}\left(4\,\mathrm{dim}\,G+\sum_{i}(\mathrm{dim}\,R_{i})(1-r_{i})(5-9(1-r_{i})^{2})\right),
\bar{S_{s}}^{ab}=\frac{2^{n-2}}{(d-2)!}\varepsilon^{ab}{}_{a_{1}a_{2}\ldotsa_{d-2}}S_{s}^{a_{1}a_{2}}S_{s}^{a_{3}a_{4}}\ldots=\frac{(-i)^{n-1}}{2}a^{a}a^{b}\,\,(a^{0}a^{1}\ldotsa^{2n}).
i{\hbar}c\frac{{\beta}(1-i{\beta})}{z_{0}^{2}}\cdot\frac{G(2-i{\beta},1-i{\beta};\frac{iz_{0}}{{\hbar}c})}{G(1-i{\beta},-i{\beta};\frac{iz_{0}}{{\hbar}c})}=\frac{1}{2{\hbar}c}+\frac{\beta}{z_{0}}.
\intd^{4}x\frac{1}{4}F_{\mu\nu}^{2}=\frac{1}{2}\intd^{4}x(-A_{0}\partial_{i}^{2}A_{0}+2A_{0}\partial_{0}\partial_{i}A_{i}-\{A_{i}\partial_{\mu}^{2}\delta_{ik}A_{k}-A_{i}\partial_{i}\partial_{k}A_{k}\}).
P_{\mu}\equiv-\epsilon_{\alpha\beta}V_{+}^{\alpha}\partial_{\mu}V_{+}^{\beta}=ie^{2i\phi}{\frac{\partial_{\mu}\bar{\tau}}{2\tau_{2}}},\qquadQ_{\mu}\equiv-i\epsilon_{\alpha\beta}V_{+}^{\alpha}\partial_{\mu}V_{-}^{\beta}=\partial\phi-{\frac{\partial_{\mu}\tau_{1}}{2\tau_{2}}},
\nabla(V,\bar{V}_{\bar{I}},\bar{V},V_{I})=(V,\bar{V}_{\bar{J}},\bar{V},V_{J})\left(\begin{matrix}{0}&{0}&{0}&{\bar{P}_{I}}\\{0}&{0}&{\bar{P}^{\bar{J}}}&{\bar{P}_{\I}^{\bar{J}}}\\{0}&{P_{\bar{I}}}&{0}&{0}\\{P^{J}}&{P_{\\bar{I}}^{J}}&{0}&{0}\\\end{matrix}\right)
{\bf8}_{\mathrm{s}}\sim({\bf2},{\bf1})^{(\frac{1}{2},\frac{1}{2})}\oplus({\bf\bar{2}},1)^{(-\frac{1}{2},-\frac{1}{2})}\oplus({\bf1},{\bf2})^{(\frac{1}{2},-\frac{1}{2})}\oplus({\bf1},{\bf\bar{2}})^{(-\frac{1}{2},\frac{1}{2})}
M_{2}^{I}\equivE^{--}\wedgef^{++I}-E^{++}\wedgef^{--I}+4i\left(E^{+q1}\wedgeE^{-\dot{q}1}+E_{q}^{+2}\wedgeE_{\dot{q}}^{-2}\right)\gamma_{q\dot{q}}^{I}=0,
\intd^{2}x\,\epsilon^{\mu\nu}\frac{\epsilon^{\rho\lambda}}{\sqrt{-g}}\frac{\epsilon^{\eta\xi}}{\sqrt{-g}}D_{\eta}D_{\rho}\partial_{\mu}\phi_{1}D_{\xi}D_{\lambda}\partial_{\nu}\phi_{2}=\frac{1}{2}\intd^{2}x\,\epsilon^{\mu\nu}\Bigl(\partial_{\mu}R\,D_{\nu}\partial_{\lambda}\phi_{1}\partial^{\lambda}\phi_{2}+\frac{1}{2}R^{2}\partial_{\mu}\phi_{1}\partial_{\nu}\phi_{2}\Bigr)\,,
\int{\calD}(B)expi[-{\frac{a}{4}}B_{\mu\nu}B^{\mu\nu}+{\frac{\alpha}{2}}\epsilon_{\mu\nu\lambda}B^{\mu}B^{\nu\lambda}-{\frac{1}{2g}}B_{\mu}B^{\mu}-{\frac{ae}{2}}B_{\mu\nu}A^{\mu\nu}+\alphae\epsilon_{\mu\nu\lambda}B^{\mu}A^{\nu\lambda}+L_{\mu}B^{\mu}].
\sumq^{\frac{1}{2}p^{2}}=\sum_{n}q^{\frac{15}{2}\left(2n+\frac{5m_{1}+3m_{3}}{30}\right)^{2}}=\sum_{n}q^{30\left(n+\frac{5m_{1}+3m_{3}}{60}\right)^{2}}=\theta_{5m_{1}+3m_{3},30}\:(\tau).
\int_{C_{h}}{g(z)dz}=\int_{C_{h}^{+}}{[g(z)+f(z)]dz}+\int_{C_{h}^{-}}{[g(z)-f(z)]dz}-\int_{C_{h}^{+}}{f(z)dz}+\int_{C_{h}^{-}}{f(z)dz}.
{J^{\prime}}_{0}=\left(\begin{array}{ccc}{-\frac{3}{2}D}&{\frac{1}{2}Da}&{Db}\\{\frac{1}{2}aD}&{\frac{1}{2}(Db+bD)}&{\frac{3}{2}cD+c_{x}}\\{bD}&{\frac{3}{2}Dc-c_{x}}&{(b^{2}-ac)D+D(b^{2}-ac)}\\\end{array}\right)\,,
R>L\sim\left(G_{N}\rho_{1}G_{N}\rho_{2}\right)^{-1/2}=R^{d}\left(G_{N}E_{1}G_{N}E_{2}\right)^{-1/2}=\frac{R^{d}}{G_{N}\sqrt{s}}=R_{s}(\frac{R}{R_{s}})^{d},
H_{initial}=\frac{1}{2}p_{i}^{A}p_{i}^{A}+\frac{1}{4}G_{ij}^{A}G^{ij\,A}+\frac{m^{2}}{2}A^{i\,A}A^{i\,A}+\frac{1}{2m^{2}}(\nabla^{i}p_{i}^{A})(\nabla^{i}p_{i}^{A}),
\left[\lambda_{q}(R)+\alphaR\frac{n_{1}-n_{2}+\frac{\delta_{1}-\delta_{2}}{2}}{n+\frac{\delta_{1}+\delta_{2}}{2}}\right]U_{nqm}^{n_{1}}(R;\delta_{1},\delta_{2})=\sum_{n_{1}^{\prime}=0}^{n-|m|-1}U_{nqm}^{n_{1}^{\prime}}(R;\delta_{1},\delta_{2})(M)_{n_{1}n_{1}^{\prime}},
\prod_{a=1}^{N}\Big(\tau^{2}\Big)^{n_{a}n_{a}^{\prime}+m_{a}m_{a}^{\prime}-n_{a}m_{a}^{\prime}-m_{a}n_{a}^{\prime}}=\Big(\tau^{2}\Big)^{\sum_{a}(n_{a}-m_{a})(n_{a}^{\prime}-m_{a}^{\prime})}\;.
[\Lambda\frac{\partial\phantom{a}}{\partial\Lambda}+\mu\frac{\partial\phantom{a}}{\partial\mu}+\beta_{1}\frac{\partial\phantom{a}}{\partialg_{1}}+\beta_{2}\frac{\partial\phantom{a}}{\partialg_{2}}-\gammaN_{F}]\Gamma^{(N)}(p_{1},\ldotsp_{N})\approx0,
{\calL}_{\mathrm{eff}}^{\beta,\mu}(E)={\calL}_{\mathrm{eff}}^{\beta,\mu}(0)-\frac{(eE)^{2}}{24\pi^{2}}\frac{|\mu-eA_{0}|}{\sqrt{(\mu-eA_{0})^{2}-m^{2}}}\,\theta\left(|\mu-eA_{0}|-m\right)\,
{\frac{dM^{(1)}}{dm}}={\frac{d(\deltaM)}{dm}}+{\frac{1}{2}}\sum_{n}{\frac{d\omega_{n}^{B}}{dm}}-{\frac{1}{2}}\sum_{n}{\frac{d\omega_{n}^{F}}{dm}}-\left({\frac{1}{2}}\sum_{n}{\frac{d\tilde{\omega}_{n}^{B}}{dm}}-{\frac{1}{2}}\sum_{n}{\frac{d\tilde{\omega}_{n}^{F}}{dm}}\right)
g_{tt}^{2}=4(h_{0}+{\frac{q_{0}}{r}})\left((h^{1}-{\frac{p^{1}}{r}})(h^{2}-{\frac{p^{2}}{r}})(h^{3}+{\frac{p^{3}}{r}})+a(h^{3}+{\frac{p^{3}}{r}})^{3}\right)\.
S_{wz}=-\frac{1}{2}\int_{M_{10}}A_{3}dA_{3}H_{3}-\frac{g}{2}\int_{M_{6}}da_{2}A_{3}+\frac{2\piG}{g}\left(\int_{M_{10}}H_{3}X_{7}+\frac{1}{24}\int_{M_{6}}A_{2}^{(0)}\chi_{4}\right).
\begin{array}{rcl}{\rho^{\prime}(2\pi)^{d}A^{\mu}(x)}&{\rightarrow}&{A^{\mu}(x)}\\{\rho^{\prime}(2\pi)^{d}\Phi^{m}(x)}&{\rightarrow}&{\Phi^{m}(x)}\\{\rho^{\prime}(2\pi)^{d}\psi(x)}&{\rightarrow}&{\psi(x)}\\\end{array}