harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\eta_{\mathrm{\tiny~GL~_q~(2)~}}=-q^{4}\eta^{(\mathrm{\tiny~fund~GL~_q(2)~})}=-\left(\begin{array}{cccc}{q^{-3}+q^{-5}}&{0}&{0}&{0}\\{0}&{0}&{q^{-1}}&{0}\\{0}&{q^{-3}}&{0}&{0}\\{0}&{0}&{0}&{q^{-1}+q^{-3}}\\\end{array}\right).
\tilde{C}=-\frac{3}{2}g^{2}\delta^{ab}\int\frac{d^{4}k}{(2\pi)^{4}}d^{2}\thetad^{2}{\overline{\theta}}\,\frac{1}{(k^{2}+m^{2})}\left\{V^{a}(p,\theta,{\overline{\theta}})V^{b}(-p,\theta,{\overline{\theta}})\right\}\,.
\intd^{2}\mathrm{\bf~x}\,\Big(\frac{1}{4}f_{ij}f^{ij}+\frac{1}{8}(V+Af)_{ij}^{\alpha}(V+Af)_{\alpha}^{ij}-\frac{1}{4}g^{ij}D_{i}A^{\alpha}\cdotD_{j}A_{\alpha}
{{\calG}^{-1}}_{\mua}^{\nub}\eta_{\nub}^{\mathrm{T,L}}=\frac{1}{2}\left(k^{2}-\Lambda_{l}+\frac{1}{2r^{2}}(N-1)(N-2)-\frac{N+d}{N+d-2}\Lambda\right)\eta_{\mua}^{\mathrm{T,L}},
\Gamma=\piA_{h}\frac{(\omega^{2}-m^{2})^{3/2}}{m(\omega-m\phi_{4})}\sqrt{\frac{r_{1}r_{2}r_{3}}{r_{4}}}\frac{1}{(e^{2\pim\sqrt{\frac{r_{1}r_{2}r_{3}}{r_{4}}}}-1)}\frac{1}{(e^{\frac{\omega-m}{T_{H}}}-1)}\frac{d^{3}k}{(2\pi)^{3}},
c_{0}=e^{i\phi_{0}}\left[1+a_{-1}a_{1}+\frac{1}{2\sqrt{2}}a_{-1}^{2}a_{2}+\frac{1}{2\sqrt{2}}a_{-2}a_{1}^{2}+\frac{1}{2}a_{-2}a_{2}+\cdots\right].
L_{ij}=m\left(\begin{array}{ccccc}{0}&{1}&{-2}&{2}&{-1}\\{-1}&{0}&{1}&{-1}&{1}\\{2}&{-1}&{0}&{0}&{-1}\\{-2}&{1}&{0}&{0}&{1}\\{1}&{-1}&{1}&{-1}&{0}\\\end{array}\right),
\lambda^{i}={\frac{2(D-2)}{\kappa^{D-1}}}{\frac{1}{W}}{\frac{\partialW}{\partial\phi}}e^{-{\frac{1}{2}}A}\tilde{\eta}_{-}^{i}\,\qquad\psi_{r}^{i}=-{\frac{2(D-2)}{\kappa^{D-1}}}\left({\frac{1}{W}}{\frac{\partialW}{\partial\phi}}\right)^{2}e^{-{\frac{1}{2}}A}\tilde{\eta}_{-}^{i}\,\qquad\psi_{\mu}^{i}=0\.
j^{\mu}=\frac{e}{8\pi^{2}}\epsilon^{\mu\nu\lambda\sigma}(1+\rho)\partial_{\nu}\thetaF_{\lambda\sigma}+\frac{e}{8\pi^{2}}\epsilon^{\mu\nu\lambda\sigma}(\partial_{\lambda}\rho)\,(\partial_{\nu}\theta)\,A_{\sigma}
{\check{\omega}}^{r}(\theta^{\alpha},\dot{\theta}^{\alpha})=-{\frac{1}{2}}\epsilon^{ruv}[R^{T}\dot{R}]^{uv}(\theta^{\alpha},\dot{\theta}^{\alpha})={\frac{1}{2}}\epsilon^{ruv}[{\dot{R}}^{T}R]^{uv}(\theta^{\alpha},\dot{\theta}^{\alpha}).
G_{MN}^{(11)}=\left(\begin{matrix}{e^{-{\frac{\hat{\Phi}}{3}}}G_{\hat{\mu}\hat{\nu}}^{(10)}}&{e^{{\frac{2}{3}}\hat{\Phi}}B_{M}}\\{e^{{\frac{2}{3}}\hat{\Phi}}B_{N}}&{e^{{\frac{2}{3}}\hat{\Phi}}}\\\end{matrix}\right),
\left[{\frac{d^{2}}{dt^{2}}}+{\frac{1}{2}}\sum_{k}{\frac{1}{t-a_{k}}}\;{\frac{d}{dt}}-{\frac{1}{4\hbar^{2}}}\sum_{k}{\frac{f_{k}}{t-a_{k}}}\right]\,\Psi(t)=0
\stackrel{.}{\mathbb{S}}^{\mu}=\frac{q}{2m}\left\{gF^{\mu\nu}\mathbb{S}_{\nu}+(g-2)\left(\mathbb{S}F\stackrel{.}{x}\right)\stackrel{.}{x}^{\mu}+(g-2)^{*}\left(F\mathcal{S}\right)^{\mu\alpha}\stackrel{.}{x}_{\alpha}\right\}.
\sqrt{-\hat{g}}\left[1+{\textstyle\frac{1}{2}\hat{\calH}^{2}}\right]=\sqrt{g}\left[e^{-\phi}+{\textstyle\frac{1}{2}}e^{\phi}{\calH}^{2}-{\textstyle\frac{3}{2}}e^{-\phi}{\calF}^{2}\right]\,,
\lambda_{AB}=\mathrm{~\frac{1}{2\sqrt{2}}~}\Bigr\{(\gamma^{\alpha})_{AB}\psi_{\alpha}+\mathrm{~\frac{1}{2}~}(\sigma^{\alpha\beta})_{AB}\chi_{\alpha\beta}-\epsilon_{AB}\tilde{\eta}\Bigr\},\qquad\lambda^{AB}=\epsilon^{AC}\epsilon^{BD}\lambda_{CD},
\sigma^{\mathrm{reg}}(\xi)=-\,{\frac{|\xi|}{2}}\,-\,{\frac{|\xi|}{\pi}}\sum_{n=1}^{\infty}{\frac{K_{1}(2n\pi\,|\xi|)}{n}}=-\,{\frac{|\xi|}{2}}\,-\,{\frac{|\xi|}{\pi}}G(2\pi\,|\xi|)\.
\left[L_{a}R_{b},L_{1},L_{2},L_{3},R_{1},R_{2}\right]=\left(i\hbar\right)^{2}\sum_{j}\left\{\left[L_{a}R_{b},L_{j}\right],L_{j},R_{3}\right\}=\left(i\hbar\right)^{3}\sum_{j,k}\varepsilon_{ajk}\left\{L_{k}R_{b},L_{j},R_{3}\right\}\;.
\widetilde{G}(x,y)\equiv\left(iK^{-1}\frac{\delta}{\delta\varphi}\right)_{x}\Phi(y)=\left(iK^{-1}\frac{\delta}{\delta\varphi}\right)_{x}\left(iK^{-1}\frac{\delta}{\delta\varphi}\right)_{y}iC\,.
\psi_{i}=\left(1,0,\sqrt{{\frac{1}{2}}}e^{-{\frac{\pi\omega_{i}}{\kappa}}}\,2\,_{i}{\lambda^{\left(a\right.}}_{i}\tau^{\left.b\right)},0,\sqrt{{\frac{3.1}{4.2}}}e^{-2{\frac{\pi\omega_{i}}{\kappa}}}\,4\,{_i\lambda^{\left(a\right.}}{_i\tau^{b}}{_i\lambda^{c}}{_i\tau^{\left.d\right)}},0,...\right)\;.
(\hat{g}_{IJ})=6\,\left(\begin{array}{cccccccc}{2}&{0}&{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{1}&{0}\\{0}&{1}&{0}&{0}&{0}&{0}&{0}&{0}\\{1}&{0}&{0}&{0}&{2}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{1}\\{0}&{0}&{1}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}&{0}&{0}\\\end{array}\right).
\begin{array}{rl}{}&{\intdA_{1}^{\prime*}dA_{1}^{\prime}dA_{2}^{\prime*}dA_{2}^{\prime}dZ^{\prime*}dZ^{\prime}e^{-L^{\prime}}W^{\prime}(z,z^{\prime})A^{\prime}(z)}\\{=}&{\intdA_{1}^{*}dA_{1}dA_{2}^{*}dA_{2}dZ^{*}dZe^{-L}W(z,z^{\prime})A(z)}\\\end{array}
I_{3}\equiv\frac{1}{4}(\frac{i}{2\bar{w}_{1}(\bar{w}_{1}-\tilde{w}_{1}^{+})(\bar{w}_{1}-\tilde{w}_{1}^{-})}+\frac{i}{(\tilde{w}_{1}^{+}-\bar{w}_{1})(\tilde{w}_{1}^{+}+\bar{w}_{1})(\tilde{w}_{1}^{+}-\tilde{w}_{1}^{-})}),
[{\calL}_{n,m},{\calL}_{k,l}]=\sum_{s=0}^{\mathrm{Min}(m,k)}{\frac{(m+1)!(k+1)!}{(m-s)!(k-s)!(s+1)!}}{\calL}_{n+k-s,m+l-s}-(m\leftrightarrowl,n\leftrightarrowk)\,.
\intd^{4}x\mathrm{~}f_{abc}\left(A^{b\mu}A_{\mu}^{*c}-c^{b}C^{*c}\right)=\beta\intd^{4}x\frac{\delta\Sigma}{\deltac^{a}}=\beta\intd^{4}x\;f_{abc}\left(A^{b\mu}A_{\mu}^{*c}-c^{b}C^{*c}\right)\;,
K_{c}(\theta)(-if_{c}^{ab})=(-if_{c}^{ab})K_{a}(\theta+i{\bar{\theta}_{ac}^{b}})S_{ab}(2\theta+i{\bar{\theta}_{ac}^{b}}-i{\bar{\theta}_{bc}^{a}})K_{b}(\theta-i{\bar{\theta}_{bc}^{a}}).
{\calF}={\calD}\left[\sum_{n=-{\infty}}^{\infty}\sum_{p=-{\infty}}^{\infty}A_{n}A_{p}t_{1}^{n-p}\right]\left[\sum_{n=-{\infty}}^{\infty}\sum_{p=-{\infty}}^{\infty}A_{n}A_{p}t_{2}^{n-p}\right]\cdots\left[\sum_{n=-{\infty}}^{\infty}\sum_{p=-{\infty}}^{\infty}A_{n}A_{p}t_{N}^{n-p}\right].
V_{E}+V_{M}=\frac{1}{4\pir}\;\mathrm{Re}\left\{\left(n_{E}^{(1)a}+n_{Mc}^{(1)}\,\tau^{ca}\right)g_{ab}\left(n_{E}^{(2)b}+\bar{\tau}^{bd}\,n_{Md}^{(2)}\right)\right\}\,,
\varepsilon(0)=p_{1}(0)=p_{2}(0)=\frac{1}{8\pi^{2}a^{4}}\int_{0}^{\infty}{dzz^{3}\left[\frac{K_{0}(z)}{I_{0}(z)}+\frac{K_{0}^{\prime}(z)}{I_{0}^{\prime}(z)}\right]}=-0.0168a^{-4}
f_{1}\left(e^{-\frac{\pi}{s}}\right)=\sqrt{s}f_{1}\left(e^{-\pis}\right),\;\;f_{3}\left(e^{-\frac{\pi}{s}}\right)=f_{3}\left(e^{-\pis}\right),\;\;f_{2}\left(e^{-\frac{\pi}{s}}\right)=f_{4}\left(e^{-\pis}\right).
F_{\mu\nu}(p)=\int{\frac{d^{2}k}{2(\pi)^{2}}}{\frac{\displaystylek_{\mu}k_{\nu}}{\displaystyle{\omega({\frac{k+p}{2}})}\omega({\frac{k-p}{2}})\left(\omega({\frac{k+p}{2}})+\omega({\frac{k-p}{2}})\right)}}
\Gamma[g]={\frac{1}{24}}\int_{x_{+}}^{L}({\frac{2}{\bar{\beta}g}}-{\frac{\bar{\beta}}{2}}{\frac{g^{2}}{g}})dx+{\frac{1}{12}}(\alpha+{\frac{(1-\alpha^{2})}{2\alpha}})\psi(x_{+})-{\frac{\bar{\beta}}{8}}g^{\prime}(L)+\Gamma_{0},
\gamma_{\mu}=\left(\begin{array}{cc}{\sigma_{\mu}}&{0}\\{0}&{-\sigma_{\mu}}\\\end{array}\right)\,\quad\frac{\sigma_{\mu\nu}}{2}\equiv{\frac{1}{4i}}[\gamma_{\mu},\gamma_{\nu}]=\frac{\epsilon_{\mu\nu\rho}}{2}\left(\begin{array}{cc}{\sigma_{\rho}}&{0}\\{0}&{\sigma_{\rho}}\\\end{array}\right)\equiv\epsilon_{\mu\nu\rho}J_{\rho}\,;\quad\mu,\nu,\rho=1,2,3\;
\frac{1}{i}\frac{\delta}{\deltaS_{\alpha}(y)}Z_{0}\mid_{J=0}=-\frac{1}{e^{2}\xi}\frac{(\Box\partial_{\mu})_{x}}{E_{0}^{2}}<T^{*}[A^{\mu}(x)A^{\alpha}(y)]>-\partial^{\alpha}\delta(x-y)=0.
\tau\to\frac{c+d\tau}{a+b\tau}\,\quad\left(\begin{array}{c}{m}\\{n}\\\end{array}\right)\to\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)\left(\begin{array}{c}{m}\\{n}\\\end{array}\right)\,\quadad-bc=1\.
S=\left(\begin{array}{ccccc}{0}&{\frac{1}{4}}&{\frac{i}{2}}&{-\frac{1}{4}}&{0}\\{1}&{\frac{1}{2}}&{0}&{\frac{1}{2}}&{1}\\{-i}&{0}&{0}&{0}&{i}\\{-1}&{\frac{1}{2}}&{0}&{\frac{1}{2}}&{-1}\\{0}&{\frac{1}{4}}&{-\frac{i}{2}}&{-\frac{1}{4}}&{0}\\\end{array}\right)\,.
\partial_{z}\tilde{\varphi}(z,\bar{z})=\tilde{\Phi}=\partial_{\varphi}\bar{S}(z,\bar{z},\varphi=\tilde{\varphi})\,,\quad\partial_{\bar{z}}\tilde{\varphi}(z,\bar{z})=\tilde{\bar{\Phi}}=\partial_{\varphi}S(z,\bar{z},\varphi=\tilde{\varphi}).
\eta_{1}=c_{1}\,{\frac{(z-\gamma)^{N_{2}}}{(z-\beta)^{N_{2}}}},~~\eta_{2}=c_{2}\,{\frac{(z-\alpha)^{N_{1}}}{(z-\gamma)^{N_{1}}}},~~\eta_{3}=c_{3}\,{\frac{(z-\beta)^{N_{1}N_{2}}}{(z-\alpha)^{N_{1}N_{2}}}}\.
{\calR}_{abc}^{d}=R_{abc}^{d}-\delta_{b}^{d}g_{ac}+\delta_{c}^{d}g_{ab}+D_{b}T_{ac}^{d}-D_{c}T_{ab}^{d}+T_{eb}^{d}T_{ac}^{e}-T_{ec}^{d}T_{ab}^{e},
{\calL}=\frac{1}{2}\mathrm{Tr}F_{\mu\nu}^{2}+\frac{1}{2}\mathrm{Tr}(D_{\mu}\phi_{p})^{2}-\frac{e^{2}}{4}\mathrm{Tr}\left([\phi_{p},\phi_{q}]\right)^{2}+\mathrm{fermion~terms}\,\,.
f^{\lambda}=\left(f_{1}e_{a}^{\lambda}e^{b\nu}+f_{2}e^{b\lambda}e_{a}^{\nu}+f_{3}g^{\nu\lambda}\delta_{b}^{a}\right)\bar{\Omega}_{~b\nu}^{a}+N_{~~~~b}^{\mu\alpha\beta~b\nu}\nabla_{\alpha}\nabla_{\beta}\bar{\Omega}_{~b\nu}^{a}
I_{eff}=\frac{1}{\pi}\intd^{2}z[\frac{k}{2}\\\frac{\partialw\overline{{\partial}}w}{w^{2}}+\frac{w^{2}}{w^{2}-4\lambda}(\partialf\overline{{\partial}}f-c^{2}\partialy\overline{{\partial}}y)+\frac{cw^{2}}{w^{2}-4\lambda}(\partialf\overline{{\partial}}y-\partialy\overline{{\partial}}f)]
f(x)=c_{1}\epsilon(x^{0})\delta((x^{0})^{2}-|{\bfx}|^{2})+c_{2}\left(((x^{0}+i0)^{2}-|{\bfx}|^{2})^{-1}+((x^{0}-i0)^{2}-|{\bfx}|^{2})^{-1}\right)+c_{3}
R_{k}=-2\partial_{2}(\partial_{1}c/\partial_{2}c)\sum_{s=1}^{k-1}sb_{2s}b_{2k-2s}-\sum_{s=1}^{k-1}P_{2s}\partial_{2}b_{2k-2s}+\sum_{s=1}^{k-1}\frac{s}{k-s}b_{2s}\partial_{2}P_{2k-2s}.
{\bar{\theta}}^{\dot{2}}=Y(x){\bar{\theta}}^{\dot{1}},\quad{\bar{\theta}}^{\dot{\alpha}}={\bar{\theta}}^{\dot{1}}{\bar{\psi}}^{\dot{\alpha}},\quad{\bar{\zeta}}^{\dot{2}}=Y(x){\bar{\zeta}}^{\dot{1}},\quad{\bar{\zeta}}^{\dot{\alpha}}={\bar{\zeta}}^{\dot{1}}{\bar{\psi}}^{\dot{\alpha}}.
\ddot{\hat{X}}^{i}=-[[\hat{X}^{i},\hat{X}^{j}],\hat{X}^{j}],\;\;\;\;\;\;\;\;\;\;\ddot{\tilde{X}}^{i}=-[[\tilde{X}^{i},\tilde{X}^{j}],\tilde{X}^{j}]\,.
T^{00}={\frac{2cosh^{2}(\gamma)}{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]
S=\int\!\!d^{4}x\int\!\!dy\Bigl[{\calL}_{\mathrm{5D}}^{SU(5)}+\delta(y){\calL}_{\mathrm{4D}}^{SU(5)}+\delta(y-\piR){\calL}_{\mathrm{4D}}^{3-2-1}\Bigr].
S(\phi)=\sum_{x}\left[\frac{a^{d-2}}{2}\sum_{\mu=1}^{d}(\phi_{x+\hat{\mu}}-\phi_{x})^{2}+a^{d}\left(\frac{m^{2}}{2}\phi_{x}^{2}+\frac{\lambda}{4}\phi_{x}^{4}\right)\right]\quad.
{\calL}_{p}=\epsilon_{\mu_{1}\mu_{2}...\mu_{2p-1}\mu_{2p}}\epsilon^{\nu_{1}\nu_{2}...\nu_{2p-1}\nu_{2p}}R_{\\\nu_{1}\nu_{2}}^{\mu_{1}\mu_{2}}...R_{\\,nu_{2p-1}\nu_{2p}}^{\mu_{2p-1}\mu_{2p}}
\left\{\psi_{i}(x),\psi_{j}(y)\right\}=-\left\{{\overline{\psi}}_{i}(x),{\overline{\psi}}_{j}(y)\right\}=-\delta_{ij}\,\delta^{\prime}(x-y),\\\\left\{\psi_{i}(x),{\overline{\psi}}_{j}(y)\right\}=0,
f({\bfk^{\prime}},{\bfk})=\frac{1}{\sqrt{2\pik}}\sum_{l,L}\,(S_{\scriptscriptstyle{l,L}}-\delta_{\scriptscriptstyle{l,L}})\,e^{il\theta}\,e^{-iL\Theta}\,e^{i\pi(W_{L}-W_{l}-1)/4}\,.
Z=\int{\calD}A_{i}^{a}(x)\;{\calD}e_{i}^{a}(x)\;exp\left\{\intd^{3}x(-\frac{1}{2}[e_{i}^{a}(x)]^{2}+\frac{i}{\kappa}e_{i}^{a}(x)B_{i}^{a}(x))\right\}.
Q_{\alpha}^{i}=\phi_{Q\alpha}^{i}+\sqrt{2}\theta\psi_{Q\alpha}^{i}+\theta^{2}F_{Q\alpha}^{i},\qquad{\tilde{Q}}^{\alphaj}=\phi_{\tilde{Q}}^{\alphaj}+\sqrt{2}\theta\psi_{\tilde{Q}}^{\alphaj}+\theta^{2}F_{\tilde{Q}}^{\alphaj},
\Phi(z,\vec{x},t)\,=\,=\sum_{p=1}^{\infty}\,\,\int{\frac{d^{n-1}k}{(2\pi)^{n-1}}}\,{\frac{z^{n/2}\,J_{\nu}(u_{p}z)}{Rw_{p}(\vec{k})\,J_{\nu\,+\,1}(u_{p}R)}}\lbrace{a_{p}(\vec{k})\}e^{-iw_{p}(\vec{k})t+i\vec{k}\cdot\vec{x}}\,\,+\,\,c.c.\rbrace
e_{\hat{0}}^{\;\;\mu}e_{\alpha}^{\;\;\nu}\left.\frac{\partial\Gamma_{R}}{\partialg^{\mu\nu}}\right|_{b_{\mu\nu}}=-{\bar{e}}_{\alpha}^{\;\;i}{e}_{\hat{0}}^{\;\;0}\frac{\partial\Gamma_{R}}{\partialv^{i}},
\frac{1}{2}(2\pi\alpha^{\prime})^{2}\mathrm{Tr}(G^{-1}\hat{F})_{\mathrm{arbitrary}}^{4}-\frac{1}{8}(2\pi\alpha^{\prime})^{2}(\mathrm{Tr}(G^{-1}\hat{F})^{2})_{\mathrm{arbitrary}}^{2},
T(0)=\prod_{j=3}^{P}\;\prod_{i=2}^{j-1}R_{ij}^{\iota^{P-j+i-1}}((\epsilon^{-j}-\epsilon^{-i})\theta^{\prime})\;\;\prod_{j=2}^{P-1}\;\prod_{i=j+1}^{P}R_{ij}^{\iota^{P-j+i}}((\epsilon^{-j}-\epsilon^{-i})\theta^{\prime})
i\frac{d}{dt}\langle\psi^{(0)}|\hat{x}^{m}\hat{p}^{n}|\psi^{(0)}\rangle=\langle\psi^{(0)}|\left[\hat{x}^{m}\hat{p}^{n},\frac{1}{2\mu}\hat{p}^{2}+V(\hat{x})\right]|\psi^{(0)}\rangle=0.
\hat{C}_{(2n)}^{+}\;=\;C_{(2n)}^{+}\,-\,C_{(2n-1)}^{+}\left(d\underline{{y}}+A^{(2)}\right)\,+\,\underline{{y}}\G_{(0)}^{+}\textstyle{\frac{1}{n!}}\hat{B}^{n}\;,
V(y_{i},\theta)=-V_{p+1}(i)^{p}\int_{0}^{\infty}\frac{dt}{t}(8\pi^{2}\alpha^{\prime}\,t)^{-\frac{p+1}{2}}e^{-\frac{y^{2}}{2\pi\alpha^{\prime}}t}16t^{4}\simeqy^{p-7}\,.
(\theta^{ij})=\left(\begin{array}{cccc}{\theta_{1}{\bf\epsilon}}&{0}&{\dots}&{0}\\{0}&{\theta_{2}{\bf\epsilon}}&{\dots}&{0}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{0}&{0}&{\dots}&{\theta_{p}{\bf\epsilon}}\\\end{array}\right)
M=\textstyle{\frac{3}{2}}\mu+\sum_{i=1}^{3}q_{i}=\textstyle{\frac{3}{2}}\left(r_{+}^{2}+r_{+}^{4}g^{2}\prod_{i=1}^{3}H_{i}\right)+\sum_{i=1}^{3}q_{i}\,
\Psi_{{\bfp}}^{(+)}({\bfx},t)=\mathrm{e}^{-iw_{{\bfp}}t}\,\Psi_{{\bfp}}^{(+)}({\bfx})=\frac{1}{{2\pi}}\frac{w_{{\bfp}}+m}{\sqrt{4mw_{{\bfp}}}}\left(\begin{array}{c}{1}\\{\frac{m-w_{{\bfp}}}{m+w_{{\bfp}}}}\\\end{array}\right)\mathrm{e}^{\,i{\bfp}\cdot{\bfx}-iw_{{\bfp}}t}\;,
\delta\varphi_{r}(t_{1})dt_{1}\int_{C_{b}^{(r)}}f_{r}(t_{2})dt_{2}K(t_{2},t^{\prime})+\int_{\hat{C}_{r}^{(-)}}K_{(o)}(t,t_{1})\varphi_{(o)r}(t_{1})dt_{1}\int_{C_{r}}\deltaf_{r}(t_{2})dt_{2}K(t_{2},t^{\prime})
{\mathsfL}_{++}^{(1)}=g_{ab}^{^{(1)}}\left(\overline{{\bfM}}_{a}\gamma^{A}{\bfM}_{b}+\frac{1}{2}\overline{{\bfM}}_{aij}\gamma^{A}{\bfM}_{b}^{ij}+\overline{{\bfM}}_{a}^{i}\gamma^{A}{\bfM}_{bi}\right){\mathsfG}_{A}.
F^{k_{1}^{\prime}k_{2}^{\prime}k_{3}^{\prime}\;0}=\left(\frac{(12)}{x_{12}^{2}}\right)^{\!\frac{1}{2}(k_{1}+k_{2}-k_{3}-2)}\left(\frac{(13)}{x_{13}^{2}}\right)^{\!\frac{1}{2}(k_{1}+k_{3}-k_{2}-2)}\left(\frac{(23)}{x_{23}^{2}}\right)^{\!\frac{1}{2}(k_{2}+k_{3}-k_{1}-2)}\;\!\!\!\!f(x_{0},\ldots,x_{4})\;,
\lambda_{N_{c}}+\bar{\lambda}_{\tilde{N}_{c}}:N_{c}{\calP}(SU(N_{c}))-\tilde{N}_{c}{\calP}(SU(\tilde{N}_{c}))\rightarrowN_{f}(\frac{N_{c}-\tilde{N}_{c}}{N_{f}}){\calP}(SU(N_{f}))
U_{k_{1}\,\,k_{2}}^{\,\,l_{1}\,\,l_{2}}\left(\begin{array}{ccc}{\lambda}&{\mu}&{\rho}\\{k_{1}}&{k_{2}}&{m}\\\end{array}\right)\left(\begin{array}{ccc}{\lambda}&{\mu}&{\rho^{\prime}}\\{l_{1}}&{l_{2}}&{m^{\prime}}\\\end{array}\right)^{\ast}=U(\lambda\mu\rho)\delta_{\rho}^{\,\,\rho^{\prime}}\delta_{m}^{\,\,m^{\prime}},
\begin{array}{ll}{A:\qquadu=U_{k}(\sigma^{+}),}&{\quadv=V_{k}(\sigma^{-})}\\{B:\qquadu=U_{k}(\sigma^{-}),}&{\quadv=V_{k}(\sigma^{+})}\\{C:\qquadu=u_{k},}&{\quadv=W[\alpha_{k}(\sigma^{+})+\beta_{k}(\sigma^{-}),\,\,u_{k}]}\\{D:\qquadu=\bar{W}[\alpha_{k}(\sigma^{-})+\beta_{k}(\sigma^{+}),\,\,v_{k}],}&{\quadv=v_{k}\,\,\,,}\\\end{array}
I(m,q,g)=\beta_{H}(|g-1|+\delta_{g,1})\left(-\frac{s(r_{s}^{3}-r_{H}^{3})}{l^{2}}+\frac{q^{2}}{r_{s}r_{H}}(r_{s}-r_{H})-2\pi\sigmar_{s}^{2}\sqrt{N(r_{s})}\right)
\Sigma_{(f)}=M^{2}\{{\frac{\hat{\lambda}}{2\epsilon}}+{\hat{\lambda}}^{2}({\frac{1}{2\epsilon^{2}}}-{\frac{1}{4\epsilon}})\}+{\frac{{\hat{\lambda}}^{2}}{24\epsilon}}p^{2}+{\frac{{\hat{\lambda}}^{2}}{2\epsilon}}M^{2}{\frac{\partial}{\partial{M^{2}}}}G+{\frac{3\hat{\lambda}}{2\epsilon}}G.
A_{\chi}=\frac{2iL^{2}\tau^{\prime}\widehat{B}_{E}}{A^{2}\epsilon_{E}G^{\prime}(\xi_{3})}\left[1+\frac{G^{\prime\prime}(\xi_{3})}{G^{\prime}(\xi_{3})}(\chi-1)\epsilon_{E}+\frac{Bq\chi\epsilon_{E}}{L^{1/2}}\right],
\tilde{F}_{\eta}(p)=-2\\frac{\int_{0}^{\infty}\!d\mu^{2}\\tilde{c}(\mu^{2},m,g)\frac{p^{2}}{(p^{2}+\mu^{2})^{2}}}{\int_{0}^{\infty}\!d\mu^{2}\\tilde{c}(\mu^{2},m,g)\frac{1}{p^{2}+\mu^{2}}}+2\rightarrow0\\\mathrm{~for~}\p\rightarrow\infty,
A=\frac{(l+m)!}{(l-m)!}\frac{(-1)^{m^{-}}}{(1-x)^{m^{+}}(1+x)^{m^{-}}}\sum_{j=0}^{l-m}{\binom{l-m}{j}}\frac{d^{j}}{dx^{j}}(1-x)^{l^{+}}\frac{d^{l-m-j}}{dx^{l-m-j}}(1+x)^{l^{-}}.
\vec{E}=-\vec{\nabla}\left([\vec{m}\cdot\vec{\nabla}]\psi\right)+\vec{m}\;\ddot{\psi}=-\vec{\nabla}\left([\vec{m}\cdot\vec{\nabla}]\psi\right)+\vec{m}\;\nabla^{2}\psi=\vec{\nabla}\times(\vec{\nabla}\times[\vec{m}\;\psi]),
L=\left(\prod_{k}e^{\tilde{\alpha}_{k}}\right)\left[\sum_{i}\dot{\tilde{\alpha}}_{i}^{2}-\dot{\tilde{\alpha}}_{i}\sum_{j}\dot{\tilde{\alpha}}_{j}+{\frac{1}{D-2}}\dot{\varphi}^{2}+\sum_{a}\dot{\alpha}_{a}^{2}-{\frac{Q^{2}}{2}}e^{-\tilde{\Phi}}\right]
K_{NP}^{(2)}\,=\,\frac{1}{16\pi^{2}}\,\int_{0}^{\infty}\,\frac{d\alpha}{\alpha}\,\int_{0}^{1}dx\,\int_{0}^{\infty}\,dz\,e^{-i(\alpha+z)|m+g\Phi|^{2}}\,\int\frac{d^{4}k}{(2\pi)^{4}}\,e^{i[\alphax(1-x)+z]k^{2}\,+\,i\frac{k\circk}{4\alpha}}\,.
\begin{array}{c}{D(\beta,R)-i\mu\gamma^{0}=e^{\mu\betat}D(\beta,R)\e^{-\mu\betat},}\\{K_{bc}(t,\vec{x};t^{\prime},\vec{x^{\prime}})=e^{\mu\betat}k(t,\vec{x};t^{\prime},\vec{x^{\prime}})\e^{-\mu\betat\prime},}\\\end{array}
\left|\mu_{j}(w^{2},m_{1}^{2},m_{2}^{2})\right|^{2}=\frac{2m_{1}^{2}m_{2}^{2}w^{2}}{\sqrt{\lambda(1,(\frac{m_{1}}{w})^{2},(\frac{m_{2}}{w})^{2})}}
\xi=a^{\mu}P_{\mu}+\epsilon^{\alpha}Q_{\alpha}+\frac{1}{2}a^{\mu\nu}M_{\mu\nu}+\frac{1}{2}\epsilon^{ij}J_{ij}+k^{\mu}K_{\mu}+\eta^{\alpha^{\prime}}S_{\alpha^{\prime}}+\lambdaD,
\int\frac{(q^{2})^{n}}{(q^{2}+a^{2})^{m}}d^{{D}}q=\pi^{D/2}a^{D+2n-2m}\,\frac{\Gamma(n+D/2)\,\Gamma(m-n-D/2)}{\Gamma(D/2)\,\Gamma(m)}\;.
\begin{array}{rclrclrcl}{d_{21}^{+}}&{=}&{-60;}&{d_{30}^{+}}&{=}&{2160;}&{d_{31}^{+}}&{=}&{1080;}\\{d_{32}^{+}}&{=}&{360;}&{d_{33}^{+}}&{=}&{885/4;}&{d_{34}^{+}}&{=}&{{315}/2;}\\{d_{35}^{+}}&{=}&{150;}&{d_{36}^{+}}&{=}&{{2041}/{128};}&{d_{37}^{+}}&{=}&{{417}/{32};}\\{d_{40}^{+}}&{=}&{{231}/8,}&{}&{}&{}&{}&{}&{}\\\end{array}
s^{I}(z,\vec{x})=\frac{\Gamma(2k)}{A\pi^{3}\Gamma(2k-3)}\intd\vec{x}^{\prime}\,\left(\frac{z}{z^{2}+|\vec{x}-\vec{x}^{\prime}|}\right)^{2k}s_{0}^{I}(\vec{x}^{\prime}).
\left(^{\ast}\Lambda^{\mu\nu}\right)-\frac{D}{a}\Lambda^{\mu\nu}=\frac{8DM^{2}}{e^{2}}\partial_{\theta}T^{\left(\mu\nu\right)\theta}-\frac{4aM^{2}}{e^{2}}\epsilon_{\;\;\;\alpha\beta}^{\mu\nu}\partial_{\theta}T^{(\alpha\beta)\theta}.
A_{\mu\nu}^{\lambda}(x)=\sum_{\kappa,\sigma,\tau=0}^{3}C_{\mu}^{\tau}C_{\nu}^{\sigma}(C^{-1})_{\kappa}^{\lambda}A_{\tau\sigma}^{\kappa}(y)+\sum_{\kappa=0}^{3}\frac{\partialC_{\nu}^{\kappa}}{\partialx^{\mu}}(C^{-1})_{\kappa}^{\lambda}.
\biggl(\sum_{l:l\neqj}{\frac{\theta^{\prime}(t_{j}-t_{l})}{\theta(t_{j}-t_{l})}}\alpha_{c(l)}-{\frac{\theta^{\prime}(t_{j})}{\theta(t_{j})}}\Lambda+2\pii\xi,\alpha_{c(j)}\biggr)=0,\qquadj=1,\dots,n.
\frac{1}{g_{{\iteff}}^{2}(v)}=\frac{1}{g_{h}^{2}(M)}-\frac{b_{0}}{16\pi^{2}}\mathrm{ln}\frac{v^{2}}{M^{2}}+c\left(\frac{Me^{8\pi^{2}/b_{0}g_{h}^{2}(M)}}{v}\right)^{4}+{\calO}\left(e^{-16\pi^{2}/g_{h}^{2}(M)}\right)
A_{lm}^{\alpha\beta}=\frac{i}{|P|}\sum_{k}~C_{k}^{\alpha\beta}(v)\mathrm{Tr}(\gamma_{k}^{\alpha}\lambda_{l}^{(0)})\mathrm{Tr}((\gamma_{k}^{\beta})^{-1}(\lambda_{m}^{(0)})^{2})\;,
ds^{2}\sim(1-\frac{Q_{p}}{r^{7-p}}\times\frac{7-p}{8})\eta_{\alpha\beta}dx^{\alpha}dx^{\beta}+(1+\frac{Q_{p}}{r^{7-p}}\times\frac{p+1}{8})\delta_{ij}dx^{i}dx^{j}
I(s)\equiv\int_{1}^{\infty}{\frac{(t^{2}-1)^{-s}dt}{\mathrm{e}^{2\piat}-1}}=\int_{1}^{t_{0}}{\frac{(t^{2}-1)^{-s}dt}{\mathrm{e}^{2\piat}-1}}+\int_{t_{0}}^{\infty}{\frac{(t^{2}-1)^{-s}dt}{\mathrm{e}^{2\piat}-1}}\,,
S=-\frac{1}{2}\int_{0}^{L}dy\intd^{4}x\sqrt{-G}\left(G^{MN}\partial_{M}\Phi\partial_{N}\Phi+V(\Phi)\right)+\intd^{4}x\sqrt{-G}G^{55}\Phi\partial_{5}\Phi\Bigr|_{0}^{L}\.
\Gamma_{\phi^{-}}^{12}(p)=\frac{-e^{\beta(\mu_{U}-\mu_{D})/2}\,(p^{2}-\bar{M}^{2})R_{UD}(p)}{\{p^{2}[K_{UD}(p)+H_{UD}(p)]+E_{UD}(p)\}^{2}+(p^{2}-\bar{M}^{2})^{2}[S_{UD}^{2}(p)-R_{UD}^{2}(p)]}=e^{\beta(\mu_{U}-\mu_{D})}\,\Gamma_{\phi^{-}}^{21}(p)
\langleg^{\mu\nu}({\bfx}_{F})g^{\sigma\rho}({\bfy}_{F})\rangle=-D^{\mu\nu;\sigma\rho}({\bfx}_{F}-{\bfy}_{F})=-C^{\mu\nu;\sigma\rho}\vert{\bfy}_{F}-{\bfx}_{F}\vert^{-4\gamma}
\Bigl(\Gamma_{a_{1}a_{2}}^{(2)}(k)\Bigr)=\Gamma_{ar}^{(2)}(k){\binom{1}{-1}}{\binom{1+n(k^{0})}{-n(k^{0})}}-\Gamma_{ra}^{(2)}(k){\binom{n(k^{0})}{-(1+n(k^{0}))}}{\binom{1}{-1}}\,.
E={\frac{T}{2\pi}}\rho_{0}^{2}\int_{1}^{\infty}dy\left[{\frac{y^{n/6}(y-\lambda)^{\delta/4}}{\sqrt{y^{2n/3}(y-\lambda)^{\delta}-(1-\lambda)^{\delta}}}}-{\frac{1}{\sqrt{y}}}\right]
n_{1}^{\mu}=\frac{q_{2}^{\mu}}{\sqrt{q_{2}^{2}}},~~~~n_{2}^{\mu}=\sqrt{\frac{q_{2}^{2}}{g}}\left(q_{3}^{\mu}-\frac{q_{2}q_{3}}{q_{2}^{2}}q_{2}^{\mu}\right),~~~~n_{3}^{\mu}=\frac{\varepsilon^{\mu}(q_{2}q_{3})}{\sqrt{g}},
|\eta(q)|^{2}=\frac{1}{2\sqrt{\tau_{2}}}(\sqrt{6}\sum_{n,m}e^{-\frac{6\pi}{\tau_{2}}|n+m\tau|^{2}}-\sqrt{\frac{3}{2}}\sum_{n,m}e^{-\frac{3\pi}{2\tau_{2}}|n+m\tau|^{2}}).
P=P_{0}(l)\left(\frac{2}{\lambda}\right)^{2l+3}\frac{(l+1)!^{2}\Gamma^{4}(1+l/4)}{\pi}=P_{0}(l)\left(\frac{2\piT}{\omega}\right)^{2l+3}\frac{(l+1)!^{2}\Gamma^{4}(1+l/4)}{\pi}\,,
<s_{+}(x)s_{+}(y)>_{+2}^{\mu=0}=<s_{-}(x)s_{-}(y)>_{-2}^{\mu=0}=\frac{1}{2}<\bar{\psi}\psi(x)\bar{\psi}\psi(y)>_{\vertn\vert=2}^{\mu\ne0}
\renewcommand{\arraystretch}{1.2}\begin{array}{rcl}{\chi_{0}^{\alpha}\omega}&{=}&{\bar{\chi}_{0}^{\alpha}\omega=-\theta^{\alpha}\,,}\\{\chi_{0}^{\alpha}\theta^{\beta}}&{=}&{\bar{\chi}_{0}^{\alpha}\theta^{\beta}=d^{\alpha\beta}\Omega\,,}\\{L_{0}\omega}&{=}&{\bar{L}_{0}\omega=\Omega\,.}\\\end{array}
\delta\bar{\psi}_{\rho}=\bar{\epsilon}\left(\overleftarrow{\partial}_{\rho}-\frac{1}{2}\,\omega_{\rhomn}\,\sigma^{mn}+\frac{1}{2}\,\alpha^{a}A_{\rho}^{a}\right)-\frac{1}{2\sqrt{2}}\,e^{\phi}\,\bar{\epsilon}
\times\;\Big\langle\prod_{b=1}^{N}\prod_{i=1}^{n_{b}^{\prime}}\overline{{\psi}}^{(b)}({x^{\prime}}_{i}^{(b)})P_{+}\psi^{(b)}({x^{\prime}}_{i}^{(b)})\prod_{i=1}^{m_{b}^{\prime}}\overline{{\psi}}^{(b)}({y^{\prime}}_{i}^{(b)})P_{-}\psi^{(b)}({y^{\prime}}_{i}^{(b)})\Big\rangle_{0}\;.
\left(\begin{array}{c}{\left[\frac12AA\right]_{\mathrm{R}}}\\{[i\bar{C}C]_{\mathrm{R}}}\\\end{array}\right)=\left(\begin{array}{cc}{Z_{1}}&{Z_{2}}\\{Z_{3}}&{Z_{4}}\\\end{array}\right)\left(\begin{array}{c}{\left[\frac12AA\right]}\\{[i\bar{C}C]}\\\end{array}\right).
\langle\psi_{m;0}|\psi_{0;m}^{\prime}\rangle=\int\,dqd\etad\eta^{\times}(\psi(q)\eta^{m}\cdots\eta^{1})^{\ast}(\psi^{\prime}(q)\eta^{\times1}\cdots\eta^{\timesm})=\int\,dq\psi^{\ast}(q)\psi^{\prime}(q)
f_{UUU}=8\pi^{2}\frac{(T-{\bar{T}})}{(U-{\bar{U}})^{2}}\int\frac{d^{2}\tau}{\tau_{2}^{2}}{\bar{F}}({\bar{\tau}})\partial_{\bar{\tau}}\left(\tau_{2}^{2}\partial_{\tau}(\tau_{2}^{2}\sum_{P_{L},P_{R}}P_{L}P_{R}P_{R}P_{R}e^{\pii\tau|P_{L}|^{2}}e^{-\pii{\bar{\tau}}|P_{R}|^{2}})\right).
<{\calP}(x_{1})\sigma_{d_{1}}^{(0)}(x_{1})\cdots{\calP}(x_{n})\sigma_{d_{n}}^{(0)}(x_{n})>\,\approx\,<\int_{\Sigma_{g}}\sigma_{d_{1}}^{(2)}(x_{1})\cdots\int_{\Sigma_{g}}\sigma_{d_{n}}^{(2)}(x_{n})>.
\deltav^{\ell}(\omega)=-e\left(\Phi_{p},\left(L^{\ell}{\frac{1}{\omega-(H-E)}}L^{k}-L^{k}{\frac{1}{\omega+H-E}}L^{\ell}\right)\Phi_{p}\right){\frac{A_{k}^{e}(\omega)}{Z_{3}^{\,2}}}~~.
P\psi=P\left(\begin{array}{c}{\psi_{0}}\\{\psi_{1}}\\{\psi_{2}}\\{\psi_{3}}\\{\psi_{4}}\\\end{array}\right)=\left(\begin{array}{c}{0}\\{0}\\{0}\\{0}\\{\psi_{4}}\\\end{array}\right);\P_{\mu}\psi=\left(\begin{array}{c}{0}\\{0}\\{0}\\{0}\\{\psi_{\mu}}\\\end{array}\right)\\left(\mu=0,...,3\right).
<\psi(x_{1})\ldots\psi(x_{n})>_{0}=\frac{\int[d\psi]\psi(x_{1})\ldots\psi(x_{n})e^{\int\frac{1}{2}\bar{\psi}(\gamma\cdot\partial+M)\psi}}{\int[d\psi]e^{\int\frac{1}{2}\bar{\psi}(\gamma\cdot\partial+M)\psi}}
\nabla^{B}\nabla_{\\\dot{A}}^{C}W_{ABC}=-2i\nabla_{\\\dot{A}}^{B}\nabla_{\underline{{A}}}^{\\\dot{B}}G_{\underline{{B}}\dot{B}}+G_{\\\dot{A}}^{B}\nabla_{\underline{{A}}}^{\\\dot{B}}G_{\underline{{B}}\dot{B}}+\frac{11}{2}i\left(\nabla^{B}G_{\\\dot{A}}^{C}\right)W_{ABC}
\psi=\left[\begin{array}{c}{\psi^{\uparrow}(\vec{t})}\\{\psi^{\downarrow}(\vec{t})}\\\end{array}\right]=\sum_{n=0,\pm1,\cdots}\left[\begin{array}{c}{\frac1{\sqrt{2\pi}}\psi_{n}^{\uparrow}(t)e^{in\theta}}\\{\frac1{\sqrt{2\pi}}\psi_{n}^{\downarrow}(t)e^{in\theta}}\\\end{array}\right],
\phi_{1}=\frac{m}{\sqrt\lambda}\left\{-\frac{3}{4}m^{2}t^{2}{\eta}_{0}-\frac{1}{4}{\eta}_{k{_0}}+a_{1}e^{i\omega_{1}t}\frac{~\mathrm{{sh}}~z}{\mathrm{{ch}}^{2}~z}+\sum_{k=0}^{\infty}a_{k}{\widetildeC}_{k}(t)\eta_{k}(x)\right\}
\varepsilon_{\nu\tau}p^{\tau}(p_{\rho}p_{\sigma}-g_{\rho\sigma}p^{2})=\varepsilon_{\rho\tau}p^{\tau}(p_{\nu}p_{\sigma}-g_{\nu\sigma}p^{2})=\varepsilon_{\sigma\tau}p^{\tau}(p_{\nu}p_{\rho}-g_{\nu\rho}p^{2})
{\mathfrakm}=\left(\begin{matrix}{1}&{0}&{0}&{0}&{0}\\{5(m_{1}-2m_{2})}&{-1}&{1}&{-5}&{10}\\{5(m_{2}-3m_{1})}&{1}&{-2}&{15}&{-5}\\{0}&{0}&{-1/5}&{1}&{0}\\{0}&{-1/5}&{0}&{0}&{1}\\\end{matrix}\right).