harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}{z_{i}^{\prime}}&{{}=e^{\xi_{i}},}\\{\xi_{i}^{\prime}}&{{}=\sum_{k\nei}e^{\xi_{k}}\frac{\wp^{\prime}(z_{i})+\wp^{\prime}(z_{k})}{\wp(z_{i})-\wp(z_{k})},\quadi=1,\dots,n.}\\\end{aligned}
\begin{array}{ccc}{A^{\sigma}(\mu)}&{\longrightarrow}&{A^{\sigma}(\mu+\nu)}\\{\downarrow}&{\,}&{\downarrow}\\{{\bfC}[\sigma^{\vee}](\mu+\chi)}&{\longrightarrow}&{{\bfC}[\sigma^{\vee}](\mu+\nu+\chi)}\\\end{array}
\begin{array}{rl}{\displaystyle\langle\psi_{s_{1}}\vert\psi_{s_{2}}\rangle}&{\displaystyle=\alpha\int_{0}^{2V}t^{-s_{12}+2k-1}dt}\\{}&{\displaystyle=\frac{2\alpha}{l}Z\left[\frac{2}{l}(2k-s_{12})\right],}\\\end{array}
\alpha_{m}^{(A)}\rightarrow\frac{1}{\sqrt{m}}\alpha_{m}^{(A)}\,\,,\,\,A_{r}\rightarrow\frac{1}{\sqrt{r}\gamma}A_{r}\,\,,\,\,B_{s}\rightarrow\frac{1}{\sqrt{s}\gamma}B_{s}\,\,,\,\,
\overline{\zeta}(\Omega;\,\omega_{0};\,\omega_{1};\,\omega_{2};\,\omega_{3};\,\omega_{4};\,\omega_{5})=\zeta(\Omega;\,\omega_{0};\,\omega_{5};\,\omega_{4};\,\omega_{3};\,\omega_{2};\,\omega_{1})
\begin{array}{rclc}{\left(0,1,0,1,e^{\phi}\right)e_{\mu}}&{=}&{e_{\mu}}&{.}\\{\left(V_{1},1,0,1,1\right)e_{\mu}}&{=}&{e_{\mu}}&{.}\\{\left(0,1,V_{2},1,1\right)e_{\mu}}&{=}&{-\sqrt{2}V_{2}^{\mu}\cdotu_{+}+e_{\mu}}&{.}\\{\left(0,C_{1},0,1,1\right)e_{\mu}}&{=}&{e_{\mu}}&{.}\\{\left(0,1,0,C_{2},1\right)e_{\mu}}&{=}&{\sum_{b=n-m+1}^{n-1}\left(C_{2}\right)_{\mu\nu}\cdote_{\nu}}&{.}\\\end{array}
\int\frac{\mathrm{d}\rho^{2}}{\rho^{2}}\,\mathrm{d}^{2}\bar{\beta}\,\mathrm{d}^{2}\bar{\theta}_{0}\,F(\rho^{2}[1+4i\bar{\beta}\bar{\theta}_{0}])=\int\frac{\mathrm{d}\rho^{2}}{\rho^{2}}\,16\rho^{4}F^{\prime\prime}(\rho^{2})=16\,F(\rho^{2}=0)\,.
{\frac{\partial^{R}{\calH}}{\partial\Pi_{c}^{a}}}-\partial_{t\mu}{\frac{\partial^{R}{\calH}}{\partial\left[{\partial_{t\mu}\Pi_{c}^{a}}\right]}}+\partial_{n}c^{a}=0,\qquad{\frac{\partial^{R}{\calH}}{\partialc^{a}}}-\partial_{t\mu}{\frac{\partial^{R}{\calH}}{\partial\left[{\partial_{t\mu}c^{a}}\right]}}+\partial_{n}\Pi_{c}^{a}=0,
{\calS}_{PCS}=\frac{k}{4\pi}\int\,\mathrm{d}g\,\epsilon^{abc}\,(A_{a}\{x_{b},A_{c}\}-\frac{1}{2}A_{a}f_{bc}^{~~d}A_{d}+\frac{1}{3}A_{a}\{A_{b},A_{c}\})~,
H_{n}=\intd{\bfx}\intdkdk^{\prime}e^{ikx}e^{ik^{\prime}x}\phi(k)\phi(k^{\prime})(k_{0}^{2}-k_{0}k_{0}^{\prime})\sum_{s+t=n-1}(i^{2})^{s+t+1}(k^{2})^{s}(k^{2})^{t}-\intd{\bfx}{\calL}_{n}
S=\left(\begin{array}{cc}{0}&{C}\\{C}&{0}\\\end{array}\right),\;\;\;\;\;\tilde{N}=\left(\begin{array}{cc}{\hat{N}^{22}(x,x)}&{\hat{N}^{23}(x,x)}\\{\hat{N}^{32}(x,x)}&{\hat{N}^{33}(x,x)}\\\end{array}\right),\;\;\;\;\;\tilde{X}=\left(\begin{array}{cc}{\hat{X}^{22}(x,x)}&{\hat{X}^{23}(x,x)}\\{\hat{X}^{32}(x,x)}&{\hat{X}^{33}(x,x)}\\\end{array}\right)
\langle\Sigma|\eta_{0}|\varphi\rangle=\left\langlef_{\Sigma}\circ\left(\oint_{C}\frac{dz}{2\pii}\eta(z)\cdot\varphi(0)\right)\right\rangle_{\mathrm{UHP}}=\left\langle\oint_{C^{\prime}}\frac{dz}{2\pii}\eta(z)\cdotf_{\Sigma}\circ\varphi(0)\right\rangle_{\mathrm{UHP}}=0,
{N}_{[0][f,\psi][f,\pm\psi]}={\frac{1}{2}}({N}_{0ff}\pm\breve{N}_{0ff})\;\;\;\;,\;\;\;\;\;Y_{[f,\psi][0][0]}={\frac{1}{2}}({Y}_{f00}+\epsilon{Y}_{f0J})\;\;\;\;.
\phi=\left(\begin{array}{cc}{\phi_{\mathrm{max}}I_{N_{1}}}&{0}\\{0}&{\phi_{*}I_{\infty-N_{1}}}\\\end{array}\right)\,\quadC_{i}=\left(\begin{array}{cc}{S_{i}}&{0}\\{0}&{V_{i}}\\\end{array}\right).
b_{0}^{\ast}(\overline{{K}},t)b_{0}^{\ast}(\overline{{P}},t)|0\rangle=\int\int\overline{{\kappa}}_{\overline{{K}},\overline{{P}}}^{(+)}(K,P)b^{\ast}(K,t)b^{\ast}(P,t)|0\rangledKdP
\bar{H}_{\mathrm{G}}=H_{\mathrm{G}}(Q^{r},P^{r},V)+\Lambda_{a}^{(1)r_{a}}\Pi_{a}^{r_{a}}+\Lambda_{a}^{(2)r_{a}}\Psi_{a}^{r_{a}}+\Lambda_{a}^{(3)}\Theta_{a}+\Lambda_{a}^{(4)}\Gamma_{a},
\mathrm{~\frac{1}{2}~}(S_{m(0)},S_{m(0)})^{a}+V_{m}^{a}S_{m(0)}=0,\qquad\mathrm{~\frac{1}{2}~}\{S_{m(0)},S_{m(0)}\}_{\alpha}+V_{\alpha}S_{m(0)}=0.
\bar{\psi}({\vec{x}})=\sum_{n_{1},n_{2},n_{3}=-N}^{+N}\frac{1}{X^{3/2}}\sum_{s}\left(a_{\vec{n}}^{s\dagger}\bar{u}_{\vec{n}}^{s}e^{-2\pii{\vec{n}}\cdot{\vec{x}}/X}+b_{\vec{n}}^{s}\bar{v}_{\vec{n}}^{s}e^{+2\pii{\vec{n}}\cdot{\vec{x}}/X}\right).
{\mathrm{p.v.}}\int_{0}^{\infty}{F(x)Y_{\nu}(x)dx}=r_{3\nu}[F(z)]-\frac{i}{\pi}\int_{0}^{\infty}{K_{\nu}(x)\left[e^{-\nu\pii/2}F(xe^{\pii/2})-e^{\nu\pii/2}F(xe^{-\pii/2})\right]dx}
{\calR}_{z\doteqz_{n}=\cdots=z_{0}}F={\calR}_{z_{n}=\cdots=z_{0}}\frac{1}{\pi}\int_{\Sigma}F\,d^{2}\!z-\frac{1}{\pi}\int_{\Sigma}{\calR}_{z_{n}=\cdots=z_{0}}F\,d^{2}\!z.
Q_{total}=\left(\begin{matrix}{1}&{-1}&{-1}&{-1}&{-1}&{1}&{0}\\{0}&{0}&{-1}&{1}&{0}&{0}&{\lambda_{5}+\lambda_{6}}\\{0}&{0}&{0}&{-1}&{1}&{0}&{\lambda_{3}+\lambda_{4}}\\{0}&{1}&{0}&{0}&{-1}&{0}&{\lambda_{1}+\lambda_{2}}\\\end{matrix}\right)
\xi_{\alpha\beta}(\mu_{1}^{2},\mu_{2}^{2},\mu_{3}^{2},p^{2},q^{2})=\int_{0}^{1}dz\int_{0}^{1-z}dy\frac{z^{\alpha}y^{\beta}}{Q(y,z,\mu_{1}^{2},\mu_{2}^{2},\mu_{3}^{2},p^{2},q^{2})},
A=\int_{\partial{\calV}}d^{4}x\partial_{c}P^{c}=\int_{0}^{\beta}dt\int_{\partial{\calV}}d^{3}x\nabla\cdot{\bfP}=\beta\int_{\partial\partial{\calV}}d^{2}x_{\perp}\hat{\bfn}\cdot{\bfP}
a_{tree}(s,b)=<0|\int\frac{d\sigma_{u}d\sigma_{d}}{(2\pi)^{2}}:a_{tree}(s,\vec{b}+\hat{\vec{X}}^{u}(\sigma_{u})-\hat{\vec{X}}^{d}(\sigma_{d})):|0>=<0|\hat{\delta}|0>
A_{\mu}=\mp\,{\frac{2i}{g}}\sum_{I=1}^{K}\,\biggl(H_{I}\,{\frac{v_{\mu}^{I}}{\rho_{I}}}+g\,\kappa\,E_{I\,K+1}^{\pm}\,R_{\mu}^{I}\,\prod_{I=1}^{K-1}\delta(R^{K}\cdotR^{I})\biggr).
\frac{i}{2}\bar{\Omega}^{iA}\hat{\calD}\Omega_{iA}=\frac{i}{2}\bar{\lambda}^{i}\gamma^{\mu}[D_{\mu}-\frac{3}{4}U_{\mu}]\lambda_{i}+\frac{i}{2}\bar{\Omega}^{ia}\gamma^{\mu}[D_{\mu}-\frac{1}{4}U_{\mu}]\Omega_{ia}-\frac{i}{\sqrt{2}}\bar{\lambda}^{i}\gamma^{\mu}\Phi\partial_{\mu}z^{a}\Omega_{ia}
g^{2}\int\frac{d^{6}p}{(2\pi)^{6}}\,\frac{d^{6}k}{(2\pi)^{6}}\,frac{e^{-i(\theta_{12}(p_{1}\,k_{2}-p_{2}\,k_{1})+\theta_{34}(p_{3}\,k_{4}-p_{4}\,k_{3}))}}{p^{2}\,k^{2}\,(p+k)^{2}}=g^{2}\frac{1}{\theta_{12}+\theta_{34}}\,\frac{1}{\theta_{12}\,\theta_{34}}.
I_{1}(l_{\psi},l_{\phi_{1}})=\frac{1}{(1-f^{2})\sqrt{2-f^{2}}}\biggl[(2-f^{2})E\left(\frac{\pi}{2},\sqrt{\frac{1-f^{2}}{2-f^{2}}}\right)-F\left(\frac{\pi}{2},\sqrt{\frac{1-f^{2}}{2-f^{2}}}\right)\biggr].
\zeta\,\left(\,t;{\bfx},{\bfy}\,|\,M\,\right)\equiv\frac{1}{\Gamma\,(s)}\:\int_{0}^{\infty}dt\,t^{\,s-1}\,\left[\,h\,\left(\,t;{\bfx},{\bfy}\,|\,M\,\right)-P^{\,(0)}\,({\bfx}\,,\,{\bfy})\,\right]\;\;\;,
S_{cl}=-\frac{1}{2}\intd^{d}\vec{x}\sqrt{|h|}\intd\bar{\eta}d\etan^{\mu}\Phi^{0}(x,\eta)\partial_{\mu}\Phi^{0}(x,\eta)+\intd^{d+1}x\sqrt{|g|}\intd\etad\bar{\eta}\frac{\lambda_{n}(\eta)}{n!}(\Phi^{0}(x,\eta))^{n}
\left.\hat{C}_{\hat{a}}\right.^{\hat{d}}\hat{{\calH}}_{\hat{d}\hat{b}\hat{c}}={\frac{\sqrt{-\mathrm{det}\hat{g}}}{3!}}\epsilon_{\hat{a}\hat{b}\hat{c}\hat{d}\hat{e}\hat{f}}\hat{{\calH}}^{\hat{d}\hat{e}\hat{f}}\,.
H_{nm}^{\{{\bfR}\}}(0,0)=\frac{m!n!(-1)^{\frac{m+n}2}}{2^{\frac{|m-n|}2}\left(\frac{m+n}2\right)!}\left[R_{11}^{n}R_{22}^{m}\left(r^{2}-1\right)^{\mu_{mn}}\right]^{\frac12}P_{\mu_{mn}}^{(\frac{|m-n|}2,\frac{|m-n|}2)}\left(\fracr{\sqrt{r^{2}-1}}\right),
f(z)_{a}\equiv\sum_{n=0}^{\infty}(-1)^{n}{\frac{a^{2n}z^{2n+1}}{(2n+1)^{2}}}=z\F\left(1,{\frac{1}{2}},{\frac{1}{2}};{\frac{3}{2}},{\frac{3}{2}};-a^{2}z^{2}\right),
R\left(\frac{Q}{2}+ip\right)=S\left(p\right)=-\left(\frac{\pi\mu\Gamma\left(b^{2}\right)}{\Gamma\left(1-b^{2}\right)}\right)^{-\frac{2ip}{b}}\frac{\Gamma\left(1+2ip/b\right)\Gamma\left(1+2ipb\right)}{\Gamma\left(1-2ip/b\right)\Gamma\left(1-2ipb\right)}\.
\widehat{T_{\k}^{j}}=\Big(1-{\frac{c_{_\mathrm{S}}^{2}}{c^{2}}}\Big)n^{j}\mu_{j}-{\frac{c_{_\mathrm{S}}^{2}}{c^{2}}}(\rhoc^{2}+P)g_{\k}^{j}\,
\xi_{a}^{N}\xi_{1}^{2}\xi_{2}^{2}\cdots\xi_{l}^{2}=(-Q)^{N-2p+2}\prod_{b\neqa=l+1}^{l+p}\left\{-\frac{1+\xi_{a}\xi_{b}+(Q+Q^{-1})\xi_{a}}{1+\xi_{a}\xi_{b}+(Q+Q^{-1})\xi_{b}}\right\}
G_{\mu\nu}=-\Lambda_{4}(\Psi)g_{\mu\nu}+8\pi\bar{G}_{N}(\Psi)\tau_{\mu\nu}+\kappa_{5}^{4}\pi_{\mu\nu}-E_{\mu\nu}+{{\frac{2\kappa_{5}^{2}}{3}}}\hat{T}_{\mu\nu}(\Psi)\,
\frac{1}{a_{1}^{\alpha}a_{2}^{\beta}}=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\int_{0}^{1}\dx\frac{(1-x)^{\alpha-1}x^{\beta-1}}{[a_{1}(1-x)+a_{2}x]^{\alpha+\beta}}
\left(\theta_{4,12}-\theta_{8,12}\right)\left[\left(\theta_{1,12}+\theta_{11,12}\right)^{2}+\left(\theta_{5,12}+\theta_{7,12}\right)^{2}\right]=\left(\theta_{0,12}-\theta_{12,12}\right)\left(\theta_{3,12}+\theta_{9,12}\right)^{2}.
\alpha={\frac{M^{2}}{g^{2}\left(\lambda_{1}^{2}-\lambda_{2}^{2}\right)}}\left({\frac{M^{2}}{g^{2}}}F\left(\lambda_{1}^{2},\lambda_{2}^{2},\Lambda^{2}\right)+ln{\frac{\lambda_{1}^{2}+m_{0}^{2}}{\lambda_{2}^{2}+m_{0}^{2}}}\right)
\begin{array}{lcr}{\partial_{\sigma}^{2}X^{9}+\sum_{\nu=1}^{8}{\frac{1}{2}}(-)^{\nu+1}(\{\partial_{\sigma}X^{\nu},X^{\nu+(-)^{\nu+1}}\}+\{X^{\nu},\partial_{\sigma}X^{\nu+(-)^{\nu+1}}\})=0,}\\\end{array}
0=\frac{1}{4}[\frac{16\pi^{4}\theta^{2}}{g^{2}\eta^{2}}z\int\frac{d^{2}p}{(2\pi)^{2}}[p^{-2}G^{-2}(p)-\frac{4\theta^{2}g^{2}}{\eta^{2}}(1-K^{2}(p)(p^{2}+\lambda\eta^{2}))]+(k^{2}+\lambda\eta^{2})-K^{-2}(k)]
\left[-\frac{\partial^{2}}{\partialr^{2}}+\frac{2ne_{L}/e-1}{a}e^{-\frac{B}{2}}\frac{\partial}{\partialr}+\frac{m(m+1-2ne_{L}/e)}{a^{2}}e^{-B}\right]\psi_{L}^{m}=e^{-A}m_{F}^{2}\psi_{L}^{m}~,
ds^{2}=-\left({\frac{r}{\mu}}\right)^{2(D-3)}dt^{2}+\left({\frac{\mu}{r}}\right)^{2}dr^{2}+\mu^{2}d\Omega_{D-2}^{2},\\\\\A_{t}=-\left({\frac{r}{\mu}}\right)^{D-3}.
\begin{array}{c}{A_{2\rightarrow2+n}=s\frac{(s_{1}/m^{2})^{\alpha(q_{1}^{2})}}{(q_{1}^{2}-m^{2})}\cdots\frac{(s_{n+1}/m^{2})^{\alpha(q_{n+1}^{2})}}{(q_{n+1}^{2}-m^{2})}}\\{\Gamma_{AD_{0}}^{i_{1}}\gamma_{i_{1}i_{2}}^{D_{1}}\cdots\gamma_{i_{n}i_{n+1}}^{D_{n}}\Gamma_{BD_{n+1}}^{i_{n+1}}}\\\end{array}
\sum_{I}C_{i_{1}i_{2}}^{I}C_{j_{1}j_{2}}^{I}=\frac{1}{2}\{\delta^{i_{1}j_{1}}\delta^{i_{2}j_{2}}+\delta^{i_{1}j_{2}}\delta^{i_{2}j_{1}}-\frac{1}{3}\delta^{i_{1}i_{2}}\delta^{j_{1}j_{2}}\}
\Bigl(\frac{{\dot{a}}}{a}\Bigr)^{2}=\frac{8\piG_{N}}{3}\rho-\frac{K}{a^{2}}+\frac{\Lambda_{4}}{3}+\frac{\kappa_{5}^{4}}{36}\rho^{2}+\frac{\mu}{a^{4}},
\begin{array}{l}{P={\frac{4\pi}{h^{3}\beta}}[{\frac{\sqrt\pi}{2}}({\frac{2m}{\beta}})^{\frac{3}{2}}z+{\sqrt\pi}({\frac{2m}{\beta}})^{\frac{3}{2}}\delta(p,q)z^{2}+...].}\\\end{array}
\left[-a^{-p}{\frac{\partial}{\partiala}}a^{p}{\frac{\partial}{\partiala}}+\left({\frac{3\pi}{2G}}\right)^{2}{\frac{1}{k^{3}}}\left(ka^{2}-{\frac{8\piG}{3}}\rhoa^{4}\right)\right]\psi(a)=0.
\frac{\ddot{a}_{0}}{a_{0}}+\Big(\frac{\dot{a}_{0}}{a_{0}}\Big)^{2}+\frac{k}{a_{0}^{2}}=\frac{1}{6M^{3}}\Big(\Lambda+\frac{\sigma^{2}}{12M^{3}}\Big)-\frac{1}{144M^{6}}\Big(\sigma(3p-\rho)+\rho(3p+\rho)\Big)-\frac{1}{6M^{3}}\tilde{T}^{y}~_{y}.
changedQ_{B}\int_{T_{0}}^{\infty}dT\langle\widetilde{V}_{2}|b_{0}e^{-TL_{0}}|V_{3}\rangle=\langle\widetilde{V}_{2}|e^{-T_{0}L_{0}}|V_{3}\rangle=\sum_{n}^{}c_{n}(T_{0})e^{(1-n)T_{0}},
{\calH}=\frac{a^{2}\mu^{2}}{2M^{2}}K_{1}^{2}(Mr)-\frac{a^{2}\mu^{2}}{2M^{3}r}K_{1}(Mr)+\frac{a^{2}(2m^{2}-\mu^{2})}{2M^{2}}K_{1}^{2}(Mr),
\begin{array}{lcr}{Region\,I\qquad}&{\psi^{b}=s\left(\begin{array}{c}{E+m}\\{-\kappa}\\\end{array}\right)e^{\kappax}}&{\qquadx\leq-\xi}\\{Region\,III\qquad}&{\psi^{b}=s^{\prime}\left(\begin{array}{c}{E+m}\\{\kappa}\\\end{array}\right)e^{-\kappax}}&{\qquadx\geq\xi}\\\end{array}
\mathrm{with}\;\;\;\;{\cal{X}}^{\alpha}({\bf{r}})={\textstyle\frac{\partial_{i}}{\partial^{2}}}A_{i}^{\alpha}({\bf{r}})\;\;\;\;\mathrm{and}\;\;\;\;\overline{{{\calY}^{\alpha}}}({\bfr})={\textstyle\frac{\partial_{j}}{\partial^{2}}\overline{{{\calA}_{j}^{\alpha}}}({\bfr})}.
M^{-}(z)=\sum_{{n\geq0}\atop{m>0}}M_{nm}^{-}(z)b_{n}^{\prime}\cdot\bar{b}_{-m}+\sum_{{n\geq0}\atop{m>0}}M_{nm}^{-}(z)\bar{b}_{n}^{\prime}\cdotb_{-m}\,,
\theta=\left(\begin{array}{ccccc}{0}&{-\theta_{1}}&{}&{}&{}\\{\theta_{1}}&{0}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{0}&{-\theta_{\frac{d}{2}}}\\{}&{}&{}&{\theta_{\frac{d}{2}}}&{0}\\\end{array}\right).
H=\sum_{\vec{k}}A_{\vec{k}}^{\dagger}A_{\vec{k}}=\sum_{\vec{k}}S_{\vec{k}}^{2}=\sum_{\vec{k}}\sqrt{P_{\vec{k}}^{2}+m_{q}^{2}+{\vec{k}}^{2}}+C(\vec{k})\,\,,
F_{j}=\sum_{k_{1}=0}^{\left[\frac{j}{2}\right]}\cdots\sum_{k_{2N-1}=0}^{\left[\frac{j}{2N}\right]}\frac{1}{k_{1}!\cdotsk_{2N-1}!}\frac{\left(j-\sum_{l=1}^{2N-1}lk_{l}\right)!}{\left(j-\sum_{l=2}^{2N}lk_{l-1}\right)!.}
{\calS}\equiv\sum_{\{n_{\bfk}\}}\left(\prod_{\bfk}N_{j_{\bfk}j_{\bfk}^{\prime}}^{m_{\bfk}n_{\bfk}^{\prime}}N_{i_{\bfk}i_{\bfk}^{\prime}}^{n_{\bfk}n_{\bfk}^{\prime}}i_{\bfk}!i_{\bfk}^{\prime}!j_{\bfk}!j_{\bfk}^{\prime}!\right)\langle\{n_{\bfk}\}\mid\,\rho\,|\{m_{\bfk}\}\rangle\,.
\{A,D\}_{(3,1)}=-4d{\partial}_{1}a\cdot{\gamma}^{2}-4{\gamma}^{2}d\cdot{\sum_{k=1}^{\infty}}{\gamma}^{2k}{\partial}_{1}a_{2k}-2(1-4{\gamma}^{2}){\partial}_{1}d\cdot{\sum_{k=1}^{\infty}}k{\gamma}^{2k}a_{2k},
S_{D}=\int\mathrm{d}^{D}x\;\sqrt{g}\,(d_{p}N)^{\frac{D-2}{\tilde{d}}}e^{\delta\phi}\left\{R+\gamma(\partial\phi)^{2}-{\frac{1}{2(d_{p}N)^{2}(\tilde{d}+1)!}}F_{\tilde{d}+1}^{2}\right\}\,.
\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
\begin{array}{c}{H\left(a^{+}\Psi\right)=a^{+}\left(H+\omega\right)\Psi=\left(E+\omega\right)a^{+}\Psi}\\{H\left(a^{-}\Psi\right)=a^{-}\left(H-\omega\right)\Psi=\left(E-\omega\right)a^{-}\Psi}\\\end{array}
\begin{array}{rll}{P\,\,}&{\,\,p_{1}+p_{2}\,\,}&{\,\,-(p_{3}+p_{4})\,,}\\{R\,\,}&{\,\,p_{1}+p_{3}\,\,}&{\,\,-(p_{2}+p_{4})\,,}\\{Q\,\,}&{\,\,p_{1}+p_{4}\,\,}&{\,\,-(p_{2}+p_{3})\,.}\\\end{array}
\left[l^{k-1}\left\{\int_{0}^{l}dl^{\prime}{\frac{\delta^{2}}{\deltaj(l^{\prime})\deltaj(l-l^{\prime})}}+g\int_{0}^{\infty}dl^{\prime}l^{\prime}j(l^{\prime}){\frac{\delta}{\deltaj(l+l^{\prime})}}\right\}+C\delta^{(k)}(l)\right]Z[j]=0,
(3d)=-\frac{\lambda^{2}}{2^{2}.3}i^{5}\int{\calD}k_{1}{\calD}k_{2}{\calD}k_{3}{\calD}k_{4}\frac{1}{k_{1}^{2}k_{2}^{2}k_{3}^{2}k_{4}^{2}(p+k_{1}+k_{2}-k_{3}-k_{4})^{2}}\;.
i\delta\tilde{Y}_{a}=\deltaY_{ab}U^{b}={\binom{-\deltaz_{\alpha\beta}u^{\beta}+\deltax_{\alpha\dot{\alpha}}\bar{u}^{\dot{\alpha}}}{\delta\tilde{x}^{\dot{\alpha}\alpha}u_{\alpha}-\delta\bar{z}^{\dot{\alpha}\dot{\beta}}\bar{u}_{\dot{\beta}}}}.
G_{\pm}(\alpha_{\pm})=e^{\alpha_{\pm}\bar{T}_{\pm}}=1+\alpha_{\pm}\bar{T}_{\pm}=\left(\begin{array}{ccc}{1}&{\frac{\alpha_{\pm}}{\sqrt{2}}}&{\mpi\frac{\alpha_{\pm}}{\sqrt{2}}}\\{0}&{1}&{0}\\{0}&{0}&{1}\\\end{array}\right)
C_{i_{1}...i_{s}}^{g}=\int_{M_{g,s+1}}\langle\oint_{C_{z_{1}}}\!\!G^{-}{\bar{G}}^{-}\phi_{i_{1}}...\oint_{C_{z_{s}}}\!\!G^{-}{\bar{G}}^{-}\phi_{i_{s}}\prod_{j,{\bar{j}}=1}^{3g-3}G^{-}(\chi_{j}){\bar{G}}^{-}({\bar{\chi}}_{\bar{j}})\rangle
\hat{p}(x)*\hat{q}(x)=e^{\frac{i}{2}\theta_{\mu\nu}\partial_{\sigma_{\mu}}\partial_{\xi_{\nu}}}\hat{p}(x+\sigma)\hat{q}(x+\xi)\mid_{\sigma=\xi=0}=\hat{p}(x)\hat{q}(x)+\frac{i}{2}\theta^{\rho\sigma}\partial_{\rho}\hat{p}(x)\partial_{\sigma}\hat{q}(x)+~O(\theta^{2}).
\langleX_{1}^{n}\rangle_{log}=5(5\sqrt{5})^{g-1}\mu^{5\{k+2(1-g)\}}\left[\left({\frac{\sqrt{5}-1}{2}}\right)^{5(k-g+1)}+(-1)^{k}\left({\frac{\sqrt{5}+1}{2}}\right)^{5(k-g+1)}\right]
x_{1}^{12}+x_{2}^{12}+x_{3}^{6}+x_{4}^{6}+x_{5}^{2}+{\tilde{a}_{1}}a_{1}x_{1}^{6}x_{2}^{6}+{\tilde{a}_{2}}a_{2}x_{1}x_{2}x_{3}x_{4}x_{5}=0.
\left(\begin{array}{cccc}{a}&{d_{1}}&{d_{2}}&{d_{3}}\\{d_{1}}&{b_{1}}&{c_{3}}&{c_{2}}\\{d_{2}}&{c_{3}}&{b_{2}}&{c_{1}}\\{d_{3}}&{c_{2}}&{c_{1}}&{b_{3}}\\\end{array}\right).
f\circ\mathrm{V}_{v}^{\mu}(w,k)=[f^{\prime}(w)]^{\alpha^{\,\prime}k^{2}}\,\mathrm{V}_{v}^{\mu}(f(w),k)+\left[\frac{\alpha^{\,\prime}}{2}\right]^{\frac12}k^{\mu}[f^{\prime}(w)]^{\alpha^{\,\prime}k^{2}}\frac{f^{\prime\prime}(w)}{f^{\prime\,2}(w)}c(f(w))e^{2ik\cdotX(f(w))}
\left[{\frac{dn_{s}(\omega|D)}{d\omega}}\right]_{\tiny\mathrm{div}}={\frac{\Gamma\left(1-\fracD2\right)}{(4\pi)^{D/2}}}\,\,{\frac{m^{D-4}}{\kappa}}\,\,\int_{\Sigma}\left[2\left(m^{2}-\left(\frac16-\xi\right)R\right)-{\frac{\omega^{2}}{\kappa^{2}}}{\calP}\right]~~~,
\begin{array}{ll}{I(g,A)=I(g)}&{+\frac{k}{2\pi}\intd^{2}z\\tr[{\bf{A}}\overline{{\partial}}gg^{-1}+\overline{{\bf{A}}}g^{-1}\partialg+{\bf{A}}\overline{{\bf{A}}}+{\bf{A}}g\overline{{\bf{A}}}g^{-1}]}\\{}&{+\frac{k}{2\pi}\intd^{2}z[\frac{4c}{k}(A\overline{{\partial}}f+\overline{{A}}\partialf)+\frac{8c^{2}}{k}A\overline{{A}}]+\frac{1}{\pi}\intd^{2}z\partialf\overline{{\partial}}f}\\\end{array}
F^{a}=\frac{\partialX}{\partialJ_{0}^{a}}\;,\qquadG^{a}=\frac{\partialX}{\partialJ_{1}^{a}}\;,\qquad\frac{\partial^{2}X}{\partialJ_{0}^{a}\,\partialJ_{1}^{b}}=\frac{\partial^{2}X}{\partialJ_{0}^{b}\,\partialJ_{1}^{a}}\;.
q=\frac{1}{8\pia^{4}}\sum_{l=1}^{\infty}(2l+1)\sum_{k=1}^{\infty}\sum_{\lambda=0}^{1}\gamma_{\nu,k}^{(\lambda)4}T_{\nu}^{AB}(\eta,\gamma_{\nu,k}^{(\lambda)})\psi_{\mu}(\gamma_{\nu,k}^{(\lambda)}/a)f_{\lambdal}^{(q)}(\gamma_{\nu,k}^{(\lambda)},\gamma_{\nu,k}^{(\lambda)}x),\quadq=\varepsilon,p_{\perp},
\zeta(s_{1},\ldots,s_{k};\sigma_{1},\ldots,\sigma_{k})=\sum_{n_{j}>n_{j+1}}\prod_{j=1}^{k}\frac{\sigma_{j}^{n_{j}}}{{n_{j}}^{s_{j}}}=\sum_{m_{j}\ge1}\prod_{j=1}^{k}\frac{\tau_{j}^{m_{j}}}{(m_{1}+m_{2}+\cdots+m_{j})^{s_{j}}}.
\hat{k}^{\hat{a}\hat{b}}={\frac{1}{16}}{\frac{(1-{\frac{2}{3}}\hat{k}^{2})^{2}}{1+{\frac{2}{3}}\hat{k}^{2}}}\left[(\hat{\calH}^{2})^{\hat{a}\hat{b}}-{\frac{1}{6}}\hat{\calH}^{2}\,\hat{\eta}^{\hat{a}\hat{b}}\right]
V_{l,k}^{m,n}(z)=:\prod_{i=-l/2}^{l/2}V_{1,0}^{m_{i},0}\left(x^{2i(\xi+1)}z\right)\prod_{j=-k/2}^{k/2}V_{0,1}^{0,n_{j}}\left(x^{2j\xi}z\right):
T_{\mu\nu}=\phi_{,\mu}\phi_{,\nu}-\frac12g_{\mu\nu}\left(\phi_{,\rho}\phi^{,\rho}+m^{2}\phi^{2}\right)+\xi\left[(R_{\mu\nu}-\frac12g_{\mu\nu}R)\phi^{2}+g_{\mu\nu}(\phi^{2})_{~;\rho}^{,\rho}-(\phi^{2})_{;\mu\nu}\right]\,.
M_{k}=\sum_{i,j=1}^{3}\epsilon_{ijk}(x_{1}^{i}\frac{dx_{1}^{j}}{dt}-x_{1}^{j}\frac{dx_{1}^{i}}{dt})(1-\frac{1}{c^{2}}|\frac{d{\bfx}_{1}}{dt}|^{2})^{-1/2}
\tilde{\lambda}_{\mu}(e,\psi)={\frac{1}{2i\kappa}}(i\kappa^{2}\gamma_{a}\gamma^{\rho}\psi(x)\partial_{[\rho}e{^a}_{\mu]}-\gamma_{a}\tilde{\Lambda}{^a}_{\mu})=-{\frac{i\kappa}{2}}{{\omega}_{\mu}}^{ab}(e)\sigma_{ab}\psi,
(\mu^{2}+\delta\mu_{(0)}^{2}+(1+\deltaZ_{\phi}^{(0)})\,\partial^{2})\frac{\deltaG^{(0)}}{\deltaj(x)}+4(\lambda+\delta\lambda_{(0)})\frac{\delta^{3}G^{(0)}}{\deltaj^{3}(x)}=0
\Psi(p)=\left(\begin{array}{c}{1}\\{0}\\\end{array}\right)\otimes\left(\begin{array}{c}{\bar{z}}\\{1}\\\end{array}\right)\phi(z,\bar{z},|{\bfp}|)+\left(\begin{array}{c}{0}\\{1}\\\end{array}\right)\otimes\left(\begin{array}{c}{1}\\{-z}\\\end{array}\right)\psi(z,\bar{z},|{\bfp}|)
\left(\begin{array}{ll}{p_{0}-m_{e}}&{\partial_{r}-\frac{m-\beta\Theta(R-r)}{r}}\\{-\partial_{r}-\frac{m+1-\beta\Theta(R-r)}{r}}&{p_{0}+m_{e}}\\\end{array}\right)\left(\begin{array}{c}{g_{1}(r)}\\{g_{2}(r)}\\\end{array}\right)\=\0\.
{\frac{d^{2}x^{\rho}}{d\tilde{s}^{2}}}+\Gamma_{\mu\nu}^{\rho}{\frac{dx^{\mu}}{d\tilde{s}}}{\frac{dx^{\nu}}{d\tilde{s}}}=-{\frac{d^{2}\tilde{s}/ds^{2}}{(d\tilde{s}/ds)^{2}}}{\frac{dx^{\rho}}{d\tilde{s}}}.
\{F,G\}=\sum\int_{\Omega}\int_{\Omega}{\frac{\partialf}{\partial\phi_{A}^{(J)}(x)}}{\frac{\partialg}{\partial\phi_{B}^{(K)}(y)}}\{D_{J}^{(x)}\phi_{A}(x),D_{K}^{(y)}\phi_{B}(y)\},
S_{A_{1}A_{2}\cdotsA_{k}}^{B_{1}B_{2}\cdotsB_{n}}:=\langleV_{B_{1}}(\theta_{1})V_{B_{2}}(\theta_{2})\cdotsV_{B_{n}}(\theta_{n})|V_{A_{1}}(\theta_{1})V_{A_{2}}(\theta_{2})\cdotsV_{A_{k}}(\theta_{k})\rangle
V_{z\ggb}\toE[1-{\frac{z}{Q_{p}}}(1-{\frac{3EQ_{F}}{2m_{f}(x^{2}+4Q_{p}z)^{\frac{n}{2}}}})]+{\frac{J_{\perp}^{2}}{2m_{f}Q_{p}^{2}}}[1-{\frac{2EQ_{F}}{m_{f}(x^{2}+4Q_{p}z)^{\frac{n}{2}}}}],
E=\int_{0}^{2\pi}d\phi\,r\left\{\frac{1}{r^{2}}\frac{\partialf^{*}}{\partial\phi}\frac{\partialf}{\partial\phi}-\mu^{2}f^{*}f+\lambda(f^{*}f)^{2}\right\}=2\pir\left\{\frac{1}{r^{2}}\alpha^{2}v^{2}-\mu^{2}v^{2}+\lambdav^{4}\right\}.
(81)\begin{array}{rlr}{A_{r}:}&{\epsilon_{j}-\epsilon_{j+1}}&{(1\leqj\leqr);}\\{B_{r}:}&{\epsilon_{j}-\epsilon_{j+1},\;\epsilon_{r}}&{(1\leqj<r);}\\{C_{r}:}&{\epsilon_{j}-\epsilon_{j+1},\;2\epsilon_{r}}&{(1\leqj<r);}\\{D_{r}:}&{\epsilon_{j}-\epsilon_{j+1},\;\epsilon_{r-1}+\epsilon_{r}}&{(1\leqj<r);}\\{G_{2}:}&{\epsilon_{1}-2\epsilon_{2}+\epsilon_{3},\;\epsilon_{2}-\epsilon_{3}.}&{}\\\end{array}
m_{G}^{2}\;=\;-2e^{2}\int_{{\mathcalR}}\frac{d^{2}p}{(2\pi)^{2}}\,\frac{\partial}{\partialp_{\mu}}(\frac{p_{\mu}}{p^{2}})+2e^{2}\int_{{\mathcalR}}\frac{d^{2}p}{(2\pi)^{2}}\,\frac{\partial}{\partialp_{\mu}}(\frac{p_{\mu}}{p^{2}+\Lambda^{2}})\;,
nB_{2}+lA_{2}=nU_{l}^{n}Str\left(\ldots\phi^{i_{2k}^{\prime}}\ldots\underbrace{\phi^{i_{2k-1}^{\prime}}\phi^{i_{n}}}\ldots\partial_{a_{l}}\phi^{j_{l}}\ldots\right)+lU_{l}^{n}Str\left(\ldots\phi^{i_{2k}^{\prime}}\ldots\underbrace{\phi^{i_{2k-1}^{\prime}}\partial_{a_{l}}\phi^{j_{l}}}\ldots\right)
{\calL}=u_{1}{}^{i}u_{1}{}^{j}u_{1}{}^{k}u_{1}{}^{l}L_{ijkl}+u_{1}{}^{(i}u_{1}{}^{j}u_{1}{}^{k}u_{1}{}^{l}u_{1}{}^{m}u_{2}{}^{n)}L_{ijklmn}+\ldots
{\partial}^{-}a_{+}=n_{-}{\pi}^{+}-\frac{{\partial}_{+}}{{\partial}_{\bot}^{2}}(\sum_{i=1}^{2}{\partial}_{i}{\pi}^{i})+\frac{n_{+}^{2}}{{\partial}_{\bot}^{2}+n^{2}{\partial}_{+}^{2}}{\partial}_{+}B,
(\vec{\nabla}_{1}+\mathrm{i}\vec{\nabla}_{2})\psi=\vec{n}_{1}\!\left(\partial_{1}-\frac{x}{r_{1}}\partial_{x}\right)\!\psi+\vec{n}_{2}\,\frac{1}{r_{1}}\,\partial_{x}\,\psi+\mathrm{i}\vec{n}_{2}\!\left(\partial_{2}-\frac{x}{r_{2}}\partial_{x}\right)\!\psi+\mathrm{i}\,\vec{n}_{1}\,\frac{1}{r_{2}}\,\partial_{x}\,\psi=0
\hat{\gamma}_{i}=\left(\begin{matrix}{\sigma_{i}}&{0}\\{0}&{-\sigma_{i}}\\\end{matrix}\right),\quad\hat{\gamma}_{0}=\left(\begin{matrix}{0}&{\mathbf{1}}\\{\mathbf{1}}&{0}\\\end{matrix}\right),\quad\hat{\gamma}_{-1}=\left(\begin{matrix}{0}&{\mathbf{1}}\\{-\mathbf{1}}&{0}\\\end{matrix}\right),
\begin{array}{ll}{\Psi_{i}^{\left(s\right)}(\mathbf{x})=}&{\sqrt{E_{i}}\sum_{l}\left(i\right)^{l}\Psi_{ls}(\mathbf{x})}\\{\Psi_{f}^{\left(s\right)}(\mathbf{x})=}&{\sqrt{E_{f}}\sum_{l}\left(i\right)^{l}e^{-il\theta}\Psi_{ls}(\mathbf{x})\.}\\\end{array}
S=\intd^{2}x\,\mathrm{tr}\Bigg[-{\frac{1}{4g^{2}}}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}D_{\mu}\phiD^{\mu}\phi+i\bar{\Psi}\gamma^{\mu}D_{\mu}\Psi\nonumber-2ig\phi\bar{\Psi}\gamma_{5}\Psi\Bigg],
e^{-F(\vec{x}_{i})/T}={\frac{\intdA_{\mu}d\psid\bar{\psi}e^{-\int_{0}^{1/T}({\frac{1}{4}}F_{\mu\nu}^{2}+\bar{\psi}(\gamma\cdot(\nabla-ieA)+m)\psi)}e^{i\sume_{i}\int_{0}^{1/T}d\tauA_{0}(\tau,\vec{x}_{i})}}{\intdA_{\mu}d\psid\bar{\psi}e^{-\int_{0}^{1/T}({\frac{1}{4}}F_{\mu\nu}^{2}+\bar{\psi}(\gamma\cdot(\nabla-ieA)+m)\psi)}}}
\int_{0}^{\infty}\int_{0}^{\infty}dp\,dq\,p^{2}q^{2}f^{J}(p)f^{J}(q)Q_{J+1}\left(z_{1}\right)=\pi\left(\frac{\alpha\mu}{n}\right)^{3}\left(\frac{2}{J+1}-\frac{1}{n}\right).
Q=\left(1+\frac{N+1}{\ell}\right)^{N}\left(1+\frac{N}{\ell+1}\right)^{-\frac{1}{2}}\left[\left(1+\frac{N}{\ell+1}\right)^{\ell+1}e^{-N}\right]e^{J(N+\ell+1)-J(\ell+1)},
\left(\!\,\left(\!\,{a}\,{{Q}_{1}}\,\mathrm{\&*}\,{b}\,{{Q}_{2}}\,\!\right)\,\mathrm{\&*}\,\left(\!\,{{Q}_{2}}+{{f}_{1}}+{{f}_{2}}\,\!\right)\,\!\right)\,\mathrm{\&*}\,{{f}_{2}}
\begin{array}{l}{K^{\mu}=i\overline{{\epsilon}}_{1}^{i}\gamma^{\mu}\epsilon_{2i}}\\{D_{\mu}K_{\nu}=aM_{ij}\overline{{\epsilon}}_{1}^{i}\gamma_{\mu\nu}\epsilon_{2j}=-D_{\nu}K_{\mu}}\\{D_{\mu}D_{\nu}K_{\rho}=a^{2}(g_{\mu\rho}K_{\nu}-g_{\mu\nu}K_{\rho}).}\\\end{array}
P_{{\calS}}(A)=\sum_{j=1,2}P_{{\calS}_{j}/{\calS}}P_{{\calS}_{j}}(A)+2\sqrt{P_{{\calS}_{1}/{\calS}}P_{{\calS}_{1}}(A)P_{{\calS}_{2}/{\calS}}P_{{\calS}_{2}}(A)}\lambda(A;{\calS},{\calS}_{j})\;.
\left.\frac{da}{dT}\right|_{T=T_{c}^{+}}=\left.\frac{da_{1}}{dT}\right|_{T=T_{c}^{-}}=-\frac{1}{4(1+c_{1})^{2}}\,;\qquad\left.\frac{db}{dT}\right|_{T=T_{c}^{+}}=\left.\frac{db_{2}}{dT}\right|_{T=T_{c}^{-}}=\frac{1}{4(1-c_{1})^{2}}\,,
\Sigma_{n}(E)=\frac{\eta}{\pi\hbar}\sum_{m}|\langlem|q|n\rangle|^{2}\int_{0}^{\Lambda}\frac{(E^{\prime}-E_{m})(\frac{E^{\prime}-E_{m}}{\hbar})^{2}\theta(E^{\prime}-E_{m})dE^{\prime}}{E^{\prime}-E-i\epsilon}
r^{2}\frac{d}{dr}\left\langlef^{\prime}(r^{\prime})f^{\prime}(r)\right\rangle=\left\langlef^{\prime}(r^{\prime})f^{3}(r)\right\rangle-\left\langlef^{\prime}(r^{\prime})f(r)\right\rangle\left(1+g^{2}(r)\right)
{\calD}_{m_{1}m_{1}^{\prime}}^{j_{1}}(L_{p}^{-1})\left(\begin{array}{lll}{j_{1}}&{j_{2}}&{j_{3}}\\{m_{1}^{\prime}}&{m_{2}}&{m_{3}}\\\end{array}\right)={\calD}_{m_{2}m_{2}^{\prime}}^{j_{2}}(L_{p}){\calD}_{m_{3}m_{3}^{\prime}}^{j_{3}}(L_{p})\left(\begin{array}{lll}{j_{1}}&{j_{2}}&{j_{3}}\\{m_{1}}&{m_{2}^{\prime}}&{m_{3}^{\prime}}\\\end{array}\right).