harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{lclcl}{X_{n}^{(I)}(0)}&{=}&{Y_{n}^{(I)}(0)}&{=}&{0}\\{}&{}&{}&{}&{}\\{{\dot{X}}_{r}^{(I)}(0)}&{=}&{{\dot{X}}_{-r}^{(I)}(0)}&{=}&{{\displaystyle\frac{1}{2}M}}\\{}&{}&{}&{}&{}\\{{\dot{Y}}_{r}^{(I)}(0)}&{=}&{-{\dot{Y}}_{-r}^{(I)}(0)}&{=}&{{\displaystyle-\frac{i}{2}M}}\\{}&{}&{}&{}&{}\\{{\dot{X}}_{n}^{(I)}(0)}&{=}&{{\dot{Y}}_{n}^{(I)}(0)}&{=}&{0~~\mathrm{for}~n\neq\pmr}\\\end{array}
e=\frac{i}{2}\left\{d\tau+\frac{GS}{2i}\left({\tt\frac{1}{z}}dz-{\tt\frac{1}{\bar{z}}}d\bar{z}\right)\right\}\left(\begin{array}{rr}{1}&{0}\\{0}&{-1}\\\end{array}\right)+\frac{i}{2}\left(\begin{array}{rr}{0}&{dz}\\{d\bar{z}}&{0}\\\end{array}\right)\,,
W^{\Omega_{z_{1}},\wedge^{m}(\Omega_{z_{2}})}(v^{(d,a)}\otimesv^{(d_{m},d)})=\sum_{a_{m}}v^{(d_{m},a_{m})}\otimesv^{(a_{m},a)}W^{(1,m)}\left(\left.\begin{array}{cc}{d_{m}}&{d}\\{a_{m}}&{a}\\\end{array}\right|z_{1}-z_{2}\right),
\dot{s}=3\left({\frac{c_{\mathrm{grav}}^{2}-c^{2}}{c^{2}c_{\mathrm{grav}}^{2}}}p{\frac{\dot{a}}{a}}+\rho{\frac{\dot{c}_{\mathrm{grav}}}{c_{\mathrm{grav}}}}\right){\frac{cc_{\mathrm{grav}}a^{3}}{T}}.
p_{j}=-i\sqrtj{\frac{\partial}{\partialz_{j}}},~~~p_{j}^{*}=-i\sqrtj{\frac{\partial}{\partial\bar{z}_{j}}},~~~\phi_{j}={\frac{z_{j}}{\sqrtj}}.~~~\phi_{j}^{*}={\frac{\bar{z}_{j}}{\sqrtj}}
{\check{Q}}_{i}=e_{i}\check{\theta}_{i}^{\ast}(\tau)\check{\theta}_{i}(\tau)=Q_{i}=e_{i}\theta_{i}^{\ast}(\tau)\theta_{i}(\tau)~~~~~\big[\dot{Q}_{i}(\tau)=0\RightarrowQ_{i}(\tau)\equivQ_{i}\big].
g^{2}M=\left(\begin{array}{cccccccc}{\rho}&{}&{}&{}&{\kappa_{1}}&{}&{}&{}\\{}&{\ddots}&{}&{}&{}&{\ddots}&{}&{}\\{}&{}&{\ddots}&{}&{}&{}&{\kappa_{N_{f}-N_{c}}}&{}\\{}&{}&{}&{\rho}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{}&{}&{}\\\end{array}\right),
P_{+}(X^{0})=-N\intd\tilde{\sigma}\frac{e^{3H(X^{0}-\tilde{\sigma})}-H^{2}L^{2}e^{-H(X^{0}-\tilde{\sigma})}}{\sqrt{e^{4H(X^{0}-\tilde{\sigma})}e^{2H(X^{0}-\tilde{\sigma})}+H^{2}L^{2}}},
\left[\left(\sum_{n=1}^{\infty}\mathcal{R}_{2k-1,2n}x_{2n}^{\mu}\right),\left(\sum_{m=1}^{\infty}\mathcal{T}_{2m,2l-1}p_{2m}^{\nu}\right)\right]_{\star}=i\delta_{k,l}\eta^{\mu\nu}.
\deltag_{\mu\nu}=h_{\mu\nu}(z=\betat,t)=h_{\mu\nu}(z=\betat+\ell,t)={\frac{16\pik\ell_{P}^{2}\ell^{2}}{8\delta^{4}}}\left(\eta_{\mu\nu}-4n_{\mu}n_{\nu}\right).
\partial_{i}B^{i(2)}\equiv+\sqrt{L_{4}L_{5}}\partial_{i}((1/2!)\epsilon^{imn045}F_{mn5})\equiv+\sqrt{L_{4}L_{5}}\partial_{i}\,^{*}\!F^{i04}=+g\sqrt{\frac{L_{4}}{L_{5}}}\delta^{(3)}({\bfx}),
\langle\vec{E}(\tau,\vec{x})\cdot\vec{E}(0,\vec{x})\rangle=\langle\vec{B}(\tau,\vec{x})\cdot\vec{B}(0,\vec{x})\rangle={\frac{\hbar}{(2\pi)^{2}}}{\frac{12}{(c\tau)^{4}}}.
\begin{array}{c}{\Xi(T,L,\mu)=e^{-\betaG(T,L,\mu)}}\\{=\int{\calD}\bar{\psi}\{\calD}\psi\e^{\int_{0}^{1}dt\int_{0}^{1}dx\bar{\psi}\left(D(\beta,L)-i\mu\gamma^{0}\right)\psi}}\\{\simDet\left(D(\beta,L)-i\mu\gamma^{0}\right)_{bc}}\\\end{array}
\frac{G_{\mu\nu}^{\mathrm{od}q}}{\alpha^{\prime}}=\frac{1}{\tilde{g}^{\frac{2}{q+1}}\tilde{\ell}^{2}}\Big(\frac{\tilde{r}}{\tilde{R}}\Big)^{\frac{5-q}{q+1}}\eta_{\mu\nu}\,\quad\Theta_{\mathrm{od}q}^{-3\ldots(q+2)}=-\tilde{g}\tilde{\ell}^{q+1}\.
p_{\theta2}-\frac{p^{1}+ip^{2}}{\sqrt{2}p^{+}}p_{\theta1}=0,\qquad{p_{\bar{\theta}}}^{\dot{1}}+\frac{p^{1}-ip^{2}}{\sqrt{2}p^{+}}{p_{\bar{\theta}}}^{\dot{2}}=0.
\Sigma_{\psi}(p)=-\frac{ie^{2}}{\theta}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\gamma^{\mu}(\not\!k+\not\!p+m)\gamma^{\nu}}{(k+p)^{2}-m^{2}}\frac{\epsilon_{\mu\nu\rho}{\bar{k}}^{\rho}}{({\vec{k}})^{2}}\,.
\begin{array}{rcl}{\tilde{G}}&{=}&{\partiala-\frac{1}{6^{3}}\,\epsilon^{mnpqrs}\,\partiala_{mnp}\,a_{qrs}\,,}\\{}&{}&{}\\{\tilde{H}_{m}}&{=}&{{\calM}_{mn}\,\left[2\partialN^{n}-\frac{1}{36}\epsilon^{npqrsu}{\calF}_{pq}a_{rsu}+a\,F^{n}(A)\right]\,.}\\\end{array}
a\left(\eta\right)=a_{\mathrm{min}}\,\sqrt{1+\gamma^{2}\,\eta^{2}}\;\;,\;\;a_{\mathrm{min}}=\sqrt{\frac{3\,Q^{2}\,\omega_{4}}{16\,M}}\;,\;\gamma=\frac{4\,M}{\sqrt{3}\,Q}\,\,.
P^{j}=p^{j}\cdot{\bf1}+\sum_{\bfk}\pi^{\perpj}({\bfk})\mu({\bfk})^{-1}(\mathrm{dim}(E))^{-1}2^{-1/2}(a(-{\bfk})-a^{*}({\bfk}))Z_{\bfk}
\begin{array}{rcl}{\left(d\psi_{+}\right)^{2,2}}&{=}&{{\calW}_{1}^{+}\,J\wedgeJ+{\calW}_{2}^{+}\,\wedgeJ\,,}\\{\left(d\psi_{-}\right)^{2,2}}&{=}&{{\calW}_{1}^{-}\,J\wedgeJ+{\calW}_{2}^{-}\,\wedgeJ\,.}\\\end{array}
q(a,r)=-\frac{1}{2\pia^{3}r^{n}S_{D}}\sum_{l=0}^{\infty}D_{l}\int_{ma}^{\infty}dz\,z^{3}\frac{\bar{K}_{\nu}(z)}{\bar{I}_{\nu}(z)}\frac{F_{\nu}^{(q)}\left[I_{\nu}(zr/a)\right]}{\sqrt{z^{2}-m^{2}a^{2}}},\quadq=\varepsilon,\,p,\,p_{\perp},\quadr<a,
\hat{\hat{S}}={\textstyle\frac{1}{2}}\intd^{11}x\,sqrt{\hat{\hat{g}}}\\left[-\hat{\hat{R}}+a_{1}\hat{\hat{G}}^{2}+a_{2}\frac{1}{\sqrt{\hat{\hat{g}}}}\hat{\hat{\epsilon}}\hat{\hat{G}}\hat{\hat{G}}\hat{\hat{C}}\right]\,.
\left\{\begin{array}{c}{\gamma}\\{\alpha\overline{{\beta}}}\\\end{array}\right\}=-g^{\gamma\overline{{\delta}}}H_{\alpha\overline{{\beta}}\overline{{\delta}}}\\,\\\left\{\begin{array}{c}{\overline{{\gamma}}}\\{\alpha\overline{{\beta}}}\\\end{array}\right\}=g^{\overline{{\gamma}}\delta}H_{\delta\alpha\overline{{\beta}}}\\,
S_{0}=\frac{1}{4e^{2}}\mathrm{Re}\intd^{4}x\,d^{2}\theta\,W_{a}C^{ab}W_{b}+\frac{1}{4}\intd^{4}x\,d^{4}\theta\,\Big(\phi^{*}e^{2V}\phi+\tilde{\phi}^{*}e^{-2V}\tilde{\phi}\Big).
\begin{array}{lcr}{\partial_{\sigma}^{2}X^{9}+\sum_{\nu=1}^{8}{\frac{1}{2}}(\{X^{\nu+(-)^{\nu+1}},\{X^{\nu+(-)^{\nu+1}},X^{9}\}\}+\{X^{\nu},\{X^{\nu},X^{9}\}\})=0,}\\\end{array}
F=T^{a}P_{a}+K^{(0)\alpha}S_{\alpha}^{(0)}+\frac{1}{2}R^{ab}J_{ab}+K_{a}^{(1)\alpha}S_{\alpha}^{(1)a}+K_{ab}^{(2)\alpha}S_{\alpha}^{(2)ab}+\cdots.
\begin{array}{rcl}{\left({\calV}^{(n)}(u){\calV}^{(n)}(u+v)\right)_{\alpha,\alpha^{\prime}}}&{=}&{\displaystyle\sum_{\gamma}{\calV}^{(n)}(u)_{\alpha,\gamma}{\calV}^{(n)}(u+v)_{\gamma,\alpha^{\prime}}}\\{}&{=}&{\displaystyle\sum_{\beta\bar{\beta}}\prod_{i=1}^{n}X(\beta_{i},\bar{\beta}_{i}|\beta_{i+1},\bar{\beta}_{i+1}|\alpha_{i},\alpha_{i}^{\prime})~,}\\\end{array}
I_{eff}^{{\mathbbH}^{11}}=-\frac1{1995840\pi^{5}a^{11}}\left(-6336+5412a^{2}b-1287a^{4}b^{2}+110a^{6}b^{3}-3a^{8}b^{4}\right)\left(a^{2}b\right)^{3/2},
\frac{1}{|\Omega_{p}||\Omega_{q}|}\sum_{z\in\Omega}\sum_{x,y\in\Omega_{p}\cap\Omega_{zq}}\overline{{\phi}}(x,y)\psi^{z}(x^{a}y^{c},x^{b}y^{d})\prod_{\xi\in\mathcal{O}(x,y)}\omega_{p_{\xi}}^{\frac{a\mu_{\xi}}{c\lambda_{\xi}}}\Lambda_{p_{\xi}}^{(zq)_{\xi}}\left(\frac{a\mu_{\xi}+c\kappa_{\xi}}{c\lambda_{\xi}}\right)\omega_{(zq)_{\xi}}^{\widetilde{\mu}_{\xi}}
23I_{4}^{A}[h^{3}]=\intd^{4}x({\calE}_{4}+\textstyle{\frac{2}{3}}\,\Box_{0}R)[1-2\Box_{0}^{-2}\partial_{\mu}(R^{\mu\nu}-\textstyle{\frac{1}{3}}\:\eta^{\mu\nu}\:R)\partial_{\nu}\!+\!...]\Box_{0}^{-2}({\calE}_{4}+\textstyle{\frac{2}{3}}\Box_{0}R)\;.
S_{1}^{(bm)}=\frac{(d-3)L^{2}}{2}\sum_{n=-\infty}^{\infty}\int_{-\infty}^{+\infty}\frac{dk_{1}dk_{2}}{(2\pi)^{2}}\left[\lambda_{1}(k_{1}^{2}+k_{2}^{2})+\frac{4\pi^{2}\lambda_{0}n^{2}}{\beta^{2}t^{2}}\right].
{\calN}(t)=\langle\psi_{\mathrm{int}}(t)|N_{\mathrm{int}}|\psi_{\mathrm{int}}(t)\rangle=\sum_{k}\langle\psi_{\mathrm{int}}(t)|a_{k}^{\dagger}a_{k}|\psi_{\mathrm{int}}(t)\rangle,
\int_{m}^{\infty}dk\(k^{2}-m^{2})^{1-s}\partial_{k}\frac{t^{j}}{l^{n}}=-m^{2-2s}\frac{\Gamma(2-s)\Gamma\left(s+\fracj2-1\right)l^{j-n}}{\Gamma\left(\fracj2\right)(Rm)^{j}\left(1+\left(\frac{l}{mR}\right)^{2}\right)^{s+\fracj2-1}}\,
\Psi^{\alpha_{1}^{1}....\alpha_{1}^{m},\alpha_{2}^{1}....\alpha_{2}^{m}}(X_{1},X_{2})=\sum_{R_{1}f_{1},R_{2}f_{2}}\psi_{R_{1}f_{1},R_{2}f_{2}}^{\alpha_{1}^{1}\alpha_{1}^{2},\alpha_{2}^{1}\alpha_{2}^{2}}(x_{1},x_{2})\phi_{R_{1}f_{1}}^{\alpha_{1}^{3}....\alpha_{1}^{m}}(y_{1})\phi_{R_{2}f_{2}}^{\alpha_{2}^{3}....\alpha_{2}^{m}}(y_{2}).
W_{\psi_{0}}(\psi_{1}\otimes\psi_{2}\otimes\psi_{3};z_{i})\sim\left(z_{1}-z_{2}\right)^{-\bar{h}t_{12}}\left(z_{1}-z_{3}\right)^{-\bar{h}(t_{13}+t_{23})}W_{\psi_{0}}^{1}(\psi_{1}\otimes\psi_{2}\otimes\psi_{3})
G_{N}=\lambda\kappa_{5}^{4}/48\pi,~~\Lambda_{4}=\kappa_{5}^{2}(\Lambda_{5}+\kappa_{5}^{2}\lambda^{2}/6)/2,~~E_{\mu\nu}=C_{\phantom{A}BCD}^{A}n_{A}n^{C}q_{\phantom{B}\mu}^{B}q_{\phantom{D}\nu}^{D}
J^{0}(z)J^{\pm}(w)=\frac{\pmJ^{\pm}(w)}{z-w},\quadJ^{+}(z)J^{-}(w)=\frac{k}{(z-w)^{2}}+\frac{2J^{0}(w)}{z-w},\quadJ^{0}(z)J^{0}(w)=\frac{k/2}{(z-w)^{2}},
H_{a}\left(\frac{dS_{a}}{d\lambda_{a}},\lambda_{a}\right)=-2A(\lambda_{a})\left(\frac{dS_{a}}{d\lambda_{a}}\right)^{2}-\frac{1}{2}\left(\lambda_{a}^{N+1}-(\alpha-1)\lambda_{a}^{N}+(1-\alpha+\beta)\lambda_{a}^{N-1}\right),
{\calL}_{HEC}^{(1)}(B)={\frac{(am)^{2}}{2\pi^{2}a^{4}}}\sum_{n=1}^{\infty}{\frac{(-1)^{n-1}}{n^{2}}}K_{2}(2amn)+{\frac{1}{2}}\biggl[1-{\frac{1}{\mu(am)}}\biggr]B^{2}+{\calL}_{HE}^{(1)\prime}(a,B)\,,
\left(\begin{array}{cc}{e^{\alpha}}&{a}\\{0}&{1}\\\end{array}\right)\left(\begin{array}{cc}{e^{\beta}}&{b}\\{0}&{1}\\\end{array}\right)=\left(\begin{array}{cc}{e^{\alpha+\beta}}&{a+e^{\alpha}b}\\{0}&{1}\\\end{array}\right),
R_{MN}-{\frac{1}{2}}G_{MN}R+{\frac{1}{2}}G_{MN}\Lambda+{\frac{1}{2}}{\frac{\sqrt{-{\widehatG}}}{\sqrt{-G}}}{\delta_{M}}^{\mu}{\delta_{N}}^{\nu}{\widehatG}_{\mu\nu}{\widetildef}\delta(z)=0~,
\begin{aligned}{{\calD}_{M}E_{N}{}^{r}}&{{}=\partial_{M}E_{N}{}^{r}+\Omega_{M}{}^{r}{}_{s}E_{N}{}^{s}\,,}\\{{\calD}_{M}E_{N}{}^{a}}&{{}=\partial_{M}E_{N}{}^{a}+\frac{1}{4}(-)^{bN}\Omega_{Mrs}(\Gamma^{rs})^{a}{}_{b}\,E_{N}{}^{b}\,.}\\\end{aligned}
\left[R_{1s}^{KG}\right]^{2}=\frac{\int{|\chi|}^{2}d\vec{x}}{\int{|\varphi|}^{2}d\vec{x}}=1-4{\left[2+{\left(y+\frac{1}{2}\right)}^{\frac{1}{2}}+\frac{{(y+\frac{1}{2})}^{\frac{3}{2}}}{2y}\right]}^{-1}
z^{A}=\left(A_{i}^{a},B_{a}^{0i},B_{a}^{ij},\pi_{ij}^{a},\varphi_{a},\pi^{a},\varphi_{a}^{\left(1\right)},\pi^{(1)a},\varphi_{a}^{\left(2\right)},\pi^{(2)a}\right).
{\calV}=-{\textstyle\frac{1}{2}}\frac{|\tau|^{2}}{\Im\mathrm{m}(\tau)}\left\{\left[\mathrm{Tr}\,\left(Q\mathcal{M}\right)\right]^{2}-2\mathrm{Tr}\,\left(Q\mathcal{M}Q\mathcal{M}\right)\right\}\;.
{\calG}_{\alpha_{1},\ldots,\alpha_{n}}^{(A)}(x_{1},\ldots,x_{n})=\left({\frac{A}{\pi}}\right)^{(\sum\alpha_{i}-Q)/b}{\frac{F_{\alpha_{1},\ldots,\alpha_{n}}(x_{1},\ldots,x_{n})}{\Gamma((Q-\sum\alpha_{i})/b)}}\,.
{\calP}_{M}(\zeta)=\left(\begin{array}{ccccc}{1}&{\langlex\rangle}&{\langlex^{2}\rangle}&{...}&{\langlex^{M}\rangle}\\{\langlex\rangle}&{\langlex^{2}\rangle}&{\langlex^{3}\rangle}&{...}&{\langlex^{M+1}\rangle}\\{\langlex^{M}\rangle}&{}&{}&{}&{\langlex^{2M}\rangle}\\\end{array}\right)
D^{\alpha\betaI}|_{h^{a\alpha}=v^{\alpha}}=(v^{*\alpha}v^{\beta}+v^{*\beta}v^{\alpha})\sum_{a=1}^{16}q^{I}{}_{a}=(v^{*\alpha}v^{\beta}+v^{*\beta}v^{\alpha})\;Q^{I}=0\.
T_{\\mu\rho\sigma}^{\mu}=-\frac{1}{24\pi}\Biggl[\left(p_{\rho}p_{\sigma}-p^{2}g_{\rho\sigma}\right)\mp\frac{1}{4}\left(\varepsilon_{\rho\lambda}p^{\lambda}p_{\sigma}+\varepsilon_{\sigma\lambda}p^{\lambda}p_{\rho}\right)\Biggl],
\left(\begin{array}{c}{\ddot{\xi}^{t}}\\{\ddot{\xi}^{*}}\\\end{array}\right)-\left(\begin{array}{c}{\xi^{t}}\\{\xi^{*}}\\\end{array}\right)+2{\calA}\left(\begin{array}{c}{\dot{\xi}^{t}}\\{\dot{\xi}^{*}}\\\end{array}\right)+{\calB}\left(\begin{array}{c}{\xi^{t}}\\{\xi^{*}}\\\end{array}\right)=\left(\begin{array}{c}{U^{t}}\\{U^{*}}\\\end{array}\right),
S\sim\oint_{-\pir_{c}}^{\pir_{c}}dy\intd^{4}x\sqrt{-G}\left(2G^{55}\bar{\Phi}^{2}(|y|)\delta^{2}(y)-2G^{55}\bar{\Phi}^{2}(|y|)\delta^{2}(y-\pir_{c})\right)\.
\begin{array}{lcl}{\phi^{\prime}*\phi^{\prime\prime}}&{=}&{(b_{i}^{\prime}+b_{i}^{\prime\prime},l_{\pm(i,j)}^{\prime}e^{-(h/2)b_{j;n-1}^{\prime\prime}}+e^{-(h/2)b_{i;n-1}^{\prime}}\;l_{\pm(i,j)}^{\prime\prime}+}\\{}&{}&{+(1-e^{\pmh})\sum_{k=i+1}^{j-1}l_{\pm(i,k)}^{\prime}l_{\pm(k,j)}^{\prime\prime}),}\\\end{array}
\sum_{f=0}^{d}\sum_{{\sigma_{f}\inP_{f},\sigma_{d-f}\inP_{d-f}}\atop{\sigma_{f}\cup\sigma_{d-f}=\sigma_{d}}}(-1)^{l(\sigma_{f})}{}_{l(\sigma_{f})-1}C_{l(\sigma_{d})-1}S(\sigma_{f})S(\sigma_{d-f})=(-1)^{l(\sigma_{d})}S(\sigma_{d})
\frac{d{\calP}^{p_{1}\top_{2}}}{d^{3}{\bfk}_{1}d^{3}{\bfk}_{2}}\;=\;\sum_{\sigma_{1}=\pm}\sum_{\sigma_{2}=\pm}\vert{\calA}_{{\bfk}_{1}{\bfk}_{2}}^{\sigma_{1}\sigma_{2}}\vert^{2}\;,
{\Gamma^{2}}_{a}=\left\{\left(\begin{matrix}{\sigma_{2}}&{0}\\{0}&{\sigma_{2}}\\\end{matrix}\right),\quad\left(\begin{matrix}{\sigma_{3}}&{0}\\{0}&{-\sigma_{3}}\\\end{matrix}\right),\quad\left(\begin{matrix}{0}&{\sigma_{3}}\\{\sigma_{3}}&{0}\\\end{matrix}\right)\right\},
Q_{N}(V^{\mu},\beta^{\mu})={\frac{1}{N!}}\int_{V^{\mu}}e^{-\beta^{\mu}p_{\mu}}\prod_{i=1}^{N}\theta(p_{i}^{o})\delta(\phi_{i})\delta(\chi_{i})det\,|\,\lbrace\phi_{i},\chi_{j}\rbrace|\,{\frac{d^{4}x_{i}d^{4}p_{i}}{(2\pi)^{3}}},
\left[z^{r}e^{\lambdaD},z^{s}e^{\muD}\right]=\left(e^{\lambdas}-e^{\mur}\right)z^{r+s}e^{(\lambda+\mu)D}+\delta_{r,-s}\frac{e^{-\lambdar}-e^{-\mus}}{1-e^{\lambda+\mu}}C.
g_{2}=\frac{M^{2}}{12}+\frac{Mr_{m}^{2}}{l^{2}}-\frac{r_{m}^{4}}{l^{4}},\;\;\;\;g_{3}=-\frac{M^{3}}{216}+\frac{M^{2}r_{m}^{2}}{6l^{2}}-\frac{Mr_{m}^{4}}{6l^{4}},
\deltaM^{\mu\nu}=\frac{2\alpha}{3\pi}\;\mathrm{log}\;q^{2}M_{0}^{\mu\nu}-\frac{4\alpha}{3\pi}e^{2}\mathrm{log}\;q^{2}T_{2}^{\mu\nu}+\frac{2\alpha}{3\pi}e^{2}\mathrm{log}\;q^{2}T_{3}^{\mu\nu}\\.
i\Pi_{(\beta,\mu)}^{\mu\nu}(0,\overline{p})=-\frac{4g^{2}}{\beta}\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{{\mathit{M}}^{\mu\nu}.sin^{2}(\frac{\tilde{p}k}{2})}{((k-p)^{2}+m^{2}+\omega_{n}^{2})(k^{2}+m^{2}+\omega_{n}^{2})}~~~,
+(\gamma_{m})_{\alpha\beta}(2D\Theta^{\lambda}E_{\lambda}^{\alpha}DX^{\underline{{n}}}E_{\underline{{n}}}^{\beta}DX^{\underline{{k}}}E_{\underline{{k}}}^{m}+DX^{\underline{{m}}}E_{\underline{{m}}}^{\alpha}DX^{\underline{{n}}}E_{\underline{{n}}}^{\beta}D\Theta^{\lambda}E_{\lambda}^{m})
\int_{\infty}x^{k}dz=\sum_{i}(\int_{A_{i}}x^{k}dz+\int_{A_{i}^{*}}x^{k}dz)=\sum_{i}\int\Big[(x_{0}^{(i)}+i\frac{m}{2})^{k}-(x_{0}^{(i)}-i\frac{m}{2})^{k}\Big]dz\,
{\calL}_{\psi\bar{\psi}H}=-\bar{\psi}[i\gamma^{\mu}\sigma^{\nu\lambda}\left\{\frac{1}{M_{P}e^{kr_{c}\pi}}H_{\mu\nu\lambda}^{0}+\frac{1}{\Lambda_{\pi}}\frac{2J_{0}(x_{n})}{\pix_{n}}\sum_{n=1}^{\infty}H_{\mu\nu\lambda}^{n}\right\}]\psi
D_{\alpha}\:=\\sum_{A,\beta}\Bigl(P_{j}^{A}\,\left(\gamma^{j}\right)_{\alpha}^{\beta}\;+\;\frac12\f_{BC}^{A}\,X^{iB}\,X^{jC}\,\left[\gamma_{i},\gamma_{j}\right]_{\alpha}^{\beta}\Bigr)\,\theta_{\beta}^{A}
\lambda(R)=-{\frac{1}{2A_{2}^{2}}}\left(1+gA_{2}^{2}-\sum_{p\ge1}pA_{p}\left(1-{\frac{p-2}{p+2}}gA_{2}^{2}\right)(c_{p}-c_{2}+\bar{\tau}\cdot\bar{g}_{p})\right)\.
S(\alpha)=\frac{1}{e^{2}}\intd^{4}x\,\sqrt{g}\,\Bigr\{(\nabla^{\mu}T_{\mu\rho}^{-})(\nabla_{\nu}T_{+}^{\nu\rho})+\alpha(T_{\mu\rho}^{-}T_{+}^{\nu\rho})(T_{-}^{\mu\sigma}T_{\nu\sigma}^{+})\Bigr\},
\overline{{S}}-\left\langleJ_{\mu}A^{\mu}\right\rangle=\intd^{2}x\left\{\frac1{2e^{2}}\phi\Box\left(\Box-m^{2}\right)\phi-\frac\lambda2\rho\Box\rho+J_{L}\rho+J_{T}\phi\right\}\textstyle{.}
\begin{array}{ll}{D=1:}&{Z_{E}(\tau,z)=0\,,}\\{D=2:}&{Z_{E}(\tau,z)=Z_{T}(\tau,z)P(\tau,z)^{r-2}\,,}\\{D=3,5:}&{Z_{E}(\tau,z)=\frac{\chi(E)}{\chi(T)}Z_{T}(\tau,z)P(\tau,z)^{r-D}\,.}\\\end{array}
h_{ij}=\delta_{ij}\frac{\mu^{2}\kappa_{11}^{2}p^{+}}{\pi^{4}}\frac{1}{96}\left[-15\frac{\sum_{k=1}^{3}(x^{k})^{2}}{|\vec{x}|^{7}}+\frac{1}{2}\frac{1}{|\vec{x}|^{5}}\right]+\frac{\mu^{2}\kappa_{11}^{2}p^{+}}{\pi^{4}}\frac{15}{64}\frac{x^{i}x^{j}}{|\vec{x}|^{7}}
m_{1}^{2}=q^{2}\phi_{cl}^{2}\qquadm_{2}^{2}=\mu^{2}+\frac{3}{4}\lambda\phi_{cl}^{2}\qquadm_{3}^{2}=\mu^{2}+\frac{1}{4}\lambda\phi_{cl}^{2}+\xiq^{2}\phi_{cl}^{2}\qquad
F\sim-\frac{2}{3\pi}\int\frac{dE}{e^{\betaE}-1}\int^{L}dr\frac{9e^{-3\delta}}{\Lambda^{2}r^{2}}\left[E^{2}+\frac{1}{3}\mu^{2}e^{2\delta}\Lambdar^{2}\right]^{\frac{3}{2}}.
Q=T_{3}+\frac{Y}{2}=\left(\begin{array}{cc|cc}{\frac{2}{3}}&{0}&{0}&{0}\\{0}&{-\frac{1}{3}}&{0}&{0}\\\hline{0}&{0}&{\frac{2}{3}}&{0}\\{0}&{0}&{0}&{-\frac{1}{3}}\\\end{array}\right),
\frac{1}{\lambda_{N^{\prime}}-\lambda_{N}}[\int_{C}d^{n}{\eta}Y_{\alpha}^{N}(\eta)MY_{\alpha^{\prime}}^{N^{\prime}}(\eta)-\int_{C}d^{n}{\eta}Y_{\alpha^{\prime}}^{N^{\prime}}(\eta)MY_{\alpha}^{N}(\eta)]
\left[A_{k}^{i},A_{m}^{j\dagger}\right]=\left[\tilde{A}_{k}^{i},\tilde{A}_{m}^{j\dagger}\right]=\left[B_{k}^{i},B_{m}^{j\dagger}\right]=\left[\tilde{B}_{k}^{i},\tilde{B}_{m}^{j\dagger}\right]=\delta_{km}\delta^{ij},
\mathrm{tr}\left[e^{-t{\calM}^{2}}f_{n}(t,A)\right]=\frac{t^{n}}{n!}\sum_{i_{1},i_{2},\ldotsi_{n}}^{N_{f}}c_{i_{1}i_{2}\ldotsi_{n}}(t)\frac{1}{n}\sum_{perm}\mathrm{tr}(A_{i_{1}}A_{i_{2}}\ldotsA_{i_{n}}),
E_{\mathrm{cas}}[\sigma]=\sum_{\ell}N_{\ell}\int_{m}^{\infty}\frac{dt}{2\pi}\,\frac{t}{\sqrt{t^{2}-m^{2}}}\,\left[\beta_{\ell}(t)\right]_{N}+E_{\mathrm{FD}}^{N}+E_{\mathrm{CT}}\,,
\frac{\lambda(-z-\frac{n}{2}w)}{\lambda(z)}=\frac{1}{\bar{\kappa}(u^{2})}\frac{(q^{n}u^{-2};t^{2})_{\infty}(t^{2}q^{-n}u^{2};t^{2})_{\infty}}{(u^{2};t^{2})_{\infty}(t^{2}u^{-2};t^{2})_{\infty}}.
\frac{\partial}{\partialt_{0}}R_{n+1}[u]=\frac{n}{2n+1}\left(\frac{1}{2}\lambda^{2}\frac{\partial^{3}}{\partialt_{0}^{3}}+2u(t_{0})\frac{\partial}{\partialt_{0}}+\frac{\partial}{\partialt_{0}}u(t_{0})\right)R_{n}[u].
\delta_{02}=\int_{r_{\mathrm{min}}}^{\infty}dr\left[\,\sqrt{D_{2}E^{2}-p_{11}^{2}-{\frac{J^{2}}{r^{2}}}}-\sqrt{E^{2}-p_{11}^{2}-{\frac{J^{2}}{r^{2}}}}\,\right]
\delta(B.I.)=-\tau_{p}\intd^{p+1}x\,e^{\beta\Phi}\sqrt{-\tilde{G}}\Bigl(-e_{1}K_{a}^{ai}K_{bi}^{b}+(\betae_{1}-e_{2})K_{a}^{ai}\partial_{i}\Phi+\betae_{2}(\partial_{i}\Phi)^{2}\Bigr)~,
I=-\frac{1}{16\piG}\intd^{3}x\sqrt{g}(R-4(\partial\phi)^{2}+2\Lambdae^{b\phi})+\frac{1}{8\piG}\intd^{2}x\sqrt{h}K+\frac{1}{8\piG}\intd^{2}x\sqrt{h}\frac{c_{0}}{l_{\mathrm{eff}}},
\psi_{1}=f_{1}(x_{1}+\frac{2i\gamma_{1}\hbar}{\theta}x_{2})e^{-\frac{3x_{1}^{2}}{8\gamma_{1}\hbar}-\frac{\gamma_{1}\hbarx_{2}^{2}}{2\theta^{2}}+\frac{ix_{1}x_{2}}{2\theta}},
\Gamma^{a_{1}\cdotsa_{N}}[k_{1},\varepsilon_{1};\ldots;k_{N},\varepsilon_{N}]=(-ig)^{N}\mathrm{tr}(T^{a_{1}}\cdotsT^{a_{N}}){\frac{{(4\pi\mu^{2})}^{-{\frac{\epsilon}{2}}}}{32{\pi}^{2}}}\sum_{\mathrm{diagrams}}D_{i}
\nabla_{\mu}e_{\nu}^{\!~~Aa}\equiv\partial_{\mu}e_{\nu}^{\!~~Aa}-\Gamma_{\mu\nu}^{~~~\rho}e_{\rho}^{\!~~Aa}+\omega_{\mu~~B}^{\!~~A}e_{\nu}^{\!~~Ba}+\omega_{\mu\!~~b}^{\!~~a}e_{\nu}^{\!~~Ab}=0,
\phi_{ia,kb}(\theta)=\phi_{ia,kb}(0)-\frac{4\pi^{2}}{|\beta^{2}|}\lambda_{i}^{\mathrm{v}}\cdot\lambda_{k}^{\mathrm{v}}-2h|\beta^{2}|\sum_{N\in\calE}\frac{P_{i}^{-N}(\theta_{i})P_{k}^{N}(\theta_{k})}{\mu^{2N}N^{2}}
{\cal{G}}_{k_{\perp}}^{\pm(\eta),sc}(t,t^{\prime})=-{\frac{\Gamma(-\nu)}{4\sqrt{\pi}}}e^{-{\frac{i\pi}{4}}}{\frac{1}{k_{\perp}}}{\frac{\partial\Omega}{\partialk_{\perp}}}S_{0}^{\prime}D_{\nu}\left({\frac{1+i}{\sqrt{\hbar}}}S_{0}(t)\right)D_{\nu}\left(-{\frac{1+i}{\sqrt{\hbar}}}S_{0}(t^{\prime})\right).
g=\left\langle\!\left\langle0\right|\right.\tilde{A}_{k}\cdot\tilde{A}_{k}^{\dagger}\left.\left|0\right\rangle\!\right\rangle=\left\langle\!\left\langle0\right|\right.\tilde{B}_{k}\cdot\tilde{B}_{k}^{\dagger}\left.\left|0\right\rangle\!\right\rangle.
\pi_{i}^{(2)a}=-\,\frac{1}{2m^{2}}\,\eta^{1b}\,G_{1i}^{(1)ba}+\frac{g}{4m^{2}}\,\bar{\eta}^{bc}\,(D_{i}\pi^{i})^{b}\,G_{1i}^{(1)ca}-\frac{1}{2}\,\eta^{2b}\,G_{2i}^{(1)ba}\,,
\Gamma_{1}={\frac{1}{\lambda_{1}}}{\frac{1}{\lambda_{2}}}\left({\frac{2}{\lambda_{1}+\lambda_{2}}}\right)^{2},\quad\Gamma_{2}={\frac{1}{2}}{\frac{1}{\lambda_{1}^{2}}}\left({\frac{2}{\lambda_{1}+\lambda_{2}}}\right)^{2},\quad\Gamma_{3}={\frac{1}{\lambda_{1}^{3}}}{\frac{2}{\lambda_{1}+\lambda_{2}}}.
G_{s}^{\overbrace{\kappa\cdots\kappa}^{m}\overbrace{w\cdotsw}^{n}\mu_{1}\cdots\mu_{s-n-m}}=\kappa^{m-2}\left(-w\right)^{n}g_{s,(s-n-m)}^{\mu_{1}\cdots\mu_{s-n-m}},
\begin{matrix}{A^{\alpha}\,\cap\,A^{\beta}\,=\,0}&{A^{\alpha}\,\cap\,B_{\gamma}\,=\,\delta_{\gamma}^{\alpha}}\\{B_{\mu}\,\cap\,A^{\beta}\,=-\,\delta_{\mu}^{\beta}}&{B_{\mu}\,\cap\,B_{\gamma}\,=\,0}\\\end{matrix}
S_{A}=-\frac{1}{4}\intd^{6}x\sqrt{^6g}g^{AB}g^{MN}F_{AM}F_{BN}\simeq-\frac{1}{4}\epsilon^{2}\int_{1}^{a}d\phi\intd^{4}x\sqrt{-\eta}\eta^{\mu\nu}\eta^{\alpha\beta}F_{\mu\alpha}F_{\nu\beta}.
[\frac{\alpha^{2}}{4}+(q^{\alpha)}+r^{\alpha)})t^{(\alpha)}]h_{i}^{(\alpha)2}C_{i}+[\frac{\alpha^{2}}{4}+q^{\alpha)}t^{\alpha)}-r^{(\alpha)}u^{(\alpha)}]\sum_{j\neqi}h_{j}^{(\alpha)2}C_{j}=C_{i},~~~i=1,...n
S_{P}^{4}=\intd^{4}x\Bigl[{\frac{1}{4}}\varepsilon^{\mu\nu\lambda\rho}B_{\mu\nu}F_{\lambda\rho}-{\frac{1}{4}}B_{\mu\nu}B^{\mu\nu}-{\frac{\mu^{2}}{2}}A_{\mu}A^{\mu}\Bigl]\;,
\frac{3}{2}\partial^{\mu}\partial_{\mu}F-6\frac{K^{\prime}}{K}F^{\prime}+\bigg[12\bigg(\frac{K^{\prime}}{K}\bigg)^{2}-(\phi_{0}^{\prime})^{2}\bigg]F-\phi_{0}^{\prime}\varphi^{\prime}=0
{\calG}^{(2)}=2\partiald^{(1)}+\frac{1}{2\pi\alpha^{\prime}}(i_{h}i_{k}C^{(4)})+2(2\pi\alpha^{\prime})\frac{(k.h)}{h^{2}}\partial\omega^{(0)}\partial{\tilde{\omega}}^{(0)}\,,
Q^{(2)}=\sum_{\sigma}\int\frac{d^{3}p}{(2\pi)^{3}}\,\left[a_{\sigma}^{\dagger}(\vec{p})c_{\sigma}(\vec{p})-b_{\sigma}(\vec{p})d_{\sigma}^{\dagger}(\vec{p})+c_{\sigma}^{\dagger}(\vec{p})a_{\sigma}(\vec{p})-d_{\sigma}(\vec{p})b_{\sigma}^{\dagger}(\vec{p})\right]\,,
\int_{g}\int_{g^{\prime}}dg\,dg^{\prime}\,\biggl({}_{\nu\bar{\nu}}\langleO_{2}|\times{}_{\eta\bar{\eta}}\langleO_{2}|\timesf^{*}(Q)\biggr)_{t=\epsilon}{g^{\prime}}^{\dagger}\,||\,g\biggl(f(Q)\times{|O_{2}\rangle}_{\eta\bar{\eta}}\times{|O_{k}\rangle}_{\nu\bar{\nu}}\biggr)_{t=0}
\tilde{\Psi}(p)=\sqrt{m}\left(\begin{array}{c}{\tilde{\Phi}(p)}\\{\tilde{\chi}^{\cdot}(p)}\\\end{array}\right),\,\mathrm{\thinspace~with\thinspace~\thinspace~\thinspace~}\left(p^{\mu}\gamma_{\mu}-m\right)\tilde{\Psi}=0\,\,\,
S[\hat{n},a_{\mu},c_{\mu}]~=~\intd^{3}x~{\frac{1}{g^{2}}}\partial_{\mu}\hat{n}\partial_{\mu}\hat{n}~+~j_{\mu}^{top}(2i\thetaa_{\mu}+c_{\mu})~+~i\theta\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}
L=\displaystyle\frac{1}{2}\,\displaystyle\frac{u^{2}-v^{2}}{u^{2}-\Omega^{2}}\,\dot{u}\,\dot{u}+\displaystyle\frac{1}{2}\,\displaystyle\frac{u^{2}-v^{2}}{\Omega^{2}-v^{2}}\,\dot{v}\,\dot{v}-\displaystyle\frac{u^{2}-\Omega^{2}}{u^{2}-v^{2}}\,f(u)-\displaystyle\frac{\Omega^{2}-v^{2}}{u^{2}-v^{2}}\,g(v)
\Lambda_{c}=\Bigg(\frac{1}{e^{2}}\Bigg)^{3/11}m_{\lambda}^{2/11}\Bigg(Ce^{-8\pi^{2}/e^{2}}M^{3N_{c}-N_{f}}\Big(\mathrm{det}(m_{q})_{i}^{j}\Big)^{2/3}\Big(\mathrm{det}(m_{sq})_{i}^{j}\Big)^{1/3}\Bigg)^{3/(11N_{c})},
S_{T\mu}~=~\frac{1}{1+\mathrm{e}^{-\frac{\sqrt{p_{3}^{2}+m^{2}}+|\mu|}{T}}}-\frac{\theta(|\mu|-\sqrt{p_{3}^{2}+m^{2}})}{1+\mathrm{e}^{-\frac{|\mu|-\sqrt{p_{3}^{2}+m^{2}}}{T}}}-\theta(\sqrt{p_{3}^{2}+m^{2}}-|\mu|)\left[1-\frac{1}{1+\mathrm{e}^{-\frac{\sqrt{p_{3}^{2}+m^{2}}-|\mu|}{T}}}\right]_{.}
\sigma^{\mu\nu}=\frac{i}{2}\left[\gamma^{\mu}\;,\;\gamma^{\nu}\right]\;,\;\;\left(F(x)K(x)\right)^{\ast\mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}\left(F(x)K(x)\right)_{\alpha\beta}\;,
\Big\{-\frac{\hbar^{2}}{2}\left[\partial_{r}^{2}+\frac{1}{r}\partial_{r}+\frac{1}{r^{2}}\partial_{\theta}^{2}+\partial_{z}^{2}\right]\,+\,\frac{1}{2}\omega^{2}r^{2}\Big\}\psi(r,\theta,z)=E\psi(r,\theta,z)\\\,
|\tilde{Z}_{1,2}|^{2}=\frac{1}{8}\left[\tilde{\calQ}^{2}+\tilde{\calP}^{2}\pm2\left(\tilde{\calQ}^{2}\tilde{\calP}^{2}-(\tilde{\calQ}\tilde{\calP})^{2}\right)^{\frac{1}{2}}\right]\,
S_{NS}(P)=G_{NS}\left(\frac{Q}{2}+iP\right)=-\left(\frac{\pi\mu}{2}\gamma\left(\frac{1+b^{2}}{2}\right)b^{1-b^{2}}\right)^{-\frac{2iP}{b}}\frac{\Gamma(1+iPb)\Gamma\left(1+\frac{iP}{b}\right)}{\Gamma(1-iPb)\Gamma\left(1-\frac{iP}{b}\right)},
Z_{\,\,{\overline{\Psi}}^{\,\,\prime}}\,\,[J_{A}]\,-\,\,Z_{\,{\overline{\Psi}}}\,[J_{A}]\,\,\,={\frac{i}{\hbar}}\int[d\varphi^{I}]\,\,\epsilon^{a}J_{r}\,{\overline{R}}_{a}^{r}\,\,exp{\frac{i}{\hbar}}\Big(S_{_\Sigma}+\hbarM_{1}\,+\,J^{A}\,\phi_{A}\Big)
\begin{matrix}{\left(\Psi^{a}={\frac{1}{\sqrt{2}}}\left[\begin{matrix}{\psi_{x}}\\{-i\bar{\psi}_{y}}\\\end{matrix}\right],\Phi=-\partialX,\Pi=-i\partial\bar{Y}\right),}\\{\left({\frac{1}{\sqrt{2}}}\left[\begin{matrix}{\psi_{y}}\\{i\bar{\psi}_{x}}\\\end{matrix}\right],-\partialY,i\partial\bar{X}\right)\,.}\\\end{matrix}
V=\sqrt{\frac{N_{2}N_{\bar{2}}}{N_{6}N_{\bar{6}}}},\\\\R_{4}=\sqrt{\frac{N_{5}N_{\bar{5}}}{g_{s}^{2}N_{6}N_{\bar{6}}}},\\\\R_{4}R_{9}^{2}=\sqrt{\frac{g_{s}^{2}N_{R}N_{L}}{N_{2}N_{\bar{2}}}}.
\phi(U,x)\stackrel{U\to\infty}{\approx}\frac{{U_{T}}^{4}}{8\piR^{6}U^{4}}\sum_{n=1}^{\infty}\frac{e^{-m_{n}r}}{m_{n}rw^{\prime}(m_{n})}\intdu^{\prime}y_{2}(u^{\prime},m_{n})du^{\prime}.
\frac{y^{2}}{\Lambda^{2(n_{c}-\tilde{n}_{c})}}=\prod_{a=1}^{r}(x-m_{a})^{2}\prod_{k=1}^{\tilde{n}_{c}-r}\left(x+\frac{1}{\tilde{n}_{c}-r}\sum_{i=r+1}^{n_{f}}m_{i}-\delta\phi_{k}\right)^{2}+\frac{4}{\Lambda^{n_{c}-\tilde{n}_{c}}}\prod_{i=1}^{n_{f}}(x+m_{i}).