harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{2\pi}\Omega=\left(\begin{array}{ccccc}{0}&{x_{1}}&{}&{}&{}\\{-x_{1}}&{0}&{}&{}&{}\\{}&{}&{0}&{x_{2}}&{}\\{}&{}&{-x_{2}}&{0}&{}\\{}&{}&{}&{}&{\ddots}\\\end{array}\right).
^{(4)}G_{\nu}^{\mu}=\kappa_{4}^{2}\,^{(4)}T_{\nu}^{\mu}-\Big(\Lambda_{4}+\frac{3}{2}\alpha^{2}\Big)\,\delta_{\nu}^{\mu}+\alpha\Big(L_{\nu}^{\mu}+\frac{L}{2}\,\delta_{\nu}^{\mu}\Big)\,,
(1-\tilde{P})\left\{a^{\dagger}\tilde{T}-[a^{\dagger},P]\tilde{T}\right\}\frac{1}{\tilde{T}^{\dagger}\tilde{T}}\tilde{T}^{\dagger}=0\quad\mathrm{and}\quad(1-\tilde{P})\left\{\partial_{t}\tilde{T}-i\gamma[a,P]\tilde{T}\right\}\frac{1}{\tilde{T}^{\dagger}\tilde{T}}\tilde{T}^{\dagger}=0,
V_{tor}\simeqka_{1}a_{2}+\frac{(d-3)\pi}{2}\left[-\frac{\zeta(3)}{\pi^{2}}\left(\frac{a_{1}}{a_{2}^{2}}+\frac{a_{2}}{a_{1}^{2}}\right)\frac{1}{\pi}\sqrt{k}\rho-\frac{1}{6\pi^{2}}k^{3/2}\rho^{3}a_{1}a_{2}\right].
\mathcal{A}_{4}^{cl}=(2\pi)^{2}16g_{c}^{2}(2\alpha^{\prime})^{(d-4)/2}B\left(-\frac{1}{2}\alpha\left(s\right),-\frac{1}{2}\alpha\left(t\right),-\frac{1}{2}\alpha\left(u\right)\right).
\left<T_{\mu\nu}(x)\right>_{\mu^{2}}^{{\mathbbH}^{5}}=-\frac{m^{2}}{120\pi^{2}a^{3}}\left(3-20\xi+a^{2}m^{2}\right)\sqrt{4-20\xi+a^{2}m^{2}}\,\,g_{\mu\nu}^{{\mathbbH}^{5}}(x).
(\vec{\alpha}_{1}\!\cdot\vec{\alpha}_{2})^{T}\,=\,(\vec{\alpha}_{1}\!\cdot\vec{\alpha}_{2})\,-\,{\frac{(\vec{\alpha}_{1}\!\cdot\vec{k})\,(\vec{\alpha}_{2}\!\cdot\vec{k}\,)}{\vec{k}\,^{2}}}.
\textstyle{\frac{1}{2\piR}}\int\mathrm{d}^{6}x\,\textstyle{\frac{1}{12}}H_{\lambda\mu\nu}H^{\lambda\mu\nu}-\textstyle{\frac{1}{24}}\epsilon^{\mu\nu\lambda\rho\sigma}\partial_{5}B_{\mu\nu}H_{\lambda\rho\sigma}\,.
\check{R}(u)=P_{j=k}+\frac{y^{2}-q^{2k}}{1-y^{2}q^{2k}}P_{j=k-1}+\ldots+\frac{y^{2}-q^{2k}}{1-y^{2}q^{2k}}\frac{y^{2}-q^{2(k-1)}}{1-y^{2}q^{2(k-1)}}\ldots\frac{y^{2}-q^{2}}{1-y^{2}q^{2}}P_{j=0}\;,
H_{2}={\frac{Q}{|\vec{z}-\vec{z}_{0}|^{6-n}}},\\\\H_{W}=1+{\frac{Q_{W}}{[x^{2}+{\frac{4Q}{(n-4)^{2}}}|\vec{z}-\vec{z}_{0}|^{n-4}]^{\frac{8-n}{2(n-4)}}}}.
P^{c}={\frac{1}{4\pi{\calA}_{P}}}\left(2g^{cb}\partial_{b}\sqrt{-g}+\sqrt{-g}\partial_{b}g^{bc}\right)={\frac{\sqrt{-g}}{4\pi{\calA}_{P}}}\left(g^{ck}\Gamma_{km}^{m}-g^{ik}\Gamma_{ik}^{c}\right)
\int{\calD}v\;\delta^{n}\left(\intvd\tau\right)F_{qG}[v,I]=\int{\calD}v\;\delta^{n}\left(\int\phid\tau\right)F_{qG}[\phi,I]\;\mathrm{Det}\;\frac{\delta\phi_{\tau}(v)}{\deltav(\tau^{\prime})}\;,
S_{KR}=\int\Bigl\{e^{-{\bf\Phi}}{\bfH}{^{*}{\bfH}}+2{\bfQ}\Bigl({\bfH}-d{\bfB}-\frac{\alpha^{\prime}}{16}({\bf\Omega}_{L}-{\bf\Omega}_{YM})\Bigr)\Bigr\}
\mathrm{d}s^{2}=M_{ij}\mathrm{d}{\vec{x}}^{i}\cdot\mathrm{d}{\vec{x}}^{j}+\frac{1}{4\pi^{2}}(M^{-1})_{ij}(\mathrm{d}\xi^{i}+{\vec{W}}_{ik}\cdot\mathrm{d}{\vec{x}}^{k})(\mathrm{d}\xi^{j}+{\vec{W}}_{jl}\cdot\mathrm{d}{\vec{x}}^{l})
G^{JJ^{\prime}}(x)=\frac{\delta^{JJ^{\prime}}}{|x|^{2J}}\left(\frac{g_{\mathrm{YM}}^{2}}{8\pi^{2}}\right)^{J}\langle\mathrm{Tr}Z^{J}~\mathrm{Tr}\bar{Z}^{J}\rangle
\left({\calV}^{(n)}(u){\calV}^{(n)}(u+v)\right)_{\alpha,\alpha^{\prime}}=\mathrm{Tr}X(\alpha_{1},\alpha_{1}^{\prime})X(\alpha_{2},\alpha_{2}^{\prime})\cdotsX(\alpha_{n},\alpha_{n}^{\prime})~,
p=\dot{\phi}^{2}(t)+\int\!\frac{d^{3}k}{(2\pi)^{3}}\,\frac{1}{2\omega_{k}^{0}}\left(|\dot{U}_{k}^{+}(t)|^{2}+\frac{\vec{k}^{2}}{3}|U_{k}^{+}(t)|^{2}\right)-{\calE}\;.
\begin{array}{rcl}{\overline{{D_{m}(k_{1}+p)}}}&{=}&{\displaystyleD_{m}(k_{1})\left[-2k_{1}\cdotpD_{m}(k_{1})\right.}\\{}&{}&{\displaystyle\left.-p^{2}D_{m}(k_{1})+4(k_{1}\cdotp)^{2}D_{m}^{2}(k_{1})+O(p^{3})\right].}\\\end{array}
\frac{1}{R^{4}}\sum_{a=0}^{2l}\sum_{b=0}^{2l}\sum_{m_{a}=-a}^{a}\sum_{m_{b}=-b}^{b}=\frac{1}{{\pi}^{2}}\int_{\Lambda}d^{2}\vec{p}_{a}\int_{\Lambda}d^{2}\vec{p}_{b}~,~{\Lambda}=\frac{2}{\theta},
\langle0|a_{p}H_{\mathrm{zm}}a_{q}^{\dagger}a_{r}^{\dagger}a_{s}^{\dagger}|0\rangle=-\frac{18\delta_{p,q+r+s}}{g\sqrt{pqrs}}\frac{1}{r+s}+\left(q\leftrightarrowr\right)+\left(q\leftrightarrows\right)
S_{\mathrm{quadr}}[a_{i}^{a},\phi^{a}]={\frac{1}{2}}\intd^{3}x\left(\begin{matrix}{a_{i}^{a},}&{\phi^{a}}\\\end{matrix}\right)\left(\begin{matrix}{{\calM}_{ij}^{ab}}&{-{\calM}_{i}^{ab}}\\{{\calM}_{j}^{ab}}&{{\calM}^{ab}}\\\end{matrix}\right)\left(\begin{matrix}{a_{j}^{b}}\\{\phi^{b}}\\\end{matrix}\right)\;,
L_{CPV}=\frac{1}{2e}(\dot{x}_{\mu}+iv\epsilon_{\mu\nu\lambda}\xi^{\nu}\xi^{\lambda})^{2}-\frac{e}{2}m^{2}-imv\theta_{1}\theta_{2}-\frac{i}{2}\xi_{\mu}\dot{\xi}{}^{\mu}+\frac{i}{2}\theta_{a}\dot{\theta}_{a}
L_{\mu\nu}=\left(\begin{array}{cccc}{0}&{k_{1}-k_{3}}&{-k_{1}+k_{2}}&{k_{3}-k_{2}}\\{-k_{1}+k_{3}}&{0}&{k_{1}}&{-k_{3}}\\{k_{1}-k_{2}}&{-k_{1}}&{0}&{k_{2}}\\{-k_{3}+k_{2}}&{k_{3}}&{-k_{2}}&{0}\\\end{array}\right),
G_{\nu}^{\mu}(\{\})=8\piG(2\alphag^{\mu\alpha}\partial_{\alpha}\Lambda\partial_{\nu}\Lambda-\alpha\delta_{\nu}^{\mu}g^{\alpha\beta}\partial_{\alpha}\Lambda\partial_{\beta}\Lambda+T_{\nu}^{\mu})=8\piG\tilde{T}_{\nu}^{\mu},
\rho={\frac{(\xi_{t}^{2}+\xi_{x}^{2}+\xi_{y}^{2})\dot{R}^{2}}{[1+R^{2}]^{2}}}+{\frac{(\psi_{t}^{2}+\psi_{x}^{2}+\psi_{y}^{2})R^{2}}{[1+R^{2}]^{2}}},
\!{\calK}_{p}(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime},N|x^{\prime},y^{\prime},z^{\prime},0)=\frac{\lambda_{p}(-2N)}{\left|4^{\frac13}N\right|_{p}^{\frac32}}\chi_{p}\left(-\bar{S}(x^{\prime\prime},y^{\prime\prime},z^{\prime\prime},N|x^{\prime},y^{\prime},z^{\prime},0)\!\right).
a^{2}\frac{\partial^{2}V}{\partial\varphi^{2}}=\biggl[{\calH}\frac{\varphi^{\prime\prime}}{\varphi^{\prime}}-6{\calH}^{2}+\frac{\varphi^{\prime\prime\prime}}{\varphi^{\prime}}+3{\calH}^{\prime}\biggr].
{\calL}=\frac{1}{2g}[\mathrm{tr}\dot{X}_{mn}^{i}\dot{X}_{mn}^{i}+\frac{1}{2}\mathrm{tr}(X_{mq}^{i}X_{qn}^{j}-X_{mq}^{j}X_{qn}^{i})(X_{nr}^{i}X_{rm}^{j}-X_{nr}^{j}X_{rm}^{i})].
\Phi(t)=\sum_{r\ge1}\int_{0}^{t}dt_{1}\cdots\int_{0}^{t_{r-1}}dt_{r}\sum_{\sigma\in\S_{r}}{\frac{(-1)^{d(\sigma)}}{r}}{\binom{r-1}{d(\sigma)}}^{-1}\psi(t_{\sigma(r)})\cdots\psi(t_{\sigma(1)})\.
\delta_{\epsilon}\theta_{\hat{\beta}}^{s}=\epsilon^{\hat{\alpha}t}\Bigl({\frac{1}{2}}\sigma_{\hat{\alpha}\hat{\beta}}^{i}(\gamma_{b^{\prime}})_{t}{}^{s}\partial_{i}X^{b^{\prime}}-{\frac{1}{6}}(\sigma_{\hat{\alpha}\hat{\beta}}^{ijk}\delta_{t}{}^{s}h_{ijk}\Bigr)
\langle0|T_{\mu}^{\mu}|0\rangle\cong4V(\varphi_{0})=-{\frac{3V_{0}^{2}}{16\pi^{2}}}\mathrm{for~SU(3)},\left(-{\frac{v^{2}}{8\pi^{2}}}\mathrm{for~SU(2)}\right).
R(\zeta)v_{\varepsilon_{1}}\otimesv_{\varepsilon_{2}}=\sum_{\varepsilon_{1}^{\prime},\varepsilon_{2}^{\prime}=\pm}v_{\varepsilon_{1}^{\prime}}\otimesv_{\varepsilon_{2}^{\prime}}R(\zeta)_{\varepsilon_{1}\varepsilon_{2}}^{\varepsilon_{1}^{\prime}\varepsilon_{2}^{\prime}},~~~~R(\zeta)=\frac{1}{\kappa(\zeta)}\overline{{R}}(\zeta),
B(u,v)=\left\{\left(\begin{array}{c}{u}\\{0}\\\end{array}\right),\left(\begin{array}{c}{0}\\{v^{\ast}}\\\end{array}\right)\right\},\\\\left\{\left(\begin{array}{c}{u}\\{0}\\\end{array}\right),\left(\begin{array}{c}{v}\\{0}\\\end{array}\right)\right\}=0
\frac{\partial}{\partial\lambda_{1}}\cdots\frac{\partial}{\partial\lambda_{p+q}}\int_{-\infty}^{\infty}\:ds\:\eta\left.T_{\bullet}^{(a_{1})}\cdotsT_{\bullet}^{(a_{p})}\:\overline{{T_{\bullet}^{(b_{1})}}}\cdots\overline{{T_{\bullet}^{(b_{q})}}}\right._{|\lambda_{1}=\cdots=\lambda_{p+q}=0}\;.
\tilde{\mathcal{C}}^{-1}\Gamma_{s}(P_{a})^{T}\tilde{\mathcal{C}}=-\Gamma_{s}(P_{a})\,,\,\,\,\,\Rightarrow(u\tilde{u})^{-1\,T}\tilde{\mathcal{C}}=\tilde{\mathcal{C}}u\tilde{u}\,,
S_{1}=\displaystyle\intd\tau\left\{\pi_{{\bar{a}}\mu}\dot{x}_{\bar{a}}^{\mu}-\frac12\sum^{\prime}\phi_{\bar{a}\bar{b}}(\pi_{{\bar{a}}\mu}\pi_{\bar{b}}^{\mu}+c_{\bar{a}}^{2}\delta_{\bar{a}\bar{b}})\right\},
g_{m_{a},m_{b};\mu}(s)=\sqrt{{\frac{\pm[(m_{a}+m_{b})^{2}-\mu^{2}](m_{a}+m_{b})^{2}}{[k^{2}(s)-(m_{a}+m_{b})^{2}]s}}}g_{m_{a}m_{b};\mu}(0)
C_{\pm}=-m^{\mu}l_{\pm}^{\nu}(\frac{1}{\kappa}\xi_{\mu;\rho}l_{\pm}^{\rho})_{;\nu}=-\frac{1}{\kappa}m_{\pm}^{\nu}l_{\pm}^{\rho}\xi_{\mu;\rho;\nu}=-\frac{1}{\kappa}R_{\rho\sigma\mu\nu}m^{\mu}l_{\pm}^{\nu}l_{\pm}^{\rho}\xi^{\sigma},
\frac{1}{8\pi}\int^{r}d^{3}x(K-K_{0})=-\frac{\pi}{\kappa}\Biggl[r-M+\frac{r(r-r_{+})(r-r_{-})}{R^{2}}-2[(r-r_{+})(r-r_{-})]^{\frac{1}{2}}\Biggr]\.
\tilde{a}_{\varphi}^{0}=2\lambda\a_{\varphi}^{0}=-\frac{e}{4\piR}\\frac12\\left[e^{-\bigl|t-\tau-R/c\bigr|/\lambda}+e^{-\bigl|t-\tau+R/c\bigr|/\lambda}\right]
S_{np}=\frac{\lambda^{5}}{48}\Big[\frac{p^{2}}{32\pi^{2}}\int_{0}^{\Lambda^{2}}\frac{d^{4}l}{(2\pi)^{4}}\frac{G(l|p)_{small}}{l^{2}(l+p)^{2}}+\frac{p^{2}}{32\pi^{2}}\int_{\Lambda^{2}}^{\infty}\frac{d^{4}l}{(2\pi)^{4}}\frac{G(l|p)_{large}}{l^{2}(l+p)^{2}}\Big],
L=\left[{\frac{\mu}{2}}+{\frac{kb}{4(b^{2}-r^{2})}}\right]\dot{r}^{2}+{\frac{kb}{4(b^{2}-r^{2})}}\dot{b}^{2}+{\frac{br}{b^{2}-r^{2}}}\dot{b}\dot{r}-{\frac{bJ^{2}}{2r^{2}(k+\mub)}}
\begin{aligned}{I_{X}}&{{}=t_{8}t_{8}W^{4}+\frac{1}{2}\varepsilon_{10}t_{8}BW^{4}\,,}\\{I_{Z}}&{{}=\varepsilon_{10}\varepsilon_{10}W^{4}+4\,\varepsilon_{10}t_{8}BW^{4}\,.}\\\end{aligned}
X^{\mu}=\bar{X}^{\mu}+\tilde{X}^{\mu},\qquad\bar{X}^{\mu}\equiv\left(\begin{array}{llll}{x_{1}^{\mu}}&{}&{}&{}\\{}&{x_{2}^{\mu}}&{}&{}\\{}&{}&{\ddots}&{}\\{}&{}&{}&{x_{N}^{\mu}}\\\end{array}\right).
P=P_{g}+P_{b}=\frac{-1}{9\times960\pi^{2}}(\Lambda_{betw}^{2}-\Lambda_{out}^{2})-\frac{\eta^{4}\Lambda_{betw}^{2}J_{3}(B_{1},B_{2})}{36\pi^{3/2}a^{4}\Gamma(3/2)}.
\begin{array}{lcr}{\sum_{j}q_{a}^{j}(|X_{j}|^{2}-|Y_{j}|^{2})\eta_{j}^{+}\overline{{\eta}}_{j}^{+}}&{=r_{a}^{2}\eta_{a}^{+}\overline{{\eta}}_{a}^{+}\qquad}&{(a)}\\{\sum_{j}q_{a}^{j}(X_{j}\overline{{Y}}_{j})\eta_{j}^{+}{\eta}_{j}^{+}}&{=0\qquad}&{(b)}\\{\sum_{j}q_{a}^{j}(\overline{{X}}_{j}{Y_{j}})\overline{{\eta}}_{j}^{+}\overline{{\eta}}_{j}^{+}}&{=0.\qquad}&{(c)}\\\end{array}
S_{1}=\intd^{4}x\left[\frac{\partial}{\partial\theta}\left\{\bar{K}^{\alphai}\bar{A}_{i}^{\alpha}+\zeta^{\alpha}\left[\partial^{\mu}A_{\mu}^{\alpha}(\bar{x})+\frac{1}{2\eta_{0}}\zeta_{,\theta}^{\alpha}(\bar{x})+t^{\alpha}(\bar{x})\right]\right\}+\frac{\partial}{\partial\lambda}[\bar{L}^{\alphai}\bar{A}_{i}^{\alpha}]+\partial\cdotL\zeta\right]
\begin{array}{r}{\sum_{\mu,\nu\in{\calR}}(\mu\cdot\nu)(q\cdot\mu)\ell_{\mu}^{j-1}\,\ell_{\nu}^{k}\,\psi\equivG_{j,k}\,\psi,}\\{\sum_{\mu,\nu\in{\calR}}(\mu\cdot\nu)(q\cdot\nu)\ell_{\mu}^{j}\,\ell_{\nu}^{k-1}\psi\equivG_{k,j}\,\psi,}\\\end{array}\quad\psi:\\mathrm{Coxeter~invariant}.
E=\frac{\sqrt{U_{0}^{2}+\left(\frac{r_{0}}{2}\right)^{2}}}{2\pi}\biggl[\int_{1}^{\infty}dy\left(\frac{y^{2}}{\sqrt{(y^{2}-1)(y^{2}+1-f^{2})}}-1\right)-1\biggr]
\Pi_{\mu\nu}(k)=ie^{2}\mathrm{Tr}\int\bar{d}^{3}p\;\frac{\gamma_{\mu}\gamma.p\gamma_{\nu}\gamma.(p-k)}{p^{2}(p-k)^{2}}=(-k^{2}\eta_{\mu\nu}+k_{\mu}k_{\nu})\frac{e^{2}}{8\sqrt{-k^{2}}},
e^{-1}{\calL}=-\frac{3}{2}[S_{o}{\bar{S}}_{o}e^{-\frac{1}{3}G(Z,{\bar{Z}})}]_{D}+\left([S_{o}^{3}]_{F}+h.c\right)+\frac{1}{4}\left([f_{ab}(Z)W^{a}W^{b}]_{f}+h.c\right)
\begin{array}{c}{S_{+}+S_{-}=}\\{-\frac\beta{2\pi}\left(\begin{array}{c}{\frac1z+i\mu+\pi/\beta\left(H(x^{\prime}-x)-H(x-x^{\prime})\right)+O(z/{\beta}^{2})\;\qquad0}\\{0\;\qquad\frac1{\bar{z}}-i\mu+\pi/\beta\left(H(x^{\prime}-x)-H(x-x^{\prime})\right)+O(\bar{z}/{\beta}^{2})}\\\end{array}\right).}\\\end{array}
\frac{i\vec{\bigtriangledown}_{p}}{\bigtriangledown_{p}^{2}}\Phi(\vec{p})=\frac{i}{2\pi}\int\\Phi(\vec{p}^{~\!\prime})\frac{(\vec{p}-\vec{p}^{~\!\prime})}{|\vec{p}-\vec{p}^{~\!\prime}|^{2}}d^{2}p^{\prime}.
\Omega=\frac{1}{2\gamma}tr\{\int_{0}^{2\pi}g^{-1}\,dg\partial(g^{-1}\,dg)dx+dg_{1}\,g_{1}^{-1}\,dg_{2}\,g_{2}^{-1}-M_{+}^{-1}\,dM_{+}\,M_{-}^{-1}\,dM_{-}\},
\underset{\sum_{i=1}^{3}\alpha_{i}=Q-sb}{res}\langleV_{\alpha_{1}}V_{\alpha_{2}}V_{\alpha_{3}}\rangle=\frac{\left(-\mu\right)^{s}}{s!}\langleV_{\alpha_{1}}V_{\alpha_{2}}V_{\alpha_{3}}\underbrace{\int\hat{E}V_{b}\dots\int\hat{E}V_{b}}_{s}\rangle_{\sum_{i=1}^{3}\alpha_{i}=Q-sb}
C_{1}^{2}(m_{j})={\frac{1}{2\pi{\calZ}^{2}[J_{3}^{2}(x_{3,j})-J_{2}(x_{3,j})J_{4}(x_{3,j})]}}=-{\frac{1}{2\pi{\calZ}^{2}J_{2}(x_{3,j})J_{4}(x_{3,j})}}\,.
\Psi_{l}^{\prime}(\zeta)=f(\theta_{+l}^{\prime}(\zeta),\theta_{+k}(\zeta))\Psi_{k}(\zeta),\quad{\bar{\Psi^{\prime}}}_{l}(\zeta)={\bar{\Psi}}_{k}(\zeta)f^{*}(\theta_{-l}^{\prime}(\zeta),\theta_{-k}(\zeta)).
S=-T_{3}\left[{\frac{1}{2}}\int_{\Sigma_{3}}\Big(\sqrt{-g}(g^{ij}\tilde{g}_{ij}-1)+\tilde{C}\Big)+\int_{\Sigma_{2},\xi^{3}=\pi}\tilde{B}-\int_{\Sigma_{2},\xi^{3}=0}\tilde{B}\right],
b_{4}=\frac{2}{3}\frac{\omega_{t}^{2}}{\omega_{y}^{2}}\frac{1}{\epsilon_{y}^{2}}+\frac{3\omega_{y}^{2}-\omega_{t}^{2}}{2\omega_{y}^{2}+2\omega_{t}^{2}}\,,\qquadc_{4}=-2\frac{\omega_{t}^{2}}{\omega_{y}^{2}}\frac{1}{\epsilon_{y}^{2}}\,.
\left\{{\frac{Det^{\prime}\left[-(d^{2}/d\tau^{2})+V^{\prime\prime}[x_{c2}(\tau)]\right]}{Det\left[-(d^{2}/d\tau^{2})+36\\omega^{2}\right]}}\right\}^{-1/2}=\sqrt{S_{2}}\K_{2}\\omega\d\tau_{c}
e^{-2\phi}=\left(\Bigl(e^{-2\hat{\phi}_{0}}+{\frac{2\tilde{m}}{r}}\Bigr)\Bigl(K_{0}+{\frac{2\hat{m}}{r}}\Bigr)-\left[(A_{n})_{0}+{\frac{2q_{n}}{r}}\right]^{2}\right)^{\frac{1}{2}}\.
\eta_{2}(\beta\sqrt{-\triangle})=\frac{2\sqrt{\pi}}{\beta}\int_{0}^{\infty}\frac{{\mathrmd}s}{s^{3/2}}\,{\mathrme}^{-sm^{2}}\,\frac{f(-s\triangle)-1-\frac16s\triangle}{(s\triangle)^{2}},\\beta\rightarrow0,
\left(A^{0},\,A^{1},\,A^{2},\,\pi^{0},\,\pi^{1},\,\pi^{2}\right)\Leftrightarrow\left(\xi^{1},\,\xi^{2},\,\xi^{3},\,\xi^{4},\,\xi^{5},\,\xi^{6}\right).
D\Psi\equivdx^{\mu}{\left\{{\partial_{\mu}}I_{4}-{i}\left(\begin{matrix}{A_{\mua}^{+}{\frac{\tau^{a}}{2}}}&{0}\\{0}&{A_{\mua}^{-}{\frac{\tau^{a}}{2}}}\\\end{matrix}\right)\right\}}{\left(\begin{matrix}{\phi_{R}}\\{\phi_{L}}\\\end{matrix}\right)},
\tau[t]=\sum_{n=0}^{\infty}\,\mathrm{e}^{nt_{1}}\tau_{n}[t],\quad\tau_{0}[t]=1,\quad\tau_{n}[t]={\frac{1}{n}}\sum_{{\calK}_{n}}{\frac{t_{0}^{k_{0}}}{(k_{0}-1)!}}\prod_{i=2}^{\infty}{\frac{(int_{i})^{k_{i}}}{k_{i}!}}\,\,\mathrm{for}\,n>0,
S_{\mathrm{W}_{A}}^{\mathrm{BI}}=-T_{0}\intd\tau~\mathrm{Str}\Bigl\{k^{-1}\sqrt{-P[E_{00}+E_{0i}(Q^{-1}-\delta)_{k}^{i}E^{kj}E_{j0}]\\mathrm{det}(Q_{j}^{i})}\\Bigr\},
\begin{array}{c}{\Psi_{i}[{\bfr}(\beta^{\prime});\beta_{n},\beta_{1}]=\psi_{i}({\bfr}_{n},\beta_{n})\psi_{i}({\bfr}_{n-1},\beta_{n-1})...\psi_{i}({\bfr}_{1},\beta_{1})}\\{\intD{\bfr}(\beta^{\prime})\cdot\Psi_{i}^{*}[{\bfr}(\beta^{\prime});\beta,\beta_{0}]\cdot\Psi_{j}[{\bfr}(\beta^{\prime});\beta,\beta_{0}]=\delta_{ij}}\\\end{array}
\tilde{\xi}_{(m,n)}^{\gamma}=F_{\alpha\beta}^{\gamma}\xi_{(m)}^{\beta}\xi_{(n)}^{\alpha},\;\;\;\tilde{\xi}_{((m,n),l)}^{\gamma}=F_{\alpha\beta}^{\gamma}\tilde{\xi}_{(m,n)}^{\beta}\xi_{(l)}^{\alpha};
\alpha_{-n}^{-}=-\frac{1}{4}\sum_{l=-\infty}^{+\infty}\left[\alpha_{n-l}^{i},\alpha_{l}^{i}\right]_{+}-\frac{1}{4}\sum_{r=-\infty}^{+\infty}\left(r-\frac{n}{2}\right)\left[b_{n-r}^{i},b_{r}^{i}\right]-\frac{a}{2}
S_{int}^{(e,\mu)}=i\int\left[\omega^{\mu}(d)eA_{\mu}+{\theta}^{\alpha}(d)\left(eA_{\alpha}+i\mu_{1}W_{\alpha}\right)+{\bar{\theta}}_{\dot{\alpha}}(d)\left(e{\bar{A}}^{\dot{\alpha}}+i\mu_{1}{\bar{W}}^{\dot{\alpha}}\right)\right].
\sum_{n=0}^{\infty}\left\{\frac{1}{(n+\frac{1}{2})^{3}}\sum_{i_{1}=0}^{n}\frac{1}{n!}\\frac{(\frac{1}{2})_{(n+1)}}{(\frac{1}{2}+i_{1})}\right\}-\sum_{n=1}^{\infty}\left\{\frac{1}{(n+\frac{1}{2})^{2}}\underbrace{\sum_{j_{1}=0}^{n}\sum_{j_{2}=0}^{j_{1}-1}}_{j_{1}>j_{2}}\frac{1}{n!}\,frac{(\frac{1}{2})_{(n+1)}}{(\frac{1}{2}+j_{1})(\frac{1}{2}+j_{2})}\right\}=0\.
F(x^{-},x^{+})=\left|{\frac{L}{\pi}}\langle0|T^{++}(k)|n\rangle\right|^{2}\left({\frac{x^{+}}{x^{-}}}\right)^{2}{\frac{M_{n}^{4}}{8\pi^{2}k^{3}}}K_{4}\left(M_{n}\sqrt{2x^{+}x^{-}}\right)
\left[J_{int}^{\mu}(x),J_{int}^{\nu}(y)\right]_{E.T.}=\left[J^{\mu}(x),J^{\nu}(y)\right]_{E.T.}=-\sum_{n=0}^{\infty}e^{n}\sum_{i=0}^{n}\left[J_{i}^{\mu}(x),J_{n-i}^{\nu}(y)\right].
A^{-1}=c_{0}+\sum_{l=1}^{L}\frac{1}{2}(\bar{k}_{l}z+k_{l}\bar{z})\theta(\frac{e^{-i\varphi_{l-1}}z-e^{i\varphi_{l-1}}\bar{z}}{2i})\theta(\frac{e^{-i\varphi_{l}}z-e^{i\varphi_{l}}\bar{z}}{-2i})
\overline{{\psi}}^{(b)}P_{+}\psi^{(b)}=\overline{{\psi}}_{1}^{(b)}\psi_{1}^{(b)}\;\;\;,\;\;\;\overline{{\psi}}^{(b)}P_{-}\psi^{(b)}=\overline{{\psi}}_{2}^{(b)}\psi_{2}^{(b)}\;.
\delta_{B}^{r}L_{B}^{r}=\eta\;\frac{d}{d\tau}\;\Bigl[q^{1/2}\lambda\;p_{\mu}\dot{x}^{\mu}-\;\frac{\lambdae}{1+q^{2}}\;(p^{2}-m^{2})+{\calB}\frac{d}{d\tau}(\lambda\;e)\Bigr].
V_{~~b}^{a}(x^{0},{\bfy}):=2gf_{~~bc}^{a}\frac{\partial}{\partialy^{k}}A^{kc}(x^{0},{\bfy})+g^{2}f_{~~c^{\prime}d}^{a}f_{~~ba^{\prime}}^{c^{\prime}}A_{k}^{~~d}(x^{0},{\bfy})A^{ka^{\prime}}(x^{0},{\bfy}).
K(q_{f}\midq_{i})=\int{\calD}q~{\calD}p\intd\theta\int_{0}^{\infty}\!\!dT~e^{\frac{i}{\hbar}\int_{0}^{1}d\tau\,[\,p_{\mu}\dot{q}^{\mu}-\frac{T}{2}(g^{\mu\nu}p_{\mu}p_{\nu}-m^{2})+\frac{\theta}{2}(\Gamma^{\mu}(q)p_{\mu}+m)\,]}
I=\sum_{m,\hat{m}}\intd^{4}x{\frac{2}{8\pi^{2}[x^{2}+(2m+1)^{2}\ell^{2}]^{3/2}}}(-\nabla^{2}){\frac{2}{8\pi^{2}[x^{2}+(2\hat{m}+1)^{2}\ell^{2}]^{3/2}}}\,
\phi(z,\bar{z})={\mathrme}^{\imath\gamma}\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\left[d_{kl}z^{1+Nk}(z\bar{z})^{l}+e_{kl}\bar{z}^{N(k+1)-1}(z\bar{z})^{l}\right].
\theta^{\alpha1}=v_{\dot{A}}^{\alpha+}\theta_{\dot{A}}^{-},\\theta^{\alpha2}=v_{A}^{\alpha-}\theta_{A}^{+},\\lambda_{+}^{\alpha}=v_{A}^{\alpha-}\lambda_{A},\\lambda_{-}^{\alpha}=v_{\dot{A}}^{\alpha+}\lambda_{\dot{A}}
S_{z}^{D}=\intd^{4}x\left(\intd^{4}\thetaK(z,z^{\dagger})\intd^{2}\thetalz\left\{w-U(A_{D})+\frac{\Lambda^{4}zl}{w}f\left(\frac{A_{D}^{2}}{w},\frac{\Lambda^{4}}{w^{2}}\right)\right\}+h.c.\right)
T_{\sigma_{1p}\sigma_{2p}}^{\nu_{1p}\nu_{2p}}({\bfk}(-){\bfp},{\bfp})=\xi_{\sigma_{1p}}^{\dagger}\xi_{\sigma_{2p}}^{\dagger}\,T^{(2)}({\bfk}(-){\bfp},\,{\bfp})\,\xi_{\nu_{1p}}\xi_{\nu_{2p}}\quad,
\begin{array}{ll}{R_{00}(k,l)}&{=-4\pi{\frac{4}{3}}\alpha_{\mathrm{s}}[4{\frac{p^{2}l^{2}-(pl)^{2}}{(p-l)^{4}}}+{\frac{3}{4}}]+\nonumber}\\{}&{\qquad+\sigma\intd^{3}\vec{\zeta}e^{-i(\vec{k}-\vec{l})\cdot\vec{\zeta}}\vert{\vec{\zeta}}\vert\vert(k_{4}+l_{4})\vert\sqrt{(k_{4}+l_{4})^{2}+(\vec{k}_{\mathrm{T}}+\vec{l}_{\mathrm{T}})^{2}}}\\\end{array}\,.
\overline{{\theta}}\Gamma^{N}\partial_{\nu}\theta=(\theta_{R}+\theta_{L})^{T}\Gamma^{0}\Gamma^{N}\partial_{\nu}(\theta_{R}+\theta_{L})=\theta_{R}^{T}\Gamma^{0}\Gamma^{N}\partial_{\nu}\theta_{R}+\theta_{L}^{T}\Gamma^{0}\Gamma^{N}\partial_{\nu}\theta_{L}
N_{Q5}^{ca}(p)=-\frac{i}{2}d_{Q}(R)\int\frac{d^{4}l}{{(2\pi)}^{4}}\mathrm{tr}\left[i\gamma_{5}iS^{ca}(l,m_{Q})i\gamma_{5}iS^{ac}(l+p,m_{Q})\right],
\mu_{n}^{(N)}=-{\frac{1}{2}}\int_{-\infty}^{\infty}dx\;w^{2}(x)\left({\calP}_{n}^{\prime}\right)^{2}+{\frac{1}{2}}\sum_{m=0}^{N-1}K_{nm}\quad(0\leqn\leqN-1).
\big[T_{j}^{i},Q^{k}\big]=\delta_{j}^{k}Q^{i}-{\frac{1}{N}}\delta_{j}^{i}Q^{k}\;,\qquad\big[T_{j}^{i},\bar{Q}_{k}\big]=-\delta_{k}^{i}\bar{Q}_{j}+{\frac{1}{N}}\delta_{j}^{i}\bar{Q}_{k}\;,
\|\psi\|^{2}{}~=~-{\frac{1}{2\piG}}{\bar{\calC}}^{(\bar{k})}{\calC}^{(\bar{k})}\int_{S_{\infty}^{2}}dS^{\underline{{i}}}\,V^{-1}\partial_{\underline{{i}}}V{}~=~4M_{ADM}\|c\|^{2}\,,
Z_{4}\propto\int_{-\infty}^{+\infty}\!dx\,e^{t_{0}x^{2}+t_{1}x^{3}+x^{4}-\kappa(t_{0}^{2}x^{4}+2t_{0}t_{1}x^{5}+(t_{1}^{2}+2t_{0})x^{6}+2t_{1}x^{7}+x^{8})}\,,\quad\mathrm{etc\ldots}
\hat{\Omega}=\hat{d}\,\hat{\bf\omega}^{n}=\S_{i_{1}<\cdots<i_{n}}\hat{d}\,T_{i_{1}\cdotsi_{n}}d\,\Phi^{\widehat{\imath_{1}}}\wedge\cdots\wedged\,\Phi^{\widehat{\imath_{n}}}.
\omega_{l}=\omega^{(>)}(\Theta_{l})=\frac{2i}{\pi}\,\frac{\widetilde{A}_{l}^{(+)}-\widetilde{A}_{l}^{(-)}}{(1+iQ_{l})\,\widetilde{A}_{l}^{(+)}+(1-iQ_{l})\,\widetilde{A}_{l}^{(-)}}\;,
\Psi(x)=\int[d{\bfk}]\sum_{s}[b_{s}({\bfk})\psi_{{\bfk}s}^{+}(\tau)e^{ik\eta}e^{i{\bf{p}}\cdot{\bfx}}+d_{s}^{\dagger}({\bf{-k}})\psi_{{\bf{-k}}s}^{-}(\tau)e^{-ik\eta}e^{-i{\bf{p}}\cdot{\bfx}}].
A(q,\hat{q})=\left(\begin{array}{ll}{q\hat{q}+q^{-1}\hat{q}^{-1}}&{-1}\\{-(1+\hat{q}^{2}+\hat{q}^{-2})}&{q\hat{q}^{3}+q^{-1}\hat{q}^{-3}}\\\end{array}\right)
g_{ij}={\calZ}_{,i}^{\mu}{\calZ}_{,j}^{\nu}{\calG}_{\mu\nu},\;\;{\calZ}_{,i}^{\mu}{\eta}^{\nu}{\calG}_{\mu\nu}=0,\;\;{\eta}^{\mu}{\eta}^{\nu}{\calG}_{\mu\nu}=-1,\;\;k_{ij}=-{\calZ}^{\mu}{}_{,i}\eta^{\nu}{}_{;j}{\calG}_{\mu\nu}
e^{-i\beta(\hat{C}_{+}^{\dagger}-\hat{C}_{-}^{\dagger})}e^{i\beta(\hat{C}_{+}-\hat{C}_{-})}=\hat{C}_{+}^{\dagger}\hat{C}_{+}+\hat{C}_{-}^{\dagger}\hat{C}_{-}+e^{-2i\beta}\hat{C}_{+}^{\dagger}\hat{C}_{-}+e^{2i\beta}\hat{C}_{-}^{\dagger}\hat{C}_{+}.
A(p)\delta_{kk^{\prime}}=ie^{2}(2|eH|)\sum_{k^{\prime\prime}}\sum_{\{\sigma\}}\delta_{nn^{\prime\prime}}\delta_{\tilde{n}^{\prime\prime}n^{\prime}}\int\frac{d^{4}\hat{q}}{(2\pi)^{4}}\frac{\mathrm{e}^{-\hat{q}_{\perp}^{2}}}{\hat{q}^{2}}\frac{G_{1}A(p^{\prime\prime})}{\bar{p}^{2}+m^{2}}.
M_{K}=E_{K}-E_{V}=M_{K}^{cl}+\frac{\hbar}{2}\underset{bound}{\sum}\omega_{b}^{K}+\frac{\hbar}{2}\left(\underset{-M_{-}}{\overset{M_{+}}{\sum}}\omega_{n}^{K}-\underset{-N_{-}}{\overset{N_{+}}{\sum}}\omega_{n}^{V}\right)+\deltaM(\deltam)+O(\hbar^{2}).
q^{-l}\prod_{{\stackrel{r=1}{r\not=s}}}^{l}\frac{1}{a\left(\frac{\lambda_{r}}{\lambda_{s}}\right)}\rho_{N}(\lambda_{s})=q^{l}\prod_{{\stackrel{r=1}{r\not=s}}}^{l}\frac{1}{a\left(\frac{\lambda_{s}}{\lambda_{r}}\right)}\sigma_{N}(\lambda_{s})\,.
R=-\frac{2}{g^{2}}\left(\frac{f^{\prime\prime}}{f}-\frac{f^{\prime}}{f}\frac{g^{\prime}}{g}\right)-\frac{4}{g^{2}}\left(\frac{h^{\prime\prime}}{h}-\frac{h^{\prime}}{h}\frac{g^{\prime}}{g}\right)-\frac{4}{g^{2}}\frac{f^{\prime}}{f}\frac{h^{\prime}}{h}+\frac{2}{h^{2}}\left(1-\left(\frac{h^{\prime}}{g}\right)^{2}\right)
T_{\mu\nu}=-{\sqrt{2}}\tau_{p}e^{-{\frac{T^{2}}{2\alpha^{\prime}}}}\left[\frac{\sqrt{\pi}}{\sqrt{(\partialT)^{2}}}\partial_{\mu}T\partial_{\nu}T\mathrm{erf}\left(\sqrt{(\partialT)^{2}}\right)-\eta_{\mu\nu}\left(e^{-(\partialT)^{2}}+\sqrt{\pi(\partialT)^{2}}\mathrm{erf}\left(\sqrt{(\partialT)^{2}}\right)\right)\right]\.
(f\starg)(\tilde{\bfr})=\frac{1}{(2\pi)^{2}}\sum_{P^{\prime},P^{\prime\prime}}\int_{-\pi}^{+\pi}d\theta^{\prime}d\theta^{\prime\prime}e^{[-2iA(\tilde{\bfr},\tilde{\bfr}^{\prime},\tilde{\bfr}^{\prime\prime})]}f(\tilde{\bfr}^{\prime})g(\tilde{\bfr^{\prime\prime}})
\begin{array}{lll}{\phi_{qI}\rightarrow\mathrm{e}^{i\beta}\phi_{qI},}&{q_{I}\rightarrowq_{I},}&{F_{qI}\rightarrow\mathrm{e}^{-i\beta}F_{qI},}\\{\phi_{\overline{q}}^{I}\rightarrow\mathrm{e}^{i\beta}\phi_{\overline{q}}^{I},}&{{\overline{q}}^{I}\rightarrow{\overline{q}}^{I},}&{F_{\overline{q}}^{I}\rightarrow\mathrm{e}^{-i\beta}F_{\overline{q}}^{I}.}\\\end{array}
P_{Y_{(2r+2)}}^{j_{1}j_{2}...j_{2r}j_{2r+1}j_{2r+2},i_{1}i_{2}...i_{2r}i_{2r+1}i_{2r+2}}P_{Y_{(2,2)}}^{i_{1}i_{2}i_{2r+1}i_{2r+2},j_{1}j_{2}j_{2r+1}j_{2r+2}}\prod_{a=2}^{r}P_{Y_{(2)}}^{i_{2a-1}i_{2a},j_{2a-1}j_{2a}}=0
\delta_{(1)}^{c}{\calL}_{APEGT}=\intd^{4}x{\frac{C_{2}\sigma^{2}g^{2}}{8\pi^{2}}}*B_{\mu\nu}^{i}(x)I^{\mu\nu,\alpha\beta}\left[f_{0}(\alpha)-f_{1}(\alpha){\frac{\partial_{\lambda}\partial^{\lambda}}{M_{A}^{2}}}+f_{2}(\alpha){\frac{(\partial_{\lambda}\partial^{\lambda})^{2}}{M_{A}^{4}}}\right]*B_{\alpha\beta}^{i}(x).
\frac{\partial^{2}\Phi}{\partial\nu^{2}}+\frac{6\left(-1+e^{2\nu}\right)\left(1+e^{4\nu}\right)}{\left(-1+4e^{2\nu}-e^{4\nu}\right)\left(1+e^{2\nu}\right)}\frac{\partial\Phi}{\partial\nu}+\left(-n^{2}+5-\frac{4(1+e^{2\nu})^{2}}{-1+4e^{2\nu}-e^{4\nu}}\right)\Phi=0.
(s,t)\rightarrow(\sum_{j=0}^{k}a_{j}^{1}s^{j}t^{k-j},\sum_{j=0}^{k}a_{j}^{2}s^{j}t^{k-j},\ldots,\sum_{j=0}^{k}a_{j}^{N}s^{j}t^{k-j})
\begin{array}{ll}{a_{k}^{\dagger}=}&{\frac{1}{\sqrt{2}}\left(\frac{x_{k}}{m_{k}}-i{p_{k}}\right),}\\{a_{k}=}&{\frac{1}{\sqrt{2}}\left(\frac{x_{k}}{m_{k}}+i{p_{k}}\right),}\\{U_{k}=}&{\frac{1}{2}\left({p_{k}}^{2}+\frac{x_{k}^{2}}{m_{k}^{2}}\right)=\frac{1}{2}\left\{a_{k},a_{k}^{\dagger}\right\},\quadH=\sum_{k=1}^{N}U_{k}.}\\\end{array}
S_{f}=\frac{1}{4\pi\alpha^{\prime}}\intd\tau\int_{0}^{2\pi|\alpha|}d\sigma\left\{ie(\alpha)\left[{\bar{\theta}}\partial_{\tau}\theta+\theta\partial_{\tau}{\bar{\theta}}+\theta\partial_{\sigma}\theta+{\bar{\theta}}\partial_{\sigma}{\bar{\theta}}\right]-2\mu{\bar{\theta}}\Pi\theta\right\}~.
\left(\begin{array}{ccc}{{f_{n}}^{(\pm)}(r,E)}\\{{g_{n}}^{(\pm)}(r,E)}\\\end{array}\right)=\left(\begin{array}{ccc}{f^{(0)}(E)J_{\pms\lambda}(kr)}\\{\pmg^{(0)}(E)J_{\pm(s\lambda+1)}(kr)}\\\end{array}\right),
\psi_{2\eta}(-\epsilon)\psi_{1\xi}^{\dagger}(\epsilon)=-ie^{\frac{i}{2}[-\theta(\eta)-\theta(\xi)-\frac{1}{2}R_{2}(\epsilon,\xi,\eta)]}e^{-i\pi\intd^{2}y\frac{1}{e}\partial_{i}E_{i}(y)}e^{iF_{2}(A,E)}
H_{0}=\frac{1}{l_{0}}\sum_{i}\mathrm{Tr}(\dot{\bfx}^{i}\dot{\bfx}^{i})+\frac{1}{l_{0}^{5}}\sum_{i,j}\mathrm{Tr}([{\bfx}^{i},{\bfx}^{j}]^{\dagger}[{\bfx}^{i},{\bfx}^{j}])\.