harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{lcl}{\tilde{ds}^{2}=-\frac{dt^{2}}{\left(-k+\left(\frac{t_{+}}{t}\right)^{2}\right)\left(1-\left(\frac{t_{-}}{t}\right)^{2}\right)}+t^{2}d\Omega_{k}^{2}}&{,}&{e^{-2\phi}\sim\sqrt{\left(1-\left(\frac{t_{-}}{t}\right)^{2}\right)\left(-k+\left(\frac{t_{+}}{t}\right)^{2}\right)}\.}\\\end{array}
\dot{x}^{AB^{\prime}}=o^{A}\bar{o}^{B^{\prime}}-\frac{\rho^{\sigma}}{\rho^{\tau}}\acute{x}^{AB^{\prime}},\,\,\,\,\acute{x}^{AB^{\prime}}=|q|(o^{A}\bar{\iota}^{B^{\prime}}+\iota^{A}\bar{o}^{B^{\prime}}).
\left[\!\!\begin{array}{c}{m}\\{n}\\\end{array}\!\!\right]\;=\;\left(\!\!\begin{array}{c}{m}\\{2n}\\\end{array}\!\!\right)\:(2n-1)!!\;=\;\frac{m!}{(m-2n)!}\:\frac{1}{2^{n}\:n!}\;\;\;.
J_{m}(\phi)=\sum_{n=1}^{\infty}j_{mm_{2}\ldotsm_{n}}^{(n)}\phi_{m_{2}}\ldots\phi_{m_{n}},\;\;\;\;\;\;\tilde{J}_{m}(\tilde{\phi})=\sum_{n=1}^{\infty}j_{mm_{2}\ldotsm_{n}}^{(n)}\tilde{\phi}_{m_{n}}\ldots\tilde{\phi}_{m_{2}}
V_{abc_{2}d_{2}}^{(3)}=\frac{1}{2}\rho^{3}v_{abc_{2}d_{2}}^{(3)}(\frac{1}{4}((\Sigma_{ren.}^{1})^{2}+2\Sigma_{ren.}^{1}D_{1}+D_{1}^{2})-\frac{1}{2}D_{1}\Sigma_{ren.}^{1}-\frac{1}{2}D_{1}^{2}).
S=\int_{\Sigma\timesR}dtd^{2}x\bigl[E^{ai}\dot{A}_{a}^{i}+\Pi_{n}\dot{\phi}_{n}+A_{0}^{i}\partial_{a}E^{ai}-N^{a}(E^{bi}F_{ab}^{i}+\Pi_{n}\partial_{a}\phi_{n})\bigl],
\begin{array}{l}{{n^{\prime}}^{m}=\displaystyle\frac{1}{c}\big[A^{2}\pi^{m}-\big((A\pi)+\sqrt{(A\pi)^{2}-A^{2}\pi^{2}}\big)A^{m}\big],}\\{{r^{\prime}}^{m}=\displaystyle\frac{1}{c}\big[A^{2}\pi^{m}-\big((A\pi)-\sqrt{(A\pi)^{2}-A^{2}\pi^{2}}\big)A^{m}\big],}\\\end{array}
\left[{\frac{\kappa}{4}}{\frac{\partial^{2}}{\partial\chi_{0}^{2}}}-{\frac{\kappa}{4}}{\frac{\partial^{2}}{\partial\Omega_{0}^{2}}}-{\frac{1}{2}}\sum_{i}{\frac{\partial^{2}}{\partialf_{i0}^{2}}}-4\lambda^{2}e^{{\frac{2}{\kappa}}(\chi_{0}-\Omega_{0})}-\kappa-2\right]\Psi=0\\.
\ddot{x}^{i}+\Gamma_{jk}^{i}\,\dot{x}^{j}\,\dot{x}^{k}=-g^{ij}\,{\frac{\partialg_{j}^{\l}}{\partialt}}\,\dot{x}^{l}+g^{ij}\,F_{jl}\,\dot{x}^{l}+F^{i}\left[g_{ij},g_{0i},g_{00}\right],
S[\vec{n}]=-\frac{1}{4}\intd^{4}x(\vec{\Gamma}_{\mu}\wedge\vec{\Gamma}_{\nu})(\vec{\Gamma}^{\mu}\wedge\vec{\Gamma}^{\nu})=-\frac{1}{4}\intd^{4}x(\partial_{\mu}\vec{n}\wedge\partial_{\nu}\vec{n})(\partial^{\mu}\vec{n}\wedge\partial^{\nu}\vec{n}).
x^{2}=\epsilon\left[\alpha-{\frac{B_{4}(Q^{2},Q^{2})}{B_{1}(Q^{2},Q^{2})}}\alphag^{2}-{\frac{B_{3}(Q^{2},Q^{2})}{B_{1}(Q^{2},Q^{2})}}\alpha^{2}+...\right]
T_{+p-\dot{p}}^{-\dot{q}}=\frac{\chi}{2}\tilde{\gamma}_{\dot{q}p}^{i}\Omega_{-\dot{p}}^{-2i},\T_{+q+p}^{-\dot{q}}=\frac{\chi}{2}\left(\tilde{\gamma}_{\dot{q}q}^{i}\Omega_{+p}^{-2i}+q\leftrightarrowp\right).
V_{G}(\phi_{B},\omega)=\frac{1}{2}r_{0}\phi_{B}^{2}+\frac{1}{4}\lambda_{0}\phi_{B}^{4}+I(x,x)+\frac{1}{2}\left(r_{0}-\omega^{2}+3\lambda_{0}\phi_{B}^{2}\right)G(x,x)+\frac{3\lambda_{0}}{4}G(x,x)G(x,x)
\frac{\partialU^{-1}}{\partialz_{\alpha}}+\frac{1}{2\pi\kappa}\sum_{\beta\neq\alpha}\frac{\hat{Q}^{a}(\xi_{\alpha},\bar{\xi}_{\alpha})\hat{Q}^{a}(\xi_{\beta},\bar{\xi}_{\beta})}{z_{\alpha}-z_{\beta}}U^{-1}=0
(2\piS\,T)^{2}=r_{0}^{2}\equiv{\frac{(M^{2}-|z_{1}|^{2})(M^{2}-|z_{2}|^{2})(M^{2}-|z_{3}|^{2})(M^{2}-|z_{4}|^{2})}{M^{6}}}\geq0\.
S(\nu,k)-1=\frac{\displaystyle{\sum_{n=0}^{N}g^{n}M_{n}^{(+)}(k)+g^{N}R_{N}^{(+)}(\nu,k)}}{\displaystyle{\sum_{n=0}^{N}}g^{n}M_{n}^{(-)}(k)+g^{N}R_{N}^{(-)}(\nu,k)}-1,
\chi_{\Lambda^{\prime}}^{\mathrm{new}}=C\chi_{\Lambda^{\prime}},~\psi_{\Lambda^{\prime}}^{\mathrm{new}}=C\psi_{\Lambda^{\prime}},~K^{\mathrm{new}}=\frac{K}{C^{2}},~\mu^{\mathrm{new}}=\frac{\mu}{C^{2}},~\xi^{\mathrm{new}}=A(0)B(0)\xi,
G(x,y;A,h)=\frac{1}{2\pi}\frac{1}{(x-y)^{2}}\left(\begin{array}{cc}{0}&{e^{-\big(\chi,\delta_{n}(x)-\delta_{n}(y)\big)}\;\overline{{(\tilde{x}-\tilde{y})}}}\\{e^{+\big(\chi,\delta_{n}(x)-\delta_{n}(y)\big)}\;(\tilde{x}-\tilde{y})}&{0}\\\end{array}\right)\;.
M=\left(\begin{matrix}{1}&{-1}&{0}&{0}&{\dots}\\{-1}&{2}&{-1}&{0}&{\dots}\\{0}&{-1}&{2}&{-1}&{\dots}\\{\vdots}&{\ddots}&{\ddots}&{\ddots}&{\vdots}\\{0}&{\dots}&{0}&{-1}&{1}\\\end{matrix}\right)
X=\left(\begin{matrix}{X_{1}}\\{X_{2}}\\\end{matrix}\right)\;,\;A=-\frac{q_{0}^{-1}}{6H}\left(\begin{matrix}{{(\frac{\partialV}{\partial{T_{1}}})}_{0}}\\{{(\frac{\partialV}{\partial{T_{2}}})}_{0}}\\\end{matrix}\right),
[a_{m},\tilde{a}_{n}]=[a_{m}^{\dagger},\tilde{a}_{n}^{\dagger}]=0,\;\;\;\;\;\;[\psi_{\dot{\alpha}},\tilde{\psi}_{\dot{\beta}}]_{+}=[\psi_{\dot{\alpha}}^{\dagger},\tilde{\psi}_{\dot{\beta}}^{\dagger}]_{+}=0
{\calZ}=\int\prod_{i=1}^{3}[\mathrm{d}{\phi}^{i}(\tau,x)]\prod_{i=1}^{3}[\mathrm{d}{\Pi}^{i}(\tau,x)]\delta\left(|\vec{\phi}|-1\right)e^{-S_{e}[\vec{\phi},\vec{\Pi}]}.
P_{k}=\frac{4\zeta}{y_{1}^{2}+y_{2}^{2}}\cdot(y_{1}{\partial}_{1}b_{2k-2}-y_{2}{\partial}_{1}a_{2k-2})+\frac{8(k-1)\zeta}{(y_{1}^{2}+y_{2}^{2})^{2}}\cdot[(y_{1}^{2}-y_{2}^{2})\cdotb_{2k-2}-2y_{1}y_{2}\cdota_{2k-2}],
b\left(\vec{u}\right)=2\pi\sum_{k=1}^{K}L_{k}\delta^{(2)}\left(\vec{u}-\vec{u}_{k}\right)\\,\\\vec{j}\left(\vec{u}\right)=-\vec{\partial}\times\vec{b}\left(\vec{u}\right)\\,\\\psi\left(\vec{u}\right)=f\left(\vec{u}\right)e^{i\theta_{0}(\vec{u})},
\phi^{\prime2}-2A_{1}P_{2}\zeta^{\prime}+A_{0}^{2}e^{2\phi}+(Q_{0}/A_{0})^{2}e^{2P_{2}\tau-2\phi}=P_{2}^{2}+2P_{1}P_{2}-\frac{1}{4}(A_{1}A_{2})^{2}
Z(R(S)^{C},\langle\cdot\,,\cdot\rangle_{H}^{C},\langle\cdot\,,\cdot\rangle)=\biggl(\prod_{k=0}^{N}|det(C_{k})|^{(-1)^{k+1}}\biggr)\,Z(R(S),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle)\,.
I_{N}=-\frac{32}{3\pi}\left\lbrace-\frac{3}{4}\left\lbrack-\frac{e^{\frac{z}{\lambda}}}{2z^{2}}-\frac{e^{\frac{z}{\lambda}}}{2\lambdaz}+\frac{1}{2\lambda^{2}}E_{i}\left(\frac{z}{\lambda}\right)\right\rbrack-e^{\frac{z}{\lambda}}\left(z\lambda-\lambda^{2}\right)+2E_{i}\left(\frac{z}{\lambda}\right)\right\rbrace_{z_{1}}^{z_{2}}
\begin{aligned}{\delta_{\xi}Y}&{{}=\sqrt{2}\xi\eta}\\{\delta_{\xi}\eta}&{{}=i\sqrt{2}\sigma^{m}\bar{\xi}\partial_{m}Y+\sqrt{\xi}F}\\{\delta_{\xi}F}&{{}=i\sqrt{2}\bar{\xi}\bar{\sigma}^{m}\partial_{m}\eta}\\\end{aligned}
L_{\mathrm{S}}=\mathrm{}-\frac{1}{2}\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\frac{m^{2}}{2}A_{\mu}A^{\mu}-\frac{1}{2}\partial_{\mu}B\partial^{\mu}B-\frac{m^{2}}{2}B^{2},
{\calO}_{1}{}^{i}{}_{j}\!\!\equiv\!\!\left[\begin{matrix}{-{\frac{\partial^{2}{\calW}^{*}}{\partialA_{\mathrm{cl}}^{*i}\partialA_{\mathrm{cl}}^{*j}}}}&{-i\left(-\partial_{1}+i\partial_{2}\right)\delta_{j}^{i}}\\{-i\left(\partial_{1}+i\partial_{2}\right)\delta_{j}^{i}}&{-{\frac{\partial^{2}{\calW}}{\partialA_{\mathrm{cl}}^{i}\partialA_{\mathrm{cl}}^{j}}}}\\\end{matrix}\right]
\eta=\frac{x_{12}^{2}+\left(y_{1}-y_{2}\right)^{2}}{x_{12}^{2}+\left(y_{1}+y_{2}\right)^{2}}\qquad1-\eta=\frac{4y_{1}y_{2}}{x_{12}^{2}+\left(y_{1}+y_{2}\right)^{2}}\,,
t_{k}(u)=Q\left(\textstyleu-\frac{(k+1)\pi}{2(p+1)}\right)P\left(\textstyleu+\frac{(k+1)\pi}{2(p+1)}\right)-Q\left(\textstyleu+\frac{(k+1)\pi}{2(p+1)}\right)P\left(\textstyleu-\frac{(k+1)\pi}{2(p+1)}\right)
\begin{array}{rcl}{\frac{df}{dt}}&{=}&{\frac{\partialf}{\partialt}+\{f,H_{0}\}+{\calU}^{m}\{f,\phi_{m}\}+\{f,{\calU}^{m}\}\phi_{m}}\\{}&{}&{}\\{}&{\approx}&{\frac{\partialf}{\partialt}+\{f,H_{0}\}+{\calU}^{m}\{f,\phi_{m}\}\,}\\\end{array}
\begin{array}{ll}{\kappa_{+}=(f_{x}+ig_{x})+i(f_{y}+ig_{y}),}&{\qquad\kappa_{-}=(f_{x}+ig_{x})-i(f_{y}+ig_{y}),}\\{\lambda_{+}=(f_{x}-ig_{x})+i(f_{y}-ig_{y}),}&{\qquad\lambda_{-}=(f_{x}-ig_{x})-i(f_{y}-ig_{y}).}\\\end{array}
{\frac{\ddot{a}_{0}}{a_{0}}}=-{\frac{16\pi^{2}G_{5}^{2}\sigma}{9c^{4}\sqrt{I}}}\left(\varrho+3I{\frac{\wp}{c^{2}}}\right)-{\frac{32\pi^{2}G_{5}^{2}\sigma}{9c^{6}}}\left({\frac{\dot{\Phi}^{2}}{c^{2}}}-V\right)+{\frac{4\piG_{5}}{3c^{2}}}\left(\Lambda+{\frac{4\piG_{5}}{3c^{4}}}\sigma^{2}\right)-{\frac{{\calC}c^{2}}{a_{0}^{4}}}.
\partial_{\mu}\left(ee_{{f}}^{\mu}u^{{\{f\}}_{{m}}}\right)+ipa\epsilon^{\mu_{{p}}\ldots\mu_{{1}}\mu_{{0}}}\omega_{\mu_{{p}}}^{{m}_{{p}}}\ldots\omega_{\mu_{{2}}}^{{m}_{{2}}}\partial_{\mu_{{1}}}\theta\Gamma_{{mm}_{{2}}\ldots{m}_{{p}}}C^{-{1}}\partial_{\mu_{{0}}}\theta=0
\mu^{2}\psi(x)=\frac{m^{2}}{x(1-x)}\psi(x)+\int_{0}^{1}dx\psi(x)+\int_{0}^{1}dy\frac{(\psi(x)-\psi(y))}{(x-y)^{2}}S(\{\psi_{n}\},\{\mu_{n}\},m,x,y),
{\mit\Pi}_{\mu\nu}^{\mathrm{\tiny~DREG}}(q)=\frac{\alpha_{s}}{2\pi}\left(\frac{4\pi\mu^{2}}{-q^{2}}\right)^{\varepsilon}\frac{{\mit\Gamma}(\varepsilon){\mit\Gamma}(1-\varepsilon){\mit\Gamma}(2-\varepsilon)}{{\mit\Gamma}(4-2\varepsilon)}\left(q^{2}g_{\mu\nu}^{(d)}-q_{\mu}q_{\nu}\right)\Big[N_{c}(5-3\varepsilon)-2n_{f}(1-\varepsilon)-n_{s}/2\Big],
I_{0,1}=\frac{\mu^{\epsilon}}{(2\pi)^{3-\epsilon}}\int\frac{d^{3-\epsilon}\vec{q}}{2|\vec{q}|}\left(\frac{1}{k^{2}+2q\cdotk}+\frac{1}{k^{2}-2q\cdotk}\right)\left[\frac{\vec{q}^{2}}{k^{2}}\left(1-x^{2}\right)\right]^{0,1}\left[\frac12+N\left(\frac{|\vec{q}|}{T}\right)\right].
D=\left(\begin{array}{cc}{\gamma^{a}e_{1a}^{\mu}\partial_{\mu}}&{-m\gamma^{a}e_{1a}^{\mu}a_{1\mu}+m\gamma^{5}\phi_{1}}\\{\gamma^{a}e_{2a}^{\mu}a_{2\mu}-m\gamma^{5}\phi_{2}}&{\gamma^{a}e_{2a}^{\mu}\partial_{\mu}}\\\end{array}\right)
\begin{array}{rcl}{T_{+}}&{=}&{-{\frac{\sqrt3}{2}}(X_{(11)}+X_{(-1-1)})+{\frac{1}{2}}(X_{(31)}+X_{(-3-1)})\,,}\\{T_{-}}&{=}&{{\frac{\sqrt3}{2}}(X_{(21)}+X_{(-2-1)})-{\frac{1}{2}}(X_{(01)}+X_{(0-1)})\,,}\\{T_{3}}&{=}&{-{\frac{\sqrt{3}}{2}}(X_{(10)}-X_{(-10)})+{\frac{1}{2}}(X_{(32)}-X_{(-3-2)})\,,}\\\end{array}
\psi^{1}(z;x)=\left(\begin{array}{cr}{0}\\{e^{-\phi(z)}G_{+}(z-\omega)}\\\end{array}\right),~~~~\psi^{2}(z;x)=\left(\begin{array}{cr}{e^{\phi(z)}G_{-}(z+\omega)}\\{0}\\\end{array}\right),
Z=\left(\begin{array}{c}{Z_{1}}\\{Z_{2}}\\{Z_{3}}\\\end{array}\right)=\left(\begin{array}{ccc}{e^{i\omega_{1}t}}&{0}&{0}\\{0}&{e^{i\omega_{2}t}}&{0}\\{0}&{0}&{e^{i\omega_{3}t}}\\\end{array}\right)\left(\begin{array}{c}{R_{1}e_{1}(\theta,\phi)}\\{R_{2}\;e_{2}(\theta,\phi)}\\{R_{3}\;e_{3}(\theta,\phi)}\\\end{array}\right)
D(\hat{k}_{\theta}^{U}(M))=\{\xi(z)+c\sum_{i=1}^{2}(\zeta_{-}^{i}(z)+\sum_{j=1}^{2}U_{ij}\zeta_{+}^{j}(z))|~\xi(z)\inD(\hat{k}_{\theta}(M)),c\in{\bfC}\}.
V(\Phi_{s},u)=e^{-\Phi_{s}}\left(\sum_{a=1}^{K}N_{a}\frac{\prod_{i=1}^{3}L_{a}^{i}}{R_{A}R_{B}R_{1}}-16\sqrt{2}(u+\frac{1}{4^{b}}\frac{1}{u})\right).
\begin{array}{rclcrcl}{{\calW}_{1}}&{\leftrightarrow}&{\left[dJ\right]^{(3,0)}\,,}&{\quad}&{{\calW}_{2}}&{\leftrightarrow}&{\left[d\Psi\right]_{0}^{(2,2)}\,,}\\{{\calW}_{3}}&{\leftrightarrow}&{\left[dJ\right]_{0}^{(2,1)}\,,}&{\quad}&{{\calW}_{4}}&{\leftrightarrow}&{J\wedgedJ\,,}\\{{\calW}_{5}}&{\leftrightarrow}&{\left[d\Psi\right]^{(3,1)}\,.}&{}&{}&{}&{}\\\end{array}
d\Theta^{2\underline{\mu}}=v_{q}^{\underline{\mu}-}\pi_{q}^{2+}+v_{\underline{\mu}\dot{q}}^{-}\pi_{\dot{q}}^{2+},=e^{-2}\psi_{-2q}^{2+}v_{q}^{\underline{\mu}-}+\chie_{q}^{+}v_{q}^{\underline{\mu}-}+e_{\dot{q}}^{-}v_{\dot{q}}^{\underline{\mu}+},
D_{\mu}\psi_{L}\equiv\partial_{\mu}\psi_{L}+igA_{\mu}\psi_{L}\quad\mathrm{and}\quadD_{\mu}^{\prime}\psi_{R}^{\prime}\equiv\partial_{\mu}\psi_{R}^{\prime}+ig^{\prime}A_{\mu}^{\prime}\psi_{R}^{\prime}.
\psi^{(j+n+1)}(x,t)=\frac{W\left(\phi_{0}^{(j)},\dots,\phi_{n}^{(j)},\psi^{(j)}\right)}{W\left(\phi_{0}^{(j)},\dots,\phi_{n}^{(j)}\right)}=(\partial_{x}+f_{j+n})\cdots(\partial_{x}+f_{j})\psi^{(j)}(x,t).
V(a)=-\frac{\kappa_{(4)}^{2}}{6}\rho_{X0}a-\frac{\kappa_{(4)}^{2}}{12\lambda}\rho_{X0}^{2}-\frac{\sigma_{0}^{2}}{6}a^{-4}-\frac{\Lambda_{(4)}}{6}a^{2}-\frac{{\calU}_{0}}{\lambda\kappa_{(4)}^{2}}a^{-2}+\frac{k}{2}.
\zeta\left(\begin{array}{rl}{3,}&{1}\\{\lambda^{3},}&{\lambda^{2}}\\\end{array}\right)=\int_{0}^{1}\frac{dz_{1}}{z_{1}}\int_{0}^{z_{1}}\frac{dz_{2}}{z_{2}}\int_{0}^{z_{2}}\frac{dz_{3}}{\lambda^{-3}-z_{3}}\int_{0}^{z_{3}}\frac{dz_{4}}{\lambda^{-5}-z_{4}}:=\zeta(\Omega^{2}\omega_{3}\omega_{5})
f(r)=f_{1}r^{2}+\frac{1}{8}f_{1}r^{4}+\ldots,\;\;a(r)=\frac{1}{8}r^{2}-\frac{1}{24}f_{1}^{2}r^{6}+\ldots,\;\;k(r)=r^{-2}+k_{1}r^{2}+\ldots,
\frac{\deltaS^{n}}{\delta\varphi_{n}^{\alpha}\left(\mathbf{x}\right)}=-\pi_{n}^{\alpha}\left(\mathbf{x}\right),\;\;\;\;\;\frac{\deltaS^{n}}{\delta\varphi_{n+1}^{\alpha}\left(\mathbf{x}\right)}=\pi_{n+1}^{\alpha}\left(\mathbf{x}\right)
\tilde{e}=\left(\begin{matrix}{\frac{1+\Omega}{2}}&{\frac{1-\Omega}{2}}\\{\frac{1-\Omega}{2}}&{\frac{1+\Omega}{2}}\\\end{matrix}\right)\cdot\left(\begin{matrix}{(1+v\Phiv^{t})(v^{-1})^{t}}&{}\\{}&{(1-v\Phiv^{t})^{-1}v}\\\end{matrix}\right)\cdot\left(\begin{matrix}{1}&{\Theta-(g^{-1}-\Phig\Phi)^{-1}\Phig}\\{}&{1}\\\end{matrix}\right)
{\hat{J}}_{(2)}={\frac{1}{\sqrt{2}}}({\bar{\beta}}_{A}{\bar{\beta}}^{B}-{\bar{\alpha}}_{A}{\bar{\alpha}}^{B}){\bar{\pi}}_{B}{\frac{\partial}{\partial{\bar{\pi}}_{A}}}-{\frac{1}{\sqrt{2}}}(\beta_{A^{\prime}}\beta^{B^{\prime}}-\alpha_{A^{\prime}}\alpha^{B^{\prime}})\pi_{B^{\prime}}{\frac{\partial}{\partial\pi_{A^{\prime}}}},
Z_{F}^{odd}(q,\alpha)=q^{-{\frac{1}{48}}}\prod_{m=1}^{\infty}(1-q^{m/2})^{-1}\sum_{n=-\infty}^{{\frac{1}{4}}[n+{\frac{1}{2}}+2\alpha]^{2}}\left[{\frac{1+(-1)^{n}}{2}}\right].
\epsilon\equiv\frac{M_{P}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2}\sim\frac{M_{P}^{2}}{f^{2}}\;,\qquad\eta\equivM_{P}^{2}\frac{V^{\prime\prime}}{V}\sim\frac{M_{P}^{2}}{f^{2}}
F^{\alpha_{1}...\alpha_{q+1}}=\sqrt{\frac{2}{q}}H_{1}^{-\frac{2}{D-2}}H_{2}^{-\frac{2(D-q-2)}{q(D-2)}}\epsilon^{\alpha_{1}...\alpha_{q+1}\beta}\partial_{\beta}H_{1}^{-1}.
S_{IIA}=\frac{1}{2\pi\alpha^{\prime}}\intd^{2}\sigma\left(-\frac{1}{2}\sqrt{-h}h^{mn}\Pi_{m}^{r}\Pi_{n}^{s}\eta_{rs}+\frac{1}{2}\epsilon^{mn}\Pi_{m}^{A}\Pi_{n}^{B}B_{BA}\right)~,
\mathcalL_{\mathrm{mass}}=\frac{1}{2}\left(\psi_{\muL}^{1}\\psi_{\muL}^{2}\right)\sigma^{\mu\nu}\left(\begin{array}{cc}{\omega/R}&{i\,p_{5}}\\{-i\,p_{5}}&{\omega/R}\\\end{array}\right)\left(\begin{array}{c}{\psi_{\nuL}^{1}}\\{\psi_{\nuL}^{2}}\\\end{array}\right)+h.c.
\left(\begin{matrix}{\tilde{\theta_{L}}}\\{\tilde{\phi}_{L}}\\{\tilde{\theta}_{R}}\\{\tilde{\phi}_{R}}\\\end{matrix}\right)~=~\left(\begin{matrix}{1/2}&{-1/2}&{1/2}&{-1/2}\\{-1/2}&{1/2}&{1/2}&{-1/2}\\{1/2}&{1/2}&{1/2}&{1/2}\\{-1/2}&{-1/2}&{1/2}&{1/2}\\\end{matrix}\right)~\left(\begin{matrix}{\theta_{L}}\\{\phi_{L}}\\{\theta_{R}}\\{\phi_{R}}\\\end{matrix}\right)
\begin{array}{c}{J^{(1)}=\int\frac{d^{4}k}{(2\pi)^{4}}\frac1{k^{2}-M^{2}+i\varepsilon}}\\{=-i{\int_{0}^{\infty}}d\alpha\int\frac{d^{4}k}{(2\pi)^{4}}e^{i\alpha(k^{2}-M^{2}+i\varepsilon)}}\\{=-\frac1{(4\pi)^{2}}{\int_{0}^{\infty}}\frac{d\alpha}{\alpha^{2}}e^{-i\alpha(M^{2}-i\varepsilon)}}\\\end{array}
\left\{\begin{array}{c}{lim}\\{\theta\rightarrow\theta_{0}^{+}}\\\end{array}\frac{\partial}{\partial\theta}ImI^{\prime}-\begin{array}{c}{lim}\\{\theta\rightarrow\theta_{0}^{-}}\\\end{array}\frac{\partial}{\partial\theta}ImI^{\prime}\right\}=0
\left.T_{g_{\mu\nu}(x)g_{\alpha\beta}(y)}^{g_{\rho\sigma}(z)}S_{,g_{\rho\sigma}(z)}\right\vert_{back}=-\frac{1}{2}\Lambda(\delta_{\mu\eta}\partial_{\nu}+\delta_{\nu\eta}\partial_{\mu}-\delta_{\mu\nu}\partial_{\eta})\frac{1}{\partial^{2}}(\delta_{\alpha\eta}\partial_{\beta}+\delta_{\beta\eta}\partial_{\alpha}-\delta_{\alpha\beta}\partial_{\eta})\delta^{4}(x-y)\.
\left<T_{\mu\nu}(x)\right>_{\mu^{2}}^{{\mathbbH}^{N}}={\frac{m^{2}a^{2-N}}{2^{N}\pi^{N/2-1}N\Gamma\left(\fracN2\right)}}\left(\sum_{n=1}^{\rho_{N}}(-1)^{n+1}c_{2n}^{N}(a^{2}b)^{n-\frac12}\right)g_{\mu\nu}^{{\mathbbH}^{N}}(x).
F(w)=C_{1}(e^{A(w)})^{\cdot}+C_{2}(e^{A(w)})^{\cdot}\int_{0}^{w}dw^{\prime}\frac{\ddot{A}(w^{\prime})+\dot{A}(w^{\prime})^{2}}{\dot{A}(w^{\prime})^{2}}e^{-3A(w^{\prime})}.
(\widetilde\nabla_{\alpha}V)^{a}=(e^{-{\calV}_{B}})^{ab}D_{\alpha}(e^{{\calV}_{B}})^{bc}V^{c},\qquad\{\widetilde\nabla_{\alpha},\overline{D}_{\dot{\alpha}}\}=-2i\sigma_{\alpha\dot{\alpha}}^{m}\widetilde\nabla_{m},
\langleH|S_{\mu}^{\\nu}|H\rangle\to\frac{7}{2}\left(\frac{11}{864M^{2}}\right)^{2}(\alpha^{2}-\beta^{2})\left(\begin{matrix}{-1}&{0}&{0}&{0}\\{0}&{1/3}&{0}&{0}\\{0}&{0}&{1/3}&{0}\\{0}&{0}&{0}&{1/3}\\\end{matrix}\right)
ds_{4}^{2}=(1+{\frac{4M}{r}})^{2}(dr^{2}+r^{2}d\theta^{2}+r^{2}sin^{2}\varphi^{2})-(1+{\frac{4M}{r}})^{-2}dt^{2}\,;\quade^{2\Phi}=e^{2\Phi_{0}}\,,
\intd\vec{x}\biggl(v^{k}\frac{\delta}{\deltaA^{0k}}+\frac{1}{2}\,u^{kl}\frac{\delta}{\deltaA^{kl}}+\frac{1}{2}\,\omega^{kl}\frac{\delta}{\delta\pi^{kl}}\biggr)\intd\vec{y}\,V^{(0)}=0
\frac{\partial\Sigma}{\partialg}=\frac{2\omega^{4}}{3g^{3}}\varepsilon^{\mu\nu\rho\sigma}{\calW}_{\mu}{\calW}_{\nu}{\calW}_{\rho}{\calW}_{\sigma}\intd^{4}x\mathrm{Tr}\frac{\phi^{2}}2\;+\;B_{\Sigma}\Xi^{-1}\;,
\mathcal{H}_{\mathrm{XXZ}}^{s=1/2}=\sum_{j=1}^{L}\left\{\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y}+\frac{q+q^{-1}}{2}\left(\sigma_{j}^{z}\sigma_{j+1}^{z}-1\right)\right\}\;,\quadL+1\equiv1
\nabla^{2}h=16\piG\left[T_{\mu}^{\mu}+\langleT_{\mu}^{\mu}\rangle_{\lambda=0}^{G}+\langleT_{\mu}^{\mu}\rangle_{\lambda=0}^{M}+\langleT_{\mu}^{\mu}\rangle_{a(\lambda)}+\langleT_{\mu}^{\mu}\rangle_{b(\lambda)}\right],
\Pi_{\mu\mu}^{\mathrm{thermal}}(k_{0},\vec{k})={\frac{4e^{2}}{(2\pi)^{4}i}}\intd^{3}p\int_{-i\infty+\epsilon}^{i\infty+\epsilon}dp_{0}\,n_{F}(p_{0})\left[{\frac{4m^{2}+2p^{2}+2p\cdotk}{(p^{2}+m^{2})((p+k)^{2}+m^{2})}}+(k_{0}\leftrightarrow-k_{0})\right]
w=w^{(1)}+iw^{(2)}={\frac{n_{1}+in_{2}}{1-n_{3}}}={\frac{2\phi_{1}\bar{\phi}_{2}}{(|\phi_{1}|^{2}+|\phi_{2}|^{2})-(|\phi_{1}|^{2}-|\phi_{2}|^{2})}}={\frac{\phi_{1}}{\phi_{2}}}.
k={\frac{u_{i}}{2}}\,\Big[\,{\frac{{\calE}+\sqrt{{\calE}^{2}+\mu_{i}^{2}}}{K+\sqrt{K^{2}+\mu_{i}^{2}}}}-{\frac{K+\sqrt{K^{2}+\mu_{i}^{2}}}{{\calE}+\sqrt{{\calE}^{2}+\mu_{i}^{2}}}}\,\Big]\,,
\begin{array}{l}{\Phi_{1}\Phi_{2}=p^{-1}\Phi_{2}\Phi_{1},}\\{\Phi_{1}\bar{\Phi}_{1}=1+pq\bar{\Phi}_{1}\Phi_{1}+(pq-1)\bar{\Phi}_{2}\Phi_{2},}\\{\Phi_{2}\bar{\Phi}_{2}=1+pq\bar{\Phi}_{2}\Phi_{2},}\\{\Phi_{1}\bar{\Phi}_{2}=p\bar{\Phi}_{2}\Phi_{1},}\\{\Phi_{2}\bar{\Phi}_{1}=q\bar{\Phi}_{1}\Phi_{2},}\\{\bar{\Phi}_{1}\bar{\Phi}_{2}=q\bar{\Phi}_{2}\bar{\Phi}_{1}.}\\\end{array}
\sum_{j=-\infty}^{\infty}(-1)^{j}\left(\overset{_\mathrm{(b)}}\Pi_{j}{}_{\mu\nu}(p)+\overset{_\mathrm{(d)}}\Pi_{j}{}_{\mu\nu}(p)\right)\bigg|_{\mathrm{div}}^{\Lambda^{-4}}=-\frac{g^{2}c_{v}}{8\pi^{2}}\frac{9}{154}M^{2}C_{4}\delta_{\mu\nu}.
\Gamma_{g.f.}=\int\left\{\frac{1}{2}\xiB^{2}+B\partialA-eB\left[(\hat{\varphi}_{1}-\xi_{A}\frac{m}{e})\varphi_{2}-\hat{\varphi}_{2}(\varphi_{1}-\hat{\xi}_{A}\frac{m}{e})\right]\right\}
V\left(Y(x)\right)=\left(\begin{matrix}{{\frac{1}{\sqrt{1-Y\bar{Y}}}}}&{{\frac{Y}{\sqrt{1-Y\bar{Y}}}}}\\{{\frac{\bar{Y}}{\sqrt{1-Y\bar{Y}}}}}&{{\frac{1}{\sqrt{1-Y\bar{Y}}}}}\\\end{matrix}\right)\,
\begin{array}{c}{d\tau=\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{d^{4}k_{2}}{(2\pi)^{4}}(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-k_{1}-k_{2})\pi\delta(k_{1}^{2}-M^{2})\theta(k_{1}^{0})}\\{\times\pi\delta(k_{2}^{2}-M^{2})\theta(k_{2}^{0})}\\\end{array}
{\calL}={\frac{1}{16\piG_{10}}}\sqrt{-\hat{G}}\,[{\calR}_{\hat{G}}+\hat{G}^{MN}\partial_{M}\Phi\partial_{N}\Phi-{\textstyle{\frac{1}{12}}}\hat{H}_{MNP}\hat{H}^{MNP}-{\textstyle{\frac{1}{4}}}\hat{F}_{MN}^{I}\hat{F}^{I\,MN}],
f(p)\:=\:\left[\frac{r^{2}\,[4p^{2}+(2m\mpM)^{2}]^{2}\;+\;(2m\mpM)^{4}}{m^{2}\,(2m\mpM)^{2}\;[p^{2}+(m\mp{\frac{M}{2}})^{2}]^{2}}\right]
{\calL}_{\mathrm{MCSH}}=-{\frac{1}{4e^{2}}}F_{\mu\nu}F^{\mu\nu}+{\frac{\kappa}{2}}\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}+\left(D_{\mu}\phi\right)^{*}D^{\mu}\phi-V(|\phi|)
T=\left(\begin{array}{cccccc}{0}&{-t}&{\cdot}&{\cdot}&{\cdot}&{0}\\{t}&{0}&{\cdot}&{\cdot}&{\cdot}&{0}\\{\vdots}&{}&{\ddots}&{}&{}&{}\\{0}&{}&{}&{}&{0}&{-t}\\{0}&{}&{}&{}&{t}&{0}\\\end{array}\right).
\big(\partial_{\lambda}\partial_{\rho}h^{\lambda\rho}-\partial_{\lambda}^{2}h\big)|_{0_{+}}=-3\ell^{-1}(\partial_{y}+2\,\ell^{-1})\,h|_{0_{+}}-M_{*}^{-3}\,(1-\gamma)^{-1}T_{55}(0)\,.
V=\left(\begin{array}{l}{X(u)=2a(u)=\frac{2\sqrt{2}}{\pi}\int_{-1}^{1}\,\sqrt{\frac{u-x}{1-x^{2}}}\,dx}\\{F_{A}(u)=2\,a_{D}(u)=2i\frac{\sqrt{2}}{\pi}\int_{1}^{u}\,\sqrt{\frac{u-x}{1-x^{2}}}\,dx}\\\end{array}\right)\.
S=-\frac{1}{2}\intd^{2}xdy\,\Bigl(i\bar{\psi}\gamma^{\mu}\frac{\partial\psi}{\partialx^{\mu}}-\frac{1}{2}\bar{\psi}\gamma_{5}\frac{\partial\psi}{\partialy}+\frac{\partial\phi}{\partialx^{\mu}}\frac{\partial\phi}{\partialx_{\mu}}-\frac{1}{4}\frac{\partial\phi}{\partialy}\frac{\partial\phi}{\partialy}+F^{2}\Bigr)
\intd^{3}x\varepsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}(\partial^{2})^{n}A_{\rho}=-\intd^{3}x\varepsilon^{\mu\nu\rho}F_{\mu\sigma}\partial_{\nu}(\partial^{2})^{n-1}F_{\rho}^{\;\sigma}\;,
\left[\textrm{Tr}\left(\frac{1}{H-z}\right)\right]_{\mathrm{{even}}}=-\mathrm{Tr}(H\triangleV\triangleV\triangleV\triangle)+\textrm{Tr}(H\triangleV\triangleV\triangleV\triangleV\triangle)-\textrm{Tr}(H\triangleV\triangleV\triangleV\triangleV\triangleV\triangle)+\cdots
\bar{n}_{0}(t)=[\left(|\alpha_{00}(t)|^{2}+|\beta_{00}(t)|^{2}\right)\bar{n}_{0}+|\beta_{00}(t)|^{2}+\sum_{k}\left[\left(|\alpha_{0k}(t)|^{2}+|\beta_{k0}(t)|^{2}\right)\bar{n}_{k}+|\beta_{k0}(t)|^{2}\right]\;.
\sum_{k,l=1}^{\infty}\big(\frac{2\hbarT}{k^{2}\pi^{2}}\delta_{ij}\delta^{kl}\big)\big[\frac{-(-1)^{k}}{k\pi}\big]\big[\frac{-(-1)^{l}}{l\pi}\big]=\langle\int_{-1}^{0}d\taud\tau^{\prime}\tau\tau^{\prime}\xi_{i}(\tau)\xi_{j}(\tau^{\prime})\rangle,
S=S_{a_{f}}+S_{\Phi}=\intL_{a_{f}}(\zeta_{f})\,d\zeta^{(1,1,1)}\wedge\cdots\wedged\zeta^{(2,2,3)}+\int\sqrt{g}L_{\Phi}(\zeta)\,d\zeta^{(111)}\wedge\cdots\wedged\zeta^{(223)},
\Gamma_{\pm}^{5}(p)=\intSp\gamma_{5}(\hat{k_{1}}+iM_{1})\frac{(1\pm\gamma_{5})}{2}(\hat{k_{2}}+iM_{2})\frac{\sqrt{M_{1}M_{2}}}{(k_{1}^{2}+M_{1}^{2})(k_{2}^{2}+M_{2}^{2})}\frac{d^{4}k}{(2\pi)^{4}}\stackrel{p\rightarrow0}{\simeq}\pm\frac{<\bar{\psi}\psi>}{2N_{c}}
g^{ij}=\frac{e^{2}LN}{4\piT}\left(\begin{matrix}{N-1}&{-1}&{\ldots}&{-1}\\{-1}&{N-1}&{\ldots}&{-1}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{-1}&{-1}&{\ldots}&{N-1}\\\end{matrix}\right)\\.
C=n_{+}{\pi}^{+}-n_{-}{\partial}_{+}A_{+}-\sum_{i=1}^{2}{\partial}_{i}A_{i}+n^{2}x^{-}B-\frac{n^{2}n_{+}}{{\partial}_{\bot}^{2}+n^{2}{\partial}_{+}}{\partial}_{+}B.\quad
m^{2}(\beta,L)\approxm_{0}^{2}+\frac{4\lambda\mu^{4-D}}{(\pi)^{D/2}}\Gamma(\frac{D}{2}-1)\left[(\beta^{2-D}+L^{2-D})\zeta(D-2)+2E_{2}((D-2)/2;\beta^{2},L^{2})\right]\;,
\begin{array}{lcll}{\mathrm{floor}}&{\mathrm{SU(2/1)}}&{\mathrm{field}}&{\mathrm{helicity}}\\{\mid\mathrm{gnd}>}&{(0~-\frac{3}{2})}&{A_{\mu}}&{+1}\\{\mid\mathrm{1st}>}&{(1~-\frac{3}{2})}&{2\lambda}&{+\frac{1}{2}}\\{\mid\mathrm{2nd}>}&{(0~-\frac{1}{2})}&{\phi}&{~0}\\\end{array}
M^{2}={\frac{1}{2}}\left(|m_{c}|^{2}-|n_{c}|^{2}-|Q_{ic}|^{2}\right)=\pi\left(\begin{matrix}{q^{(e)}}&{q_{(m)}}\\\end{matrix}\right)\left(\begin{matrix}{\eta}&{0}\\{0}&{\eta}\\\end{matrix}\right)\left(\begin{matrix}{q^{(e)}}\\{q_{(m)}}\\\end{matrix}\right).
v\rightarrow\widetildev={\calS}v=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{-1}\\{0}&{0}&{1}&{0}\\{0}&{1}&{0}&{0}\\\end{matrix}\right)v=\left(\begin{matrix}{1}\\{i}\\{-iz}\\{z}\\\end{matrix}\right)\.
h^{\mu\nu}=\frac{e}{4f}\left(\epsilon^{\alpha\beta\sigma\nu}\partial_{\sigma}T_{(\alpha\beta)}{}^{\mu}+\epsilon^{\alpha\beta\sigma\mu}\partial_{\sigma}T_{(\alpha\beta)}{}^{\nu}\right)-g^{\mu\nu}\frac{ke}{2\left(f+4k\right)f}\epsilon^{\alpha\beta\rho\kappa}\partial_{\rho}T_{(\alpha\beta)\kappa}.
\sum_{m=1}^{\infty}\left\langle\frac{P_{n}(x_{1})\tilde{Q}_{m}(y_{1})}{x_{1}-\phi_{a}}\right\rangle\left\langle\frac{P_{m}(x_{2})\tilde{Q}_{n}(y_{2})}{y_{2}-\chi_{b}}\right\rangle=\left\langle\frac{P_{n}(x_{1})\tilde{Q}_{n}(y_{1})}{(x_{1}-\phi_{a})(y_{1}-\chi_{b})}\right\rangle
\Gamma\lbrackA\rbrack=\!-\!2\!\int_{0}^{\infty}\!{\frac{dT}{T}}e^{-m^{2}T}\!\int\!{\calD}x{\calD}\psi\,\mathrm{exp}\biggl[\!-\!\!\int_{0}^{T}\!\!\!\!\!d\tau\Bigl({\frac{1}{4}}{\dot{x}}^{2}\!+\!{\frac{1}{2}}\psi\dot{\psi}\!+\!ieA_{\mu}\dot{x}^{\mu}\!-\!ie\psi^{\mu}F_{\mu\nu}\psi^{\nu}\Bigr)\!\biggr]
C_{\mu\nu}=\{\omega_{\mu},\omega_{\nu}\}=m\left(\begin{array}{cccc}{0}&{0}&{0}&{-m}\\{0}&{0}&{-1}&{\partial_{2}^{x}}\\{0}&{1}&{0}&{-\partial_{1}^{x}}\\{m}&{\partial_{2}^{x}}&{-\partial_{1}^{x}}&{0}\\\end{array}\right)\delta^{2}(x-y)
\langleB_{3}(\theta)|h\rangle=-\frac{\hat{\mu}}{2}\,T\,h_{MN}\,M^{MN}(\theta)\;,\;\;\langleB_{3}(\theta)|A\rangle=-\frac{\hat{\mu}}{8}\,T\,A_{MNPQ}\,M_{ab}(\theta)\,\Gamma_{ba}^{MNPQ}
V(\phi,\chi)=\sum_{i\neqj}^{3}\left(\Gamma_{iij}(\phi_{i}-\phi_{j})-X_{j}\phi_{i}\right)^{2}+\sum_{cyclic}^{3}\left(\Gamma_{ijk}(\phi_{i}-\phi_{k})-\Gamma_{ikj}(\phi_{i}-\phi_{k})-g\phi_{j}\phi_{k}\right)^{2}\,.
q_{\mui}=(x_{i}^{A^{\prime}},\bar{C}_{\hat{l}i}),\quadM^{\mu\nu}=\left(\begin{array}{c|c}{\bar{S}_{A^{\prime}B^{\prime}}}&{M^{x\hat{l}}}\\\hline{\left(M^{x\hat{l}}\right)^{T}}&{H^{\hat{l}\hat{m}}}\\\end{array}\right),