harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{2H}(\frac{\partial\varphi}{\partial\tau})_{w}=\frac{V^{1/2}|_{x=\varphi}}{2}\int_{1}^{\varphi}\frac{dx}{V^{3/2}}\left(x\left(\dot{H}-\frac{k}{a^{2}}\right)\mp\frac{2\mu}{a^{4}}\left(1+\frac{4\alpha\Lambda}{3}+\frac{8\alpha\mu}{a^{4}x^{2}}\right)^{-3/2}\right)
\begin{array}{l}{S=\displaystyle\intd\tau\left\{v_{\mu}\Pi^{\mu}-\frac12ev^{2}+n_{\mu}\dot{z}^{\mu}-\frac12\phi(n^{2}+1)\right\},}\\{\qquad\Pi^{\mu}\equiv\dot{x}^{\mu}-i(\bar{\theta}\Gamma^{\mu\nu}\dot{\theta})n_{\nu}-\xin^{\mu},}\\\end{array}
\{A_{i},\,A_{j}\}_{\pm}={\frac{\partial(\deltaa_{j})}{\partiala_{i}^{*}}}\pm{\frac{\partial(\deltaa_{i})}{\partiala_{j}^{*}}}\,,\quad\{A_{i}^{*},\,A_{j}^{*}\}_{\pm}={\frac{\partial(\deltaa_{i}^{*})}{\partiala_{j}}}\pm{\frac{\partial(\deltaa_{j}^{*})}{\partiala_{i}}}\,,
[H,\;\tilde{a}_{m}^{\dagger}]=\omega_{m}\tilde{a}_{m}^{\dagger}(\sum_{n=0}^{\infty}\;a_{m_{1}}^{\dagger}\ldotsa_{m_{n}}^{\dagger}\mid0\rangle\langle0\mida_{m_{n}}\ldotsa_{m_{1}})=\omega_{m}\tilde{a}_{m}^{\dagger}.
\Omega=\sqrt{\frac{Q^{2}}{4\pi\epsilon_{0}mr_{c}^{2}}}={\frac{e\piQ^{2}}{4\pi\epsilon_{0}J_{1}^{3}}}{\frac{1}{{\bar{\sigma}}^{2}}}={\frac{\pimQ^{4}}{2\epsilon_{0}^{2}J_{1}^{3}}}~.
{\calL}=-{\frac{1}{4}}e[{\calR}-{\frac{1}{2}}\partial_{\mu}\varphi\partial^{\mu}\varphi+{\frac{1}{4}}\partial_{\mu}g_{ij}\partial^{\mu}g^{ij}+{\frac{1}{2}}e^{\varphi}g^{ik}g^{jl}F_{\mu\nu\,ij}F_{\\\,kl}^{\mu\nu}],
\begin{array}{lll}{D=3:}&{[0010]}&{\mathrm{~\mathbf{8_s}~of~SO(8)}}\\{D=4:}&{[010]}&{\mathrm{{\bf~6}~of~SU(4)}}\\{D=5:}&{[1]}&{\mathrm{{\bf~2}~of~SU(2)}}\\{D=6:}&{[01]}&{\mathrm{{\bf~5}~of~USp(4)}}\\\end{array}
c(r^{\prime})=\frac{u_{2}(r^{\prime})}{(r^{\prime})^{3}\,g(r^{\prime})W(r^{\prime})}\quad;\quadd(r^{\prime})=-\frac{u_{1}(r^{\prime})}{(r^{\prime})^{3}\,g(r^{\prime})W(r^{\prime})}
S_{\pm}[x]\approxS_{0}[x_{qc}]\mp\varphi[x_{qc}]+\frac{1}{2}\int_{0}^{\tau\hbar/m}\mathrm{d}t\left[\dot{\eta}^{2}(t)+\frac{1}{2}{\Phi^{2}}^{\prime\prime}(x_{qc}(t))\,\eta^{2}(t)\right]\quad.
V^{n}{}_{\left\{m\right\},\left\{m^{\prime}\right\}}=\sum_{k_{1},\ldotsk_{n}}^{\prime}V_{\left\{m\right\},\left\{m^{\prime}\right\}}^{n}\left(k_{1},\ldotsk_{n}\right)\prod_{i=1}^{n}\left(a_{k_{i}}^{+}\right)^{m_{i}}\left(a_{k_{i}}\right)^{m_{i}^{\prime}},
\begin{array}{llllll}{p_{\mu}(y_{i})}&{=}&{y_{i},\qquad}&{p_{\mu}(h_{1})}&{=}&{h_{1}-\mu_{2}+\mu_{3},}\\{p_{\mu}(h_{2})}&{=}&{h_{2}+\mu_{1}-\mu_{3},\qquad}&{p_{\mu}(h_{3})}&{=}&{h_{2}-\mu_{1}+\mu_{2}.}\\\end{array}
\{\theta_{\alpha},\theta_{\beta}\}=\{\bar{\theta}^{\dot{\alpha}},\bar{\theta}^{\dot{\beta}}\}=\{\theta_{\alpha},\bar{\theta}^{\dot{\alpha}}\}=[\hat{x}^{\mu},\theta_{\alpha}]=[\hat{x}^{\mu},\bar{\theta}^{\dot{\alpha}}]=0.
\deltax^{R}=\sum_{n\neq0}[a_{n}^{R}e^{-in(\Omega_{n}\tau-\sigma)}+\tilde{a}_{n}^{R}e^{-in(\Omega_{n}\tau+\sigma)}]+A_{0}^{R}e^{-i\frac{m\alpha^{\prime}}{l}\tau}+\tilde{A}_{0}^{R}e^{i\frac{m\alpha^{\prime}}{l}\tau},
{\calL}_{1/m^{3}}^{(2+1)scal}=-\frac{\alpha}{720m^{3}}\left[6F^{\nu\lambda}F_{\nu\lambda,\mu}^{~~~~\mu}+4F^{\nu\lambda,\mu}F_{\nu\lambda,\mu}+F_{~~~\lambda}^{\nu\lambda,}F_{\nu\mu,}^{~~~\mu}\right],
H_{R}=\prod_{u}H_{Ru}(x_{a_{1}^{u}},\dots,x_{a_{n_{u}}^{u}}),~~~~~g_{a_{i}^{u}a_{i}^{u}}=g_{a_{j}^{u}a_{j}^{u}},~~~~~\eta_{MN}\partial_{M}\partial_{N}H_{Ru}=0,
\begin{array}{rcl}{A_{\xi}}&{=}&{-{\textstyle\frac{m}{2}}(2\pi\alpha^{\prime})\b_{\xi}\,,}\\{}&{}&{}\\{\eta}&{=}&{{\textstyle\frac{m}{2}}\(2\pi\alpha^{\prime})\\rho^{(0)}\,.}\\\end{array}
[\delta_{\kappa_{1}},\delta_{\kappa_{2}}]z^{M}=\delta_{b}z^{M}=-\mathrm{i}(b_{\nu}{\widetildeB}^{\nu})z^{M},\qquadb^{\mu}=-\mathrm{i}{\overline{\kappa}}_{1}\Gamma^{\mu}\kappa_{2},
\chi_{\pm}\rightarrow\chi_{\pm}^{\prime}=G_{\pm}(\tau_{\pm})\chi_{\pm}G_{\pm}^{T}(\tau_{\pm})=\chi_{\pm}+\left(\begin{array}{ccc}{\tau_{\pm}^{2}}&{\frac{\tau_{\pm}}{\sqrt{2}}}&{\frac{\pmi\tau_{\pm}}{\sqrt{2}}}\\{\frac{\tau_{\pm}}{\sqrt{2}}}&{0}&{0}\\{\frac{\pmi\tau_{\pm}}{\sqrt{2}}}&{0}&{0}\\\end{array}\right).
\langle\psi^{\mu}(z)\psi^{\nu}(w)\rangle={\frac{\eta^{\mu\nu}}{z-w}}\,\qquad\langle\psi^{\mu}(z)\bar{\psi}^{\nu}(\bar{w})\rangle={\frac{i\eta^{\mu\nu}}{1-z\bar{w}}}\.
B_{i}^{(cl)}=\left(\begin{array}{ccc}{0}&{0}&{B_{i}^{(2)(cl)}}\\{B_{i}^{(3)(cl)}}&{0}&{0}\\{0}&{B_{i}^{(1)(cl)}}&{0}\\\end{array}\right)=\left(\begin{array}{ccc}{0}&{0}&{Z_{i}}\\{Z_{i}}&{0}&{0}\\{0}&{Z_{i}}&{0}\\\end{array}\right)
Z=\intdG_{\mu}exp\i\intd^{3}x[\frac{1}{2}G_{\mu}^{2}-\frac{\lambda^{2}}{8\pi}\epsilon_{\mu\nu\lambda}G^{\mu}\partial^{\nu}G^{\lambda}+\frac{\lambda^{2}}{24\pim}(\partial_{\mu}G_{\nu}-\partial_{\nu}G_{\mu})^{2}+.....+G_{\mu}(J^{\mu}+K^{\mu})]
G({\bfx},{\bfy},E)=\frac{m}{\pi\hbar^{2}}\int_{o}^{\infty}\frac{kdk}{q^{2}-k^{2}}J_{o}(k|{\bfx}-{\bfy}|)=-i\frac{m}{2\hbar^{2}}H_{o}^{1}(q|{\bfx}-{\bfy}|).
{\frac{d^{2}x^{\rho}}{d\tilde{\lambda}^{2}}}+\tilde{\Gamma}_{\mu\nu}^{\rho}{\frac{dx^{\mu}}{d\tilde{\lambda}}}{\frac{dx^{\nu}}{d\tilde{\lambda}}}=-{\frac{1}{2}}{\frac{{\calW}^{\prime}}{\calW}}{\frac{dx^{\rho}}{d\tilde{\lambda}}}.
D_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}\equiv\partial_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}+{\hat{A}}_{\hat{\imath}}{\hat{k}}^{\hat{\mu}}-\frac{m}{2}l_{p}^{2}\,{\hat{b}}_{\hat{\imath}}{\hat{h}}^{\hat{\mu}}\,,
F^{\mu}=(-g)^{-\tau}\partial_{\nu}\left(g^{\mu\nu}(-g)^{\tau}\right)+a_{1}\bar{\Gamma}_{~\alpha\beta}^{\mu}g^{\alpha\beta}+a_{2}\bar{\Gamma}_{~\alpha\nu}^{\nu}g^{\alpha\mu}+a_{3}\bar{\Gamma}_{~\nu\alpha}^{\nu}g^{\alpha\mu}
\left(\begin{array}{ccc}{0}&{\frac{7}{16}}&{\frac{3}{2}}\\{\frac{1}{10}}&{\frac{3}{80}}&{\frac{3}{5}}\\{\frac{3}{5}}&{\frac{3}{80}}&{\frac{1}{10}}\\{\frac{3}{2}}&{\frac{7}{16}}&{0}\\\end{array}\right).
\deltaE(P)={\frac{\lambda^{2}}{3!P^{+}}}\intdk_{1}dk_{2}dk_{3}~16\pi^{3}\delta^{3}(P-k_{1}-k_{2}-k_{3})\Biggl[{\frac{P_{\perp}^{2}}{P^{+}}}-{\frac{k_{1\perp}^{2}}{k_{1}^{+}}}-{\frac{k_{2\perp}^{2}}{k_{2}^{+}}}-{\frac{k_{3\perp}^{2}}{k_{3}^{+}}}\Biggr]^{-1}\;,
\left\langleU_{\lambda}\left(w\right)\right\rangle_{w^{\prime}}=\prod_{X_{i^{\prime}}^{\prime}\in{\itG}}\left\langleU_{\lambda}\left(w\right)\right\rangle_{w^{\prime}}^{\left({\itG}_{1}\right)_{X_{i^{\prime}}^{\prime}}}\;\mathrm{~where~}
Z^{0i}\stackrel{|x|\to\infty}{\longrightarrow}\frac{2}{\sqrt{3}}\left\{{\bar{\calA}}\Pi_{h.}^{i}+\frac{1}{8\pi}{\bar{\calA}}_{D}B^{i}\right\}\displaystyle{\biggl\vert}=\frac{2}{\sqrt{3}}\left\{{\bar{\calA}}\Pi^{i}+\frac{1}{8\pi}{\bar{\calA}}_{D}B^{i}\right\}\displaystyle{\biggl\vert}.
d\xi^{m}\varepsilon_{mn}\sqrt{-g}g^{nk}\partial_{k}{\hat{X}}^{a}\vert_{\xi^{m}={\tilde{\xi}}^{m}(\tau)}=0,\quad\Leftrightarrow\quad\partial_{\perp}{\hat{X}}^{a}\vert_{\partial{\calM}^{2}}=0,
L_{s}=\,\frac1{2e}\left(\dot{x}_{\mu}-\frac{s\,\dot{q}\,f_{\mu}^{\prime}}{f\cdot\dot{x}}\right)^{2}\,+\,\frac{e\,m^{2}}2\,-\,\frac{\alpha\,m\,e\,\dot{q}}{f\cdot\dot{x}}.
Z^{ij}=\beta_{ij}^{3}|Z|=i\sigma_{2}\;\left(\begin{matrix}{-|Z|}&{0}&{0}&{0}\\{0}&{-|Z|}&{0}&{0}\\{0}&{0}&{|Z|}&{0}\\{0}&{0}&{0}&{|Z|}\\\end{matrix}\right)=|Z|\left(\begin{matrix}{0}&{-1}&{0}&{0}\\{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{0}\\\end{matrix}\right),
\left(\prod_{i}^{n_{f}}\frac{\lambda_{i}}{\mu_{m}}\right)^{1/(n_{f}-n_{c}-1)}\tilde{\Lambda}^{(3(n_{f}-n_{c}-1)-n_{f})/(n_{f}-n_{c}-1)}=m_{i}\lambda_{i}-\frac{\lambda_{i}^{2}}{\mu}.
\phi\approx\lambda^{2}\,\quad\psi\approx\sigma^{mn}\lambda_{a}F_{mn,a}\quad\mathrm{and}\quadF\approx-\frac{1}{2}F_{a}^{mn}F_{mn,a}-\frac{i}{4}\epsilon_{mnrs}F_{a}^{mn}F_{a}^{rs}\.
D_{\alpha}D_{\beta}\Omega-g_{\alpha\beta}D\cdotD\Omega+\frac{\gamma}{8}\left[g_{\alpha\beta}\frac{(D\Omega)^{2}}{\Omega}-2\frac{D_{\alpha}\OmegaD_{\beta}\Omega}{\Omega}\right]+\frac{\mu}{2}g_{\alpha\beta}\Omega^{1-\lambda}
{\frac{1}{2}}\int_{-T}^{T}|H_{5}t|^{-(1-\epsilon)}dt\intd^{3}\vec{x}\pi^{2}(\vec{x},0)={\frac{(H_{5}T)^{\epsilon}}{\epsilonH_{5}}}\intd^{3}\vec{x}\pi^{2}(\vec{x},0).
\underline{{\underline{R}}}_{a0}=-\frac1{\ell^{2}}\underline{{\Theta}}_{a}\wedge\underline{{\Theta}}_{0}-{\frac{1}{r}}\left(d\underline{{\gamma}}_{a}+\underline{{\omega}}_{ab}\wedge\underline{{\gamma}}^{b}\right)+O(r^{-3})\qquad,
\gamma^{0}=\sigma^{3}=\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right),\\\gamma^{1}=i\sigma^{1}=\left(\begin{matrix}{0}&{i}\\{i}&{0}\\\end{matrix}\right),\\\gamma^{2}=i\sigma^{2}=\left(\begin{matrix}{0}&{1}\\{-1}&{0}\\\end{matrix}\right),
Z_{N}=\frac{1}{h^{N}}\int\left(\prod_{a=1}^{N}d\mu(p_{a})dx_{a}\right)\prod_{a=1}^{N-1}\theta\left(x_{a+1}-x_{a}-\frac{m\sigma}{2p_{a}}-\frac{m\sigma}{2p_{a+1}}\right).
\frac{1}{R^{2}}\frac{d}{dr}(R^{2}A^{2}\frac{d\chi}{dr})=\frac{\sqrt{2}}{2}e^{2\sqrt{2}(\phi+\chi)}\frac{Q_{m}^{2}}{R^{4}}+\frac{\sqrt{2}}{2}e^{-2\sqrt{2}(\phi-\chi)}\frac{Q_{e}^{2}}{R^{4}}.
m_{2}=0\,\qquad(e^{-2\phi})_{\mathrm{fix}}=\left|{\frac{n_{2}}{m_{1}}}\right|=\left|{\frac{\xi}{m}}\right|=(e^{-2\phi})_{\mathrm{min}}\,\qquad(a)_{\mathrm{fix}}={\frac{n_{1}}{m_{1}}}=-{\frac{e}{m}}=(a)_{\mathrm{min}}\.
\begin{array}{rcl}{}&{}&{\mathcal{D}_{\mu}{\calF}_{mn}{}^{\mu\nu}=\frac{1}{4}m^{2}\epsilon^{\nu\rho\sigma\delta\lambda}B_{[m|\,\rho\sigma}\tilde{B}_{|n]\,\delta\lambda}\,,}\\{}&{}&{}\\{}&{}&{{}^{*}H_{m}=+m[\tilde{B}_{m}+\frac{1}{2}\epsilonV_{mn}B_{n}]\,,}\\{}&{}&{}\\{}&{}&{{}^{*}\tilde{H}_{m}=-m[B_{m}+\frac{1}{2}\epsilonV_{mn}\tilde{B}_{n}]\,,}\\\end{array}
\deltag_{\mu\nu}=[(F\xi)_{\mu\nu}+F(F^{\dagger}F)^{-1}F^{\dagger}{\frac{\partialg_{\mu\nu}}{\partiall_{i}}}\deltal_{i}]+[1-F(F^{\dagger}F)^{-1}F^{\dagger}]{\frac{\partialg_{\mu\nu}}{\partiall_{i}}}\deltal_{i}
+B_{\mu}{\frac{{\frac{(p^{2}\alpha+qa^{2})}{4}}\partial^{2}+q\alpha(\alpha+q)}{({\frac{p+a}{2}})^{2}\partial^{2}+(q+\alpha)^{2}}}\epsilon^{\mu\nu\lambda}B_{\nu\lambda}-{\frac{1}{2g}}B_{\mu}B^{\mu}.
\left\langle\left[\partial_{\muy}A_{\nu}^{m}(y)\right]\left[g\epsilon^{3mn}a^{\mu}(x)A^{n\nu}(x)\right]\right\rangle\Bigr|_{y=x}=-8iga^{\mu}(x)\partial_{\muy}\mathcalP(y,x)\Bigr|_{y=x}.
{\calL}_{Coul}^{pzm}=\partial_{i}a_{-}\partial_{i}a_{+}+\frac12\left(\partial_{+}a_{-}\right)^{2}-\frac12\left(\partial_{i}a_{j}^{Tr}\right)^{2}+J_{pzm}^{+}a_{+}+J_{pzm}^{i}a_{i}^{Tr}+J_{pzm}^{-}a_{-}\,
\Boxh_{\mu\nu}^{(2)}=-16\piG\left(T_{\mu\nu}^{\mathrm{grav}}-{\frac{1}{2}}\eta_{\mu\nu}T^{\mathrm{grav}}\right)-\partial_{\mu}\left(f(r)\partial_{\nu}f(r)\right)-\partial_{\nu}\left(f(r)\partial_{\mu}f(r)\right)
H^{\prime}-\betaN\varepsilon_{F}=\intdp\,\biggl\{-\frac{1}{2}\partial_{p}\tilde{\psi}^{\dagger}\tilde{\partial}_{p}\psi+\frac{p^{2}}{2}\tilde{\psi}^{\dagger}\tilde{\psi}+\overline{\mu}\tilde{\psi}^{\dagger}\tilde{\psi}\biggr\}\.
\psi_{m}^{\mathrm{R}}=\frac{\xi^{1/2}}{2}\left(\psi_{m}-\frac{1}{8}\Gamma_{m}\Gamma^{n}\psi_{n}+\frac{1}{8}\Gamma_{m}\lambda\right),\quad\lambda^{\mathrm{R}}=\frac{\xi^{-1/2}}{4\sqrt2}\left(\lambda-\Gamma^{m}\psi_{m}\right).
\Pi_{MM^{\prime}}^{AA^{\prime}}(m^{2})=\sum_{i=1}^{3}\Bigl(\Pi_{MM^{\prime}}^{(i)+AA^{\prime}}(m^{2})+\alpha^{M^{\prime}}\eta^{A^{\prime}}\Pi_{MM^{\prime}}^{(i)-AA^{\prime}}(m^{2})\Bigr).
\bar{W}[\bar{K},t]=\prod_{\alpha,y}\delta(K^{\alpha4}(y))\prod_{\beta,z}K_{,\theta}^{\beta4}(z)W[K_{,\theta}^{\alpha\mu};K_{,\theta}^{\alpha5};-t_{,\theta}^{\alpha};K^{\alpha\mu};K^{\alpha5};t^{\alpha}]
\Gamma_{a}(p,q)_{\mu}=-i\int\frac{d^{3}k}{(2\pi^{3})}\frac{\bar{u}(p-q)\gamma_{\nu}(p\!\!\!/-q\!\!\!/-k\!\!\!+m)\gamma_{\mu}(p\!\!\!/-k\!\!\!/+m)\gamma_{\lambda}u(p)}{[(p-k)^{2}-m^{2}][(p-k-q)^{2}-m^{2}]}G(k)^{\nu\lambda}
g=g_{R}g_{L}=\left(\begin{array}{cc}{A}&{U}\\{-V}&{B}\\\end{array}\right)=\left(\begin{array}{cc}{a\tilde{a}-u\tilde{v}}&{a\tilde{u}+u\tilde{b}}\\{-v\tilde{a}-b\tilde{v}}&{b\tilde{b}-v\tilde{u}}\\\end{array}\right)
\Lambda^{\mu\nu}+\frac{D}{a}\left(^{\ast}\Lambda^{\mu\nu}\right)=-\frac{4DM^{2}}{e^{2}}\epsilon^{\mu\nu\kappa\lambda}\partial_{\theta}T_{(\kappa\lambda)}^{\;\;\;\;\;\theta}-\frac{8aM^{2}}{e^{2}}\partial_{\theta}T^{(\mu\nu)\theta}.
B=-\frac{1}{8}\bar{D}(1-\gamma_{5})D(W_{a}^{*})^{2}=(D^{a})^{2}-\frac{1}{2}(F_{\mu\nu}^{a})^{2}-\frac{i}{2}F_{\mu\nu}^{a}\tilde{F}_{\mu\nu}^{a}+O(\theta),
L^{(0)}\rightarrowL^{(N)}=\sum_{k=0}^{N}\int\,dx\,(\dot{\phi_{k}}\phi_{k}^{\prime}-\phi_{k}^{2})+\sum_{m=k+1}^{N}\,\sum_{k=0}^{N-1}\int\,dx\,(\dot{\phi_{k}}-\phi_{k}^{\prime})\phi_{m}^{\prime}
\Gamma=\left(A^{3}\frac{\sigma^{3}}{2}+A^{+}\sigma^{+}+A^{-}\sigma^{-}\right)+\left(B^{a}T^{a}\right)+\left(\psi_{+\alpha}R^{+\alpha}+\psi_{-\alpha}R^{-\alpha}\right)\equivA+B+\Psi
\begin{array}{l}{\partial_{i}x^{j}=\delta_{i}^{j}+q\hat{\calR}_{il}^{jk}x^{l}\partial_{k}~,}\\{\tilde{\partial}_{k}x^{v}=\delta_{k}^{v}+q^{-1}\left.\hat{\calR}^{-1}\right._{kj}^{vi}x^{j}\tilde{\partial}_{i}~.}\\\end{array}
S=\frac{L}{4\pi\lambda^{2}}\sum_{\pm}\int_{z_{0}}^{\infty}dz\;\frac{z(z^{2}-z_{0}^{2})}{\sqrt{z^{2}-z_{0}^{2}+(\lambdam)^{2}}}\;\bigl[\psi(z\mpB\lambda^{2})+\psi(z+1\pmB\lambda^{2})\bigr]
\psi_{m}^{\alpha}(x^{i},t)=e_{m}^{(\mu)}\sigma_{(\mu)}^{\alpha\dot{\beta}}\bar{\lambda}_{\dot{\beta}}(x^{i})\bar{\lambda}(t),\qquad\bar{\psi}_{m}^{\dot{\alpha}}(x^{i},t)=e_{m}^{(\mu)}\sigma_{(\mu)}^{\dot{\alpha}\beta}\lambda_{\beta}(x^{i})\lambda(t),
<0|\tilde{F}_{ij}(0)|e_{\pm}^{\lambda},{\vec{p}}>=-{\frac{i}{(2\pi)^{3/2}}}\sqrt{\frac{p_{0}}{2}}\epsilon_{ijk}e_{\pm}^{k}\mathrm{lim}_{p^{2}\rightarrow0}{\frac{1}{1-\Pi(p^{2})}}
\Omega(x,y)=1-\frac{1}{6}(x^{\sigma}+y^{\sigma})\\theta^{\mu\lambda}\H_{\mu\lambda\sigma}-\frac{1}{6}(x^{\sigma}+y^{\sigma})\(G^{-1}\theta^{-1}G^{-1})^{\mu\lambda}\H_{\mu\lambda\sigma}.
\int_{M}\alphaA={\frac{1}{2}}\langle\int_{M}\alpha({\tilde{\overline{\Psi}}}_{L}{\frac{\delta{S}^{-}}{\delta{\tilde{\overline{\Psi}}}_{L}}}+{\tilde{\Psi}}_{L}{\frac{\delta{S}^{-}}{\delta{\tilde{\Psi}}_{L}}})\rangle.
{\sfS}=\sum_{\mathrm{compact\atoptopologies}}e^{(n-\chi)\Phi_{0}}\int\frac{[dX\,dg]}{V_{\mathrm{diff\timesWeyl}}}e^{-S_{\mathrm{P}}}\,\prod_{i=1}^{n}\intd^{2}\sigma_{i}\,{\calV}_{i}\.
l=\left(\begin{array}{cc}{t^{4}\frac{i}{2}\gamma_{4}+t^{p}\frac{i}{2}\gamma_{p}\frac{1+i\gamma_{4}}{2}}&{\sqrt{e}\,\Theta_{M}}\\{\sqrt{e}\,\overline{{\Theta}}^{N}}&{0}\\\end{array}\right)
\phi(x)={\frac{1}{\sqrt{2\pi}}}\left(q[T]+{\frac{1}{\sqrt{2}}}\sum_{k=1}^{\infty}{\frac{1}{\sqrt{k}}}[(a_{(+)k}[T]+a_{(-)k}^{\dagger}[T])e^{-ikx}+(a_{(+)k}^{\dagger}[T]+a_{(-)k}[T])e^{ikx}]\right),
{\mathrm{reg}}\,q_{c}(a,r)=-\frac{1}{16\pi^{2}a^{4}}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}{dk\int_{0}^{\infty}{dz\sum_{\lambda,n}\Omega_{\lambdam}^{(n)}(z)f_{m}^{(nq)}(zx)}}.
inv=u^{2}\sum_{k=1}^{m}[1,k]^{2}x_{k}^{2}+v^{2}2\sum_{k=m+1}^{m+n}(1,\hat{\bar{k}}-m)^{2}\theta_{k}\theta_{-k}\equivu^{2}x^{2}(j)+v^{2}2\theta^{2}(\omega),
{\calS}={\calS}(A_{\mu}^{p},B_{\mu\nu}^{p})=\int{d^{4}}x\{-\frac{1}{4}{\varepsilon}^{\mu\nu\rho\sigma}G_{\mu\nu}^{p}B_{\rho\sigma}^{p}+\frac{1}{2}A_{\mu}^{p}A^{p\mu}\},
ds_{5}^{2}=\frac{1}{z^{2}}\left[-e^{2T(r,z)}dt^{2}+e^{2\left(A(r,z)+B(r,z)\right)}\left(dr^{2}+dz^{2}\right)+e^{2\left(A(r,z)-B(r,z)\right)}r^{2}d\Omega_{2}^{2}\right]
{\frac{\partial\Omega\big|_{(\chi,\psi)}}{\partial\psi_{\betaL}(x)}}={\frac{\kappa}{4\pi}}\;\partial_{x}{\calL}_{\alpha}^{[\chi]}(x)\;L_{\beta\alpha}(x)\;,\quad{\frac{\partial\Omega\big|_{(\chi,\psi)}}{\partial\psi_{\betaR}(x)}}={\frac{\kappa}{4\pi}}\;\partial_{x}{\calR}_{\alpha}^{[\chi]}(x)\;R_{\beta\alpha}(x)\;.
F_{n}(z)=a_{n}(z+\frac{1}{k})^{\frac{1}{2}}N_{\alpha+\frac{1}{2}}(m_{n}(z+\frac{1}{k}))+b_{n}(z+\frac{1}{k})^{\frac{1}{2}}J_{\alpha+\frac{1}{2}}(m_{n}(z+\frac{1}{k})).
K(p)=-\frac{N}{2}\int\frac{\mathrm{d}^{D}k}{{(2\pi)}^{D}}\frac{1}{k^{2}(p-k)^{2}}=-\frac{N}{2(4\pi)^{\mu}}\frac{H^{2}(1)}{H(2\mu-2)}p^{D-4}\equivb^{-1}p^{D-4},
G_{\alpha\beta}=\left(\begin{array}{rr}{G_{00}}&{G_{0s}}\\{G_{r0}}&{-g_{rs}}\\\end{array}\right)=\left(\begin{array}{cc}{2\partial_{\tau}\theta-(\partial_{\tau}\bbox{x})^{2}}&{\quad\partial_{\tau}\theta-\partial_{\tau}\bbox{x}\cdot\partial_{s}\bbox{x}}\\{\partial_{r}\theta-\partial_{r}\bbox{x}\cdot\partial_{\tau}\bbox{x}}&{-\partial_{r}\bbox{x}\cdot\partial_{s}\bbox{x}}\\\end{array}\right)
\zeta\left(s\mid\vec{x},\vec{y}\right)=\frac{\Gamma(\frac{3}{2}-s)}{\left(4\pi\right)^{\frac{3}{2}}\Gamma(s)}\sum_{n_{1},n_{2}=-\infty}^{\infty}\left(Z_{1}+Z_{2}+Z_{3}+Z_{4}\right),
(r_{b}|r_{a})_{E,l}=-i\frac{\hbar}{2Mc}\frac{M}{\hbar\kappa}\frac{\Gamma(-\nu+\tilde{l}+1)}{(2\tilde{l}+1)!}W_{\nu,\tilde{l}+1/2}\left(2\kappar_{b}\right)M_{\nu,\tilde{l}+1/2}\left(2\kappar_{a}\right)
{\frac{d}{dt}}\left({\frac{M{\bfV}}{\sqrt{1-{\bfV}^{2}}}}\right)=g({\bfB}_{0}-{\bfV}\times{\bfE}_{0})+q({\bfE}_{0}+{\bfV}\times{\bfB}_{0}),\\\({\bfV}\equiv{\frac{d}{dt}}{\bfX})
\Phi=t_{2}I+\left(\begin{matrix}{N^{-1/2}}&{0}&{0}\\{0}&{(LM)^{1/2}}&{0}\\{0}&{0}&{0}\\\end{matrix}\right)\left(\begin{matrix}{\phi^{(1)}}&{\phi^{(3)}}&{0}\\{{\phi^{(3)}}^{\dagger}}&{\phi^{(2)}}&{0}\\{0}&{0}&{0}\\\end{matrix}\right)\left(\begin{matrix}{N^{-1/2}}&{0}&{0}\\{0}&{(ML)^{1/2}}&{0}\\{0}&{0}&{0}\\\end{matrix}\right)
\sqrt{3}\hat{T}_{R}={\frac{-{\frac{f_{1}+f_{2}}{2}}+f_{3}}{\sum_{k=1}^{3}f_{k}}}={\frac{-{\frac{f_{1}f_{2}(f_{1}+f_{2})}{2}}+1}{f_{1}f_{2}(f_{1}+f_{2})+1}}
\mathrm{C_{\sigma\beta}=\delta_{\sigma\beta}+\left(\frac{n^{2}R_{\sigma}R_{\beta}-R\cdotn\,(R_{\sigma}n_{\beta}+R_{\beta}n_{\sigma})}{-\vec{R}^{2}}\right)~,\qquadR_{\sigma}\equiv(p-k-q)_{\sigma}~;}
\frac{2}{M^{2}}\frac{1}{\sqrt{-G}}\frac{\partial}{\partialu^{a}}(\sqrt{-G}\;G^{ab}\frac{\partial\phi}{\partialu^{b}})+\frac{1}{2}\frac{\partial^{2}\phi}{\partials^{2}}+\frac{1}{2h}\frac{\partialh}{\partials}\frac{\partial\phi}{\partials}-(\phi^{2}-1)\phi=0.
\hat{\hat{T}\,}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}{}^{\hat{\hat{\rho}}}=-\left(i_{\hat{\hat{k}}_{(n)}}\hat{\hat{C}\,}\right)_{\hat{\hat{\mu}}\hat{\hat{\nu}}}Q^{nm}\hat{\hat{k}}_{(m)}{}^{\hat{\hat{\rho}}}\,.
\int_{-\infty}^{\infty}da\,\frac{1}{(a^{2}+b^{2}+r^{2})^{N/2}}=\frac{\pi\,\Gamma(N-1)}{2^{N-2}\Gamma(N/2)^{2}}\,\frac{1}{\sqrt{b^{2}+r^{2}}^{\,N-1}}.
\deltav^{\ell}(\omega)=e\left\{+\left(\delta^{kl}-{\frac{p^{k}p^{\ell}}{E^{2}}}\right)\left({\frac{1}{\omega-2E}}-{\frac{1}{\omega+2E}}\right)+i\,<\sigma_{kl}>\left({\frac{1}{\omega-2E}}+{\frac{1}{\omega+2E}}\right)\right\}A_{k}^{e}(\omega)~~.
\int_{-x_{c,N}}^{x_{c,N}}x^{2t}e^{-x^{2}}\mathrm{d}x=\left(x_{c,N}\right)^{2t+1}\sum_{k=0}^{\infty}{\frac{(-1)^{k}}{k!}}{\frac{\left(x_{c,N}\right)^{2k}}{\left(k+t+1/2\right)}}
\;\;\;\;\;\;\;\;\;\;\;\;\;=\oint_{\gamma}dx^{\mu}\oint_{\gamma^{\prime}}dy^{\nu}\Omega_{\mu\sigma}(x)\Omega_{\nu\lambda}(y)\left\langle\widehat{A}^{\sigma}(x)\widehat{A}^{\lambda}(y)\right\rangle_{\mathrm{CS}}\;,
F_{\mu\nu}^{0}+\mathrm{i}\,G_{\mu\nu}^{0},\quadF_{\mu\nu}^{1}+\mathrm{i}\,G_{\mu\nu}^{1},\quad\dots\,\quadF_{\mu\nu}^{n+1}+\mathrm{i}\,G_{\mu\nu}^{n+1}\,
F_{(4)}=\betadx^{+}\left(4d\tilde{x}^{1}d\tilde{x}^{2}dx^{7}+2d\tilde{x}^{3}d\tilde{x}^{4}dx^{7}+2d\tilde{x}^{5}d\tilde{x}^{6}dx^{7}\right).
Y_{\ell}^{(a)}(u+i)Y_{\ell}^{(a)}(u-i)=\frac{(1+Y_{\ell+1}^{(a)}(u))(1+Y_{\ell-1}^{(a)}(u))}{(1+(Y_{\ell}^{(a+1)}(u))^{-1})(1+(Y_{\ell}^{(a-1)}(u))^{-1})},
\mathrm{Z}_{\mathrm{R}}^{I}(r)=\frac{1}{2}\,\mathrm{Tr}_{\mathrm{R}}\left(r^{2L_{0}}\right)=\mathrm{Tr}_{\mathrm{R}}\left(r^{2L_{0}}{\frac{1+\Omega}{2}}\right)~~,
F_{8}^{{\mathcal{O}}_{2,0}^{1,0}|++------}=\frac{\langle{\mathcal{O}}_{2,0}^{1,0}\rangle(\sigma_{1}^{+})^{3}(\sigma_{6}^{-})^{1/2}/(\sigma_{2}^{+})^{3}}{2^{6}\,\mathrm{cosh}(\theta_{12}/2)\,\prod_{3\leqi<j\leq8}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
{\calL}=\frac{9}{\alpha}\intd^{2}\!\thetad^{2}\!{\bar{\theta}}\left(SS^{\dagger}\right)^{\frac{1}{3}}+\left\{\intd^{2}\theta\,S\left[\mathrm{ln}\left(\frac{S}{\Lambda_{SYM}^{3}}\right)^{N_{c}}-N_{c}\right]+\mathrm{H.c.}\right\}\,
{\calH}=\sum_{\sigma}\int\frac{d^{3}{\bfp}}{(2\pi)^{3}}E_{p}\,\left[a_{\sigma}^{\,\dagger}({\bfp})a_{\sigma}({\bfp})+b_{\sigma}({\bfp})b_{\sigma}^{\,\dagger}({\bfp})+c_{\sigma}^{\,\dagger}({\bfp})c_{\sigma}({\bfp})+d_{\sigma}({\bfp})d_{\sigma}^{\,\dagger}({\bfp})\right]\quad,
\langle\psi^{-}|\hat{\bboxp}j_{3},{\mathsfs}_{R}j^{-}\rangle\equiv-\frac{i}{2\pi}\ointd{\mathsfs}\langle\psi^{-}|\hat{\bboxp}j_{3}{\mathsfs}j^{-}\rangle\frac{1}{{\mathsfs}-{\mathsfs}_{R}}=\frac{i}{2\pi}\int_{-\infty_{II}}^{+\infty_{II}}d{\mathsf{s}}\langle\psi^{-}|\hat{\bboxp}j_{3}{\mathsf{s}}j^{-}\rangle\frac{1}{{\mathsf{s}}-{\mathsf{s}}_{R}}\quad\mathrm{~for~all~}\psi^{-}\in\Phi_{+}\,.
S_{ct}^{(1)}=-\frac{1}{16\pi^{2}\epsilon}\Big(-g^{2}|\phi|^{2}+\lambda^{2}{|W_{1}^{\prime}|}^{2}-\frac{1}{M^{2}}\big[\lambda^{2}g^{2}{|W_{1}^{\prime}|}^{2}|\phi|^{2}-\lambda^{4}{|W_{1}^{\prime}|}^{4}\big]\Big)
g_{0}^{2}I(p,nx)\equiv-ig_{0}^{2}\int\frac{d^{D}k/(2\pi)^{D}}{k^{2}((k+p)^{2(1+nx)}}=\frac{\alphaz^{x}}{4\pi}\frac{i_{n+1}}{(p^{2})^{nx}},
\omega^{i}=\phi^{j}\,(\mathrm{1}+\frac{1}{2!}A^{2}+\frac{1}{4!}A^{4}+\cdots)_{j}^{\,\,\,i}-d\xi^{\beta}\,(\frac{1}{2!}A+\frac{1}{4!}A^{3}+\cdots)_{\beta}^{\,\,\,\,i}
\left.i\left(\begin{matrix}{\langleU_{z},\bar{U}_{\bar{z}}\rangle}&{\langleU_{z},V\rangle}\\{\langle\bar{V},\bar{U}_{\bar{z}}\rangle}&{\langle\bar{V},V\rangle}\\\end{matrix}\right)\right|_{z=0}=\left(\begin{matrix}{1}&{1}\\{1}&{1}\\\end{matrix}\right)\,
\overline{{{\calM}}}_{0,\infty}(q)=\coprod_{n=0}^{\infty}\left(\overline{{{\calM}}}_{0,n+3}\times[0,q^{n}]\right)/\left(\overline{{{\calM}}}_{0,n},q^{n}\right)\sim\left(i_{n}\overline{{{\calM}}}_{0,n},0\right).
\left.\begin{array}{llr}{\gamma_{1,3,(\pm1,0,0)}}&{=}&{{\bf1}_{1}}\\{\gamma_{1,{\tilde{7}}_{3},(\pm1)}}&{=}&{-\alpha{\bf1}_{1}}\\{\gamma_{1,7_{4},(0)}}&{=}&{-\alpha{\bf1}_{2}}\\{\gamma_{1,7_{5},(0)}}&{=}&{\alpha^{2}{\bf1}_{3}}\\\end{array}\right.
\midj\rangle=\frac{\sqrt2}{4}\Gamma_{\alpha\beta}^{j}\mu_{\alpha}\mu_{\beta}\mid0\rangle,\mid4\rangle=\frac{1}{\sqrt2i}(1+\mu_{1}\mu_{2}\mu_{3}\mu_{4})\mid0\rangle,\;\mid5\rangle=\frac{-1}{\sqrt2}(1-\mu_{1}\mu_{2}\mu_{3}\mu_{4})\mid0\rangle,
(\varepsilon_{u,\ell}^{1,{\bfm}},\varepsilon_{u,\ell}^{2,{\bfm}})=e^{i\sigma^{2}\theta/2}e^{i\sigma^{3}\phi/2}(a_{u,\ell}^{1,{\bfm}}(r),a_{u,\ell}^{2,{\bfm}}(r)),
D_{r}=(\gamma^{\mu}\otimes\mathbf{1})(\mathbf{f}_{\mu}-\frac{1}{4}\Gamma_{\mu\nu\rho}\gamma^{\nu}\gamma^{\rho}+\frac{1}{4}A_{\mu}^{\;m}t_{mkl}\mathbf{1}\otimes\gamma^{k}\gamma^{l})-\frac{1}{8}F_{\mu\nuk}\omega\gamma^{\mu}\gamma^{\nu}\otimes\gamma^{k}+\omega\otimes\epsilon=0\,,
\begin{array}{clll}{{}}&{K^{\dagger}=K,}&{\quadX_{\pm}^{\dagger}=-X_{\mp},}&{\quad\quad\mathrm{when}\;q\in{\mathbbR}}\\{{}}&{K^{\dagger}=K^{-1},}&{\quadX_{\pm}^{\dagger}=-X_{\mp},}&{\quad\quad\mathrm{when}\;\vertq\vert=1}\\\end{array}
0=4(1+q)\left(\frac{d\phi}{dA}\right)^{2}-2\frac{P^{\prime}(A)}{P(A)}\frac{d\phi}{dA}-\frac{f_{0}^{2}}{2P^{2}(A)}+\frac{2Q+\mue^{-4(1+q)\phi}/Q}{4P(A)}
\begin{array}{c}{{\bfp}{\bfp}_{+}=0,\quad{\bfp}{\bfE}_{+}=0,}\\{{\bfp}_{+}^{2}+\alpha^{2}=0,\quad{\bfp}_{+}{\bfE}_{+}=0,\quad{\bfE}_{+}^{2}+{\alpha^{2}/{\bfp}^{2}}=0.}\\\end{array}
S_{\mathrm{}}={\frac{1}{2\pi}}\intd\taud\sigma\Bigl(G_{ij}\partial^{\alpha}X^{i}\partial_{\alpha}X^{j}-B_{ij}\epsilon^{\alpha\beta}\partial_{\alpha}X^{i}\partial_{\beta}X^{j}+A_{iI}\epsilon^{\alpha\beta}\partial_{\alpha}X^{i}\partial_{\beta}X^{I}\Bigr)\;.
H=\frac{1}{2}p_{i}^{A}p_{i}^{A}+\frac{1}{4}G_{ij}^{A}G^{ij\,A}+\frac{m^{2}}{2}(A^{i\,A}-\frac{1}{m^{2}}\partial^{i}p_{0}^{B}\Phi^{BA})^{2}+\frac{1}{2m^{2}}(\nabla^{i}p_{i}^{A})(\nabla^{i}p_{i}^{A}),
(1-1){\frac{V_{6}L}{8}}\int_{0}^{\infty}{\frac{dt}{t^{4}\sqrt{t}}}\;{\frac{f_{4}^{8}(e^{-2\pit})}{f_{1}^{5}(e^{-2\pit})}}\displaystyle\sum_{j=0}^{k/2}\chi_{j}(e^{-4\pit})
\begin{array}{l}{M_{nl}\delta_{ij}-M_{il}\delta_{nj}+M_{ij}\delta_{nl}-M_{nj}\delta_{il}=0}\\{\epsilon_{nj}(M\epsilon)_{ik}+\delta_{kn}M_{ij}=M_{in}\delta_{jk}}\\{\epsilon_{nj}\epsilon_{ik}+\delta_{kn}\delta_{ij}=\delta_{in}\delta_{jk}.}\\\end{array}
(N-2)\int_{1}^{\infty}d\lambda\,\lambda^{-N+1}(1-\lambda)\int_{0}^{\lambda}d\rho\,\left(\frac{\lambda}{\rho}-1\right)\frac{\rho^{N-2}}{(\rhox-y)^{N}}\left(-\eta_{\alpha\beta}+N\frac{x_{\alpha}x_{\beta}}{(\rhox-y)^{2}}\right)
{\calL}_{c.t.}^{(1)}\,=\,{\frac{\sqrt{-g}}{(4\pi)^{2}\epsilon}}\lbrack{\frac{1}{120}}R^{2}+{\frac{7}{20}}R_{\mu\nu}R^{\mu\nu}-{\frac{53}{180}}\epsilon^{\alpha\beta\gamma\delta}\epsilon_{\mu\nu\rho\sigma}{R^{\mu\nu}}_{\alpha\beta}{R^{\rho\sigma}}_{\gamma\delta}\rbrack
G_{\lambda}^{(4)}(p_{1},p_{2},\dot{-}p_{3},\dot{-}p_{4})=\left.\frac{\delta^{4}\bar{Z}_{D}^{(1)}\left[J\right]}{\deltaJ(\dot{-}p_{1})\deltaJ(\dot{-}p_{2})\deltaJ(p_{3})\deltaJ(p_{4})}\right|_{J=0}~,
\bar{\alpha}^{-}|\Phi_{s}\rangle|_{\alpha^{+}=0}^{\vphantom{5pt}}=(-\frac{\partial^{I}}{\partial^{+}}\bar{\alpha}^{I}+\frac{s+d-4}{\hat{\partial}^{+}}\bar{\alpha}^{z})|\Phi_{ph}\rangle\,.