harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_{n}^{\mathcal{O}|\ldots\mu_{i}\mu_{i+1}\ldots}(\ldots,\theta_{i},\theta_{i+1},\ldots)=F_{n}^{\mathcal{O}|\ldots\mu_{i+1}\mu_{i}\ldots}(\ldots,\theta_{i+1},\theta_{i},\ldots)S_{\mu_{i}\mu_{i+1}}(\theta_{i,i+1})\,.
\left[\begin{array}{l}{Q_{1}^{0\prime}-iQ_{1}^{3\prime}}\\{Q_{1}^{2\prime}-iQ_{1}^{1\prime}}\\\end{array}\right]=\left[\begin{array}{l}{Q_{1}^{0}-iQ_{1}^{3}}\\{Q_{1}^{2}-iQ_{1}^{1}}\\\end{array}\right]+\left[\begin{array}{ll}{A_{11}}&{A_{12}}\\{A_{21}}&{A_{22}}\\\end{array}\right]\left[\begin{array}{l}{Q_{1}^{0}-iQ_{1}^{3}}\\{Q_{1}^{2}-iQ_{1}^{1}}\\\end{array}\right].
F_{g}(\beta)=\int_{{\calF}_{g}}d\mu(m)\sum_{s}\Lambda_{s}(\tau,\bar{\tau})\theta\left[\begin{array}{cc}{{\bf0}}&{{\bf0}}\\{{\bfs}_{2}}&{{\bfs}_{1}}\\\end{array}\right](0|{\bf\tilde{\Omega}}_{g})
{\calL}_{\mathrm{eff}}=\frac{1}{4\pi}\sum_{\varsigma=\pm}\left[\mu_{\varsigma}^{2}m-\frac{(eB)^{2}}{4m}\right]=\frac{\mu^{2}m}{2\pi}+\frac{(eB)^{2}}{8\pim}[(2\sigma)^{2}-1],
\varphi^{N}\leftrightarrowN_{n}^{(N)}\tilde{K}_{n}^{(N)}(\underline{{\theta}})\leftrightarrowp_{n}^{(N)}(\underline{{l}})=N_{n}^{(N)}\left(\sum_{i=1}^{n}(-1)^{l_{i}}\right)^{N}
\partial_{+}\partial_{-}(e^{-2\phi})=-\lambda^{2}+\frac{\mu^{4}}{16\beta}{}~\mathrm{sech}^{4}\left[\frac{\mu}{2\sqrt2}(\alpha(x^{+}-x_{0}^{+})-\frac{1}{\alpha}(x^{-}-x_{0}^{-}))\right],
H_{HO}=\hbar\Omega\left(a_{x}^{\dagger}a_{x}^{\prime}+a_{y}^{\dagger}a_{y}^{\prime}+1\right)-\frac{M\Omega^{2}\theta}{2}\left(a_{x}^{\dagger}a_{x}^{\prime}-a_{y}^{\dagger}a_{y}^{\prime}\right).
g=-\kappa^{\frac{4}{3}}\left(\begin{matrix}{-2(\psi^{2}-{\frac{f}{\kappa^{2}}})}&{0}&{-{\frac{1}{\sqrt{2}}}\psi}\\{0}&{{\frac{2}{f\kappa^{2}}}}&{0}\\{-{\frac{1}{\sqrt{2}}}\psi}&{0}&{-{\frac{1}{4}}}\\\end{matrix}\right).
\langleB\vert\intd\vec{x}\,\partial_{\mu}\widehatJ^{\mu}(x)\vertA\rangle=f_{\pi}m_{\pi}^{2}\langleB\vert\intd\vec{x}\,\pi(x)\vertA\rangle\nonumber\,=\langleB\vert\intd\vec{x}\,f_{\pi}j_{\pi}(x)\vertA\rangle,
H^{(0)}=\frac{1}{2}\left[\frac{\left({\bfp}_{r}{\,}^{2}+m_{1}^{2}\right)}{\mu_{1}}+\frac{\left({\bfp}_{r}{\,}^{2}+m_{2}^{2}\right)}{\mu_{2}}+\mu_{1}+\mu_{2}+\sigma^{2}{\bfr}^{2}\int_{0}^{1}\frac{d\beta}{\nu}+\nu_{0}+\frac{{\bfL}^{2}}{\rho{\bfr}^{2}}\right]
ds^{2}=\alpha^{\prime}\left[\frac{u^{2}}{{\tilde{R}}^{2}}(-\tilde{f}d\tilde{x}_{0}^{2}+d\tilde{x}_{1}^{2}+d\tilde{x}_{2}^{2}+d\tilde{x}_{3}^{2})+\frac{{\tilde{R}}^{2}}{u^{2}}(\tilde{f}^{-1}du^{2}+u^{2}d\Omega_{5}^{2})\right].
\psi(z)=\frac{(x^{2r+3}z;x^{4},x^{2r})_{\infty}(x^{2r+3}z^{-1};x^{4},x^{2r})_{\infty}}{(xz;x^{4},x^{2r})_{\infty}(xz^{-1};x^{4},x^{2r})_{\infty}}.
(\psi^{-},\phi^{+})=\sum_{jj_{3}}\int\frac{d^{3}\hat{\bboxp}}{2\hat{p}^{0}}\int_{(m_{1}+m_{2})^{2}}^{\infty}d{\mathsfs}\langle\psi^{-}|\hat{\bboxp}j_{3}{\mathsfs}j^{-}\rangleS_{j}({\mathsfs})\langle\hat{\bboxp}j_{3}{\mathsfs}j^{+}|\phi^{+}\rangle
\begin{array}{l}{\bar{c}(r^{11})=c_{n_{r}}^{\prime},\quad\bar{c}(r^{21})=c_{q_{r}}^{\prime},\quadN_{11}=N,\quadN_{21}=Q,}\\{\bar{c}(r^{12})=c_{m_{r}}^{\prime},\quad\bar{c}(r^{22})=c_{t_{r}}^{\prime},\quadN_{12}=M,\quadN_{22}=T,}\\\end{array}
\hat{a}_{2}(x)=\frac{1}{180}(R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}-R_{\mu\nu}R^{\mu\nu}+\BoxR)\hat{1}+\frac{1}{2}\hat{\calQ}^{2}+\frac{1}{12}\hat{\calR}_{\mu\nu}\hat{\calR}^{\mu\nu}+\frac{1}{6}\Box\hat{\calQ},
T_{B}^{(i)}(z)=\displaystyle\sum_{\mu=0}^{n-1}\phi_{(a,a+\bar{\varepsilon}_{\mu})}(z)K^{(i)}\left(\left.a+\bar{\varepsilon}_{\mu}\begin{array}{c}{a}\\{a}\\\end{array}\right|z\right)\phi^{(a+\bar{\varepsilon}_{\mu},a)}(z)
\lambda^{\mu}\left(t\right)=\bar{X}^{\mu}+\xi^{\mu}t-\frac{1}{2}\Gamma_{\nu\rho}^{\mu}\xi^{\nu}\xi^{\rho}t^{2}-\frac{1}{3!}\Gamma_{\nu\rho\kappa}^{\mu}\xi^{\nu}\xi^{\rho}\xi^{\kappa}t^{3}+\ldots,
\frac{d^{2}\xi}{dr^{2}}+\left(\frac{\omega^{2}}{\lambda^{4}}-\frac{\frac{l(l+1)}{r^{2}}+\frac{2M}{r^{3}}-\frac{2Q^{2}}{r^{4}}+m^{2}}{\lambda^{2}}-\frac{1}{\lambda}\frac{d^{2}\lambda}{dr^{2}}\right)\xi=0.
\gamma^{2}=\frac{2GM}{b}-\frac{2G}{b}\sqrt{M^{2}-\frac{8J^{2}\alpha^{2}}{9}}\quad;\quad\frac{\omega^{2}}{\alpha^{2}}=\frac{4GM}{b}+\frac{4G}{b}\sqrt{M^{2}-\frac{8J^{2}\alpha^{2}}{9}}
eV^{\mathrm{QED_{3}}}(\vec{r})=\frac{e^{2}}{2\pi}\left(1+\frac{e^{2}}{8\pim}\right)K_{0}\left(\frac{e^{2}}{8\pi}r\right)+\frac{8}{mr^{2}}\left[1-\frac{e^{2}}{8\pi}rK_{1}\left(\frac{e^{2}}{8\pi}r\right)\right]L.
ds^{2}=\left(1-{\frac{r_{-}}{r}}\right)^{-1}(dx^{5}+{\frac{2Q}{r}}dt)^{2}-\left(1-{\frac{r_{+}}{r}}\right)dt^{2}+{\frac{1-r_{-}/r}{1-r_{+}/r}}dr^{2}+r^{2}(1-{\frac{r_{-}}{r}})^{2}d\Omega_{2}^{2},
\begin{array}{rcl}{[\rho_{i,k},\rho_{i,k\prime}]}&{=}&{T_{i}{(D^{-1})}_{i,j}{\frac{1}{2\pi}}k\delta_{k+k\prime},}\\{\rho_{e}}&{=}&{-e\sum_{i}\rho_{i},}\\{\left[\rho(x),\psi(x\prime)\right]}&{=}&{\ell_{i}(D^{-1})_{ij}\delta(x-{x\prime})\psi_{\ell}({x\prime}),}\\{\psi_{\ell}}&{\propto}&{e^{i{\frac{\ell_{i}}{T_{i}}}\phi_{i}}}\\\end{array}
S[\Phi]=\int_{t_{i}}^{t}\,d^{4}x\left[\textstyle{{\frac{1}{2}}}\partial_{\mu}\Phi\partial^{\mu}\Phi-\textstyle{{\frac{1}{2}}}m^{2}\Phi^{2}-\textstyle{{\frac{1}{4!}}}g^{2}\Phi^{4}\right],
b_{n}J_{\nu}^{\prime}(kr^{\prime})+c_{n}J_{-\nu}^{\prime}(kr^{\prime})-a_{n}J_{\nu}^{\prime}(kr^{\prime})={\frac{2\nu\Gamma\left({\frac{D-2}{2}}\right)}{4(\pir^{\prime})^{\frac{D}{2}}k}}
\widetilde{\psi}_{-}(x^{+},x^{-},\widetilde{k})=\Biggl({\frac{m-\widetilde{k}\cdot\widetilde{\gamma}}{m^{2}+\widetilde{k}\cdot\widetilde{k}}}\Biggr)\gamma^{+}i\partial_{+}\widetilde{\psi}_{+}(x^{+},x^{-},\widetilde{k})\;.
f^{2}g^{2}{\frac{\partial^{2}}{\partialr^{*2}}}t+f(fg^{\prime\prime}+f^{\prime}g^{\prime}){\frac{\partial}{\partialr^{*}}}t-\{\sqrt2ff^{\prime}-2f(1-f)\}t-{\frac{\partial^{2}}{\partialt^{2}}}t=0.
\begin{array}{c}{\omega=\mathrm{d}{\bfp}\wedge\mathrm{d}{\bfx}+\sum_{i=1}^{N}\mathrm{d}{\bfp}_{i}\wedge\mathrm{d}{\bfe}_{i},}\\{{\calH}={s}\left[{\bfp}{\bfe}_{1}+\sum_{i=1}^{N}{k}_{i-1}{\phi}_{i-1.i}+\frac{{k}_{N}}{2c}({\Phi}_{N.N}-c^{2})+\sum_{i,j=1}^{N}d_{ij}({\bfe}_{i}{\bfe}_{j}-\delta_{ij})\right],}\\\end{array}
\int_{0}^{1}dt\cdot\omega_{\mu}(t,z,\tau)+\tilde{q}^{-\mu}\int_{0}^{1}dt\cdot\omega_{-\mu}(t,z,\tau)=(1-\tilde{q}^{-\mu})\int_{0}^{-z}dt\cdot\omega_{\mu}(t,z,\tau)
S_{0}^{\prime}\,=\,\intd^{4}x\,\,\,J_{_{5\,\mu}}^{^{BPHZ}}\,\partial_{\mu}\rho\,=\,\intd^{4}x\,e\,\alpha_{1}\,\overline{\psi}\gamma^{\mu}\gamma_{5}\partial_{\mu}\rho\,\psi\,\,.
G_{l,s}^{\,\alpha}(\vec{\alpha},\vec{\lambda},\vec{p})=-\tilde{G}_{l,s}(\vec{\alpha},\vec{\lambda}^{\prime},0)\,\Theta(\alpha_{s}-\alpha)+\sum_{j=1}^{3}\sum_{\mu=0}^{3}p_{\mu,j}\\Bigl(\partial_{\mu,j}\,\tilde{G}_{l,s}\Bigl)(\vec{\alpha},\vec{\lambda}^{\prime},\lambda_{\sigma(l,s)}\,\vec{p})\,\Theta(\alpha-\alpha_{s})\quad.
G_{i}^{(sign1,sign2);\nu}=\oint_{C_{i}}dz~\left\langle\sigma^{sign1}(z_{1})\sigma^{sign2}(z_{2})\psi(z)\right\rangle_{\nu}\left\langlee^{-\fraci2\alpha_{-}\phi(z_{1})}e^{-\fraci2\alpha_{-}\phi(z_{2})}e^{i\alpha_{-}\phi(z)}\right\rangle_{\nu}
X_{\bot}^{\mu\nu}X_{\bot}^{\rho\sigma}=\frac{1}{6}\left(\,\pi^{\mu\sigma}\pi^{\nu\rho}-\varepsilon^{\mu\sigma\lambda}e_{\lambda}^{(0)}\varepsilon^{\nu\rho\tau}e_{\tau}^{(0)}\,\right)\cdotC+\frac{i}{4}\left(\,\pi^{\mu\sigma}\varepsilon^{\nu\rho\lambda}+\pi^{\nu\rho}\varepsilon^{\mu\sigma\lambda}\,\right)e_{\lambda}^{(0)}\cdotQ_{0}.
H=\int{d^{3}}x\left({-\frac{1}{2}\Pi_{i}^{a}\Pi_{ai}+\frac{1}{4}F_{ij}^{a}F^{aij}+c^{a}\left(x\right)\left({\partial^{i}\Pi_{i}^{a}+gf^{abc}A^{bi}\Pi_{i}^{c}-J^{0}}\right)}\right),
E_{-}^{(2)}=E_{-}-{\frac{\hbar^{2}}{16m^{2}}}\sum_{n=1}^{\infty}{\frac{|p_{2}(0)|^{2}}{E_{1,n}-E_{-}}}\left|\intdx\,\varphi_{1,n}(x)\partial_{x}\theta(x)\right|^{2}\varphi_{2+}(0)^{2}.
{\frac{ds^{2}}{l_{s}^{2}g_{5}^{2}N/4\pi}}={\frac{dr^{2}}{r(1-{\frac{1}{r^{6}}})}}+r^{3}(1-{\frac{1}{r^{6}}})d\tau^{2}+r^{3}\sum_{i=1}^{4}dx_{i}^{2}+rd\Omega_{4}^{2}.
\mathrm{ch}_{en}(g)=Str_{en}(g)=\mathrm{exp}i[x_{n}e+x_{e}(n-1/2)]\left[e^{-ix_{e}/2}(1-\frac{e}{2}x_{\psi^{+}}x_{\psi})-e^{ix_{e}/e}(1+\frac{e}{2}x_{\psi^{+}}x_{\psi})\right]
L_{DR}=-{\frac{a}{4}}(B+\gammaA)_{\mu\nu}(B+\gammaA)^{\mu\nu}-{\frac{m}{6\pi}}B_{\mu}B^{\mu}+\alpha\epsilon^{\mu\nu\lambda}(B+\gammaA)_{\mu}\partial_{\nu}(B+\gammaA)_{\lambda}.
\begin{aligned}{\gamma^{r}=\sqrt{2}\,\beta^{-1/2}\,\psi^{r}\,,}&{{}\quad\quad\partial_{\mu}=\beta^{-1}g^{\frac{1}{4}}\,ip_{\mu}\,g^{-\frac{1}{4}}\,,}\\{\{\psi^{r},\psi^{s}\}=\beta\,\delta^{rs}\,,}&{{}\quad\quad[x^{\mu},p_{\nu}]=i\beta\,\delta^{\mu}{}_{\nu}\,.}\\\end{aligned}
{\calL}_{F}=\bar{\psi}i\gamma^{\mu}(\partial_{\mu}-ieA_{\mu}-iB_{\mu})\psi-{\frac{1}{2g}}\midB_{\mu}\mid^{2}-m\bar{\psi}\psi-{\frac{pe^{2}}{4}}\midA_{\mu\nu}\mid^{2}+{\frac{qe^{2}}{2}}\epsilon_{\mu\nu\lambda}A^{\mu}A^{\nu\lambda}.
E_{gg}(\omega_{n})={\frac{1}{2}}\sum_{n=1}^{M-1}\omega_{n}-{\frac{1}{2}}\lambdaM\left[{\frac{a^{2}d}{2}}+R_{1}^{2}\right]^{-d/2}\!\!\!+g^{2}M\sum_{k=1}^{M}\left[{\frac{a^{2}d}{2}}+R_{k}^{2}\right]^{-d/2}\,.
T=\left(\begin{array}{cc}{a}&{\beta}\\{\gamma}&{d}\\\end{array}\right)=\left(\begin{array}{cc}{\mathrm{e}^{x}}&{\mathrm{e}^{x}\xi}\\{\zeta\mathrm{e}^{x}}&{\mathrm{e}^{\hat{x}}+\zeta\mathrm{e}^{x}\xi}\\\end{array}\right)\,.
\int\!d^{4}x\varphi_{1}(x,\theta)*\varphi_{2}(x,\theta)=\int\!d^{4}x\varphi_{1}(x,\theta)\varphi_{2}(x,\theta)=\int\!d^{4}x\varphi_{2}(x,\theta)*\varphi_{1}(x,\theta),
g(p)=\int\frac{d^{2}{\bfq}}{(2\pi)^{2}}\,\frac{1}{(p_{3}-q_{3}^{\mathrm{pole}})^{2}+\left|{\bfp-q}\right|^{2}}.\frac{g(q_{3}^{\mathrm{pole}},\left|{\bfq}\right|)}{-\delta+\frac{\left|{\bfq}\right|^{2}}{m}+2\Re\Sigma_{-}(q_{3}^{\mathrm{pole}},\left|{\bfq}\right|)}.
\partial_{t_{i}}\partial_{t_{j}}\lambda_{SW}=\sum_{k=1}^{r}{C_{ij}}^{k}(t)\partial_{t_{k}}\partial_{t_{r}}\lambda_{SW}+{\frac{1}{2\pii}}\mu\partial_{\mu}\left({\frac{x^{-2}Q_{ij}}{(\widetilde{W}_{B}^{2}-4\mu^{2})^{1/2}}}\right)dx+\partial_{x}(*)\,dx,
\gamma_{g,9}={\mathrm{diag}}(\omega{\bfI}_{2},\omega^{2}{\bfI}_{2},\omega^{3}{\bfI}_{2},\omega^{4}{\bfI}_{2},\omega^{5}{\bfI}_{2},\omega^{6}{\bfI}_{2},{\bfI}_{4})~.
\Omega^{(1)}(E)=\sum_{n=1}^{\infty}\sum_{p=0}^{1}\sum_{m=0}^{\infty}A(m,n,p)\frac{e^{\beta_{n}E}}{E^{27/2+m+p}}{\calH}\left(E-\frac{8\pin}{\beta_{H}\varepsilon}\right)
{\calL}_{\mathrm{eff}}^{\beta,\mu}(E)={\calL}_{\mathrm{eff}}^{\beta,\mu}(0)-\frac{(eE)^{2}}{24\pi^{2}}\left(1-2\pi(\frac{m}{T})^{2}\sum_{l=0}^{\infty}\frac{1}{\left((m/T)^{2}+(2l+1)^{2}\pi^{2}\right)^{3/2}}\right)\,
(\,[X_{\bfm},X_{\bfn}]\,)_{\bfkl}=\sum_{\bfj}\left[\left(\,X_{\bfm}\,\right)_{\bfkj}\,\left(\,X_{\bfn}\,\right)_{\bfjl}-\left(\,X_{\bfn}\,\right)_{\bfkj}\,\left(\,X_{\bfm}\,\right)_{\bfjl}\right]=-4\pi^{2}{\bfm}\times{\bfn}\,\left(\,X_{\bfm+n}\,\right)_{\bfkl}\,.
\Phi^{0}(z,{\bfk}_{\|},\omega)=i\sqrt{{\frac{16\pi^{3}\hbar|\omega|}{\kappa^{2}}}}\mathrm{sin}\left(|\kappa|z\right)\left[a(|\kappa|\hat{\bfz}+{\bfk}_{\|})\Theta(\omega)+a(|\kappa|\hat{\bfz}-{\bfk}_{\|})^{\dagger}\Theta(-\omega)\right]\Theta(\kappa^{2}),
\mathcal{F}=\frac{a}{2}e^{-2g^{1}}\left(2g^{1}D+\frac{4(g^{2})^{2}+6g^{2}D-2D+D^{2}}{D-2+4g^{2}}\right)\sqrt{\frac{\pi}{D-2+4g^{2}}}.
+\;{\frac{1}{8\pi^{2}}}\intd^{4}x\intd^{4}yB^{\mu}(x){\frac{\partial}{\partialx^{\mu}}}G(x,y)\left[fe^{2}\tilde{F}^{\mu\nu}(y)F_{\mu\nu}(y)+{\frac{1}{3}}\;f^{3}\;\tilde{G}^{\mu\nu}(y)G_{\mu\nu}(y)\right].
V_{gen}^{(1)}=\frac{gH}{2\pi}\sum_{l=-\infty}^{+\infty}\int_{-\infty}^{+\infty}\frac{dp_{3}}{2\pi}\sum_{n=0,\sigma=\pm1}^{\infty}log[\beta^{2}(\omega_{l}^{2}+\epsilon_{n,\sigma,p_{3}}^{2}+\Pi(n,T,H,\sigma))]
\phi(g_{1},\ldots,g_{4})=\sum_{N_{1}\ldotsN_{4}}\sum_{\Lambda}\phi_{\calA}^{N_{1}\ldotsN_{4}\,\Lambda}\sqrt{\mathrm{dim}_{N_{1}}\ldots\mathrm{dim}_{N_{4}}}\left({C}_{\beta_{1}\ldots\beta_{4}}^{N_{1}\ldotsN_{4}\,\Lambda}D_{\beta_{1}\gamma_{1}}^{(N_{1})}(g_{1})\ldotsD_{\beta_{4}\gamma_{4}}^{(N_{4})}(g_{4})\w_{\gamma_{1}}\ldotsw_{\gamma_{4}}\\right).
m_{\alpha}^{2}=V_{0}+m_{3/2}^{2}+m_{3/2}^{2}n_{\alpha}\left(1+\frac{1}{3}\tilde{\delta}_{GS}\frac{dP}{dy}\right)^{-2}|1-\frac{1}{3}\tilde{\delta}_{GS}\rho|^{2}(T+\bar{T})^{2}\left|\hat{G}(T,\bar{T})\right|^{2}
{\calD}(g)=\left(\begin{array}{ccc}{z_{1}^{\prime}}&{-\bar{z}_{2}^{\prime}}&{-\chi}\\{z_{2}^{\prime}}&{\bar{z}_{1}^{\prime}}&{-\bar{\chi}}\\{\theta^{\prime}}&{-\bar{\theta}^{\prime}}&{\lambda}\\\end{array}\right)\,,\quad\quad\sum_{i}|z_{i}^{\prime}|^{2}+\bar{\theta}^{\prime}\theta^{\prime}=1
{\cal{V}}=-\frac{\pi^{2}}{V}\frac{N-1}{N}+\frac{V}{2\pi^{2}}\left(\frac{4\pi}{V}\sum_{i=2}^{N}|A_{+}^{(i1)}|^{2}+\frac{1}{2\pi^{2}}\sum_{k}(B^{k})^{2}+\frac{1}{\pi^{2}}\sum_{i}\sum_{j>i}|X^{(ij)}|^{2}\right)
F^{\alpha\beta,\mu\nu}(\nabla)=\sqrt{-g}C^{\alpha\beta,\lambda\sigma}\left\{\Box\delta_{(\lambda}^{\mu}\delta_{\sigma)}^{\nu}-\frac{2\lambda}{1+\lambda}\delta_{(\lambda}^{(\mu}\nabla_{\sigma)}\nabla^{\nu)}+\frac{\lambda}{1+\lambda}g^{\mu\nu}\nabla_{(\lambda}\nabla_{\sigma)}+P_{\lambda\sigma}^{\mu\nu}\right\},
\left[(\frac{\partial}{\partialt}-\Omega\frac{\partial}{\partial\theta})^{2}-\frac{1}{r}\frac{\partial}{\partialr}r\frac{\partial}{\partialr}-\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}-\frac{\partial^{2}}{\partialz^{2}}\right]\varphi(t,r,\theta,z)=0
\prod_{l=1}^{p-1}[x-\mathrm{exp}(2i\pil/p]=\frac{\prod_{l=0}^{p-1}(x-\mathrm{exp}2i\pil/p)}{x-1}=\frac{x^{p}-1}{x-1}=\sum_{l=0}^{p-1}x^{l}
\left|\begin{array}{cccc}{f^{0}}&{\cdots}&{f^{n}}&{f^{A}}\\{\vdots}&{}&{\vdots}&{\vdots}\\{f^{(n+1)\,0}}&{\cdots}&{f^{(n+1)\,n+1}}&{f^{(n+1)\,A}}\\\end{array}\right|\equiv\left\{\sum_{k=0}^{n+1}U_{n+1-k}\partial^{(k)}\right\}f^{A}=0,
z=\frac{1}{a_{1}}(w-w_{0})+\sum_{m=2}^{\infty}\frac{(w-w_{0})^{m}}{m!}\frac{(-1)^{m+1}(m-2)}{a_{1}^{2m-1}}\sum_{l_{1}=0}^{\infty}\cdots\sum_{l_{m}=0}^{\infty}\delta_{(\suml_{i}-m+1,0)}\quad{}C_{l_{1}}\ldotsC_{l_{m}}
\Box\sigma_{T}+(m^{2}+\xiR)\sigma_{T}-4\kappa\sigma_{T}^{3}+\frac{3\kappa\sigma_{T}m}{\piT}+\frac{15\lambda^{2}\sigma_{T}T}{4\pim}-\frac{15\lambda^{2}\sigma_{T}^{3}m}{2\piT}+3\lambda^{2}\sigma_{T}^{5}=0
S^{+}\otimesS^{+}\,\Psi^{\mu}\,\Psi^{\nu_{1}}\Psi^{\nu_{2}}\,\Psi^{\nu_{3}}\Psi^{\nu_{4}}\to\Gamma_{\nu}\times\eta^{\nu\nu_{1}}\eta^{\nu_{2}\nu_{3}}\eta^{\nu_{4}\mu}\,.
[I(\tau,\vec{\Sigma}(\tau,\vec{\sigma}_{o}))]_{\;\check{r}}^{i}\,\frac{\partial\Sigma^{\check{r}}(\tau,\vec{\sigma}_{o})}{\partial\sigma_{o}^{\check{s}}}=\frac{\partial\alpha_{o}^{i}(\vec{\sigma}_{o})}{\partial\sigma_{o}^{\check{s}}},
\nabla^{\gamma}\nabla^{\alpha}C_{\alpha\beta\gamma\delta}=\frac{1}{2}\BoxR_{\beta\delta}-\frac{1}{2}\nabla_{\gamma}\nabla_{\delta}R_{\beta}^{\gamma}+\frac{1}{12}\nabla_{\beta}\nabla_{\delta}R-\frac{1}{12}g_{\beta\delta}\BoxR
Q(\Phi_{0})_{ij}=Q\left(\sum_{k}(e^{-\Psi_{0}})_{ik}\starQ(e^{\Psi_{0}})_{kj}\right)=-\sum_{k,l,m}(e^{-\Psi_{0}})_{ik}\starQ(e^{\Psi_{0}})_{kl}\star(e^{-\Psi_{0}})_{lm}\starQ(e^{-\Psi_{0}})_{mj}
\partial_{k}C_{ij}^{(2)}\left(-{\frac{1}{12}}\epsilon_{ijklmn}\mathrm{Tr}\;(X^{l}[X^{m},X^{n}])+{\frac{1}{4}}\mathrm{Tr}\;(X^{k}\epsilon^{abcd}(F_{ab}F_{ci}F_{dj}-{\frac{1}{4}}F_{ab}F_{cd}F_{ij}))+\cdots\right)
\begin{array}{l}{(G_{\mu})_{\mathrm{B}}=Z_{G}^{1/2}G_{\mu},~~(\phi)_{\mathrm{B}}=Z_{\phi}^{1/2}\phi,}\\{\displaystyle(g)_{\mathrm{B}}=Z_{G}^{-1/2}Z_{g}g,~~(\lambda)_{\mathrm{B}}=Z_{\phi}^{-2}Z_{\lambda}\lambda,}\\{\displaystyle(\mu^{2})_{\mathrm{B}}=Z_{\phi}^{-1}\mu^{2}\left(1+\frac{\delta\mu^{2}}{\mu^{2}}\right),~~(v)_{\mathrm{B}}=Z_{\phi}^{1/2}v\left(1+\frac{\deltav}{v}\right).}\\\end{array}
L={\frac{1}{2}}\intd^{D}{\bfx}\,\dot{\phi}^{2}-{\frac{1}{2}}\intd^{D}{\bfx}d^{D}{\bfy}\,\phi({\bfx})F({\bfx}-{\bfy})\phi({\bfy})=\int{\frac{d^{D}{\bfk}}{(2\pi)^{D}}}{\frac{1}{2}}\left[|\dot{Q}_{\bfk}|^{2}-\omega_{\bfk}^{2}|Q_{\bfk}|^{2}\right]
\left(\begin{array}{cccc}{0}&{\ldots}&{}&{0}\\{\vdots}&{}&{}&{1}\\{}&{}&{\ddots}&{\vdots}\\{0}&{-1}&{\ldots}&{0}\\\end{array}\right)\,\qquad\qquad\left(\begin{array}{cccc}{0}&{\ldots}&{}&{0}\\{\vdots}&{}&{}&{i}\\{}&{}&{\ddots}&{\vdots}\\{0}&{i}&{\ldots}&{0}\\\end{array}\right)\,
\times\prod_{I=2}^{N}\;\Bigg\langle\prod_{b=1}^{N}\;\prod_{j=1}^{n_{b}}:e^{-i2\sqrt{\pi}U_{Ib}\Phi^{(I)}(x_{j}^{(b)})}:_{M^{(I)}}\;\prod_{l=1}^{m_{b}}:e^{+i2\sqrt{\pi}U_{Ib}\Phi^{(I)}(y_{l}^{(b)})}:_{M^{(I)}}\Bigg\rangle_{Q^{(I)}}
\delta_{\epsilon}B^{(1)\mu_{1}\ldots\mu_{p-2k-2}}=\partial^{\left[\mu_{1}\right.}\epsilon^{\left.\mu_{2}\ldots\mu_{p-2k-2}\right]}-M\bar{\epsilon}^{\mu_{1}\ldots\mu_{p-2k-2}}+\partial_{\mu}\epsilon^{\mu\mu_{1}\ldots\mu_{p-2k-2}},
\left\{\phi_{a}(\vec{\sigma}),\,\phi_{b}(\vec{\sigma}^{\,\prime})\right\}_{P}=\intd^{2}\vec{\sigma}^{\,\prime\prime}\,C_{ab}^{\\\c}(\vec{\sigma},\vec{\sigma}^{\,\prime};\vec{\sigma}^{\,\prime\prime})\,\phi_{c}(\vec{\sigma}^{\,\prime\prime})\,,
\theta_{(3)}^{(2,0)}(\tau)=\sum_{(m_{1},m_{2})\in{\bfZ}^{2}}q^{\frac{4}{9}+2m_{1}+3(m_{1}^{2}+m_{2}^{2}-m_{1}m_{2})}=:q^{\frac{4}{9}}{\tilde{\theta}_{(3)}^{1}}(\tau),
\intd^{2}\vec{x}\alpha_{b}(\vec{x})\{A_{i}^{a}(\vec{x^{\prime}}),G^{b}(\vec{x})\}=-\partial_{i}\alpha^{a}(\vec{x^{\prime}})+f^{abc}\alpha_{b}(\vec{x^{\prime}})A_{ic}(\vec{x^{\prime}})
\left(\frac{\partial^{2}}{\partialr^{2}}+\frac{3}{r}\frac{\partial}{\partialr}-\frac{4}{r^{2}}\:\hat{L}^{2}-\frac{r^{2}}{g^{2}}+\frac{2}{g^{2}}\:E\right)\Psi(r,\vartheta,\psi)=0
V_{matter}^{fd}={\frac{2N_{F}^{fd}}{\pi^{2}L^{4}}}{\frac{\beta^{2}}{24}}\Biggl(\sum_{i=1}^{N-1}\Bigl(6\theta_{i}^{2}-12\pi\theta_{i}+\beta^{2}+4\pi^{2}\Bigr)+6\Bigl(\sum_{i=1}^{N-1}\theta_{i}\Bigr)^{2}-12\pi\sum_{i=1}^{N-1}\theta_{i}+\beta^{2}+4\pi^{2}\Biggr).
ds_{volcano}^{2}=\eta_{\mu\nu}dZ^{\mu}dZ^{\nu}=-dt^{2}+\frac{1+A^{2}/r^{2p-2}}{1-(r_{0}/r)^{2p-2}}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\thetad\Omega_{p-2}^{2})
L_{M}=-\frac{1}{4}e^{2\kappa\Psi}F_{\mu\nu}^{a}F^{\mu\nu,a}-\frac{1}{2}\partial_{\mu}\Psi\partial^{\mu}\Psi-\frac{1}{2}e^{-4\kappa\Psi}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}-e^{-2\kappa\Psi}V(\Phi^{a})\,
\tilde{S}=\pit\left[\left(m_{i}+C_{ijk}n^{j}p^{k}\right)^{2}+\left(n^{i}\right)^{2}\left(p^{i}\right)^{2}-\left(n^{i}p^{i}\right)^{2}\right]+2\piim_{i}\left(\tau_{1}n^{i}+\omega_{1}p^{i}\right),
T_{N}^{M}=\partial^{M}\Phi^{a}\partial_{N}\Phi^{a}-{\frac{1}{2}}\delta_{N}^{M}\partial_{A}\Phi^{a}\partial^{A}\Phi^{a}-{\frac{\sqrt{-h}}{\sqrt{-g}}}\sigma_{i}(\Phi^{a})\delta(y_{i})\mathrm{diag}(1,1,1,1,0)\,.
R=A_{1}{\frac{\rho^{z_{1}}\left[-\mu+\lambda+{\frac{2c_{2}\lambda\rho^{2\lambda}}{1-c_{2}\rho^{2\lambda}}}\right]^{2}}{[(r^{d}-r_{1}^{d})(r^{d}-r_{2}^{d})]^{1+{\frac{1}{d}}}(1-c_{2}\rho^{2\lambda})^{z_{2}}}}+A_{2}{\frac{\rho^{z_{1}+2\lambda}\left({\frac{1-c_{2}}{1-c_{2}\rho^{2\lambda}}}\right)^{z_{2}+2}}{[(r^{d}-r_{1}^{d})(r^{d}-r_{2}^{d})]^{1+{\frac{1}{d}}}}}
\displaystyle{\hat{M}_{0}(z)=\left(\begin{array}{ccc}{\hat{\omega}(z)+\frac{1}{2}\hat{\lambda}(z)}&{0}&{0}\\{-i\mathrm{b}}&{\tilde{\hat{\omega}}(z)-\frac{1}{2}\hat{\lambda}(z)}&{2\hat{\rho}^{j}(z)}\\{2\bar{\hat{\rho}}_{i}(z)}&{0}&{\hat{T}_{i}^{~j}(z)}\\\end{array}\right)}
{\hat{A}}{\hat{k}}\equiv\left(\begin{array}{ccc}{c_{3}c_{2}}&{-c_{3}c_{1}}&{\Phi_{1.3}}\\{{\Phi}_{1.3}}&{{\Phi}_{2.2}-c_{3}^{2}}&{c_{3}c_{2}}\\\end{array}\right)\left(\begin{array}{c}{k_{1}}\\{k_{2}}\\{k_{3}}\\\end{array}\right)=0
m_{\tilde{G}}={\frac{1}{3}}\langle\,|M|\,\rangle=\frac{\,1}{\,4\,}\,\langle\,|\sum_{a}b_{a}^{\prime}u_{a}-4e^{K/2}W|\,\rangle=\frac{\,1}{\,4\,}\,\langle\,|\sum_{a}b_{a}u_{a}|\,\rangle\,\approx\,{\frac{1}{4}}b_{+}\langle\,\rho_{+}\,\rangle.
\alpha_{(0)}^{\prime2}+\alpha_{(0)}^{\prime}\beta_{(0)}^{\prime}=\frac{1}{6}\phi_{(0)}^{\prime2}-\frac{1}{6}\left(b^{2}-\frac{2}{3}\right)\sigma^{2}e^{2(\gamma_{(0)}-\sqrt{2}b\phi_{(0)})},
\begin{array}{lclclcl}{Z_{ab}^{V}}&{=}&{\tilde{I}_{ab}+\deltaZ_{ab}^{V}}&{\qquad}&{Z_{f_{i}}^{R}}&{=}&{1+\deltaZ_{f_{i}}^{R}}\\{Z_{ab}^{S}}&{=}&{\tilde{I}_{ab}+\deltaZ_{ab}^{S}}&{\qquad}&{Z_{f_{i}}^{L}}&{=}&{1+\deltaZ_{f_{i}}^{L}}\\\end{array}
D^{1\alpha}D_{\alpha}^{1}(\Phi\bar{\Phi}\;W^{[a_{1},\ldots,a_{N-1}]})=\bar{D}_{N\dot{\alpha}}\bar{D}_{N}^{\dot{\alpha}}(\Phi\bar{\Phi}\;W^{[a_{1},\ldots,a_{N-1}]})=0\;.
Z_{K}(V)\approx\sqrt{\frac{\alpha}{2\pi}}\,\frac{\sqrt{K}}{d}\left(\sqrt{\frac{2\pi}{\alpha}}\,\frac{C}{(K\tauh(d))^{\beta-1}}\right)^{K}\frac{\Gamma((\beta-1)K)}{\Gamma(\betaK+1)}\V^{\betaK}\,,
\begin{array}{ll}{D_{\mu}^{i}=\nabla_{\mu}+i\frac{q}{2\sqrt{\beta}}Q_{i}\tilde{H}_{\mu},}\\{D_{\mu}^{I}=\nabla_{\mu}+i\frac{e}{2}E_{I}\tilde{A}_{\mu}-i\frac{\alphae}{2\sqrt{\beta}}E_{I}\tilde{H}_{\mu},}\\\end{array}
\left(f_{2}-\frac{\partialf_{2}}{\partialz}\right)|_{z=0}=0\,\\\left(m_{2}+\frac{\partialm_{2}}{\partialz}\right)|_{z=0}=0\,\\\left(l_{2}+\frac{\partiall_{2}}{\partialz}\right)|_{z=0}=0\.
\oint_{\infty}d\Sigma^{r}\epsilon_{0}^{\ast}\frac{\tilde{\epsilon}}{r^{2}}=\oint_{\infty}d\Sigma^{r}\lbrace\epsilon_{0}^{\ast}\gamma^{r}\gamma^{m}\Gamma_{m}\epsilon_{0}+\epsilon_{0}^{\ast}\frac{\gamma^{r}}{r^{3}}(\gamma^{\alpha}\partial_{\alpha})\tilde{\epsilon}\rbrace
\begin{array}{l}{:i<P^{+}f>g<\partial_{z}X^{-}h>:\rightarrow}\\{\quad\quad\quadi\left(\frac{p^{+}}{2z}-\frac{p^{+}}{z-w}\right):fg\left(-p^{-}\int^{w}\frac{dw^{\prime}}{k}\frac{-i}{(z-w^{\prime})^{2}}:e^{\frac{2u(w^{\prime})}{k}}:\left(\frac{w-w^{\prime}}{\sqrt{ww^{\prime}}}\right)^{\frac{2it_{2}}{k}}\right)h:}\\\end{array}
\begin{array}{ccccccccccl}{D3:}&{1}&{2}&{3}&{\_}&{\_}&{\_}&{\_}&{\_}&{\_}&{\quad\mathrm{probe}\nonumber}\\{F1:}&{\_}&{\_}&{\_}&{4}&{\_}&{\_}&{\_}&{\_}&{\_}&{\quad\mathrm{soliton}\nonumber}\\{D1:}&{\_}&{\_}&{\_}&{4}&{\_}&{\_}&{\_}&{\_}&{\_}&{\quad\mathrm{soliton.}}\\\end{array}
M_{R}=\left(\begin{array}{ccc}{1}&{r_{1}}&{r_{3}}\\{0}&{1}&{r_{2}}\\{0}&{0}&{1}\\\end{array}\right),\;\;M_{D}=\left(\begin{array}{ccc}{d_{1}}&{0}&{0}\\{0}&{d_{2}}&{0}\\{0}&{0}&{(d_{1}d_{2})^{-1}}\\\end{array}\right),\;\;M_{L}=\left(\begin{array}{ccc}{1}&{0}&{0}\\{l_{1}}&{1}&{0}\\{l_{3}}&{l_{2}}&{1}\\\end{array}\right).
e^{-\frac{e^{2}}{2\pi}\left(\varepsilon_{\mu\nu}\partial_{\mu}b_{\nu}^{(l)},K_{ll}\,\varepsilon_{\rho\sigma}\partial_{\rho}b_{\sigma}^{(l)}\right)}=\Big\langlee^{ie\frac{1}{\sqrt{\pi}}\left(\varepsilon_{\mu\nu}\partial_{\nu}\varphi^{(l)},b_{\mu}^{(l)}\right)}\Big\rangle_{B}\;,
\begin{array}{ccccccc}{\frac{1}{g_{s}}}&{}&{}&{}&{}&{}&{}\\{}&{\diagdown}&{}&{}&{}&{}&{}\\{}&{}&{\frac{R_{1}R_{2}}{g_{s}^{2}}}&{-}&{\frac{R_{1}R_{2}R_{3}}{g_{s}^{2}}}&{-\dots-}&{\frac{1}{R_{d}}}\\{}&{\diagup}&{}&{}&{}&{}&{}\\{\frac{R_{1}}{g_{s}}}&{}&{}&{}&{}&{}&{}\\\end{array}
S_{E}=\frac{1}{2\kappa^{2}}\intd^{10}x\sqrt{-g}\left(R-\frac{1}{2}\left(\partial\Phi\right)^{2}-\frac{1}{2\left(p+2\right)!}e^{\frac{3-p}{2}\Phi}F_{p+2}^{2}\right).
{\mathcalH}\;=\;-\partial^{2}+g^{2}\varphi^{2}-g\gamma_{2}\partial_{2}\varphi\;\;,\;\;{\tilde{\mathcalH}}\;=\;-\partial^{2}+g^{2}\varphi^{2}+g\gamma_{2}\partial_{2}\varphi\;.
\bar{\phi}_{ab}(z_{2})\otimes\bar{\phi}_{bc}(z_{1})\check{R}(z_{1}-z_{2})=\sum_{d}W_{z_{1}-z_{2}}\left(\begin{array}{cc}{a}&{b}\\{d}&{c}\\\end{array}\right)\bar{\phi}_{ad}(z_{1})\otimes\bar{\phi}_{dc}(z_{2}).
\begin{array}{rcl}{\omega^{x}}&{=}&{2\,f_{tt}\,df_{tx}\wedgedf_{tt}+\frac{3}{2}\,df_{x}\wedgedf_{tx},}\\{\omega^{t}}&{=}&{2\,f_{tx}\,df_{tt}\wedgedf_{tx}-\frac{1}{2}\,df_{x}\wedgedf_{xx}}\\\end{array}
\frac{\widehat{SU}(3)_{1}\times\widehat{SO}(4)_{1}}{\widehat{SU}(2)_{2}\times\widehat{U}(1)_{24}}\cong\frac{\widehat{SU}(2)_{2}\times\widehat{SO}(2)_{1}}{\widehat{U}(1)_{8}}~.
L=\Gamma\int_{D}d^{2}x\epsilon^{\mu\nu\rho}\left<[Q,D_{\mu}Q]F_{\nu\rho}\right>+\frac{\kappa}{2\pi}\int_{D}d^{2}x\epsilon^{\mu\nu\rho}\left<A_{\mu}\partial_{\nu}A_{\rho}+\frac{2}{3}A_{\mu}A_{\nu}A_{\rho}\right>,
S_{eff}=-{\frac{\kappa}{4}}\,\intd^{2}\sigma\,\left[G_{ij}\,\partial_{+}x^{i}\,\partial_{-}x^{j}+B_{ij}\,(\partial_{+}x^{i}\,\partial_{-}x^{j}-\partial_{-}x^{i}\,\partial_{+}x^{j})\right]\,
{\calZ}_{BF}={\displaystyle\int}{\calD}(n^{a},\lambda_{2}^{\prime})~({\sfdet}|-n^{2}|)^{1/2}~{\sfexp}\left({\displaystylei\int{\sfdxdt}\left[|n|^{2}(\lambda_{2}^{\prime})^{2}-\frac{\displaystyle(\partial_{1}{\widetilden}_{_{BF}}^{a})(\partial_{1}{\widetilden}_{a_{(BF)}})}{\displaystyle4}\right]}\right).
S(\varphi)=\int_{0}^{L}dz\intd^{d-1}r(\frac{1}{2}(\partial\varphi)^{2}+\frac{1}{2}m^{2}\varphi^{2}+\frac{1}{4!}\lambda\varphi^{4})+\intd^{d-1}r(c_{1}\varphi^{2}(\vec{r},0)+c_{2}\varphi^{2}(\vec{r},L)).
_C\langleL^{\prime},S^{\prime}|\tilde{q}^{H^{(c)}}|L,S\rangle_{C}=\sum_{\ell,s}N_{L,L^{\prime}}^{\ell}\chi_{\ell}^{(N-2)}(q)\chi_{s}^{SO(8)}(q)\delta_{S-S^{\prime}+s+2,0}^{(4)}~,
+e^{2}A_{\mu}{\frac{\{-4\alpha+8g(1-\alpha)\partial^{2}(2\alpha^{2}g-a+a^{2}g\partial^{2})\}}{(ag\partial^{2}-1)^{2}+4\alpha^{2}g^{2}\partial^{2}}}\epsilon^{\mu\nu\lambda}\partial_{\nu}A_{\lambda}-4J_{\mu}A^{\mu}].
Q=\left(\begin{array}{ccccccr}{0}&{-1}&{0}&{0}&{0}&{0}\\{1}&{-1}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}&{-1}&{-1}\\{0}&{0}&{1}&{1}&{-1}&{-1}\\{0}&{0}&{0}&{1}&{-1}&{0}\\{0}&{0}&{0}&{1}&{0}&{-1}\\\end{array}\right).