harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{~\cal~L~}_{\mathrm{eff}}=-\frac{1}{8\pi^{3/2}}\int_{0}^{\infty}\frac{d\tau}{\tau^{3/2}}\mathrm{e}^{-m^{2}\tau}\left(|eB|\frac{\mathrm{cosh}(2eB\sigma\tau)}{\mathrm{sinh}(|eB|\tau)}-\frac{1}{\tau}\right).
\partial_{t}P=\partial_{x}\left[\left(\frac{\partial_{x}V}{M\gamma}+\frac{\beta^{-1}}{M\gamma_{\scriptscriptstyle0}}\frac{\partial_{t}Q}{\partial_{x}Q}-\frac{\partial_{x}^{2}Q}{(\partial_{x}Q)^{3}}\right)P\right]+\frac{\beta^{-1}}{M\gamma_{\scriptscriptstyle0}}\Big(\partial_{x}\Big[(\partial_{x}Q)^{-2}\Big]\Big)\partial_{x}P\,.
\vert{{h+n},{\bar{h}+\bar{n}}}\rangle={\sum_{\stackrel{n=\sum{\alpha_{i}}n_{i}}{\bar{n}=\sum{\beta_{j}}n_{j}}}\lambda_{\{\alpha_{i}\}\{\beta_{j}\}}{(\Pi_{i}L_{-n_{i}}^{\alpha{i}})(\Pi_{j}\bar{L}_{-n_{j}}^{\beta{j}}})}\vert{h,{\bar{h}}}\rangle.
(-\partial_{0}^{2}+\nabla^{2}-2e^{2}|\phi|^{2})\chi_{\uparrow}+i\kappa\partial_{0}\chi_{\uparrow}-\sqrt{2}e(D_{-}\phi)^{*}\psi_{\downarrow}+i\kappa\partial_{+}\chi_{\downarrow}=0,
{\mathcalL}_{G}^{(3)}|=\frac{1}{\gamma^{2}}\,\,\frac{\partial_{z}v_{1}\partial_{\bar{z}}v_{1}-\partial_{z}v_{0}\partial_{\bar{z}}v_{0}}{1+v_{1}^{2}-v_{0}^{2}},~~~~~~~~{\mathcalL}_{G}^{(4)}|=\frac{1}{\gamma^{2}}\,\,\frac{\partial_{z}v_{2}\partial_{\bar{z}}v_{2}-\partial_{z}c\partial_{\bar{z}}c}{1+v_{2}^{2}-c^{2}},
S_{i\;j}^{k\;\ell}\left(\begin{array}{cc}{{*}}&{{b}}\\{{a}}&{{*}}\\\end{array}\right)(\theta)=S_{j\;\bar{\ell}}^{\bar{\imath}\;k}\left(\begin{array}{cc}{a}&{{*}}\\{{*}}&{b}\\\end{array}\right)(\mathrm{i}\pi-\theta)
\partial_{\nu}\left(\phi^{\mu\nu}+\left\langle\Phi^{\mu\nu}\right\rangle\right)=-\epsilon^{3mn}\left(\left\langleA_{\nu}^{m}F^{n\mu\nu}\right\rangle+\left\langleA_{\nu}^{m}G^{n\mu\nu}\right\rangle\right).
\psi_{0}=-\frac{C}{e^{3}}\int_{r}^{\infty}\frac{x^{4}}{(1+\frac{x^{4}}{e^{2}})^{3/2}}dx=-C\frac{\partial}{\partiale}\int_{r}^{\infty}\frac{dx}{(1+\frac{x^{4}}{e^{2}})^{1/2}}=-C\frac{\partial\phi_{c}}{\partiale}
{\mathcalA}_{cov}^{a}\left(x\right)=-\left(D^{\mu}\langlej_{\mu}\rangle\right)^{a}\left(x\right)=\fraci{2\pi}\varepsilon_{\mu\nu}{\mathrmtr}\left(t^{a}\left(\partial^{\mu}A^{\nu}+A^{\mu}A^{\nu}\right)\right)\left(x\right)
0=\alpha_{i}\beta_{j}b_{i}b_{j}+\beta_{i}^{*}\alpha_{j}^{*}b_{i}^{\dagger}b_{j}^{\dagger}+\beta_{i}^{*}\beta_{j}b_{i}^{\dagger}b_{j}+q\alpha_{i}\alpha_{j}^{*}b_{j}^{\dagger}b_{i}
\frac{\deltaS}{\delta\bar{C}}+2\partial_{\mu}\frac{\deltaS}{\deltaA_{\mu}^{*}}=-2\partial_{\mu}\psi^{\mu},\qquad\frac{\deltaS}{\delta\bar{\phi}}+2\partial_{\mu}\frac{\deltaS}{\delta\psi_{\mu}^{*}}=0,
\left|\phi\left(q,p_{1}\right)-\phi\left(q,p_{2}\right)\right|\;=\;\left|p_{1}-p_{2}\right|\left|\int_{p_{1}}^{p_{2}}\frac{dp}{\left|p_{1}-p_{2}\right|}\phi,_{p}\left(q,p\right)\right|\;\leq\;\left|p_{1}-p_{2}\right|^{1/2}\left(N^{\left|\alpha\right|}K_{\delta}\right)^{1/2}.
b_{nk}=\frac{1}{n+k}\left[\frac{(k+1)(k+2)}{2}b_{n-1,k+2}+\frac{\lambda(\lambda-1)}{2}\sum_{m=0}^{k}(-1)^{m+1}\frac{m+1}{q^{m+2}}b_{n-1,k-m}\right].
\begin{array}{c}{\Gamma\left(s\right)\zeta\left(\fracs2,\frac{D_{B}}{\mu^{2}}\right)=\frac{\Gamma\left(s\right)}{\Gamma\left(\fracs2\right)}\left[\Gamma\left(\fracs2\right)\zeta\left(\fracs2,\frac{D_{B}}{\mu^{2}}\right)\right]=}\\{=\frac{2^{s-1}}{\sqrt{\pi}}\Gamma\left(\frac{s+1}2\right)\left[\Gamma\left(\fracs2\right)\zeta\left(\fracs2,\frac{D_{B}}{\mu^{2}}\right)\right]}\\\end{array}
\begin{array}{ccc}{sA_{\mu}=-D_{\mu}c\;,}&{sc=c^{2}\;,}&{s\phi=\left[c,\phi\right]\;,}\\{s\psi_{\mu}=\left\{c,\psi_{\mu}\right\}\;,}&{s\chi_{\mu\nu}=\left\{c,\chi_{\mu\nu}\right\}\;,}&{s\eta=\left\{c,\eta\right\}\;,}\\{s\overline{{\phi}}=\left[c,\overline{{\phi}}\right]\;,}&{}&{}\\\end{array}
X_{j}(\eta)=\sqrt{2j+1}\frac{(2j_{0}-j)!}{j!(2j_{0})!}\sum_{s=0}^{2j_{0}-j}\frac{(j+s)!(2j_{0}-s)!}{(2j_{0}-j-s)!s!}\eta^{j-j_{0}}(1-\eta)^{s-j_{0}}.
d\xi=\frac{2i\pi}{a}\xidz\longrightarrowdz=\frac{a}{2i\pi}\frac{d\xi}{\xi}\longrightarrow\left(dz\right)^{2}=-\frac{a^{2}}{\left(2\pi\right)^{2}}\frac{\left(d\xi\right)^{2}}{\xi^{2}}\tag{3.1}
\Bigl[\,m^{2}+(\,\underline{{p}}+\sum_{j=1}^{b_{1}+\ldots+b_{k}}\underline{{p}}_{j})^{2}+(\,p_{0}+\sum_{j=1}^{b_{1}+\ldots+b_{k}}p_{0,j})^{2}-(k_{0}+\sum_{j=1}^{b_{1}+\ldots+b_{k}}k_{0,j})^{2}\Bigr]\,\geq\,\varepsilon\,>\,0\.
U(\rho,T)=\frac{\pi^{2}}{9}\Bigl(\frac{T^{2}-T_{cr}^{2}}{T}\Bigr)^{2}\rho+\frac{1}{N}\frac{8\pi^{2}}{3}\frac{T^{2}-T_{cr}^{2}}{T^{2}}\rho^{2}+\frac{1}{N^{2}}\frac{64\pi^{2}}{3}\frac{1}{T^{2}}\rho^{3}.
\frac{d\phi}{dr^{2}}=\frac{r_{+}r_{-}}{2}\left(\frac{1}{r^{2}}-\frac{1}{r_{\beta^{\prime}}^{2}-r^{2}}\right)\frac{1}{\sqrt{(r_{+}^{2}-r^{2})(r^{2}-r_{-}^{2})}}
\delta\lambda_{\alpha(n)}^{n}=\omega^{\nu\sigma}x_{\nu}\partial_{\sigma}\lambda_{\alpha(n)}^{n}-{\frac{n}{4}}\omega^{\nu\sigma}\varepsilon_{\nu\sigma\rho}e^{\rho|}{}_{\alpha}{}^{\beta}\lambda_{\alpha(n-1)\beta}^{n}.
\tilde{S}_{1f}^{\prime}=(\alphab_{1}+1-\alpha)\frac{1}{N}\mathrm{Tr}\,B_{1}^{4}+(\alphab_{2}+1-\alpha)\frac{1}{N}\mathrm{Tr}\,B_{2}^{4}+\alphab_{12}\frac{1}{N^{2}}\mathrm{Tr}\,B_{1}^{2}\,\mathrm{Tr}\,B_{2}^{2}
{\calZ}\,=\,\sum_{N=0}^{\infty}\frac{\eta^{2N}}{{(N!)}^{2}}\langle\,{[\intd^{2}xe^{-2\pii\theta(x)}]}^{N}{[\intd^{2}ye^{+2\pii\theta(y)}]}^{N}\,\rangle\;.
Y_{ni}=(4\pi)^{-1}\int<n|\alpha^{\mu}e^{i({\bfkx})}\frac{\nu+(E/|E|)}{|{\bfk}|+\nu(E_{n}-E_{i})}e^{-i({\bfkx})}\alpha_{\mu}|i>\frac{d^{3}k}{|{\bfk}|}.
R={\frac{1-\tau}{1+\tau}}=\left({\frac{\rho_{0}(1+t^{2})+2\tilde{\rho}_{0}t-4t\rho^{+}}{\rho_{0}(1+t^{2})+2\tilde{\rho}_{0}t+4t\rho^{-}}}\right)^{1/2}.
\partial_{\mu}\left[(1-\frac{1}{2}\theta^{\alpha\beta}F_{\alpha\beta})\,F^{\mu\nu}-\left((F^{2}\theta)^{\mu\nu}+(F\thetaF)^{\mu\nu}+(\thetaF^{2})^{\mu\nu}\right)-\frac{1}{4}\,\theta^{\mu\nu}\,F_{\lambda\rho}F^{\lambda\rho}\right]=0
a_{1}=u^{\mu}u_{\mu}~,~a_{2}=\sigma^{\mu\nu}\sigma_{\mu\nu}~,~a_{3}=u_{\mu}\sigma^{\mu\nu}\sigma_{\nu\lambda}u^{\lambda}~,~a_{4}=det\sigma={\frac{1}{16}}(\sigma^{\mu\nu}\sigma_{\mu\nu}^{*})^{2}~,~\sigma_{\mu\nu}^{*}={\frac{1}{2}}\epsilon_{\mu\nu\rho\lambda}\sigma^{\rho\lambda}.
\frac{3}{2}n\lambda_{n}\widehat{Q}_{\beta}^{1}\psi_{0}=-P_{0}\bigl(Q_{\beta}^{1}\overline{{P}}_{0}\psi_{n}+Q_{\beta}^{2}\psi_{n-1}+\ldots+Q_{\beta}^{n+1}\psi_{0}\bigr)\;,\nonumber
E_{_{B}}=\frac{6\sqrt{2}\,Q_{_{B}}^{2}}{\piz\left(\pi^{2}+z^{2}\right)}+\frac{\piz\left(\pi^{2}+z^{2}\right)}{12\sqrt{2}}+\frac{\sqrt{2}\pi}{4}\left(\frac{\pi^{2}+3z^{2}}{6}+z\right).
S_{YM}=-\,\frac{1}{g_{\mathrm{YM}}^{2}}\intd^{4}x\left\{\frac{1}{4}F_{\alpha\beta}^{A}\,F_{A}^{\alpha\beta}+\frac{1}{2}D_{\alpha}{\overline{\Psi}}^{A}\,D^{\alpha}\Psi_{A}\right\}+\frac{\theta_{\mathrm{YM}}}{32\pi^{2}}\intd^{4}x\,F_{\alpha\beta}^{A}\,{\widetilde{F}}_{A}^{\alpha\beta}
\begin{array}{rcl}{\langleg\rangle^{(m)}}&{=}&{E^{(m)}(u)|_{u=1}}\\{{}~}&{=}&{\omega^{m+1}\displaystyle\frac{(q^{2};q^{2})_{\infty}^{2}}{(t^{2};t^{2})_{\infty}^{2}}\displaystyle\frac{(t^{2}\omega;t^{2})_{\infty}(t^{2}\omega^{-1};t^{2})_{\infty}}{(q^{2}\omega;q^{2})_{\infty}(q^{2}\omega^{-1};q^{2})_{\infty}}.}\\\end{array}
f_{2}(\tau)-g_{2}(\tau)=\left({\frac{\eta(\tau)}{\eta(2\tau)}}\right)^{24}-{\frac{(\theta_{3}(\tau)\theta_{4}(\tau))^{8}+2^{-4}\theta_{2}(\tau)^{16}}{\eta(\tau)^{8}\eta(2\tau)^{8}}}\,.
\hat{A}\Psi_{0}^{(3)}=\Biggl[\frac{\partial}{\partiala^{*}}+\sum_{k=0}^{3}b_{k}^{(3)}a^{*(3-k)}\Bigl(\frac{\partial}{\partiala^{*}}\Bigr)^{k}\Biggr]\Psi_{0}^{(3)}=0.
m{\frac{d\Omega_{BH}}{dm}}\sim{\frac{m}{m_{p}^{2}t_{f}}}\sim\left({\frac{\lambda}{e^{2}}}\right)^{1/3}\left({\frac{M}{m_{p}}}\right)^{2}\left({\frac{M}{em}}\right)^{\frac{n-2}{3}}\;.
\delta\phi_{\mu\alpha}=-X^{\nu}(\Gamma_{\mu\nu}\epsilon)_{\alpha}-\phi_{\mu\nu}(\Gamma^{\nu}\epsilon)_{\alpha}+\frac{1}{6}(\overline{{\epsilon}}\Gamma_{\mu\nu}\theta)(\Gamma^{\nu}\theta)_{\alpha}+\frac{1}{6}(\overline{{\epsilon}}\Gamma^{\nu}\theta)(\Gamma_{\mu\nu}\theta)_{\alpha},
V_{-1;\chi}=e^{-\phi}\prod_{j=1}^{3}\sigma_{1-\nu}^{j}e^{i(1-\nu_{j})H_{j}}e^{ik_{\mu}X^{\mu}}\,,\\\V_{-1;\chi^{*}}=e^{-\phi}\prod_{j=1}^{3}\sigma_{\nu}^{j}e^{-i(1-\nu_{j})H_{j}}e^{ik_{\mu}X^{\mu}}\,,
{\calL}=g^{\mu\nu}\frac{\partialf^{*}}{\partialx^{\mu}}\frac{\partialf}{\partialx^{\nu}}-\frac{1}{r^{2}}\frac{\partialf^{*}}{\partial\phi}\frac{\partialf}{\partial\phi}+\mu^{2}f^{*}f-\lambda(f^{*}f)^{2}.
V_{eff}(\phi,\upsilon+\chi)=\frac{\left[sM^{4}e^{-2\alpha\phi/M_{p}}+V_{1}(\upsilon+\chi))\right]^{2}}{4\left[b\left(sM^{4}e^{-2\alpha\phi/M_{p}}+V_{1}(\upsilon+\chi)\right)-V_{2}(\upsilon+\chi)\right]}
I_{mn}^{\prime\prime}=\frac{2}{A^{4}}\int\frac{d^{d}k}{{(2\pi)}^{d}}\frac{k_{m}k_{n}dxdydz\delta(1-x-y-z)}{{(k^{2}+2kp(x+y)+a_{2}x+a_{3}(y+z))}^{3}}
\begin{array}{l}{\nabla^{2}\phi_{e}=-\rho_{e}(\mathbf{x}),~~~~\nabla^{2}\phi_{m}=\rho_{m}(\mathbf{x}),}\\{\nabla^{2}\mathbf{A_{m}}=-\mathbf{J_{e}}(\mathbf{x}),~~~~\nabla^{2}\mathbf{A_{e}}=\mathbf{J_{m}}(\mathbf{x})}\\\end{array}
(2\pi\sqrt{\alpha^{\prime}})^{2}\,\tau_{p+2}=\tau_{p}\quad\mathrm{and}\quad\int_{{\calC}_{1}}B=b=(2\pi\sqrt{\alpha^{\prime}})^{2}\,\left(\frac{1}{2}+\frac{1}{(2\pi\sqrt{\alpha^{\prime}})^{2}}\tilde{b}\right)
\begin{aligned}{ds^{2}}&{{}=2dx^{+}dx^{-}+\sum_{i,j=1}^{8}A_{ij}x^{i}x^{j}(dx^{-})^{2}+\sum_{i=1}^{8}dx^{i}dx^{i}}\\{F_{5}}&{{}=\lambda\,dx^{-}\wedge(dx^{1}\wedgedx^{2}\wedgedx^{3}\wedgedx^{4}+dx^{5}\wedgedx^{6}\wedgedx^{7}\wedgedx^{8})~,}\\\end{aligned}
L_{1}=\int_{0}^{1}dx_{3}x_{3}^{2\alpha^{\prime}\alpha_{13}-1}(1-x_{3})^{2\alpha^{\prime}\alpha_{34}}\int_{0}^{x_{3}}dx_{2}{x_{2}}^{2\alpha^{\prime}\alpha_{12}-1}(1-x_{2})^{2\alpha^{\prime}\alpha_{24}-1}(x_{3}-x_{2})^{2\alpha^{\prime}\alpha_{23}}
\frac{1}{n_{j}}\sum_{m_{j}=1}^{n_{j}}coth\frac{1}{2}K\bigl(r\pmi(t-c_{j}-2\pik_{j}^{-1}m_{j})\bigr)=coth\frac{1}{2}k_{j}\bigl(r\pmi(t-c_{j})\bigr)
\nonumber\tag{15}{\cal\widetilde{L}}_{\varphi^{4}}=\int\{\Lambda_{a}(\dot{\xi}^{a}-\omega^{ab}\delta_{b}H_{\varphi^{4}})+i\overline{{\Gamma}}_{a}(\partial_{t}\delta_{b}^{a}-\omega^{ac}\delta_{c}\delta_{b}H_{\varphi^{4}})\Gamma^{b}\}d^{3}x
\intdtG(x_{1},t_{1};0,t)G(0,t;x_{2},t_{2})=-\int\frac{d\omega}{2\pi}e^{-i\omega(t_{1}-t_{2})}e^{-i\hat{k}(x_{1}+x_{2})}\frac{1}{(i\hat{k}-2\sigma_{1})^{2}}.
\int[d\delta\phi]_{e^{\phi}{\hat{g}}}~e^{-S_{L}[\phi,{\hat{g}}]+S_{\mathrm{boundary}}[\phi,{\hat{e}}]}=\int[d\delta\phi]_{{\hat{g}}}^{\prime}\int_{-\infty}^{\infty}d\phi_{0}~e^{-S[\phi_{R},{\hat{g}}]}\quad,
R_{\mu\nu}=2\kappa^{2}\lbrack2\nabla_{\mu}\phi\nabla_{\nu}\phi+2be^{-2a\phi}(F_{\mu\rho}F_{\nu}^{~\rho}-\frac{1}{4}g_{\mu\nu}F^{2})\rbrack+\kappa^{2}(T_{\mu\nu}^{m}-\frac{1}{2}g_{\mu\nu}T^{m})~~,
J_{B\mu}^{\prime}~=~F_{\mu\nu}\partial^{\nu}\frac{1}{\partial_{i}\partial^{i}}{\overline{{\calP}}}+m^{2}(A_{\mu}+\partial_{\mu}\rho)\frac{1}{\partial_{i}\partial^{i}}{\overline{{\calP}}}+B\partial_{\mu}\frac{1}{\partial_{i}\partial^{i}}{\overline{{\calP}}}.
\mu^{2}\bar{G}_{20}(q_{1}^{+},-q_{1}^{+}-k^{+})+{\frac{\lambda}{8\piV}}\Big[{\frac{f^{2}(q_{1}^{+})}{q_{1}^{+}}}+{\frac{f^{2}(q_{1}^{+}+k^{+})}{q_{1}^{+}+k^{+}}}\Big]\bar{G}_{20}(q_{1}^{+},-q_{1}^{+}-k^{+})+{\frac{\lambda}{4\pi}}\phi_{c}(k)=0,
L^{\Lambda,\Lambda_{0}}(\varphi)=(2\pi)^{4}\sum_{n=1}^{\infty}\frac{1}{n!}\int_{p_{1}}\cdot\cdot\int_{p_{n}}\delta(p_{1}+\cdot\cdot+p_{n})\hat{\varphi}(p_{1})\cdots\hat{\varphi}(p_{n})\mathcal{L}_{n}^{\Lambda,\Lambda_{0}}(p_{1},\cdots,p_{n})
m^{\pm\mu}\=\\oint\frac{dz_{1}}{2\piiz_{1}}\oint_{|z_{2}|<|z_{1}|}\frac{dz_{2}}{2\pii}\int_{z_{2}}^{z_{1}}\!\!dw\;\partial\xi^{\pm}(w)\e^{-\varphi^{\mp}}i\psi^{\mp\mu}(z_{2}),
X^{i}(\tau,\sigma)=x^{i}+\frac{p^{i}}{2\pi}\,\tau+\frac{i}{2\sqrt{\pi}}\sum_{n\ne0}\left(\frac{1}{n}\alpha_{n}^{i}\,e^{-in(\tau-\sigma)}+\frac{1}{n}\tilde{\alpha}_{n}^{i}\,e^{-in(\tau+\sigma)}\right),
H={\frac{1}{2m}}\intdx\mathrm{Tr}|\nabla\phi|^{2}+{\frac{T_{0}^{2}}{2mN_{c}}}\intdxdyV(x-y)\mathrm{Tr}[\phi^{\dagger}(x)\phi^{\dagger}(y)\phi(y)\phi(x)]\;.
T\mapstoT_{\star}=\left(\begin{matrix}{1}\\{z^{m}}\\\end{matrix}\right)\quad\mathrm{and}\quad\tilde{T}\mapsto\tilde{T}_{\star}=\left(\begin{matrix}{\bar{z}^{m}}\\{-1}\\\end{matrix}\right)+\left(\begin{matrix}{1}\\{z^{m}}\\\end{matrix}\right)\star\frac{-2i}{1+\bar{z}^{m}\starz^{m}}\star\{m\bar{z}^{m-1}t+h_{\star}(\bar{z})\}.
C_{6}(v^{2},u_{2k})=v^{6}-\frac{u_{2}}{2}v^{4}-\left(\frac{u_{4}}{4}-\frac{u_{2}^{2}}{8}\right)v^{2}-\frac{1}{6}\left(u_{6}-\frac{3u_{4}u_{2}}{4}+\frac{u_{2}^{3}}{8}\right)
A_{\pm}A_{\mp}\psi_{n_{\pm},m_{\pm}}T_{n_{\pm},m_{\pm}}=\left(\begin{array}{c}{\scriptstylen_{+}}\\{\scriptstyle{n_{-}+1}}\\\end{array}\right)\psi_{n_{\pm},m_{\pm}}T_{\kappa_{\pm};n_{\pm},m_{\pm}}.
\omega_{\alpha\beta}\dot{\xi}_{\beta}=\frac{\partialh}{\partial\xi_{\alpha}},\qquad\mathrm{where}\qquad\big(\omega_{\alpha\beta}\big)=\left(\begin{array}{cccc}{0}&{\theta}&{1}&{0}\\{-\theta}&{0}&{0}&{1}\\{-1}&{0}&{0}&{B}\\{0}&{-1}&{-B}&{0}\\\end{array}\right).
\prod_{n=1\atopn\not\equiv0,\pmi\mathrm{(mod}2k+1)}^{\infty}(1-q^{n})^{-1}=\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k-1}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k-1}^{2}+n_{i}+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}.
\frac{f_{\Omega}}{f_{\Lambda}}=\frac{n_{\omega^{s}}(\Lambda_{i}-\ell)\bigl(n_{\omega^{s}}(\Lambda_{i}-\ell)-1\bigr)}{\bigl(n_{\omega^{s}}(\Lambda_{i})+1\bigr)\bigl(n_{\omega^{s}}(\Lambda_{j})+1\bigr)}\,.
V_{0}=\frac12<\{(K^{\tilde{M}\tilde{A}}\vec{L}^{\tilde{A}})^{2}+|{\calZ}^{\tilde{M}}K^{\tilde{M}\tilde{A}}|^{2}+2|{\calZ}^{\tilde{M}}K^{\tilde{M}\tilde{A}}\vec{L}^{\tilde{A}}|^{2}\}>.
\bar{\calL}_{b}=g\left[g^{\mu\nu}D_{\mu}\phi^{\prime\dagger}D_{\nu}\phi^{\prime}-\lambda^{2}\bar{\mu}^{2}\phi^{\prime\dagger}\phi^{\prime}+{\frac{1}{2}}\bar{b}\left(\phi^{\prime\dagger}\phi^{\prime}\right)^{2}\right].
\hat{\phi}(x^{0}\,,\,x^{1})\,=\,-\frac{\hat{\xi}}{2R}\,+\,\frac{1}{\sqrt{2\pi}}\left(\frac{\pi}{R}\right)\sum_{n>0}\frac{1}{\sqrt{\omega_{n}}}\left[\hat{\Lambda}_{n}e^{-i\omega_{n}(x^{0}+x^{1})}\,+\,\hat{\Lambda}_{n}^{\dagger}e^{i\omega_{n}(x^{0}+x^{1})}\right],
D^{2}(p+r)=D^{2}(p)+\frac{1}{2}r^{\alpha\beta}\left[\frac{\partial}{\partialr^{\alpha\beta}}D^{2}(r)\right]_{r=p}+\frac{1}{2}\left[\frac{1}{2}\frac{\partial^{2}}{\partialr_{\alpha\beta}\partialr^{\alpha\beta}}D^{2}(r)\right]_{r=p}.
T^{{\underline{a}}}={\calD}E^{{\underline{a}}}=-{\frac{i}{2}}\hat{E}^{\hat{\underline{\alpha}}}\wedge\hat{E}^{\hat{\underline{\beta}}}\Gamma_{{\underline{{\hat{\alpha}\hat{\beta}}}}}^{{\underline{a}}}=-{\frac{i}{2}}(\hat{E}\Gamma^{{\underline{a}}}\hat{E})
K^{W}\;:=\;\mathrm{diag}\left(\frac{\pi}{\pi+gN}\frac{1}{-\triangle+\frac{e^{2}N}{\pi+gN}}\;,\;\frac{1}{-\triangle+1}\;,\;.\;.\;.\;.\;,\;\frac{1}{-\triangle+1}\right)\;.
\{z^{+}\,,\:z^{-}\}_{S}-\gamma\,(\frac{\partial_{-}^{2}z^{+}}{\partial_{-}z^{+}})^{2}=(\dot{z}^{-})^{-2}\:[\:\frac{\stackrel{...}{z}^{\,+}}{\dot{z}^{+}}-\frac{3}{2}\,(\frac{\ddot{z}^{+}}{\dot{z}^{+}})^{2}-\frac{\stackrel{...}{z}^{\,-}}{\dot{z}^{-}}+\frac{3}{2}\,(\frac{\ddot{z}^{-}}{\dot{z}^{-}})^{2}-\gamma\,(\frac{\ddot{z}^{+}}{\dot{z}^{+}}-\frac{\ddot{z}^{-}}{\dot{z}^{-}})^{2}\:]\,.\nonumber\,
A\;=\;(2\pi)^{\frac{d}{2}}[\alpha]^{-{\frac{1}{2}}}\prod_{\mu=0}^{d-1}\delta\left(\sum_{r=1}^{4}p_{r}^{\mu}\right)\;\int_{\mathbb{C}}\!d^{2}x\;|z|^{\frac{p_{1}\cdotp_{2}}{2\alpha}}|1-z|^{\frac{p_{2}\cdotp_{3}}{2\alpha}}G^{\scriptscriptstyleL}(z,\bar{z})\,
{\calL}=-\frac{\epsilon_{0}}{4}F_{\mu\nu}F^{\mu\nu}-\frac{\epsilon_{0}}{4}G_{\mu\nu}G^{\mu\nu}-\frac{1}{c}{j_{e}}_{\mu}A^{\mu}-\frac{1}{c}{j_{m}}_{\mu}W^{\mu}+(gaugefixingterms)\,.
S^{T}=M_{0}^{2}\int_{0}^{1/T}dt\int_{0}^{R}dr\left[1+\frac{1}{2}{\bfu}\left(1-\frac{\alpha}{M_{0}^{2}}\triangle\right)(-\triangle){\bfu}\right]-\beta\int_{0}^{1/T}dt\,\ddot{\bfu}\acute{\bfu}
ds_{ex}^{2}=-\left(\frac{r^{2}}{\ell^{2}}-2\frac{r_{ex}^{2}}{\ell^{2}}\right)dt^{2}+\frac{\ell^{2}r^{2}}{\left(r^{2}-r_{ex}^{2}\right)^{2}}dr^{2}-2\frac{r_{ex}^{2}}{\ell}dtd\phi+r^{2}d\phi^{2}.
\frac{r_{h}}{2}>\frac{r_{h}}{2}-[\frac{b^{2}r^{3}}{3}(\sqrt{1+\frac{a^{4}}{r^{4}}}-1)+\frac{2q^{2}}{3}\int_{0}^{r_{h}}{\frac{ds}{\sqrt{a^{4}+s^{4}}}}],
M_{P}=M_{s}^{3/2}\left(\int_{-1/\alpha}^{1/\alpha}dze^{-2\sigma(z)}\right)^{1/2}\sim\frac{2}{\sqrt{3}}M_{s}\left(\frac{M_{s}}{\alpha}\right)^{1/2}=\frac{2}{\sqrt{3}}\frac{M_{s}}{g_{s}^{\frac{2-\eta}{4}}}
X=\left(\begin{array}{cccccccc}{a}&{1}\\{}&{a}&{1}\\{}&{}&{a}\\{}&{}&{}&{b}&{1}\\{}&{}&{}&{}&{b}\\{}&{}&{}&{}&{}&{c}\\{}&{}&{}&{}&{}&{}&{\ddots}\\{}&{}&{}&{}&{}&{}&{}&{z}\\\end{array}\right)
J_{ijk}=\frac{1}{6}\left(J_{i}\left\{J_{j},J_{k}\right\}+J_{j}\left\{J_{k},J_{i}\right\}+J_{k}\left\{J_{i},J_{j}\right\}\right),~~T_{ijk}=\frac{1}{3}\left(g_{ij}J_{k}+g_{jk}J_{i}+g_{ki}J_{j}\right)\,,
\begin{array}{||r|c|c|c||}\hline{}&{}&{}&{}\\{\mathrm{ghost:}}&{\overline{{c}}}&{\overline{{\phi}}}&{\overline{{\chi}}_{\mu\nu}}\\\hline{}&{}&{}&{}\\{\mathrm{L-N~field:}}&{b}&{\eta}&{b_{\mu\nu}}\\\hline{}\\\end{array}
\partial_{x}^{-1}\left({\bar{\calL}}^{i-1}({\bar{\Phi}}_{2})\,{\bar{\Psi}}_{2}\right)=\frac{\partial_{x}^{-1}\left({\calL}^{i}({\Phi}_{2})\,{\Psi}_{1}\right)}{\partial_{x}^{-1}(\Phi_{2},\Psi_{1})}
A_{i}^{\mathrm{vp}}(x_{j})=\Bigl(\frac{\Phi\alpha}{4\pi^{3}}\epsilon^{ij}\,\partial_{j}\Bigr)\,\int_{0}^{1}dz\frac{z^{2}}{1-z^{2}}\,\biggl(1-\frac{1}{3}z^{2}\biggr)\,\intd^{2}qe^{i\vec{q}_{t}\cdot\vec{x}_{t}}\,\frac{1}{\frac{4m^{2}}{1-z^{2}}+q_{t}^{2}}J_{0}(R\,q_{t}),
\left[\nabla^{2}(\nabla^{2}-s^{2})-(\overrightarrow{v}.\overrightarrow{\nabla}^{\ast})(\overrightarrow{v}.\overrightarrow{\nabla}^{\ast})\right]A_{0}=-\nabla^{2}\rho-s\overrightarrow{\nabla}\times\overrightarrow{j}-(\overrightarrow{v}.\overrightarrow{\nabla}^{\ast})J,
\chi_{\tilde{\beta}}^{\tilde{B}}(\Gamma(f))=\int_{\tilde{B}}d\zeta\int_{N_{\tilde{B}}}dn\int_{N\capK}dk\,f(x_{\zeta}g_{\tilde{B}}x_{\zeta}^{-1},x_{\zeta}nx_{\zeta}^{-1}k)\chi_{\tilde{\beta}}(n).
\delta{\calE}_{2}^{1,0}=\frac{8K}{a_{0}^{6}}\left\{\left[\left(\frac{a_{0}}{\Lambda}\right)^{2}-2\left(\frac{a_{0}}{\Lambda}\right)\right]e^{-2\Lambda/a_{0}}+4E_{1}\left(\frac{2\Lambda}{a_{0}}\right)\right\}\,{.}
g_{\mathrm{mj}}\biggl(\ddot{x}^{\mathrm{j}}+\Gamma_{\mathrm{kl}}^{\mathrm{j}}\;\dot{x}^{\mathrm{k}}\dot{x}^{\mathrm{l}}\biggr)=-{\frac{\partialV}{\partialx^{\mathrm{m}}}}
\begin{array}{l}{\deltaa=-i\bar{\zeta}\psi}\\{\deltab=\bar{\zeta}\gamma_{5}\psi}\\{\delta\psi=(f+i\gamma_{5}g)+i\not\!\partial(a+i\gamma_{5}b)\zeta}\\{\deltaf=\bar{\zeta}\not\!\partial\psi}\\{\deltag=i\bar{\zeta}\gamma_{5}\not\!\partial\psi,}\\\end{array}
\begin{array}{ll}{\begin{array}{ll}{\frac{d}{dr}\sigma(r)=0}&{r=0}\\{\sigma(r)=0}&{r\rightarrow\infty}\\\end{array}}&{\begin{array}{ll}{A(r)\neq0}&{r\rightarrow\infty}\\{A(r)=1}&{r=0.}\\\end{array}}\\\end{array}
S=\intd^{d-1}x\sqrt{|g|}\\left[R+{\textstyle\frac{1}{2}}\left(\partial\varphi\right)^{2}-{\textstyle\frac{1}{4}}e^{-a\varphi}F_{(2)}^{2}+{\textstyle\frac{1}{2}}\left({\calD}\phi\right)^{2}-{\textstyle\frac{1}{2}}m^{2}e^{a\varphi}\right]\,,
\tilde{S}^{\prime}\[X^{\mu},b_{\xi}]=-|P_{y}|\intd\xi\e^{-\phi}\sqrt{|g_{\xi\xi}|}+P_{y}\intd\xi\left(C^{(1)}{}_{\xi}-{\textstyle\frac{m}{2}}\(2\pi\alpha^{\prime})\b_{\xi}\right)\,,
U=\hat{U}_{12}\cdot\cdot\cdot\hat{U}_{1n}\hat{U}_{23}\cdot\cdot\cdot\hat{U}_{2n}\cdot\cdot\cdot\hat{U}_{i,i+1}\cdot\cdot\cdot\hat{U}_{in}\cdot\cdot\cdot\hat{U}_{n-1,n}\\,
I=\frac{1}{16\piG_{N}}\intd^{4}x\sqrt{-g}\left[R+a_{1}R^{2}+a_{2}(3R_{\nu}^{\mu}R_{\mu}^{\nu}-R^{2})+a_{3}(R_{\\\rho\sigma}^{\mu\nu}R_{\\\mu\nu}^{\rho\sigma}-4R_{\nu}^{\mu}R_{\mu}^{\nu}+R^{2})\right],
\phi\left(x\right)=\int\frac{d^{d}k\,\,\theta\left(k^{0}\right)}{\left(2\pi\right)^{d-1}}\,\,\delta\left(k^{2}\right)\,\left[a\left(k\right)\,e^{ik\cdotx}+a^{\dagger}\left(k\right)\,e^{-ik\cdotx}\right].
S[\phi,h,B]=\frac{1}{\pi}\intd^{2}x\{-\frac{1}{2}\bar{\partial}\phi^{i}\partial\phi^{i}+\frac{1}{2}\partial\phi^{i}h\partial\phi^{i}+\frac{1}{3}Bd^{ijk}\partial\phi^{i}\partial\phi^{j}\partial\phi^{k}\}.
\int_{0}^{\infty}dl\\int_{m}^{\infty}dk\(k^{2}-m^{2})^{1-s}\partial_{k}\frac{t^{j}}{l^{n}}=-\frac{m^{2-2s}}{2}\frac{\Gamma(2-s)\Gamma\left(\frac{1+j-n}{2}\right)\Gamma(s+\frac{n-3}{2})}{(Rm)^{n-1}\Gamma(j/2)}\.
H_{D}=\intd^{3}\sigma\lbrack\lambda^{\mu}(\tau,\vec{\sigma}){\calH}_{\mu}(\tau,\vec{\sigma})+\lambda_{\tau}(\tau,\vec{\sigma})\pi^{\tau}(\tau,\vec{\sigma})-A_{\tau}(\tau,\vec{\sigma})\Gamma(\tau,\vec{\sigma})].
{\calF}=if\,dz\wedged{\bar{z}}-if\,dv\wedged{\bar{v}}+\mathrm{e}^{i\alpha}\,\sqrt{\frac{1}{4}-f^{2}}\,dz\wedged{\bar{v}}-\mathrm{e}^{-i\alpha}\,\sqrt{\frac{1}{4}-f^{2}}\,dv\wedged{\bar{z}}\;.
H=-\sum_{i=1}^{N}\frac{\partial^{2}}{\partial{x_{i}}^{2}}+\sum_{i<j}^{N}\frac{g}{(x_{i}-x_{j})^{2}}+{\frac{1}{4N}}\sum_{i<j}^{N}(x_{i}-x_{j})^{2}\,\,\,,
{\calL}=\bar{\Psi}i\gamma^{\hat{\mu}}\partial_{\hat{\mu}}\Psi+\bar{\Psi}M\Psi=\bar{\Psi}(i\gamma^{\mu}\partial_{\mu}+\sqrt{\kappa}\chi_{\mu}\gamma^{\mu}{\frac{\partial}{\partialx^{5}}})\Psi+\bar{\Psi}M\Psi+\mathrm{gravitational\\pieces}
F_{,\rho}+F_{,\dot{\phi}_{k}}{\frac{\partial\dot{\phi}_{k}}{\partial\rho}}~=~{\frac{\partial^{2}S}{\partial\rho^{2}}}+{\frac{\partial}{\partial\rho}}\left({\frac{\deltaS}{\delta\phi_{k}}}\right)\dot{\phi}_{k}+S_{,\phi_{k}}{\frac{\partial\dot{\phi}_{k}}{\partial\rho}}
B(q^{2})=b_{0}(q^{2})g_{0}^{3}+g_{0}^{5}\left[{\frac{b_{1}(q^{2})}{\epsilon}}+b_{2}(q^{2})\right]+g_{0}^{7}\left[{\frac{b_{3}(q^{2})}{\epsilon^{2}}}+{\frac{b_{4}(q^{2})}{\epsilon}}+b_{5}(q^{2})\right]+...
m_{\beta}m_{\nu}\:\frac{1}{\Box}X^{\nu\beta}=2m_{\beta}m_{\nu}\nabla^{\nu}\frac{1}{\Box}X^{\beta}+2m_{\beta}m_{\nu}\frac{1}{\Box}\Bigl([\nabla^{\nu},\Box]\frac{1}{\Box}X^{\beta}\Bigr)\;.
\langleP[\overline{{\psi}},\psi,A,h]\rangle=\frac{1}{Z}\langleP[\overline{{\psi}},\psi,A,h]e^{-S_{M}[\overline{{\psi}},\psi]}\rangle_{0}=\frac{1}{Z}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\langleP[\overline{{\psi}},\psi,A,h]\;S_{M}[\overline{{\psi}},\psi]^{n}\rangle_{0}\;.
\hat{l}=\left(\begin{array}{clcr}{\frac{\partial^{2}}{\partialz_{0}^{2}}}&{0}&{0}&{0}\\{0}&{\frac{\partial^{2}}{\partialz_{0}^{2}}}&{0}&{0}\\{0}&{0}&{\frac{\partial^{2}}{\partialz_{0}^{2}}}&{0}\\{0}&{0}&{0}&{\frac{\partial^{2}}{\partialz_{0}^{2}}}\\\end{array}\right),
\int\omega_{i}\wedge\bar{\omega}_{j}=\sum_{k=1}^{g}\left(\oint_{a_{k}}\omega_{i}\oint_{b_{k}}\bar{\omega}_{j}-\oint_{b_{k}}\omega_{i}\oint_{a_{k}}\bar{\omega}_{j}\right)=-2i\left(\Omega_{2}\right)_{ij}
\langle\dot{T}^{2}\rangle={\frac{\int\dot{T}^{2}\,dt}{\intdt}}={\frac{\oint(1-V^{2}(T)/\varepsilon^{2})^{1/2}\,dT}{\oint(1-V^{2}(T)/\varepsilon^{2})^{-1/2}\,dT}}=\frac{2}{3}\,
\sum_{i=1}^{M}\left[\frac{d}{dt}\left(\Lambda_{i}(\xi,\dot{\xi})\frac{\partial\Lambda_{i}(\xi,\dot{\xi})}{\partial\dot{\xi}_{n}}\right)-\Lambda_{i}(\xi,\dot{\xi})\frac{\Lambda_{i}(\xi,\dot{\xi})}{\partial\xi_{n}}\right]=0,\quadn=1,\dots,M.
\left.\left(\frac{-8M^{2}}{k^{2}}\right)^{-s}\frac{\Gamma(s)}{\Gamma(\frac14+\frac{s}{2})\Gamma(\frac34+\frac{s}{2})}F(-\frac{s}{2}+\frac{3}{4},-\frac{s}{2}+\frac{1}{4};1-s;1-\frac{k^{4}}{(k^{2}+8M^{2})^{2}})\right]\,.
\mathcal{O}_{\lambda}=1\,,\qquad\mathcal{C}_{\lambda_{1},\lambda_{2},\lambda_{3}}=\mathcal{C}_{\lambda_{1}^{*},\lambda_{2}^{*},\lambda_{3}^{*}}=\sqrt{\frac{\Gamma(\lambda_{1})\Gamma(\lambda_{2})\Gamma(\lambda_{3})}{\Gamma(\lambda_{1}^{*})\Gamma(\lambda_{2}^{*})\Gamma(\lambda_{3}^{*})}}\,,
{\calE}_{KE}[\tilde{A}_{0};R]=\frac{2\pi}{R}\sum_{n\in{\bfZ}}n^{2}|a_{0}^{(n)}|^{2},\quad{\calE}_{KE}[\tilde{A}_{\pm};R]=\frac{2\pi}{R}\sum_{n\in{\bfZ}}[(n\pm\alpha)^{2}-\alpha^{2}]|a_{\pm}^{(n)}|^{2}.
\partial_{\bar{\imath}}F^{g}=\frac{1}{2}C_{\bar{\imath}\bar{\jmath}\bar{k}}e^{2K}G^{i\bar{\imath}}G^{j\bar{\jmath}}[~\sum_{g^{\prime}=1}^{g-1}D_{j}F^{g^{\prime}}D_{k}F^{g-g^{\prime}}+D_{j}D_{k}F^{g-1}~]
<\partial^{\mu}j_{\mu}^{5}(y))>_{v}\equiv<\frac{\alpha_{j^{5}}}{2}\,(\partial_{\mu}\partial^{\mu}\alpha(y))>_{v}=\frac{e^{2}}{16\,\pi^{2}}\,\epsilon^{\mu\nu\alpha\beta}\,F_{\mu\nu}\,F_{\alpha\beta}.
H^{\prime}\left(\begin{array}{c}{\psi_{+}}\\{\psi_{-}}\\\end{array}\right)=\left(\begin{array}{c}{P_{-}P_{+}+p_{3}^{2}\qquad0}\\{0\qquadP_{+}P_{-}+p_{3}^{2}}\\\end{array}\right)\left(\begin{array}{c}{\psi_{+}}\\{\psi_{-}}\\\end{array}\right).
\begin{array}{l}{R^{(1)}T_{1}T_{2}=T_{2}T_{1}R^{(1)}\quad,}\\{\,}\\{T_{1}^{\dagger}R^{(2)}T_{2}=T_{2}R^{(2)}T_{1}^{\dagger}\quad,}\\{\,}\\{T_{2}^{\dagger}R^{(3)}T_{1}=T_{1}R^{(3)}T_{2}^{\dagger}\quad,}\\{\,}\\{R^{(4)}T_{1}^{\dagger}T_{2}^{\dagger}=T_{2}^{\dagger}T_{1}^{\dagger}R^{(4)}\quad.}\\\end{array}
\sum_{i=1}^{N-1}\epsilon_{i}\,{\frac{\mu_{i}^{N-1-j}\,d\mu_{i}}{\displaystyle2\sqrt{h_{N}-\mu_{i}}\,\prod_{j=1}^{N-1}(\mu_{i}-h_{j})}}=\delta_{j,1}\,dx,\quad1\leqj\leqN-1,\quad\epsilon_{i}=\pm1.