harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rcl}{S[B_{m},\tilde{B}_{m},V_{mn}]}&{=}&{{\displaystyle\int}d^{5}x\left\{{\textstyle\frac{1}{2\cdot3!}}H_{m}H_{m}+{\textstyle\frac{1}{4}}({}^{*}\tilde{H}_{m}+H_{p}V_{pm}){\calP}_{mn}^{-1}({}^{*}\tilde{H}_{n}+H_{q}V_{qn})\right.\,}\\{}&{}&{}\\{}&{}&{\left.-{\textstyle\frac{1}{4}}{\calF}_{mn}{\calF}_{mn}+{\textstyle\frac{1}{12}}\epsilon\tilde{H}_{m}B_{m}\right\}\,.}\\\end{array}
{\delta_{\rho}}{\hat{G}_{K\varepsilon}}(\xi^{\prime},\xi^{\prime})=4\rho(\xi^{\prime})-{\frac{1}{\oint{d}\hat{s}(\xi^{\prime\prime\prime})}}\oint{d}\hat{s}(\xi^{\prime\prime})\rho(\xi^{\prime\prime}){\hat{G}_{K}}(\xi^{\prime\prime},\xi^{\prime}).
M(C^{a},c^{a})=\left(\begin{matrix}{I}&{\gamma\Sigma_{a}c^{a}}\\{0}&{I}\\\end{matrix}\right)\left(\begin{matrix}{\Sigma_{a}C^{a}}&{0}\\{0}&{\Sigma_{a}c^{a}}\\\end{matrix}\right)\left(\begin{matrix}{I}&{-\gamma\Sigma_{a}c^{a}}\\{0}&{I}\\\end{matrix}\right),
F(x)G(x)=\left[\sum_{k=0}^{n_{c}}(-1)^{k}s_{2k}(m)x^{n_{c}-k}\right]^{2}+\left[\sqrt{x}\sum_{k=0}^{n_{c}-1}(-1)^{k}s_{2k+1}(m)x^{n_{c}-k-1}\right]^{2}.
(-x)^{n+N(f)+1}\frac{d^{n+1}f}{dx^{n+1}}(x)=(-1)^{n+N(f)+1}(x)^{N(f)+1}\bigl(\prod_{j=1}^{n}(x\frac{d}{dx}+1-j)\bigr)\frac{df}{dx}(x).
\left[\omega_{(0)}^{2}(z)\right]^{i}\;\;,\;\;\left[\omega_{(0)}^{2}(z)\right]^{i}\omega_{(0)}^{1,\pm}(z)\;\;,\;\;\left[\omega_{(0)}^{2}(z)\right]^{i}\omega_{(0)}^{1,+}(z)\omega_{(0)}^{1,-}(z)
T_{oo}=\frac{g^{4}\left(\frac{m}{q^{+}}+\frac{m}{p^{+}}\right)^{2}}{4\pi\left(q^{+}-p^{+}\right)D_{1}^{2}}\int_{0}^{p^{+}}\!\!\!\!\!dk^{+}\left[\frac{\left(\frac{m}{p^{+}}+\frac{m}{k^{+}}\right)^{2}}{\left(p^{+}-k^{+}\right)D_{2}}+\frac{1}{k^{+}}\right],
\mathrm{~\,~\frac{1}{2}~\,~}(1-\gamma_{3})\,\Gamma^{AS}\,\mathrm{~\,~\frac{1}{2}~\,~}(1+\gamma_{3})=\mathrm{~\,~\frac{1}{2}~\,~}(1-\gamma_{3})\left(\begin{array}{c}{\gamma_{4}}\\{\gamma_{5}}\\\end{array}\right)g(q_{3},\mathrm{~\left|~{\bf~q}~\right|~}).
F_{{\sfGI}\,ij}^{a}({\bfr})=\partial_{j}A_{{\sfGI}\,i}^{a}({\bfr})-\partial_{i}A_{{\sfGI}\,j}^{a}({\bfr})-gf^{abc}A_{{\sfGI}\,i}^{b}({\bfr})A_{{\sfGI}\,j}^{c}({\bfr})
f_{u}^{\prime\prime}+\frac{1}{r}f_{u}^{\prime}-\left[-\frac{1}{2}+\frac{3}{2}(F^{\mathrm{ANO}}(r))^{2}+\frac{1}{r^{2}}(\chi^{\mathrm{ANO}}(r))^{2}\right]f_{u}=F^{\mathrm{ANO}}(r)\alpha^{2}(r),
\xi=\ell\int_{\delta}^{\infty}e^{-{\frac{2\upsilon^{2}}{3M^{3}}}\zeta_{\infty}z}dz={\frac{2\delta\ell}{\zeta_{\infty}^{2}}}{\frac{1-\epsilon}{\epsilon}}e^{-{\frac{\epsilon\zeta_{\infty}^{2}}{2(1-\epsilon)}}},
<A_{p}^{(n)}(0,0)A_{p}^{(n)}(L,\tau)>=\int\frac{d^{4}p}{(2\pi)^{4}}\frac{e^{-ip_{0}\tau+i\vec{p}\cdot\vec{L}}}{p_{0}^{2}+p^{2}+m_{n}^{2}}
{\calL}=-{\frac{1}{4}}F^{\mu\nu}F_{\mu\nu}+{\frac{\kappa}{4}}\epsilon^{\alpha\beta\gamma}F_{\alpha\beta}A_{\gamma}+i\bar{\psi}{\gamma}^{\mu}{\partial}_{\mu}\psi-m\bar{\psi}\psi+eA_{\mu}(J^{\mu}+lG^{\mu})+{\frac{1}{2}}g(\bar{\psi}\psi)^{2},
\omega^{(>)}(\Theta_{l})=\frac{\widetilde{A}_{l}^{(+)}\,\Theta_{l}\,H_{\epsilon/4}^{(1)\,\prime}(\Theta_{l})+\widetilde{A}_{l}^{(-)}\,\Theta_{l}\,H_{\epsilon/4}^{(2)\,\prime}(\Theta_{l})}{\widetilde{A}_{l}^{(+)}\,H_{\epsilon/4}^{(1)}(\Theta_{l})+\widetilde{A}_{l}^{(-)}\,H_{\epsilon/4}^{(2)}(\Theta_{l})}\;.
\delta_{\epsilon}A_{\mu}^{a}=\left(D_{\mu}\right)_{\;\;b}^{a}\epsilon^{b},\;\delta_{\epsilon}B_{\mu\nu}=\partial_{\left[\mu\right.}\epsilon_{\left.\nu\right]}+2g\epsilon_{a}\partial_{\left[\mu\right.}A_{\left.\nu\right]}^{a},
\left(\frac{(13)}{(x_{13}^{2})^{\frac{D-2}{2}}}\right)^{m-j-k}\left(\frac{(23)}{(x_{23}^{2})^{\frac{D-2}{2}}}\right)^{n-j-k}\left[\frac{(13)}{(x_{13}^{2})^{\frac{D-2}{2}}}\\frac{(23^{-})}{(x_{23}^{2})^{\frac{D-2}{2}}}-1\leftrightarrow2\right]^{j}\.
{\vec{\alpha}}_{2}\,{\vec{R}}^{(2)}\,=\,\alpha_{2}\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_{t}(x)=\frac{1}{L_{1}L_{2}}\sum_{n_{i}}\frac{e^{\frac{2\pii}{L_{1}}(n_{1}x_{1}+|\tau|^{-1}n_{2}x_{2})}}{\left(\frac{2\pi}{L_{1}}\right)^{2}[(n_{1}-\tilde{t}_{1})^{2}+|\tau|^{-2}(n_{2}-\tilde{t}_{2})]}.
g^{\mu\nu}(t,\vec{x})\equiv{\frac{1}{c^{2}}}\left[\begin{matrix}{-1}&{\vdots}&{-v\;\hat{r}^{j}}\\{\cdots\cdots}&{\cdot}&{\cdots\cdots\cdots\cdots}\\{-v\;\hat{r}^{i}}&{\vdots}&{(c^{2}\;\delta^{ij}-v^{2}\;\hat{r}^{i}\;\hat{r}^{j})}\\\end{matrix}\right].
\Gamma_{4}^{non-planar}=-\frac{\lambda^{2}}{16\pi}\:\mathrm{cos}\frac{{\tilde{p}}_{1}p_{2}}{2}\:\mathrm{cos}\frac{{\tilde{q}}_{1}q_{2}}{2}\left(\mathrm{ln}\frac{1/{\tilde{P}}^{2}}{E-\frac{P^{2}}{4}}+i\pi\right).
H_{f}=-{\frac{i}{2}}\sum_{\vec{x},j}\left(\psi_{\sigma}^{\dagger}(\vec{x}+\vec{j})\alpha_{\sigma\sigma^{\prime}}^{j}\psi_{\sigma^{\prime}}(\vec{x})-\psi_{\sigma}^{\dagger}(\vec{x})\alpha_{\sigma\sigma^{\prime}}^{j}\psi_{\sigma^{\prime}}(\vec{x}+\vec{j})\right)
\stackrel{\wedge}{\Sigma}_{(2)}(p,m,\lambda,\Lambda_{0})=\frac{i\lambda^{2}}{384\pi^{3}}\int_{0}^{1}dx\int_{0}^{1}dy\fracy{\left[yC\right]^{3/2}}F(p^{2},m^{2},x,y,\Lambda_{0})
\{\Psi^{a},\Psi^{b}\}=\frac{\partial\Psi^{a}}{\partial\Pi^{i}}\frac{\partial\Psi^{*b}}{\partial\Pi^{*i}}-\frac{\partial\Psi^{*a}}{\partial\Pi^{*i}}\frac{\partial\Psi^{b}}{\partial\Pi^{i}}
\tilde{H}(\tilde{A}_{0},\tilde{A}_{i},\tilde{\pi}_{i})=H_{c}(A_{0},A_{i},\pi_{i})_{\Bigg\vert\begin{array}{c}{A_{0}\rightarrow\tilde{A}_{0}}\\{A_{i}\rightarrow\tilde{A}_{i}}\\{\pi_{i}\rightarrow\tilde{\pi}_{i}}\\\end{array}}\,.
\intD\PsiD\overline{{\Psi}}DA\frac{\delta}{\delta\overline{{\Psi}}(x)}e^{i\intd^{2}x\left[{\calL}(x)+\overline{{\eta}}(x)\Psi(x)+\overline{{\Psi}}(x)\eta(x)+J^{\mu}(x)A_{\mu}(x)\right]}=0\;.
X^{1}=a\left(\begin{array}{ccccc}{\ddots}&{\ddots}&{\ddots}&{}&{}\\{\ddots}&{1}&{0}&{\ddots}&{}\\{\ddots}&{0}&{0}&{0}&{\ddots}\\{\ddots}&{\ddots}&{0}&{-1}&{\ddots}\\{}&{}&{\ddots}&{\ddots}&{\ddots}\\\end{array}\right)
\Theta_{\Gamma}\left(-1/\tau,\alpha,\beta;P,\frac{\gamma_{+}}{\tau}+\frac{\gamma_{-}}{\bar{\tau}}\right)=\sqrt{\frac{\vert\Gamma\vert}{\vert\Gamma^{*}\vert}}(-i\tau)^{b_{+}/2}(i\bar{\tau})^{b_{-}/2}\Theta_{\Gamma^{*}}(\tau,\beta,-\alpha;P,\gamma),
\lambdaf(\omega)\;=\;{\frac{(\omega^{1/2}-\omega^{-1/2})}{2}}\int_{0}^{\infty}{\frac{d\omega^{\prime}(\omega^{1/2}-\omega^{-1/2})}{(\omega^{\prime}+\omega)(\omega^{\prime}+\omega^{-1})}}{f(\omega^{\prime})}\;.
3\left(z-{\frac{\mu^{2}}{z}}\right)^{2}-x^{8}+2ux^{6}-\left[u^{2}+\left(z+{\frac{\mu^{2}}{z}}\right)\right]x^{4}+\left[v+2u\left(z+{\frac{\mu^{2}}{z}}\right)\right]x^{2}=0.
\frac{2}{(4\piL)^{2n}}e^{n^{2}I^{(L)}}+O(1/L)=\frac{2}{(2\pi)^{2n}}\left(\frac{e^{2}N}{4\pi}\right)^{\frac{n^{2}}{N}}e^{\frac{n^{2}}{N}(2\gamma+1)}{\left(\frac{1}{2L}\right)}^{2\frac{nN-n^{2}}{N}}\;+O(1/L)\;.
{\calL}(y,\lambda)\equiv\left(\begin{array}{cc}{i\phi^{\prime}(y)}&{\lambda}\\{\lambda}&{-i\phi^{\prime}(y)}\\\end{array}\right)\,,\quad\bar{\calL}(\bar{y},\lambda^{-1})\equiv\left(\begin{array}{cc}{i\bar{\phi}^{\prime}(\bar{y})}&{\lambda^{-1}}\\{\lambda^{-1}}&{-i\bar{\phi}^{\prime}(\bar{y})}\\\end{array}\right)\,.
A_{i}^{\prime}\,\left(A+\delta_{\lambda}A\right)-A_{i}^{\prime}\,\left(A\right)-\partial_{i}\lambda^{\prime}-i\left[\lambda^{\prime},A_{i}\right]-i\left[\lambda,A_{i}^{\prime}\right]=-\frac{1}{2}\theta^{kl}\left(\partial_{k}\lambda\partial_{l}A_{i}+\partial_{l}A_{i}\partial_{k}\lambda\right)+o\left(\theta^{2}\right),
\alpha_{j}(A)\equiv\sum_{m_{1},\ldots,m_{j}\neq0}\epsilon^{m_{1}+\cdots+m_{j}}\oint\frac{dt_{1}}{2\pii}\cdots\oint\frac{dt_{j}}{2\pii}\mathrm{e}^{-N(\widetilde{\Phi}_{m_{1}}(t_{1})+\cdots+\widetilde{\Phi}_{m_{j}}(t_{j}))}M_{12}M_{23}\cdotsM_{j1}
S=Tr[\frac{m_{1}^{2}}{2}(\phi^{1})^{2}+\frac{m_{2}^{2}}{2}(\phi^{2})^{2}+\frac{\lambda_{1}}{4N}(\phi^{1})^{4}+\frac{\lambda_{2}}{4N}(\phi^{2})^{4}+g\phi^{1}\phi^{2}]
\partial_{z}\partial_{\bar{z}}\theta^{L}-2\frac{(\partial_{z}s)(\partial_{\bar{z}}\bar{s})}{(s+\bar{s})^{2}}\theta^{L}=0\,,\quad\Rightarrow\quad\partial_{s}\partial_{\bar{s}}\theta^{L}(s,\bar{s})-2\frac{1}{(s+\bar{s})^{2}}\theta^{L}(s,\bar{s})=0
\begin{array}{c}{S=\frac14\intd^{4}x{}\epsilon^{\mu\nu\alpha\beta}{}F_{\mu\nu}^{\tau\lambda}{}F_{\alpha\beta}^{\sigma\rho}{}\epsilon_{\tau\lambda\sigma\rho}+8\intd^{4}x{}h^{\mu\nu}(F_{\mu\nu}-\frac12\eta_{\mu\nu}F)}\\{-8\intd^{4}x{}(h^{2}-h^{\mu\nu}h_{\mu\nu}).}\\\end{array}
\delta{\bfF}=\Xi{\bfF}=\left(\begin{array}{cc}{m}&{b}\\{\bar{b}}&{n}\\\end{array}\right)\left(\begin{array}{c}{{\bfQ}}\\{\bar{\bfD}}\\\end{array}\right)=\left(\begin{array}{cc}{m{\bfQ}}&{b\bar{{\bfD}}}\\{\bar{b}{\bfQ}}&{n\bar{{\bfD}}}\\\end{array}\right)=\left(\begin{array}{c}{\delta{\bfQ}}\\{\delta\bar{\bfD}}\\\end{array}\right),
L_{Yuk}(\overline{{\Psi^{\prime}}}\Psi^{\prime}\varphi\,;\,\phi)=-\frac{h}{\sqrt{2}}\overline{{\Psi}}^{\prime}\Psi^{\prime}\varphi\frac{e^{\gamma\phi/M_{p}}}{\chi^{3/2}}=-\frac{h}{4}\overline{{\Psi^{\prime}}}\Psi^{\prime}\varphi\left[\frac{M^{4}+V_{1}(\phi)}{V_{2}(\phi)}\right]^{3/2}e^{\gamma\phi/M_{p}}.
S_{_{\mathrm{T}}}=\intd^{2}x\,\left[{\overline{\psi}}i\gamma_{\mu}\partial^{\mu}\psi+m{\overline{\psi}}\psi-\frac{g}{2}{\overline{\psi}}\gamma_{\mu}\psi{\overline{\psi}}\gamma^{\mu}\psi\right]\,.
\Omega_{+2}^{-2i}=-\frac{1}{4(1+\chi^{2})}\nabla_{+q}\gamma_{q\dot{p}}^{i}\psi_{+2\dot{p}}^{1-},\\Omega_{-2}^{+2i}=-\frac{1}{4(1+\chi^{2})}\nabla_{-\dot{p}}\tilde{\gamma}_{\dot{p}q}^{i}\psi_{-2q}^{2+},
=\sum_{\gamma_{i}\beta_{i}^{\prime\prime}\bar{\beta}_{i}^{\prime\prime}}w(\bar{\beta}_{i},\beta_{i},\beta_{i}^{\prime\prime},\bar{\beta}_{i}^{\prime\prime}|v)w(\beta_{i}^{\prime\prime},\alpha_{i},\beta_{i}^{\prime},\gamma_{i}|u+v)w(\bar{\beta}_{i}^{\prime\prime},\gamma_{i},\bar{\beta}_{i}^{\prime},\alpha_{i}^{\prime}|u)~.
\left(\partial_{t_{i}}a_{I}\partial_{t_{j}}a_{J}-\sum_{k=1}^{r}{C_{ij}}^{k}(t)\partial_{t_{k}}a_{I}\partial_{t_{r}}a_{J}\right)\partial_{a_{I}}\partial_{a_{J}}\Pi+P_{ij}^{I}\partial_{a_{I}}\Pi=0,
\{f,g\}_{0}=\sum_{i=1}^{N}\left(\frac{\partialf}{\partialx^{i}}\frac{\partialg}{\partialx^{i+N}}-\frac{\partialf}{\partialx^{i+N}}\frac{\partialg}{\partialx^{i}}\right)+\sum_{\alpha=1}^{M}\epsilon_{\alpha}\frac{\partial^{R}f}{\partial\theta^{\alpha}}\frac{\partial^{L}g}{\partial\theta^{\alpha}},\;\;\epsilon_{\alpha}=\pm1.
S_{\phi}=\intdtd^{3}x\biggl\{\frac{1}{4}[-(\partial_{i}\theta_{j}-\partial_{j}\theta_{i})^{2}+2(\partial_{0}\theta_{i}-\partial_{i}\chi)^{2}]+\mu^{2}\chi^{2}+\mathrm{cubic\,and\,quartic\,terms\,in\,\chi,\theta}+\mathrm{const.}\biggr\}.
E_{eff(3+1)}=-L_{z}\sum_{\{n\}}\int_{-\infty}^{+\infty}\frac{dk_{3}}{2\pi}\left(\sqrt{k_{3}^{2}+E_{\{n\}}+m_{f}^{2}}-\sqrt{k_{3}^{2}+m_{f}^{2}}\right)
\hat{\lambda}_{~A}^{(a)}={\frac{\sigmaa^{\frac{3}{2}}}{2^{\frac{1}{4}}}}\lambda_{~A}^{(a)},~~\hat{\bar{\lambda}}_{~A^{\prime}}^{(a)}={\frac{\sigmaa^{\frac{3}{2}}}{2^{\frac{1}{4}}}}\bar{\lambda}_{~A^{\prime}}^{(a)}~,
h_{flat}\,=\,2^{\frac{n}{2}}\int_{0}^{\infty}\frac{2\,dkk^{n-1}}{(4\pi)^{\frac{n}{2}}\Gamma(\frac{n}{2})}e^{-s(k^{2}+m^{2})}=\frac{2^{\frac{n}{2}}e^{-sm^{2}}}{(4\pis)^{\frac{n}{2}}}
\hat{x}_{12}^{\alpha\dot{\alpha}}=x_{12}^{\alpha\dot{\alpha}}+{\frac{4i}{(12)}}[(1^{-}2)\theta_{1}^{+}\bar{\theta}_{1}^{+}+(2^{-}1)\theta_{2}^{+}\bar{\theta}_{2}^{+}+\theta_{1}^{+}\bar{\theta}_{2}^{+}+\theta_{2}^{+}\bar{\theta}_{1}^{+}]^{\alpha\dot{\alpha}}\;.
\begin{array}{l}{<u(z)\,S(w)>=\left(\fraci{z-w}+\fraci{2w}\right)\left(\frack2-1\right)}\\{\left[J^{0}(z)-J^{1}(z)\right]\timesX^{+}(w)\rightarrow\fraci{z-w}i:ie^{2u(w)/(k-2)}}\\\end{array}
M=\frac{2\,\Gamma\left(\frac{p}{2}\right)}{\sqrt{\pi}\,\Gamma\left(\frac{p+1}{2}\right)}\left(\frac{\pi\,\Gamma\left(\frac{1}{p+1}\right)}{\Gamma\left(\frac{p}{p+1}\right)}\,\mu\right)^{\frac{p+1}{2}},
{\calP}_{a}^{x}\,=\,\frac{1}{2}tr(J^{x}a+J^{x}qt^{t}a)\quad;\quad{\calP}_{b}^{x}\,=\,-\frac{1}{2}tr(J^{x}qb^{t})\quad;\quad{\calP}_{c}^{x}\,=\,-\frac{1}{2}tr(J^{x}qcq^{t})
{\tilde{t}}{\,}_{reg}^{jj}(s)=\frac1{(4\pi)^{\fracd2}}\frac{\Gamma(s+1-\fracd2)}{\Gamma(s+1)}\int_{0}^{\infty}dk\,\,k\left(k^{2}+m^{2}\right)^{\fracd2-s-1}{\Sigma}_{0}(kr)\,,
A_{KB}^{(1)}(p_{1},\cdots,p_{M})\sim\,\displaystyle{}\prod_{i=1}^{M}\left[{\calN}_{i}\!\!\!\\!\!\!\!\!\int_{|x_{i}|\leq1}{}\displaystyle\frac{d^{2}x_{i}}{|x_{i}|^{4}}\right]
H=-\sum_{i=1}^{N}\frac{\partial^{2}}{\partial{x_{i}}^{2}}+\sum_{i<j}^{N}\frac{g}{(x_{i}-x_{j})^{2}}-\frac{g^{\prime}}{\sum_{i<j}(x_{i}-x_{j})^{2}}+U(\sqrt{\sum_{i<j}(x_{i}-x_{j})^{2}})\,\,\,,
\left[{\frac{d^{2}}{d{\tilde{a}}^{2}}}+{\frac{1}{\tilde{a}}}{\frac{d}{d\tilde{a}}}+{\tilde{a}}^{2}-\tilde{\mu}{\tilde{a}}^{4}-{\tilde{\beta}}^{2}\right]\psi(\tilde{a})=0.
\phi_{n}^{\prime}(\tau,\sigma)=e^{-i\omega_{n}^{\prime}\tau-in\sigma}~,~~~\tilde{\phi}_{n}^{\prime}(\tau,\sigma)=e^{-i\omega_{n}^{\prime}\tau+in\sigma}~,~~~\omega_{n}^{\prime}=\mathrm{sign}(n)\sqrt{\left(\frac{m}{6}\right)^{2}+n^{2}}~.
\langle{\bfp}\,\vert{\bf0},0\rangle=\langle{\bfp}\,\vertU(B(p))\vert{\bf0},0\rangle=\sqrt{{\frac{m}{\omega_{m}({\bfp}\,)}}}\langle{\bfp}={\bf0}\vert{\bfx}={\bf0},t=0\rangle=N\sqrt{{\frac{m}{\omega_{m}({\bfp}\,)}}}
Re\left[\left(*_{6}G-iG\right)_{pqr}\right]\,\overline{{\Theta}}^{\prime}\,\Gamma^{pqr}\Theta^{\prime}=Re\left[e^{-2i\delta}\left(*_{6}G-iG\right)_{pqr}\right]\,\overline{{\Theta}}\,\Gamma^{pqr}\Theta
\begin{array}{lcr}{{L}_{q}}&{=}&{-\int{\frac{dz}{2i\pi}}J^{q}(z)=-J_{0}^{q};\quadq=0,\pm1}\\{{\bar{L}}_{q}}&{=}&{\int{\frac{d{\bar{z}}}{2i\pi}}\bar{J}^{q}(\bar{z})=-{\bar{J}}_{0}^{q};\quadq=0,\pm1.}\\\end{array}
vol_{p^{\ast}}(\mathcal{\bar{M}}_{g,n})=\frac{1}{d!}\int_{\mathcal{\bar{M}}_{g,n}}(p_{1}^{2}c_{1}(\lambda_{1})+\cdot\cdot\cdot+p_{n}^{2}c(\lambda_{n}))^{d}\tag{3.16a}
S=\frac{-1}{4\pi\alpha^{\prime}}\intd\sigmad\tau\sqrt{-g}g_{ab}\partial^{a}X^{\mu}\partial^{b}X_{\mu}+\frac{1}{2}\int_{\partial\Sigma}d\tauF_{\mu\nu}X^{\nu}\partial_{\tau}X^{\mu}|_{\sigma=a}
\Theta_{\alpha\(j_{1}\cdotsj_{k})}^{(i_{1}\cdotsi_{k})\J=k}=Tr\Big[\underbrace{\Theta_{\j_{1}}^{i_{1}}\otimes_{h.w.}\cdots\otimes_{h.w.}\Theta_{\j_{k}}^{i_{k}}}_{k\\mathrm{objects}}\otimes_{h.w.}\Theta_{\alpha}^{\J=0}\Big]\,.
J(Q^{2})=v_{4}k^{-2}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+2)!(n+1)}\left(\frac{Q^{2}}{k^{2}}\right)^{n+1}\left\{\int_{0}^{\frac{1}{2}}dz\frac{z^{n+1}}{(1-z)^{2}}-\int_{\frac{1}{2}}^{\frac{2}{3}}dz\frac{z^{n+1}}{(1-z)^{2}}\right\}
\begin{array}{c}{4dx\wedgedy=ds\wedgeds^{*}+d\tilde{v}\wedged\tilde{v}^{*}+ds\wedged\tilde{v}+ds^{*}\wedged\tilde{v}^{*}=}\\{=ds\wedgeds^{*}+{\frac{1}{\alpha^{2}}}d\tilde{V}\wedged\tilde{V}^{*}+{\frac{1}{\alpha}}\left(ds\wedged\tilde{V}+ds^{*}\wedged\tilde{V}^{*}\right)}\\\end{array}
2\pi^{2}\,\epsilon_{0}^{\dagger}\epsilon_{0}^{\vphantom{\dagger}}[{\frac{GM}{2}}-Z]=\int_{\calM}\nabla_{\mu}N^{\mu\nu}d\Sigma_{\nu}=2\pi^{2}\,\epsilon_{0}^{\dagger}\epsilon_{0}^{\vphantom{\dagger}}[Pe^{-\phi_{0}}+Qe^{\phi_{0}}]\,
{\calT}_{11}(q)=i\int_{0}^{\infty}\frac{d{q_{0}^{\prime}}^{2}}{2\pi}\rho(q_{0}^{\prime},\vec{q})\left[\frac{i}{{q_{0}}^{2}-{q_{0}^{\prime}}^{2}+i\epsilon}+2\pi\delta({q_{0}^{\prime}}^{2}-{q_{0}}^{2})\frac{1}{e^{\beta|q_{0}|}-1}\right]
V^{1}(\chi,\sigma)=\int\!{\frac{d^{3}p}{(2\pi)^{3}}}\mathrm{ln}\left[1+2A\int\!{\frac{d^{3}q}{(2\pi)^{3}}}{\frac{1}{(p+q)^{2}+\sigma}}{\frac{1}{q^{2}+\sigma}}\right]+{\frac{1}{2}}Tr\mathrm{ln}\Gamma_{\mu\nu},
\psi_{s}(x,y)=\sum{c^{(s)}}_{a_{1}a_{2}a_{3}a_{4}}(x_{a_{1}}x_{a_{2}}-\frac{1}{9}x_{a}x_{a}\delta_{a_{1}a_{2}})(y_{a_{3}}y_{a_{4}}-\frac{1}{9}y_{a}y_{a}\delta_{a_{3}a_{4}})
=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}\left(\frac{4\pi}{gL}\right)^{2n}n_{\nu_{1}}b_{\mu_{1}}(y)\cdotsn_{\nu_{2n}}b_{\mu_{2n}}(y)\int{\calD}z\mu[z]{\calP}_{\mu_{1}\nu_{1}}[z]\cdots{\calP}_{\mu_{2n}\nu_{2n}}[z],
\frac{\deltaS_{0}^{L}}{\deltaB_{\mu\nu}^{a}}\equiv-\frac12F_{a}^{\mu\nu}=0,\;\frac{\deltaS_{0}^{L}}{\deltaA_{a}^{\mu}}\equivA_{\mu}^{a}+\left(D^{\lambda}\right)_{\;\;b}^{a}B_{\lambda\mu}^{b}=0.
Tr[\,Q_{2}\lambda_{SU(2)}^{2}\,]=-\frac{2}{3}\sum_{k=1}^{3}Tr[\,\gamma_{k,7_{4},(0)}\lambda_{U(1)_{2,(0)}}\,]\,Tr[\,\gamma_{k,7_{4},(0)}^{-1}\lambda_{SU(2)}^{2}\,]
\frac{\partial}{\partial\varphi(x,\bar{\theta})}=-\frac{\partial}{\partialc(x)}+\bar{\theta}^{\dot{\alpha}}\frac{\partial}{\partialb^{{\dot{\alpha}}}(x)}-{\mathrm{\small{~\frac12~}}}\bar{\theta}^{2}\frac{\partial}{\partiala(x)}\,
\sum_{A=2}^{m}\phi_{A}^{a}f_{11}^{A}=-\sum_{A=2}^{m}[2\phi_{A1}^{a}f_{1}^{A}+\sum_{B=2}^{m}(\phi_{AB}^{a}f_{1}^{B})f_{1}^{A}]-\phi_{11}^{a},\;\;\;a=1,2,...,m-1
{\frac{(\sqrt{v_{1}}-y_{1})(\sqrt{v_{1}}-y_{2})}{(\sqrt{v_{1}}+y_{1})(\sqrt{v_{1}}+y_{2})}}=e^{2\sqrt{v_{1}}(x-x_{1})},\quad{\frac{(y_{1}-\sqrt{v_{2}})(\sqrt{v_{2}}-y_{2})}{(\sqrt{v_{2}}+y_{1})(\sqrt{v_{2}}+y_{2})}}=e^{2\sqrt{v_{2}}(x-x_{2})}.
K_{2}=\sum_{l=0}^{\infty}\int_{\partial{\calM}}dz\S(z)\partial^{2l}S(z)\frac{t^{-\frac{m-1}2+1+2l}}{(4\pi)^{\frac{m-1}2}}\frac{\Gamma(l+\frac12)\,\Gamma(l+\frac32)}{\pil!\Gamma(2l+2)}\,.
u\starv=\sum_{s=0}^{\infty}{\frac{\kappa^{s}}{2^{s}s!}}\omega^{i_{1}j_{1}}\ldots\omega^{i_{s}j_{s}}{\frac{\partial^{s}u}{\partialx^{i_{1}}\ldots\partialx^{i_{s}}}}{\frac{\partial^{s}v}{\partialx^{j_{1}}\ldots\partialx^{j_{s}}}}\,.
(J_{2}-K_{1}-1)~D_{x_{1}x_{2},y_{1}y_{2}}^{J_{1}J_{2},K_{1}K_{2};0}+(J_{2}-K_{2}-1)~D_{x_{1}x_{2},y_{1}y_{2}}^{J_{1}J_{2},K_{1}K_{2};J_{1}}.
\omega=\omega_{\underline{{i_{1}\dotsi_{\ell}}}}\,V^{\underline{{i_{1}}}}\wedge\dots\wedgeV^{\underline{{i_{\ell}}}}\quad\Leftrightarrow\quad\star\omega=\frac{1}{(D-\ell)!}\,\epsilon_{\underline{{a_{1}\dotsa_{D-\ell}b_{1}\dotsb_{\ell}}}}\omega^{\underline{{b_{1}\dotsb_{\ell}}}}\,V^{\underline{{a_{1}}}}\wedge\dots\wedgeV^{\underline{{a_{D-\ell}}}}
U(\tau)=:\sum_{\mathrm{\tiny~{\cal~N}~}}e^{i\beta_{\mathrm{\tiny~{\cal~N}~}}(\tau)}|{\mathrm{\tiny~{\cal~N}~}};\tau\rangle\langle{\mathrm{\tiny~{\cal~N}~}};\tau|\:,~~~~~~\beta_{\mathrm{\tiny~{\cal~N}~}}(\tau)\in[0,2\pi)\:.
M_{\hat{\mu}\hat{\nu}}^{\mathrm{cov.}}=G_{\hat{\mu}\hat{\nu}}+i{\frac{G_{\hat{\mu}\hat{\rho}}G_{\hat{\nu}\hat{\lambda}}}{\sqrt{-G(\partiala)^{2}}}}\tilde{H}_{\mathrm{cov.}}^{\hat{\rho}\hat{\lambda}}.
x_{+}=\left[\frac{(3w_{X}-2)\Omega_{X,0}+\sqrt{[(2-3w_{X})\Omega_{X,0}]^{2}-16(1-3w_{X})\Omega_{\Lambda_{(4)},0}\Omega_{\lambda,0}}}{(3w_{X}-1)\Omega_{\lambda,0}}\right]^{-1/3w_{X}}.
\Box\bar{h}_{\nu}^{\mu}+l_{T\pm}^{2}\Box^{2}\bar{h}_{\nu}^{\mu}+\frac{1}{3}(l_{T\pm}^{2}+l_{S\pm}^{2})(\partial^{\mu}\partial_{\nu}\Box-\delta_{\nu}^{\mu}\Box^{2})\bar{h}=-16\piG_{N\pm}\bar{S}_{\pm\nu}^{\mu},
\begin{array}{lcl}{X^{(III)}(\sigma,\tau)={\displaystyleX_{0}^{(III)}(\tau)}}\\{+{\displaystyle\frac{2\pi-\sigma_{0}}{2\pi}~k~\sum_{n\neq0}^{}\left\{\frac{1}{|n|}~F_{\nu}\left(\frac{2\pi}{2\pi-\sigma_{0}}|n|\tau,\frac{2\pi}{2\pi-\sigma_{0}}|n|\tau_{0}\right){\phi_{n}}^{\ast}(2\pi-\sigma_{0}){}~e^{i\frac{2\pi}{2\pi-\sigma_{0}}n(\sigma-\sigma_{0})}\right\}}}\\\end{array}
a\equiva_{D}=4\left\{(\omega_{n}+i\mu_{U})^{2}+{\stackrel{\rightharpoonup}{l}}^{2}+m_{U}^{2}+(\omega_{n}+i\mu_{U})[\Omega_{m}+i(\mu_{D}-\mu_{U})]+\stackrel{\rightharpoonup}{l}\cdot\stackrel{\rightharpoonup}{p}+m_{U}^{2}(m_{D}^{2}-m_{U}^{2})/(m_{U}^{2}+m_{D}^{2})\right\}.
Z=\beta^{N/2}\Gamma((N-1)/2)^{-1}{\LARGE\int_{0}^{\infty}}d\sigma{\LARGE\int_{-\infty}^{\infty}}d\sigma_{1}(4\beta^{2}\sigma^{2}+(2\beta\sigma_{1}+a)^{2}+m^{2})^{M}\sigma^{N-2}exp[-\beta(\sigma^{2}+\sigma_{1}^{2})]
{\calL}=\sqrt{-g}\biggr[R-\frac{1}{2}\sum(\partial\eta_{i})^{2}-\frac{1}{4}\left(e^{-\eta_{1}-\eta_{2}-\eta_{3}}F^{1}F^{1}+e^{-\eta_{1}-\eta_{2}+\eta_{3}}F^{2}F^{2}+e^{-\eta_{1}+\eta_{2}+\eta_{3}}F^{3}F^{3}+e^{-\eta_{1}+\eta_{2}-\eta_{3}}F^{4}F^{4}\right)\biggr].
U_{\mathrm{s}}\;=\;{\frac{4}{3n_{\mathrm{s}}(k^{2})^{2}}}\int_{0}^{k}d\zeta\,\rho_{\mathrm{s}}(\zeta)(k^{2}-\zeta^{2})-{\frac{1}{3}}-{\frac{2}{3}}{\frac{d_{\mathrm{s}}(k^{2})-p_{\mathrm{s}}(k^{2})}{n_{\mathrm{s}}(k^{2})^{2}}}\;.
N=8:\quadA={\frac{1}{2}}\left(\begin{matrix}{1}&{2}&{1}\\{1}&{0}&{-1}\\{-1}&{-2}&{-1}\\\end{matrix}\right);\quadB={\frac{1}{2}}\left(\begin{matrix}{1}&{1}&{2}\\{-1}&{-1}&{-2}\\{1}&{-1}&{0}\\\end{matrix}\right)
\left(\begin{array}{l}{f_{n}(r)}\\{g_{n}(r)}\\\end{array}\right)=\left(\begin{array}{c}{C_{n}^{(1)}(E)J_{n-\Phi^{(0)}+\Upsilon}(kr)+C_{n}^{(2)}(E)Y_{n-\Phi^{(0)}+\Upsilon}(kr)}\\{{\frac{ik}{E+m}}\bigl[C_{n}^{(1)}(E)J_{n+1-\Phi^{(0)}+\Upsilon}(kr)+C_{n}^{(2)}(E)Y_{n+1-\Phi^{(0)}+\Upsilon}(kr)\bigr]}\\\end{array}\right),
\tilde{\Omega}_{D}^{\,^{\bullet}}({\mathcalA})\;:=\;\displaystyle\bigoplus_{k=0}^{\infty}\;\tilde{\Omega}_{D}^{k}({\mathcalA}),\\tilde{\Omega}_{D}^{k}({\mathcalA})\;:=\;\Omega^{k}({\mathcalA})\big/\left(K^{k}+\deltaK^{k-1}\right)\;\cong\;\tilde{\Omega}^{k}({\mathcalA})/\deltaK^{k-1}\.
\frac{1}{4\beta}F_{\nu,\alpha\beta}^{\infty}(x,y)=-\int_{1}^{\infty}\frac{d\alpha}{\alpha^{2}}j_{\nu,\alpha\beta}^{\infty}(\alphax,y)-\frac{1}{2}\frac{\partial}{\partialx^{\nu}}\int_{1}^{\infty}\frac{d\alpha}{\alpha}\left(1-\frac{1}{\alpha}\right)x^{\rho}j_{\rho,\alpha\beta}^{\infty}(\alphax,y)
\sum_{d_{1},\cdots,d_{m}}P_{a_{1},\cdots,a_{m}}^{d_{1},\cdots,d_{m}}Q_{d_{1},\cdots,d_{m}}^{c_{1},\cdots,c_{m}}=\sum_{b_{1},\cdots,b_{m}}Q_{a_{1},\cdots,a_{m}}^{b_{1},\cdots,b_{m}}P_{b_{1},\cdots,b_{m}}^{c_{1},\cdots,c_{m}}.
\begin{array}{lll}{L_{IJ}^{(+)}L_{JK}^{(+)}}&{-}&{(-1)^{P_{J}(P_{I}+P_{K})}L_{JK}^{(+)}L_{IJ}^{(+)}=}\\{}&{}&{(q^{-1}-q)(-1)^{P_{I}P_{J}}L_{JJ}^{(+)}L_{IK}^{(+)},(I<J<K)}\\\end{array}
[\hat{q}^{i},\hat{\Pi}_{j}]_{-}=\hbar\delta_{j}^{i},\quad[\hat{\psi}^{i},\hat{\psi}^{j}]_{+}=\hbar\delta^{ij},[\hat{\varphi}_{m},\hat{\varphi}_{n}]_{+}=\hbar\delta_{mn}
\widetilde{\eta}_{n_{r}}=\left(\frac{y}{z_{2}}\right)^{2}=p^{2}\,\left\{1+2\,C_{n_{r}}\,p^{-2/3}-2\left(\lambda_{l}^{(\ast)}\right)^{1/2}\,c\,p^{-1}+O\left(p^{-4/3}\right)\right\}\;.
N^{(l)}(L)\;:=\;\sum_{\{\alpha_{a},\beta_{a}\}}\prod_{b=1}^{l}\left[\psi_{\alpha_{b}}^{(b)}(-L,0)\;U\Big({\calC}(L)\Big)\;\overline{{\psi}}_{\beta_{b}}^{(b)}(+L,0)\right]\;,
Z^{I}=Z_{\infty}^{I}+\sum_{a=1}^{a=5}{\frac{q^{I}{}_{a}}{|\vec{x}-\vec{\underline{{x}}}_{a}|}}\,\,\qquadQ^{5}=\sum_{a=1}^{a=5}q^{5}{}_{a}=Q^{7}=\sum_{a=1}^{a=5}q^{7}{}_{a}=0\,
M_{X}^{2}\=\\frac{\sum_{k}\delta_{X}^{k}\delta_{X}^{k}+2X\sum_{k}\delta_{X}^{k}\langlem_{k}\rangle}{\langle\mathrm{Re}f_{p}\rangle+\sum_{k}c_{X}^{k}\langlem_{k}\rangle}\M_{Pl}^{2}
\begin{array}{l}{d{\hat{s}}^{2}=2F(x)\,du\left[dv-\frac{1}{2}K(x)\,du+\omega_{I}dx^{I}\right]-dx^{I}dx^{I}}\\{\hat{B}=-2F(x)\,du\wedge\left[dv+\omega_{I}dx^{I}\right]\qquad,\qquade^{2\hat{\phi}}=F(x)\,,}\\\end{array}
\frac{d}{d\tilde{\tau}}{\tilde{\zeta}}^{\mu}(z_{o},\tilde{\tau})={\tilde{U}}^{\mu}({\tilde{\zeta}}^{\mu}(z_{o},\tilde{\tau})),\qquad{\tilde{\zeta}}^{\mu}(z_{o},0)=z_{o}^{\mu}.
Str\left(D_{a_{1}}\phi^{j_{1}}\ldotsD_{a_{l}}\phi^{j_{l}}\phi^{i_{1}}\ldots\phi^{i_{n}}\underbrace{\phi^{i_{2(k+r)}^{\prime}}\phi^{i_{2(k+r)-1}^{\prime}}}\dots\underbrace{\phi^{i_{2}^{\prime}}\phi^{i_{1}^{\prime}}}\ldotsF_{a_{p}a_{p-1}]}\right),
\left(\begin{array}{c}{\tilde{U}^{t}}\\{\tilde{U}^{*}}\\\end{array}\right)=\frac{2E^{2}}{lm^{2}}[(\frac{\partial\deltax^{\parallel}}{\partial\tau})^{2}-(\frac{\partial\deltax^{\parallel}}{\partial\sigma})^{2}]\left(\begin{array}{c}{\sqrt{E^{2}-m^{2}}/m}\\{E/m}\\\end{array}\right)+{\calO}(1/r),
\bigl<n;t|i\hbar\frac{\partial}{\partialt}|n;t\bigr>=\intdq\bigl<n;t|q\bigr>\Bigl(i\hbar\frac{\partial}{\partialt}\Bigr)\bigl<q|n;t\bigr>=(n+\frac{1}{2})\frac{\hbar}{2\sqrt{\kappa}}\frac{\dot{B}(t)A(t)-B(t)\dot{A}(t)}{A(t)},
\begin{array}{lll}{d_{v}}&{=}&{[\sum_{i=1}^{k-1}4i]+k(N-2k+3)+(k+1)(N-2k)}\\{}&{=}&{(2k+1)N-k(2k+1)}\\{d_{H}}&{=}&{2\sum_{i=1}^{k-1}[\frac{(2i+1)2i}{2}+\frac{2i(2i+3)}{2}-2i(2i+1)]}\\{}&{+}&{(2N-4k)\frac{2k(2k+2)}{2}-(N-2k+1)k(2k+1)-(N-2k)(k+1)(2(k+1)-1)}\\{}&{+}&{2\frac{2k}{2}+2\frac{2k(2k+1)}{2}-2k(2k+1)}\\{}&{+}&{N}\\{}&{=}&{[2k(k-1)]+[-(N-2k)-k(2k+1)]+[2k]+[N]}\\{}&{=}&{k}\\\end{array}
x_{r}(t)=a\,{\frac{\mathrm{dn}(at,k)}{\mathrm{sn}(at,k)}},\quady_{r}(t)=-a^{2}{\frac{\mathrm{cn}(at,k)}{\mathrm{sn}^{2}(at,k)}},\quadz_{r}(t)=-b^{2}\wp(bt),
\hat{Q}=\intdx\,\colon\left[\gamma^{\mu}\gamma^{0}\hat{\vec{\psi}}(x^{\mu})\partial_{\mu}\hat{\vec{\psi}}(x^{\mu})+i\gamma^{0}\hat{\vec{\psi}}(x^{\mu})\vec{\nabla}\hat{W}(x^{\mu})\right]\colon\qquad.
\delta_{f}L_{3}=-\frac{6\kappa_{1}^{2}}{r}\left((1-f^{2})f_{rr}-f\:f_{r}^{2}-\frac{1}{r}(1-f^{2})f_{r}+\frac{1}{r^{2}}(1-f^{2})^{2}f\right)
R_{j_{1}j_{2}}^{k_{1}k_{2}}(u)R_{k_{1}j_{3}}^{l_{1}k_{3}}(u+v)R_{k_{2}k_{3}}^{l_{2}l_{3}}(v)=R_{j_{2}j_{3}}^{k_{2}k_{3}}(v)R_{j_{1}k_{3}}^{k_{1}l_{3}}(u+v)R_{k_{1}k_{2}}^{l_{1}l_{2}}(u).
S=S^{\prime}+\intd\tau\left(\sum_{a}q_{s}^{*}\delta_{a}q^{s}\eta_{a}-\sum_{ab}\alpha_{b}^{*}\left(\zeta^{-1}\right)_{ab}\eta_{a}+\sum_{abc}\frac{1}{2}C_{ab}{}^{c}\eta_{c}^{*}\eta_{a}\eta_{b}+\sum_{a}\bar{\eta}_{a}^{*}b_{a}\right).