harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
a_{0}=-\frac{6}{g}\Sigma_{3}+\frac{6}{g^{2}}\sum_{n\neq0}{\frac{1}{|n|}}\left(\Sigma_{3}a_{n}a_{-n}+a_{n}a_{-n}\Sigma_{3}+a_{n}\Sigma_{3}a_{-n}-\frac{3\Sigma_{3}}{2}\right)+\ldots\;.
\begin{aligned}{\hat{g}_{\mu\nu}}&{{}=e^{a\phi+b\varphi}g_{\mu\nu}\,,}\\{\hat{g}_{xx}}&{{}=e^{-7(a\phi+b\varphi)}\,,\qquad}\\\end{aligned}\qquad\begin{aligned}{\hat{\phi}}&{{}=c\phi+d\varphi\,,}\\{\hat{A}_{x}}&{{}=-2\chi\,.}\\\end{aligned}
\deltaZ_{\chi}=\frac{\lambda^{2}h^{2}}{3}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{k^{2}}{(k^{2}+M_{H}(h)^{2})^{4}}=\frac{\lambda^{2}h^{2}}{M_{H}(h)^{3}}\frac{1}{192\pi}.
\begin{array}{l}{\delta_{\mu}A_{\nu}^{a}=0\;,}\\{\delta_{\mu}\phi^{a}=-\varepsilon_{\mu\nu}\partial^{\nu}\overline{{c}}^{a},}\\{\delta_{\mu}c^{a}=-A_{\mu}^{a}\;,}\\{\delta_{\mu}\overline{{c}}^{a}=0\;,}\\{\delta_{\mu}b^{a}=\partial_{\mu}\overline{{c}}^{a}.}\\\end{array}
\int{\frac{d^{N}zd^{N}p}{(2\pi)^{N}}}e^{-{\calH}_{z}/T_{H}}=\intd^{N}z\left[(\Omega_{\mathrm{hor}}T_{H}/2\pi)\left(\phi^{\dagger}T_{i}T_{j}\phi\right)\right]^{1/2}
Z^{0i}=\frac{i}{4\pi}{\bar{\calA}}\left\{\sigma_{\alpha\beta}^{0i}D^{\alpha\beta}{\calA}_{D}-{\bar{\sigma}}_{\dot{\alpha}\dot{\beta}}^{0i}{\bar{D}}^{\dot{\alpha}\dot{\beta}}{\bar{\calA}}_{D}\right\}|-\frac{i}{4\pi}{\bar{\calA}}_{D}\left\{\sigma_{\alpha\beta}^{0i}D^{\alpha\beta}{\calA}-{\bar{\sigma}}_{\dot{\alpha}\dot{\beta}}^{0i}{\bar{D}}^{\dot{\alpha}\dot{\beta}}{\bar{\calA}}\right\}|.
\langlen|\,H^{(2)}|n\rangle={\frac{1}{2}}\sum_{i}\nu_{i}\left[g({\vec{p}}_{i})+{\frac{{\vec{p}}_{i}+m^{2}+{\frac{\lambda}{2}}\phi_{0}^{2}}{g({\vec{p}}_{i})}}\right]+{\frac{\lambda}{8}}\int{\frac{d^{\nu}k}{(2\pi)^{\nu}}}{\frac{1}{g(\vec{k})}}\sum_{i}\nu_{i}{\frac{1}{g({\vec{p}}_{i})}}\;.
W_{ALE}^{(2)}=\epsilon\left[\frac{1}{12}(\zeta+\frac{1}{\zeta})+\frac{y_{3}^{2}}{2}+\frac{y_{5}^{2}}{2}+\frac{y_{4}^{3}}{3}-\frac{\tilde{u}_{0}^{4}}{4}y_{4}-\frac{\tilde{u}_{1}+\tilde{u}_{0}^{6}}{12}\right]\,
a=\frac{1}{2l_{s}^{2}}\kappa_{e}^{1/2}V^{e}\tau^{-1}\bar{V}^{e}\kappa_{e}^{1/2},\;\;a^{-1}=2l_{s}^{2}\kappa_{e}^{-1/2}V^{e}\tau\bar{V}
{\bfp}_{1}\,=\,M_{0}^{2}\,[\varepsilon(r)\,+\,\alpha\,r_{s}^{2}\,\Box]\,\dot{\bfu}\,{,}\quad{\bfp}_{2}\,=\,-\,\alpha\,r_{s}^{2}\,M_{0}^{2}\,\Box\,{\bfu}\,=\,M_{0}^{2}\,{\bfu}_{2}\,{.}
e^{-2\phi(x^{+},x^{-})}=v_{+}(x^{+})+\frac{\displaystyle1-\frac{\Lambda}{8}F_{+}F_{-}}{\displaystyle1+\frac{\Lambda}{8}F_{+}F_{-}}\int^{x^{+}}\!dy\frac{F_{+}^{\prime}(y)}{F_{+}(y)}v_{+}(y)+\frac{a}{2}+(+\leftrightarrow-).
\Theta=\frac{1}{2g^{3}}\tilde{\beta}(g)(F_{\mu\nu}^{a})^{2}+\frac{1}{32\pi^{2}}\tilde{b}(g)B_{\mu\nu}^{2}+\frac{\tilde{c}(g)}{16\pi^{2}}(W_{\mu\nu\rho\sigma})^{2}-\frac{a(g)}{16\pi^{2}}(\tilde{R}_{\mu\nu\rho\sigma})^{2}+\frac{\tilde{c}(g)}{6\pi^{2}}V_{\mu\nu}^{2}\,,
\left(D_{M}+i\frac{e^{\phi}}{4}\partial_{M}C\right)^{(1)}\varepsilon^{(0)}+\frac{i}{480}\left(\Gamma^{M_{1}...M_{5}}F_{M_{1}...M_{5}}\Gamma_{M}\right)^{(1)}\varepsilon^{(0)}=0
\psi_{a_{1}a_{2}}^{\sigma_{1}}(x_{1},x_{2})=\sum_{\alpha,\beta=+,-}[\,e^{i(\alphak_{1}x_{1}+\betak_{2}x_{2})}B_{a_{1}a_{2}}(\alpha,\beta)-e^{i(\alphak_{1}x_{2}+\betak_{2}x_{1})}A_{a_{2}a_{1}}(\alpha,\beta)\,].
S_{(5)}=\frac{1}{2\kappa_{(5)}^{2}}\intd^{4}x\intdy\sqrt{-g_{(5)}}\left[R^{(5)}+12k^{2}\right]-\intd^{4}x\left[\sqrt{-g_{(+)}}\lambda_{(+)}+\sqrt{-g_{(-)}}\lambda_{(-)}\right],
\left(X^{\mu}\right)(\tau,\sigma^{1},\sigma^{2})=\left(\begin{array}{c}{\tau}\\{X^{1}(\tau,\sigma^{1},\sigma^{2})}\\{X^{2}(\tau,\sigma^{1},\sigma^{2})}\\{X^{3}(\tau,\sigma^{1},\sigma^{2})}\\\end{array}\right),
H_{c}={\frac{1}{2}}\sum_{i=1}^{N}p_{i}^{2}+{\frac{1}{2}}\sum_{i>j}{\frac{\lambda(\lambda-1)}{(x_{i}-x_{j})^{2}}}+{\frac{1}{2}}\omega^{2}\sum_{i=1}^{N}x_{i}^{2}
H(\pi,q)=\intd^{3}x\,(N(-q^{1/2}R+q^{-1/2}(\tilde{\pi}^{ab}\tilde{\pi}_{ab}-{\textstyle{\frac{1}{2}}}\tilde{\tilde{\pi}}^{2}))-2N^{b}D_{a}\tilde{\pi}_{b}^{a})
\overline{{Z}}_{\Psi}=\int\prodD\Phi^{a}\prodD\vartheta^{b}\prod\delta(d^{\gamma})exp\left({\frac{i}{\hbar}}W(\Phi^{a},{\frac{\partial\overline{\Psi}}{\partial\Phi^{a}}},\vartheta^{b},{\frac{\partial\overline{\Psi}}{\partial\vartheta^{b}}}\right)
\left(\mathrm{m}_{i}F_{p}\right)\left(\phi_{0},\cdots,\hat{\phi}_{i-1},\phi_{i},\cdots,\phi_{p}\right)=F_{p}\left(\phi_{0},\cdots,\phi_{i-2},\phi_{i},\phi_{i},\cdots,\phi_{p}\right)+O(\hbar)
\frac{1}{N^{2}}\langleH_{QM}\rangle\;\;\equiv\;\;f\;\;=\;\;\frac{1}{3}E\;-\;\frac{1}{3}\sum_{n=1}^{\infty}\Big[\lambda_{n}|c_{n}|^{2}\;+\;2(\;\kappa_{n}c_{n}\;+\;\bar{\kappa}_{n}c_{-n}\;)\Big]-\frac{\gamma}{96}\;.
Re\omega_{\phi}=\Bigl\{\begin{array}{c}{-2\frac{\alpha}{\pi}\sum\frac{t^{2}}{B^{2}n^{2}}K_{2}(nBt)}\\{\alpha\sum_{n=1}^{\infty}\frac{\midt\mid^{2}}{n^{2}B^{2}}Y_{2}(nB\midt\mid)}\\\end{array}\Bigr\}.
\hat{h}^{c}\psi_{\varkappa,n}^{c}=\epsilon_{\varkappa,n}^{c}\psi_{\varkappa,n}^{c}\,,\;\psi_{\varkappa,n}^{c}=-\sigma_{3}\psi_{-\varkappa,n}^{\ast}\,,\;\epsilon_{\varkappa,n}^{c}=-\epsilon_{-\varkappa,n},\;\left(\psi_{k,n}^{c}\,,\psi_{\varkappa^{\prime},n^{\prime}}^{c}\right)=\varkappa\delta_{\varkappa,\varkappa^{\prime}}\delta_{n,n^{\prime}}\,.
na_{1}(\vec{q}\,^{\prime},\vec{q})+\sum_{\nu=1}^{3}(q_{\nu}^{\prime}-q_{\nu})\frac{\partiala_{1}(\vec{q}\,^{\prime},\vec{q})}{\partialq_{\nu}^{\prime}}=a_{1}(\vec{q}\,^{\prime},\vec{q}\,^{\prime})=-V(\vec{q}\,^{\prime}),
\langlec_{p\mathbf{k}}\midc_{p\mathbf{k}^{^{\prime}}}\rangle=\langle\underline{{c}}_{p\mathbf{k}}\mid\underline{{c}}_{p\mathbf{k}^{^{\prime}}}\rangle=\langlec_{p\mathbf{k}}\midH_{0}\midc_{p\mathbf{k}^{^{\prime}}}\rangle=\langle\underline{{c}}_{p\mathbf{k}}\midH_{0}\mid\underline{{c}}
C_{1}(\tau)_{free}\;=\;s^{\prime}\;\left(\frac{1}{2\pi}\right)^{2M}\Big(\tau^{2}\Big)^{\sum_{b=1}^{N}(n_{b}-m_{b})(n_{b}^{\prime}-m_{b}^{\prime})}\;\left(1+O\Big(\frac{1}{\tau}\Big)\right)\;.
Q_{\pm}^{\prime}=Q_{\pm}\mpe\lambda\epsilon_{0}^{(0)}\\,\\A_{\pm,n}^{\prime}=A_{\pm,n}\mpe\lambda\sqrt{n}\,\epsilon_{\mpn}^{(0)}\\,\\{A^{\dagger}}_{\pm,n}^{\prime}=A_{\pm,n}^{\dagger}\pme\lambda\sqrt{n}\,\epsilon_{\pmn}^{(0)}\,
S=-T_{2}\int_{{\calM}_{3}}d^{3}z\sqrt{-detg_{mn}}-T_{2}\int_{{\calM}_{3}}dz^{m_{1}}\wedgedz^{m_{2}}\wedgedz^{m_{3}}A_{m_{1}m_{2}m_{3}}^{(3)}(x(z))\quad(m=0,1,2),
\lambda_{n}=\frac{4\pi}{3}\int_{0}^{\infty}dr~r^{2}[\frac{F_{\pi}^{2}}{2n^{2}}sin^{2}nF+\frac{2}{e^{2}}\frac{sin^{2}nF}{n^{2}}(2F^{2}+\frac{sin^{2}nF}{n^{2}r^{2}})]
\langle{\bfx}\cdot\nablaV({\bfx})\rangle=\langle{\frac{{\bfp}^{2}}{\sqrt{{\bfp}^{2}+m_{1}^{2}}}}+{\frac{{\bfp}^{2}}{\sqrt{{\bfp}^{2}+m_{2}^{2}}}}\rangle.
\int_{S^{4}}x^{i_{1}}\cdotsx^{i_{2m}}=\frac{3\omega_{4}}{2^{3m+2}\Gamma(m+\frac{5}{2})}\bigl(\delta^{i_{1}i_{2}}\cdots\delta^{i_{2m-1}i_{2m}}+\mathrm{perms.}\bigr),
\frac{1}{\lambda_{r}}\sigma_{0}-\frac{2}{\pi}\sigma_{0}+\frac{1}{\pir}\frac{\displaystyle\Gamma\left(\frac{3}{2}+imr\right)\Gamma\left(\frac{3}{2}-imr\right)}{\displaystyle\Gamma\left(1+imr\right)\Gamma\left(1-imr\right)}=0\,,
{\calL}_{m}[B,\Lambda]={\frac{1}{4\kappa}}(B_{\mu\nu}+\partial_{\mu}\Lambda_{\nu}-\partial_{\nu}\Lambda_{\mu})^{2}-{\frac{1}{4\eta^{2}}}B_{\mu\nu}\partial_{\lambda}\partial^{\lambda}B^{\mu\nu}+{\frac{1}{4\gamma^{4}}}B_{\mu\nu}(\partial_{\lambda}\partial^{\lambda})^{2}B^{\mu\nu}.
\tilde{G}_{\mu\nu}^{0A}=\tilde{G}_{\mu\nu}^{0L}+\biggl[\biggl(k_{\mu}k_{\nu}\Sigma_{1}+\eta_{\mu}k_{\nu}\Sigma_{2}\biggr)ln\Sigma_{3}+(k\to-k;\mu\leftrightarrow\nu)\biggr]
{\calL}=\frac{1}{12}\,H_{\mu\nu\rho}H^{\mu\nu\rho}-\frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\,m\,\epsilon_{\mu\nu\rho\lambda}\,A^{\mu}\partial^{\nu}B^{\rho\lambda}
\tilde{A}(p)=3g^{2}N(N^{2}-1)\delta_{ab}\int\frac{d^{4}k}{(2\pi)^{4}}d^{4}\theta\,\left\{\frac{m^{2}\left[V^{a}(p,\theta,{\overline{\theta}})V^{b}(-p,\theta,{\overline{\theta}})\right]}{(k^{2}+m^{2})[(p-k)^{2}+m^{2}]}\right\}.
\langleT_{UU}\rangle\sim\frac{1}{r^{2}F^{2}}\int_{r_{+}}^{r}dr\;F\,\biggl\{\frac{d}{dr}[2r^{2}\langleT_{\theta}^{\theta}\rangle-r^{2}\langleT_{\alpha}^{\alpha}\rangle]+4r\langleT_{\theta}^{\theta}\rangle\biggr\}.
\begin{array}{rcl}{\frac{\partial^{n}}{\partial\lambda^{n}}\psi}&{=}&{(-1)^{n}\,n!\,L(\lambda)^{-n}\,\psi\,,}\\{\frac{\partial^{n}}{\partial\lambda^{n}}\bar{\psi}}&{=}&{(-1)^{n}\,n!\,L(\lambda)^{-n}\,\bar{\psi}\,,\quadn\geq1\,,}\\\end{array}
\sum_{j=-\infty}^{\infty}(-1)^{j}M_{j}^{2r}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{k_{\mu_{1}}\cdotsk_{\mu_{n}}}{(k^{2}+M_{j}^{2})^{N}(q^{2}+M_{j}^{2})^{N}}.
A_{+}=\left(\begin{array}{cc}{\frac{\omega_{+}}2}&{\frac{\sigma_{\+}^{-}}r}\\{\frac{r}{\ell^{2}}+\frac{\sigma_{\+}^{+}-\sigma_{\-}^{-}}{2r}}&{-\frac{\omega_{+}}2}\\\end{array}\right)\quad,\quad\tilde{A}_{-}=\left(\begin{array}{cc}{\frac{\omega_{-}}2}&{\frac{-r}{\ell^{2}}+\frac{\sigma_{\+}^{+}-\sigma_{\-}^{-}}{2r}}\\{\frac{-\sigma_{\-}^{+}}r}&{-\frac{\omega_{-}}2}\\\end{array}\right)\quad,
\begin{array}{rcl}{{\calB}_{\pm}}&{\equiv}&{M^{2}+N^{2}(\aleph-\aleph^{2}a^{2}g^{2}+4g^{2}N^{2})-\left[(\aleph+\aleph^{2}a^{2}g^{2}+6g^{2}N^{2})\right.}\\{}&{}&{}\\{}&{}&{\left.\pm2g\sqrt{a^{2}(1+\aleph-\aleph^{2})-N^{2}\left(\aleph-3\aleph^{2}a^{2}g^{2}+3g^{2}N^{2}\right)}\right]Z^{2}\,,}\\\end{array}
C_{l}=\left|\begin{array}{ccccc}{a_{2}}&{a_{1}}&{0}&{\ldots}&{0}\\{a_{3}}&{a_{2}}&{a_{1}}&{\ldots}&{0}\\{\vdots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{a_{l}}&{a_{l-1}}&{a_{l-2}}&{\ldots}&{a_{1}}\\{a_{l+1}}&{a_{l}}&{a_{l-1}}&{\ldots}&{a_{2}}\\\end{array}\right|
\begin{array}{ll}{I_{(a)}+I_{(b)}+I_{(c)}+I_{(d)}}&{\sim\left[G_{++}^{(0)}(k+p_{2})-G_{-+}^{(0)}(k+p_{2})\right]G_{R}(k-p_{1})}\\{{}}&{=G_{A}(k+p_{2})G_{R}(k-p_{1}).}\\\end{array}
\Omega=-{\calH}\,\mathrm{d}z\wedge\mathrm{d}\bar{z}+\bar{p}\,\mathrm{d}z\wedge\mathrm{d}\varphi+p\,\mathrm{d}\varphi\wedge\mathrm{d}\bar{z}\stackrel{!}{=}\,\mathrm{d}\bar{S}\wedge\mathrm{d}z-\mathrm{d}S\wedge\mathrm{d}\bar{z}\,,
R_{(f)s}^{(1)}(z,z^{\prime})=\frac{\partial_{z}\{\Theta[1/2,1/2](J_{(1)}|\omega_{s}^{(1)})\}}{\Theta[1/2,1/2](J_{(1)}|\omega_{s}^{(1)})}\sqrt{\frac{\partial_{z^{\prime}}J_{(o)s}^{(1)}(z^{\prime})}{\partial_{z}J_{(o)s}^{(1)}(z)}}
F_{0}^{BBA}\,=-\frac{R_{3}^{0}}{2R_{2}^{0}}F_{10}^{hBA}\,,\quadF_{0}^{\bar{c}cB}\,=-\frac{R_{3}^{0}}{R_{2}^{0}}F_{0}^{\bar{c}ch}\,,\quadF_{0}^{AAhh}\,=\frac{R_{5}^{0}}{R_{3}^{0}}F_{10}^{AABB}\,.
\left(\begin{matrix}{\Gamma^{\phi_{P}^{0}11}(p)}&{\Gamma^{\phi_{P}^{0}12}(p)}\\{\Gamma^{\phi_{P}^{0}21}(p)}&{\Gamma^{\phi_{P}^{0}22}(p)}\\\end{matrix}\right)=M_{P}^{-1}\left(\begin{matrix}{\Gamma_{R}^{\phi_{P}^{0}}(p)}&{0}\\{0}&{\Gamma_{R}^{\phi_{P}^{0}*}(p)}\\\end{matrix}\right)M_{P}^{-1}
S_{\mathrm{eff}}^{d=2}|_{m=0}={\frac{4}{\left(\beta\sqrt{\pi}\right)^{3}}}\sum_{k=1}^{\infty}{\frac{{\calB}_{2k}(\beta^{2}B/2)^{2k}\Gamma\left(\frac{3}{2}-2k\right)}{(2k)!}}\sum_{l=0}^{\infty}{\frac{(\beta\mu)^{2l}\eta(3-4k-2l)}{(2l)!}}
N^{\calP}~=~\sum_{m_{1}=-j_{1}}^{j_{1}}\cdots\sum_{m_{p}=-j_{p}}^{j_{p}}\left[~{\bar{\delta}}_{(\sum_{n=1}^{p}m_{n}),0}~-~\frac12~{\bar{\delta}}_{(\sum_{n=1}^{p}m_{n}),1}~-~\frac12~{\bar{\delta}}_{(\sum_{n=1}^{p}m_{n}),-1}~\right],
\Sigma^{0}(p_{f})={\frac{e^{2}}{\beta}}\int{\frac{d^{2}k}{(2\pi)^{2}}}{\frac{{\calM}({\bfk},p_{0f})}{{\bfk}^{2}+p_{0f}^{2}+{\calM}^{2}({\bfk},p_{0f})}}[D_{L}^{0}({\bfq})+D_{T}^{0}({\bfq})]
\left(\partial^{\gamma}\partial_{\alpha}T^{(\alpha\nu)\beta}-\partial^{\beta}\partial_{\alpha}T^{(\alpha\nu)\gamma}\right)=-\frac{D^{2}}{ae^{2}}\partial^{2}T^{(\beta\gamma)\nu}+\frac{D}{2e^{2}}\partial^{2}\epsilon^{\beta\gamma}{}_{\kappa\lambda}T^{(\kappa\lambda)\nu}.
ds_{11}^{2}=dx^{+}dx^{-}+\frac{l_{p}^{9}}{R^{2}}\sum_{n=1}^{N}\frac{Q_{n}}{|\vec{r}-\vec{r}_{n}|^{7}}dx^{-}dx^{-}+dx_{1}^{2}+\\cdots\+dx_{9}^{2}.
\tilde{\calE}_{q}[\phi(x)]=\frac{1}{2}\sum_{j}(\tilde{\omega}_{j}-\tilde{m})+\int_{0}^{\infty}\frac{dk}{2\pi}(\tilde{\omega}(k)-\tilde{m})\frac{d}{dk}\tilde{\delta}(k)+\tilde{\calE}_{ct}[\phi(x)]
O^{T}TO=\left(\begin{array}{cccc}{0}&{1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\{0}&{0}&{0}&{-1}\\{0}&{0}&{1}&{0}\\\end{array}\right),~O^{T}QO=\left(\begin{array}{cccc}{0}&{-1}&{0}&{0}\\{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{-1}\\{0}&{0}&{1}&{0}\\\end{array}\right).
(\bar{M}^{-1})_{\gamma}^{\beta}=(q^{{\frac{1}{2}}}-\bar{q}^{{\frac{3}{2}}})\bar{a}_{+}^{\beta}\bar{a}_{-}^{\sigma}{\calE}_{\sigma\gamma}+\bar{q}^{\bar{N}+{\frac{3}{2}}}\delta_{\gamma}^{\beta}
\langle\|\frac{1}{2}\pi^{a}\pi^{a}|\\rangle=\langle\|\left(\frac{1}{2}p^{\dagger}p+\frac{1}{2(g\chi)^{2}}\sum_{a=1,2}G_{\mathrm{rad}}^{a}G_{\mathrm{rad}}^{a}\right)|\\rangle\.
(f\lhda)(\beta)=\sum_{s\in{\calS}^{p-3}}f(\beta-s)\;\stackrel{a}{K}{}\!\!_{0,p}^{(\pm)}\left(\begin{array}{c}{\alpha}\\{\beta-s,s}\\\end{array}\right)\frac{\Xi_{0,p}[\alpha,\beta-s]{\calP}[\beta-s]}{\Xi_{0,p}[\alpha,\beta]{\calP}[\beta]}.
f(x,p)\starg(x,p)=\sum_{s=0}^{\infty}\sum_{i=0}^{s}{\frac{\theta^{s}}{s!}}(-)^{i}c_{s}^{i}(\partial_{x}^{i}\partial_{p}^{s-i}f)(\partial_{x}^{s-i}\partial_{p}^{i}g)
\begin{aligned}{a}&{{}=c_{0}^{2}\left(c_{0}^{2}f_{1}\dotsf_{8}-\frac{1}{3}g^{2}\right),}\\{b}&{{}=\frac{1}{27}c_{0}^{3}g\left(2g^{2}-9c_{0}^{2}f_{1}\dotsf_{8}\right).}\\\end{aligned}
X_{8}:\qquadds^{2}=dr^{2}+a(r)^{2}\sum_{a=1}^{3}(\widetilde\sigma_{a})^{2}+b(r)^{2}\sum_{a=1}^{2}(\sigma_{a}^{\prime}-{\frac{1}{2}}\widetilde\sigma_{a})^{2}+c(r)^{2}(\sigma_{3}^{\prime}-{\frac{1}{2}}\widetilde\sigma_{3})^{2}+c^{\prime}(r)^{2}(\sigma_{4}^{\prime})^{2}\,.
\eta=-{\frac{1}{10}}\intd^{3}x^{\prime}\int_{-\infty}^{0}\!\!\!\!dt\int_{-\infty}^{t}\!\!\!\!dt^{\prime}\left\langle\pi^{\mu\nu}(0)\pi_{\mu\nu}({\bfx}^{\prime},t^{\prime})\right\rangle^{\mathrm{ret}}\,,
\Psi_{\varkappa,n}^{c}=\gamma^{2}\Psi_{-\varkappa,n}^{\ast}\;,\;\;\epsilon_{\varkappa,n}^{c}=-\epsilon_{-\varkappa,n}\,,\;\left(\Psi_{\varkappa,n}^{c},\Psi_{\varkappa^{\prime},n^{\prime}}^{c}\right)=\delta_{\varkappa,\varkappa^{\prime}}\delta_{n,n^{\prime}},\;\;\varkappa=\pm\;.
F_{1}(p,q)=-{\frac{1}{(4\pi^{2})}}\left[{\frac{Z_{0}}{4}}((p+q)^{2};m^{2})-{\frac{1}{4}}-{\frac{m^{2}\xi_{00}(p,q)}{2}}+{\frac{q^{2}\xi_{01}(p,q)+p^{2}\xi_{10}(p,q)}{4}}\right]
y^{2}=\Bigl(x^{2}-[U_{(0,\,r-1)}^{2}+\Lambda_{1}^{4}\Omega_{(2,\,r-1)}+\Lambda_{r}^{4}\Omega_{(1,\,r-2)}-M_{0}\Omega_{(1,\,r-1)}]\Bigr)^{2}-4\prod_{i=1}^{r}\Lambda_{i}^{4}.
(\mathrm{Mass})^{2}={\frac{1}{4\lambda_{2}^{(0)}}}\left[{\frac{1}{2}}(\hat{\alpha}_{L})^{2}+N_{L}-1\right]={\frac{1}{4\lambda_{2}^{(0)}}}\left[{\frac{1}{2}}(\hat{\alpha}_{R})^{2}+N_{R}-\delta\right]~.
\left(r^{2}-r_{H}^{2}\right)\partial_{r}^{2}R(r)+\frac{1}{r}\left(3r^{2}-r_{H}^{2}\right)\partial_{r}R(r)+\frac{\omega^{2}x_{+}^{4}}{(r^{2}-r_{H}^{2})}R(r)=0,
Q_{5}=\frac{1}{NR_{11}}\int_{-\infty}^{\infty}dx_{4}\int_{-\infty}^{\infty}dx_{5}\int_{-\infty}^{\infty}dx_{6}\frac{v_{0}^{2}B}{(x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+v_{0}^{2})^{5/2}}=\frac{Q}{2\piNR_{11}}\,,
\begin{array}{l}{\displaystylex_{j}^{2}=\frac{\prod_{i=1}^{D}(\rho_{i}-e_{j})}{\prod_{i\not=j}^{D}(e_{i}-e_{j})}\enspace,\qquad(j=1,\dots,D)}\\{\displaystylee_{1}<\rho_{1}<e_{2}<\dots<e_{D}<\rho_{D}\enspace.}\\\end{array}
\left(1-r^{2}\omega^{2}\right)\frac{\dot{{\calH}}}{1+\dot{{\calH}}^{2}}\frac{d\Phi_{D}}{d{\calH}}-\frac{1}{2}\frac{\partial}{\partialr}\left[\left(1-r^{2}\omega^{2}\right)\frac{\dot{{\calH}}}{1+\dot{{\calH}}^{2}}\right]=0\,.
K_{ik}\,K_{kj}~=~K_{ij}~,~~~~t_{0}^{kl}\,K_{ij}~=~e^{\,{\frac{i}{2}}\,(\,\alpha_{ik}+\alpha_{kj}+\alpha_{jl}+\alpha_{li}\,)\,}~K_{ij}\,t_{0}^{kl}~.
\intd\vartheta_{1}d\vartheta_{2}e^{{\frac{i}{2l_{11}^{3}}}\sum_{I=1}^{b_{3}}\sum_{\alpha=1}^{2}\sum_{i=1}^{8}({\bar{\chi}}(x)\sigma^{(i)})_{\alpha}\vartheta_{\alpha}\omega_{I}^{(i)}}=-{\frac{1}{4l_{11}^{3}}}\sum_{I=1}^{b_{3}}\sum_{i<j=1}^{8}\omega_{I}^{(i)}\omega_{I}^{(j)}({\bar{\chi}}\sigma^{(i)})_{1}({\bar{\chi}}\sigma^{(j)})_{2},
\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{4}=-u_{1}+u_{4}\rightarrowB_{4}^{1}=\left[B_{\bar{1}}^{0},B_{4}^{0}\right]=-D_{10}^{\bar{8}}+D_{12}^{\bar{4}}-D_{13}^{\bar{3}}
<PP>=\frac{1}{\pi}\mathrm{Re}\sum_{n=odd}\frac{1}{\frac{1}{2\sqrt{\alpha^{\prime}}}(2n+1)+i\beta\mu}=-\frac{\sqrt{\alpha^{\prime}}}{2\pi}\mathrm{Re}\psi(\frac{3+2\sqrt{\alpha^{\prime}}\beta\mui}{4}).
E=-\frac{\xi^{2}}{\pia}\sum_{l=1}^{\infty}\nu\int_{0}^{\infty}dx\,\lambda_{l}^{2}(x)-\frac{\xi^{4}}{2\pia}\sum_{l=1}^{\infty}\nu\int_{0}^{\infty}dx\,\lambda_{l}^{4}(x),
\begin{array}{rcl}{ds^{2}}&{=}&{\left(\sigma^{-1}\eta_{ij}-\sigma^{-2}\eta_{ik}\eta_{jl}x^{k}x^{l}\right)dx^{i}dx^{j}}\\{}&{=}&{-\displaystyle\frac{dxJ(I-x^{\prime}xJ)^{-1}dx^{\prime}}{1-xJx^{\prime}}~,}\\\end{array}
t-t_{0}=\int\left(9V^{4/3}+3\LambdaV^{2}+3k_{4}^{2}\rho_{0}V^{2-\gamma}+\frac14k_{5}^{4}\rho_{0}^{2}V^{2-2\gamma}+C\right)^{-1/2}dV,
\oint_{\gamma}{\frac{\mathrm{d}z}{2\pii}}\oint_{\delta}{\frac{\mathrm{d}w}{2\pii}}z^{m+q/3}w^{n-q/3}(z-w)^{p+2/3}G^{+}(z)G^{-}(w)\chi_{q}(0)~,
\delta_{\xi}g_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)},\\\delta_{\xi}\Gamma_{\rho\sigma}^{\tau}=\frac{1}{2}g^{\tau\nu}\left(\nabla_{\rho}\delta_{\xi}g_{\sigma\nu}+\nabla_{\sigma}\delta_{\xi}g_{\rho\nu}-\nabla_{\nu}\delta_{\xi}g_{\rho\sigma}\right).
\Gamma_{\mathrm{fixed}}={\frac{\sigma_{\mathrm{abs}}}{e^{\frac{\omega^{\prime}}{T_{H}}}-1}}={\frac{1}{\omega}}{\frac{e^{2\sqrt{s-1}\pi}-1}{\left(e^{{\omega_{R}^{\prime}/2T_{R}}+\sqrt{s-1}\pi}+1\right)\left(e^{{\omega_{L}^{\prime}/2T_{L}}+\sqrt{s-1}\pi}+1\right)}}.
\left\{{\frac{\partial^{2}}{\partialr^{2}}}+{\frac{5}{r}}{\frac{\partial}{\partialr}}+\tilde{\omega}^{2}\left(1+{\frac{\tilde{R}_{\frac{\pi}{2}}^{4}}{r^{4}}}\right)-{\frac{2\tilde{R}_{\frac{\pi}{2}}^{8}}{r^{10}\tilde{f}^{2}}}\right\}\hat{\phi}\simeq0
\begin{array}{ll}{\left(\frac{z\bar{m^{\prime}}+w}{zm^{\prime}+w}\right)^{1/2}}&{<\gammam^{\prime},\overline{{\gammam^{\prime}}};\tau_{2}|m,\bar{m};\tau_{1}>}\\{}&{=\left(\frac{-zm+x}{-z\bar{m}+x}\right)^{1/2}<m^{\prime},\bar{m^{\prime}};\tau_{2}|\gamma^{-1}m,\overline{{\gamma^{-1}m}};\tau_{1}>.}\\\end{array}
\sum_{\{k\}}R_{a_{1}a_{2}}^{k_{1}k_{2}}(\lambda-\mu)L_{k_{1}(l)}^{b_{1}}(\lambda)L_{k_{2}(l)}^{b_{2}}(\mu)e^{i\theta\hat{k}_{2}(\hat{b}_{1}-\hat{k}_{1})}=\sum_{\{k^{\prime}\}}e^{i\theta\hat{a}_{1}(\hat{k}_{2}^{\prime}-\hat{a}_{2})}L_{a_{2}(l)}^{k_{2}^{\prime}}(\mu)L_{a_{1}(l)}^{k_{1}^{\prime}}(\lambda)R_{k_{1}^{\prime}k_{2}^{\prime}}^{b_{1}b_{2}}(\lambda-\mu),\ll{bqybelany}
S_{G}=\intd^{2}\thetaG\Bigl(\Phi^{\dagger}(\theta)\left(iL^{a}e_{a}^{\alpha}\nabla_{\alpha}+\kappa\right)\Phi-R-{\frac{1}{4}}G^{2}F_{\alpha\beta}F^{\alpha\beta}+6m^{2}\Bigr),
{\mathcal{F}_{D}^{1}}[\hat{g}]={\mathcal{F}_{D}^{0}}[\hat{g}]-\alpha{Q}\int{d^{2}}\xi\sqrt{\hat{g}(\xi)}{\hat{R}}(\xi){\hat{G}_{D}}(\xi,\xi^{\prime})+4\pi{\alpha^{2}}{\hat{G}_{D}}(\xi^{\prime},\xi^{\prime}).
I={\frac{1}{16\piG}}\left[\int_{\calM}\sqrt{-g}d^{4}xR+2\int_{\partial{\calM}}K\sqrt{-h}d^{3}x\right]-{\frac{1}{4\pi}}\sum_{k}\hat{\mu}_{k}\int\sqrt{\sigma_{k}}\,d^{2}\zeta_{k}~~~.
q_{i}^{\epsilon}=\sum_{j=i}^{N}\left(\prod_{k=i+1}^{j}\epsilon_{k}\right)p_{j}=p_{i}+\epsilon_{i+1}p_{i+1}+\epsilon_{i+1}\epsilon_{i+2}p_{i+2}+\cdots+\epsilon_{i+1}\epsilon_{i+2}\cdots\epsilon_{N}p_{N}~,
{\calL}_{M}=-\frac{1}{4}G_{\mu\nu}^{a}G^{\mu\nu,a}-{\frac{1}{2}}\kappa_{1}^{2}D_{\mu}\phi^{\tilde{a}}D^{\mu}\phi^{\tilde{a}}-{\frac{1}{2}}\kappa_{2}^{4}D_{[\mu}\phi^{\tilde{a}}D_{\nu]}\phi^{\tilde{b}}D^{[\mu}\phi^{\tilde{a}}D^{\nu]}\phi^{\tilde{b}}-V(\phi^{4})\,
{\calL}=-{\frac{1}{4}}E^{(11)}[{\calR}^{(11)}+{\frac{1}{12}}F_{MNPQ}^{(11)}F^{(11)\,MNPQ}-{\frac{8}{12^{4}}}\varepsilon^{M_{1}\cdotsM_{11}}F_{M_{1}\cdotsM_{4}}^{(11)}F_{M_{5}\cdotsM_{8}}^{(11)}A_{M_{9}M_{10}M_{11}}^{(11)}],
q^{\mu/2}\;\sum_{k=0}^{\infty}\frac{q^{2k(k+\mu)}}{(q)_{2k+\mu}}\times\frac{1}{(q)_{\infty}}\sum_{j=-\infty}^{\infty}\left\{q^{20j^{2}+j}-q^{20j^{2}+9j+1}\right\}=q^{1/48+7/240}\chi_{1+\mu,1}^{(4)}(q)\:\chi_{1,1}^{(5)}(q).
\int\frac{dw}{2\pi}e^{-iw(t^{\prime}-t)}\frac{1}{2\bar{k}}(e^{i\bar{k}|x^{\prime}-x|}+K_{a}(w)e^{-i\bar{k}(x^{\prime}+x)}),~~\bar{k}=\sqrt{w^{2}-m_{a}^{2}}.
\tilde{A}_{\mathrm{m}}^{\mu}=e_{\mathrm{m}}^{\mathrm{j}}(X_{\mathrm{p}}^{\mathrm{k}}){\frac{\partial\chi^{\alpha}}{\partialX^{\mathrm{j}}}}_{\mathrm{p}}\xi_{\alpha}^{\mu}(x_{\mathrm{p}}^{\mathrm{k}})
\langle\phi^{\prime}|\phi\rangle=\inte^{iS_{c}}{\calD}\{\eta\}{\calD}\{a\}{\calD}\{c\}{\calD}\{\bar{c}\}e^{i\int\eta\hat{M}\etadx}e^{i\left\{\frac{1}{2}\left[\int\Psi_{0}\etadx\right]^{2}+\bar{c}c\int\left(\Psi_{0}^{2}-\Psi_{0}\frac{d\eta}{dx}\right)dx\right\}}.
S_{\mathrm{FP}}=\mathrm{tr}\intd^{8}z\,\left((\bar{c}^{\prime}c-c^{\prime}\bar{c})-\left(c^{\prime}[\Phi^{i},{\frac{\lambda}{\Box_{+}}}[\bar{c},\bar{\Phi}_{i}]]+\bar{c}^{\prime}[{\frac{\bar{\lambda}}{\Box_{-}}}[c,\Phi^{i}],\bar{\Phi}_{i}]\right)\right).
R_{ab}^{\frac{1}{2}}\left(\theta\right)\timesQ_{\left|n\right\rangle}\left(\eta,\vartheta,\theta\right)\qquad\mathrm{or}\qquadR_{ab}^{\frac{1}{2}}\left(\theta\right)\timesP_{\left|n\right\rangle}^{\pm}\left(\eta,\vartheta,\theta\right)\;.
L^{2}b^{2}\partial_{\alpha}(g_{ind}^{\alpha\beta}\partial_{\beta}y^{i})+\partial_{\beta}(g_{ind}^{\alpha\beta}g_{\alphai})-(\partial_{y_{i}}g_{\alphai})g_{ind}^{\alpha\beta}\partial_{\beta}y^{i}-4b^{2}\epsilon^{ij}\partial_{\tau}y^{j}=0\.
{\frac{d\tilde{n}^{(\lambda)}}{d\omega}}\simeq{\frac{2\omega^{D-2}}{(4\pi)^{(D-1)/2}}}\sum_{n=0}^{\infty}\left[{\frac{\tilde{a}_{n}(\lambda)}{\Gamma\left({\frac{D-1}{2}}-n\right)}}\omega^{-2n}+{\frac{\tilde{b}_{n}(\lambda)}{\Gamma\left({\frac{D-2}{2}}-n\right)}}\omega^{-(2n+1)}\right]~~~,
\Psi_{s}^{out}\left(x_{\bot}\right)=N\left(\begin{array}{c}{\left[1+\left(p^{3}+s\widetilde{M}\right)/M\right]g_{0}\left(\varphi\right)\psi_{\omega}^{out}\left(r\right)}\\{\left[-1+\left(p^{3}+s\widetilde{M}\right)/M\right]g_{0}\left(\varphi\right)\psi_{\omega}^{out}\left(r\right)}\\\end{array}\right)\;,
\begin{aligned}{{}}&{{}|\Re(\alpha_{1}-\alpha_{2})|<Q/2;\qquad|\Re(Q-\alpha_{1}-\alpha_{2})|<Q/2;}\\{}&{{}|\Re(\alpha_{3}-\alpha_{4})|<Q/2;\qquad|\Re(Q-\alpha_{3}-\alpha_{4})|<Q/2,}\\\end{aligned}
G(\alpha,-\alpha)={\frac{\displaystyle{\sum_{n=0}^{\infty}}{\frac{1}{n!}}{\left(\frac{\lambda}{2\pi}\right)}^{2n}{\binom{2n}{n}}\intd^{2}w_{1}\cdotsd^{2}w_{2n}\omega(\hat{\beta},w_{i})\tau(\alpha,\hat{\beta},w_{i})}{\displaystyle{\sum_{n=0}^{\infty}}{\frac{1}{n!}}{\left(\frac{\lambda}{2\pi}\right)}^{2n}{\binom{2n}{n}}\intd^{2}w_{1}\cdotsd^{2}w_{2n}\omega(\hat{\beta},w_{i})}}
(\bigtriangleupA)_{r}=\left({\frac{\partial^{2}}{\partialr^{2}}}+{\frac{d}{r}}{\frac{\partial}{\partialr}}+{\frac{1}{r^{2}}}{}^{(d)}\bigtriangleup-{\frac{d}{r^{2}}}\right)A_{r}-{\frac{2}{r^{3}}}A_{p}^{\;\;\midp},
S^{(3)}=-\frac{e^{3}}{4}\int_{-\infty}^{\infty}{\mathrmd}^{4}x_{2}\int_{-\infty}^{\infty}{\mathrmd}^{4}x_{3}\int_{-\infty}^{\infty}{\mathrmd}^{4}x_{1}T(\{I\}+\{II\})
A_{M}=\left(\begin{array}{c}{a_{m}}\\{\psi_{\dot{\alpha}}}\\\end{array}\right),\;\;\;\;\;\;A_{M}^{\dagger}=\left(\begin{array}{c}{a_{m}^{\dagger}}\\{\psi_{\dot{\alpha}}^{\dagger}}\\\end{array}\right),\;\;\;\;\;\;m,\;\dot{\alpha}=1\ldots2^{d-1},\;\;\;\;\;\;M=1\ldots2^{d}.
W^{\ast}=\frac{1}{6}x_{0}^{6}+\frac{1}{6}x_{3}^{6}+\frac{1}{6}x_{4}^{6}+\frac{1}{2}x_{5}^{2}+\hat{c}(z;b,c)^{-1/6}x_{0}x_{3}x_{4}x_{5}.
\begin{array}{c}{lim_{Q\toq}{\frac{1}{k}}[Q^{-N_{B}}B^{-},[Q^{-N_{B}}B^{-},[\dots[Q^{-N_{B}}B^{-},{B^{+}}^{k}]_{Q^{2k}}\dots]_{Q^{4}}]_{Q^{2}}]}\\{=lim_{Q\toq}{\frac{Q^{\frac{k(k-1)}{2}}}{[k]!}}[Q^{-N_{B}}{(B^{-})^{k}},(B^{+})^{k}]}\\{=q^{\frac{k(k-1)}{2}}.}\\\end{array}
{\calSH}=\mathrm{Tr}\big[\underbrace{\Theta_{\Sigma}\otimes\Theta_{\Sigma}\otimes\Theta_{\Sigma}}_{J=0}\big]=\mathrm{Tr}\left[\Theta_{\Sigma}^{+}\Theta_{\Sigma}^{\,0\,}\Theta_{\Sigma}^{-}\right]\,,
Z_{v}^{X}\left(-\frac{1}{\tau}\right)=2^{-\frac{b_{2}(X)}{2}}\left(\frac{\tau}{i}\right)^{-\frac{\chi(X)}{2}}\cdot\sum_{u\inH^{2}(X,Z_{2})}(-1)^{u\cdotv}Z_{u}^{X}(\tau).
L_{D}=\int_{\calM}\varepsilon^{ijk}\bigg[{\frac{k}{4\pi}}(A_{i}^{a}\widehat{D}_{j}A_{ka}+{\frac{1}{3}}f_{abc}A_{i}^{a}A_{j}^{b}A_{k}^{c})+{\frac{1}{2\pi}}\chi_{i}^{a}F_{jka}(B)\bigg],
\psi=\left|\begin{array}{c}{E^{3}+iH^{0}}\\{E^{1}+iE^{2}}\\{iH^{3}+E^{0}}\\{-H^{2}+iH^{1}}\\\end{array}\right|=U\mathcal{E},\quadU=\left|\begin{array}{cccc}{0}&{0}&{C_{+}}&{C_{-}}\\{C_{+}}&{iC_{+}}&{0}&{0}\\{0}&{0}&{C_{-}}&{C_{+}}\\{C_{-}}&{iC_{-}}&{0}&{0}\\\end{array}\right|,\quadC_{\mp}\equiv\frac12(C\mp1),
(n+k)d_{kl}^{n}\pm(k+1)d_{k+1,l}^{n}=\left(\frac{(k+1)(k+2)}{2}-g\right)d_{k+2,l}^{n-1}-g\sum_{m=k+n-1}^{\infty}s_{m}d_{k-m,l}^{n-1}.
ds^{2}=H_{1}^{1/4}H_{5}^{3/4}\left[{\frac{du}{H_{1}H_{5}}}(-dv+Kdu+2A_{i}dy^{i}+2A_{a}dx^{a})+{\frac{dy^{i}dy_{i}}{H_{5}}}+dx^{a}dx_{a}\right]
K(T):=\textrm{Tr}\;e^{-\frac{i}{\hbar}{\cal{H}}T}=\intdq_{0}K(q_{0},T\midq_{0})=\intdq_{0}\int_{q_{0},t^{\prime}}^{q_{0},t^{\prime\prime}}{\cal{D}}\;q(t)e^{\frac{i}{\hbar}S[q(t),T]}.