harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{0}^{R}\mathrm{d}r\;u_{kl}^{2}\left(r\right)=\frac{1}{2k}\left[u_{kl}^{\prime}\left(R\right)\partial_{k}u_{kl}\left(R\right)-u_{kl}\left(R\right)\partial_{k}u_{kl}^{\prime}\left(R\right)\right]\,.
A_{0}(\rho,\varphi)=-4i\gamma\sum_{n}{\int{\frac{kdk}{2\pi}}\frac{k\sqrt{\kappa_{0}}}{v^{2}}J_{n}(k\rho)\left[\alpha_{Q,n}(k)e^{in\varphi}-\alpha_{Q,n}^{\star}(k)e^{-in\varphi}\right]},
\langleT_{i}^{k}\rangle_{N}^{(1b)}(\xi_{1},\xi)=-A_{d}\delta_{i}^{k}\int_{0}^{\infty}dkk^{d}\int_{0}^{\infty}d\omega\frac{I_{\omega}^{\prime}(k\xi_{1})}{K_{\omega}^{\prime}(k\xi_{1})}F^{(i)}[K_{\omega}(k\xi)]
\bbox{28}\oplus\bbox{\overline{{28}}}\rightarrow(\bbox{1},\bbox{15})\oplus(\bbox{1},\bbox{\overline{{15}}})\oplus(\bbox{2},\bbox{6})\oplus(\bbox{2},\bbox{\overline{{6}}})\oplus(\bbox{1},\bbox{1})\oplus(\bbox{1},\bbox{1}).
r_{0}=R_{0}\lambda_{EQ}=\left[\frac{12\,t_{EQ}}{H_{\infty}^{2}}\right]^{1/3}\congH_{\infty}^{-1}\left(\frac{8\,t_{EQ}}{t_{\Lambda}}\right)^{1/3}=\frac{8^{1/3}H_{\infty}^{-1}}{(1+z)_{EQ}^{1/2}}
\phi^{(\alpha)}=C^{\alpha\beta}\int_{0}^{\ell}dx\;\biggl(L_{\beta\lambda}^{-1}(x){\frac{\delta}{\delta\xi_{\lambdaL}(x)}}-R_{\beta\lambda}^{-1}(x){\frac{\delta}{\delta\xi_{\lambdaR}(x)}}\biggr)\quad.
\{4\pi(\pi_{i}^{a}+\frac{1}{8\pi}\epsilon_{ik}A^{k,a})(x),4\pi(\pi_{j}^{b}+\frac{1}{8\pi}\epsilon_{jl}A^{l,b})(x^{\prime})\}=4\pi\delta^{ab}\epsilon_{ij}\delta(x-x^{\prime})
S_{GF-ghost}=\intd^{d}x\left\{-\frac{1}{2\xi}\partial_{\mu}A^{\mu}(x)\star\partial_{\nu}A^{\nu}(x)+\partial^{\mu}\overline{{c}}(x)\star\left(\partial_{\mu}c(x)-i[A_{\mu}(x),c(x)]_{M}\right)\right\}
q^{i}\equiv\frac{{\bfP}^{i}}{{\bfP}^{L}}\,,\qquad\rho\equiv\frac{{\bfP}^{i}{\bfP}^{i}+2{\bfP}^{R}{\bfP}^{L}}{2({\bfP}^{L})^{2}}\,,\qquad\frac{{\bfP}^{R}}{{\bfP}^{L}}=\rho-\frac{q^{2}}{2}\,.
\^{*}F^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}=2\epsilon_{abc}\phi_{a}\partial^{\mu}\phi_{b}\partial^{\nu}\phi_{c}+\partial^{\mu}{\mathcalB}^{\nu}-\partial^{\nu}{\mathcalB}^{\mu}
\biggl(J_{AC}J_{BD}{\calB}^{[AB]}{\calB}^{[CD]}-{\frac{R_{s}^{2}}{4k^{2}}}J_{AC}J_{BD}M^{\{AB\}}M^{\{CD\}}\biggr)|T_{[A_{1}B_{1}]\cdots[A_{k-2}B_{k-2}]}\rangle=R_{s}^{2}|T_{[A_{1}B_{1}]\cdots[A_{k-2}B_{k-2}]}\rangle
\sum_{k=1}^{n}\left(\partial_{k}^{2}+\frac{1}{x_{k}}\partial_{k}\right)\Psi(x_{1},\ldots,x_{n};y_{1},\ldots,y_{n})=\left(y_{1}^{2}+y_{2}^{2}+\ldots+y_{n}^{2}\right)\Psi(x_{1},\ldots,x_{n};y_{1},\ldots,y_{n})
\left(\frac{\partial}{\partialx^{k}}\frac{\delta}{\deltaA_{k}^{a}(x^{0},{\bfx})}-gf_{ab}^{~~c}A_{k}^{b}(x^{0},{\bfx})\frac{\delta}{\deltaA_{k}^{c}(x^{0},{\bfx})}\right)\Psi_{W}(A)=0
{{M_{T}}^{\alpha\beta}}_{\rho\sigma}=-\delta_{\rho}^{(\alpha}\delta_{\sigma}^{\beta)}\Box+a_{1}\delta_{(\rho}^{(\alpha}\nabla^{\beta)}\nabla_{\sigma)}+a_{2}g_{\rho\sigma}\nabla^{(\alpha}\nabla^{\beta)}+a_{3}g^{\alpha\beta}\nabla_{(\rho}\nabla_{\sigma)}-{P^{\alpha\beta}}_{\rho\sigma}\,
\langle\prod_{i=1}^{n}S(x_{i})\,S(\tilde{y}_{i})\rangle_{w}=\langle1\rangle_{w}\,(4\pi\,r_{o})^{n}n!\prod_{i=1}^{n}{\frac{1}{4\pi\midx_{i}-x_{o}\mid}}{\frac{1}{4\pi\mid\tilde{y}_{i}-\tilde{y}_{o}\mid}}\,\,.
N={\calN}|_{\pi^{*}{\calS}}={\calO}_{\pi^{*}{\calS}}((n(\frac{1}{2}+\lambda)\sigma+(\frac{1}{2}-\lambda)\pi^{*}\eta+(\frac{1}{2}+n\lambda)\pi^{*}c_{1}(B))\cdot\pi^{*}{\calS}).
\begin{array}{c}{\displaystyle{z_{0}^{-1}\delta\left(\frac{z_{1}-z_{2}}{z_{0}}\right)Y(u,z_{1})Y(v,z_{2})-z_{0}^{-1}\delta\left(\frac{z_{2}-z_{1}}{-z_{0}}\right)Y(v,z_{2})Y(u,z_{1})}}\\{\displaystyle{=z_{2}^{-1}\delta\left(\frac{z_{1}-z_{0}}{z_{2}}\right)Y(Y(u,z_{0})v,z_{2})}}\\\end{array}
+i[({\frac{a}{4\alpha^{2}}}-{\frac{1}{4\alpha^{2}g\partial^{2}}})g^{\mu\nu}\delta(x-y)-({\frac{a}{4\alpha^{2}}}+g){\frac{\partial^{\mu}\partial^{\nu}}{\partial^{2}}}\delta(x-y)+{\frac{1}{\alpha\partial^{2}}}\epsilon^{\mu\nu\lambda}\partial_{\lambda}\delta(x-y).
S=\frac{1}{2}\int[(e^{-\Phi}G_{0}^{+}e^{\Phi})(e^{-\Phi}\tilde{G}_{0}^{+}e^{\Phi})-\int_{0}^{1}dt(e^{-t\Phi}\partial_{t}e^{t\Phi})\left\{e^{-t\Phi}G_{0}^{+}e^{t\Phi},e^{-t\Phi}\tilde{G}_{0}^{+}e^{t\Phi}\right\}].
\psi(\theta^{a})=\sum_{i=0,1,..,3,5,..,d}\quad\sum_{\{a_{1}<a_{2}<...<a_{i}\}\in\{0,1,..,3,5,..,d\}}\alpha_{a_{1},a_{2},...,a_{i}}\theta^{a_{1}}\theta^{a_{2}}\cdots\theta^{a_{i}}.
T_{H}=\frac{1}{4\piM}\left[\frac{\sqrt{(2M^{2}-\beta)^{2}-4{Q_{e}}^{2}{Q_{m}}^{2}}}{\sqrt{(2M^{2}-\beta)^{2}-4{Q_{e}}^{2}{Q_{m}}^{2}}+(2M^{2}-\beta)}\right]~~.
|\mathrm{init}\rangle_{\mathrm{anc}}=\int\prod_{\Sigma=1}^{N}\left(d\tilde{U}_{\Sigma}\right)|\tilde{U}_{1},\tilde{U}_{2},\dots,\tilde{U}_{N}\rangle\delta(\prod_{\Sigma=1}^{N}\tilde{U}_{\Sigma}-I)~,
\varphi(x)_{x^{+}=0}=\int_{0}^{\infty}dk^{+}\int_{-\infty}^{\infty}\frac{dk_{\perp}}{\sqrt{2k^{+}(2\pi)^{2}}}\left[a({\bfk})e^{-ik^{+}x^{-}+ik_{\perp}x_{\perp}}+a^{\dagger}({\bfk})e^{ik^{+}x^{-}-ik_{\perp}x_{\perp}}\right],
I_{cov}(p)=(-i)^{n+1}\intq.d^{2\omega}\int_{0}^{\infty}d\alpha_{1}\cdotsd\alpha_{n}d\beta\cdot2q^{-}.e^{i[\sum_{1}^{n}\alpha_{i}f_{i}(q,p)+\beta2q^{+}q^{-}]},
2\left\{3\mathrm{Cl}_{2}(\theta_{7})-3\mathrm{Cl}_{2}(2\theta_{7})+\mathrm{Cl}_{2}(3\theta_{7})\right\}=Z_{7}=7\left\{\mathrm{Cl}_{2}\left(\frac{2\pi}{7}\right)+\mathrm{Cl}_{2}\left(\frac{4\pi}{7}\right)-\mathrm{Cl}_{2}\left(\frac{6\pi}{7}\right)\right\}
{\calK}(t,\bar{t})=-\mathrm{log}\,\,\mathrm{i}\Bigl[2\left({\calF}-\bar{\calF}\right)-\left(\partial_{I}{\calF}+\partial_{{I^{\star}}}\bar{\calF}\right)\left(t^{I}-\bar{t}^{I^{\star}}\right)\Bigr]
{\frac{dr}{d\lambda}}={\frac{dr}{ds}}{\frac{ds}{d\widetilde{\lambda}}}{\frac{d\widetilde{\lambda}}{d\lambda}}=(4^{(1/\beta)}cosh^{\left({2/\beta}\right)}({\frac{\betar}{2L}}))(C^{*})(4^{(-2/\beta)}sech^{\left({4/\beta}\right)}({\frac{\betar}{2L}}))=Csech^{\left({2/\beta}\right)}({\frac{\betar}{2L}}),
M_{\mu\nu}^{(p)}=\eta_{\mu\nu}+F_{\mu\nu}+\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{i}-2\bar{\lambda}(\Gamma_{\mu}+\Gamma_{i}\partial_{\mu}\phi^{i})\partial_{\nu}\lambda+\bar{\lambda}\Gamma^{m}\partial_{\mu}\lambda\bar{\lambda}\Gamma_{m}\partial_{\nu}\lambda.
R_{L}^{(n)}(t,t^{\prime};\{q\})=R_{L}^{[n]}(t,t^{\prime};\{q\})-\frac{1}{2}\sum_{m,m^{\prime}}\Phi_{m}(t;L;\{q\})\hat{V}_{mm^{\prime}}^{-1}\Phi_{m^{\prime}}(t^{\prime};L;\{q\})
\left(\begin{array}{c}{a_{x}}\\{a_{y}}\\\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{1}&{-i}\\{i}&{-1}\\\end{array}\right)\left(\begin{array}{c}{a_{x}^{\prime}}\\{a_{y}^{\prime}}\\\end{array}\right),
E[\sigma]=\sum_{\ell}N_{\ell}\left[\int_{0}^{\infty}\frac{dk}{2\pi}\sqrt{k^{2}+m^{2}}\frac{d}{dk}[\delta_{\ell}(k)]_{N}+\frac{1}{2}\sum_{j}\omega_{j}\right]+\sum_{i=1}^{N}E_{\mathrm{FD}}^{(i)}+E_{\mathrm{CT}}
\sum_{g,h,i\inG}R_{a,b}^{g^{-1},i}(\lambda)R_{i^{-1},c}^{h,d}(\mu)R_{g,h^{-1}}^{f,e}(\nu)=\sum_{g,h,i\inG}R_{b,c}^{h^{-1},g}(\nu)R_{a,h}^{f,i^{-1}}(\mu)R_{i,g^{-1}}^{e,d}(\lambda).
G^{+}(x,0)=\frac{1}{2^{\beta+2}\pi(1+\sigma/l^{2})^{1+\beta}}{}_{2}F_{1}\left(\frac{1+\beta}{2},\frac{\beta+2}{2},1+\beta,\frac{1}{(1+\sigma/l^{2})^{2}}\right)
\prod_{m=1}^{M}(Z^{1\dagger})_{j_{m}}^{i_{m}}\prod_{m^{\prime}=1}^{M^{\prime}}(Z^{2\dagger})_{j_{m^{\prime}}^{\prime}}^{i_{m^{\prime}}^{\prime}}\prod_{n=1}^{Nk}(\Psi^{1\dagger})_{l_{n}}|0\rangle,
ds^{2}=R^{2}\left[{\frac{r^{2}}{R^{4}}}(-dt^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2})+{\frac{dr^{2}}{r^{2}}}+ds_{5}^{2}\right]\,\qquadR^{4}\simg_{s}N(\alpha^{\prime})^{2}.
P_{k}^{a}(x^{0},{\bfx})\equiv\frac{\deltaL_{0}}{\delta\dot{A}_{a}^{k}(x^{0},{\bfx})}=\left(\nabla_{k}(A)A_{0}\left\{{\bfA},{\dot{{\bfA}}}\right\}\right)^{a}(x^{0},{\bfx})-\dot{A}_{k}^{a}(x^{0},{\bfx}),
\dot{\widetilde{\deltap}}+\left(3{\frac{\dot{a}_{b}}{a_{b}}}+{\frac{\dot{n}_{b}}{n_{b}}}\right)\widetilde{\deltap}=k^{2}\widetilde{\delta\pi}+6\left({\frac{\rho+P}{\lambda}}\right)\left({\frac{\dot{a}_{b}}{a_{b}}}\deltap-{\frac{k^{2}}{2}}\delta\pi\right)\,.
I={\frac{1}{4\pi\alpha^{\prime}}}\intd^{2}\xi[{\sqrtg}G_{ij}(\vec{x})\partial_{\alpha}x^{i}\partial^{\alpha}x^{j}+i\epsilon^{\alpha\beta}B_{ij}(\vec{x})\partial_{\alpha}x^{i}\partial_{\beta}x^{j}+\alpha^{\prime}{\sqrtg}R^{(2)}\phi(\vec{x})]\.
ds^{2}=\left(M-\frac{r^{2}}{l^{2}}\right)dt^{2}+\frac{1}{\left(\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4r^{2}}\right)}dr^{2}+r^{2}d\phi^{2}-Jdtd\phi
\left\langleH\right\rangle_{\Omega}=\left\langle\Omega\right|\int{dx_{1}}\left({-\frac{1}{2}\Pi_{1}\left({1-\frac{{e^{2}}}{\pi}\frac{1}{{\partial_{1}^{2}}}}\right)^{-1}\Pi^{1}}\right)\left|\Omega\right\rangle.
\delta{\calL}_{1}={\frac{1}{1-\lambda_{++}\lambda_{--}}}\left\{\left(2\lambda_{++}\lambda_{--}A_{+}\partial_{-}\alpha+2A_{-}\partial_{+}\alpha\right)-\delta\left(\lambda_{--}A_{+}^{2}+\lambda_{++}A_{-}^{2}\right)\right\}
F_{[M_{1}^{1}\cdotsM_{\left(d+2\right)/2}^{1};M_{1}^{2}]\cdotsM_{\left(d+2\right)/2}^{2}];\cdots\lbrackM_{1}^{n/2}\cdotsM_{\left(d+2\right)/2}^{n/2}]}=0,
\frac{1}{\sqrt{g}}(\frac{1}{3}(g^{ij}{\frac{\deltaS}{\deltag^{ij}}})^{2}-{\frac{\deltaS}{\deltag^{ij}}}{\frac{\deltaS}{\deltag_{ij}}}-\frac{1}{2}G^{IJ}\frac{\deltaS}{\delta\phi^{I}}\frac{\deltaS}{\delta\phi^{I}})=\sqrt{g}{\cal{L}},
\omega_{\rho}^{2}(q)=q^{2}\frac{1+\frac{2}{\pi}V_{(0)}^{00}+\frac{1}{2\pi^{2}}\{(U_{(0)}^{00})^{2}-2U_{(0)}^{00}U_{(1)}^{00}\}}{1+\frac{2}{\pi}V_{(1)}^{00}-\frac{1}{2\pi^{2}}(U_{(1)}^{00})^{2}},
J_{-}(x)=\psi_{2}^{\dagger}(x)\psi_{1}(x)\equiv\frac{1}{8\pi}\intd\hat{\epsilon}e^{-i\theta(\hat{\epsilon})}\left\{\left[\psi_{2}^{\dagger}(x+\epsilon),\psi_{1}(x-\epsilon)\right],e^{ie\int_{x-\epsilon}^{x+\epsilon}dx_{i}A_{i}^{T}}\right\}_{|\eta|,|\xi|,|\epsilon|\propto1/\Lambda}
\int_{0}^{\infty}dy\,a^{-2}\,G_{K}({\bfx}_{1},y;{\bfx}_{2},0)G_{K}({\bfx}_{3},y;{\bfx}_{4},0)={\frac{1}{2(4\pi)^{2}R_{12}R_{34}}}\mu(R_{12}+R_{34}),
\begin{array}{lll}{\psi_{1}(x)}&{=A:e^{-ia\rho(x)-ib\bar{\rho}(x)}:~~~~~~~~~~\psi_{1}^{\dagger}(x)}&{=A:e^{ia\rho(x)+ib\bar{\rho}(x)}:}\\{\psi_{2}(x)}&{=-iA:e^{ib\rho(x)+ia\bar{\rho}(x)}:~~~~~~~~~\psi_{2}^{\dagger}(x)}&{=iA:e^{-ib\rho(x)-ia\bar{\rho}(x)}:}\\\end{array}
\;=\;-\intd^{3}x^{P}\,\frac{\kappa}{2}\,\epsilon^{\mu\nu\lambda}A_{\mu}^{P}(x^{P})\frac{\partial}{\partialx_{\nu}^{P}}A_{\lambda}^{P}(x^{P})\;=\;-S_{CS}[A^{P}]\;.
I\!\!{P}(A)={\widetildeA}\equiv~:e^{-\psi{\hat{\chi}}}:A~=A-\psi\{\chi,A\}+\frac{1}{2!}\psi^{2}\{\chi,\{\chi,A\}\}-\frac{1}{3!}\psi^{3}\{\chi,\{\chi,\{\chi,A\}\}\}+\ldots-\ldots
m^{2}=\sum_{m_{1},m_{2}\atopn^{1},n^{2}}\frac{1}{Y}|-TU^{\prime}n^{2}+iTn^{1}-iU^{\prime}m_{1}+3m_{2}|_{U^{\prime}=U-2i}^{2}={|{\calM}|^{2}}/{(Y/2)},
a_{0}=a_{1/2}=a_{1}=a_{2}=0,\quad\frac{a_{3/2}}{(4\pi)^{3/2}}=\frac{3}{64\,R},\quad\frac{a_{5/2}}{(4\pi)^{3/2}}=\frac{153}{8192}\frac{c^{2}}{R^{3}}.
L_{\mathrm{eff}}={\frac{f^{2}}{2\,g_{YM}^{2}}}\left[1+\sum_{L,m=1}^{\infty}c_{Lm}\left({\frac{g_{YM}^{2}f^{2}}{x^{7-p}}}\right)^{L}~\left({\frac{f^{2}}{x^{4}}}\right)^{m-L}\right]
r(z,\lambda,\mu)~=~-\,{\frac{2\lambda\mu}{(1\!-\!\lambda\mu)(\lambda\!-\!\mu)}}\,C\,-\,{\frac{2(1\!+\!\lambda\mu)(\lambda\!-\!\mu)}{(1\!-\!\lambda\mu)(1\!-\!\lambda^{2})(1\!-\!\mu^{2})}}\,j(z)~~~,
\stackrel{P}{\calF}{}\!\!_{0,p}^{(\pm)}\left(\begin{array}{c}{\!\!\alpha\!\!}\\{\!\!\beta\!\!}\\\end{array}\right)=\;\sum_{s\in{\calS}^{p-3}}\stackrel{P}{K}{}\!\!_{0,p}^{(\pm)}\left(\begin{array}{c}{\!\!\alpha\!\!}\\{\!\!\beta,s\!\!}\\\end{array}\right)\stackrel{\alpha}{\omega}_{0,p}\!\!\lbrack\beta+s\rbrack\;\;,
G_{\hat{\mu}\hat{\nu}}^{11}=\left(\begin{array}{cc}{e^{-\frac{2}{3}\phi}G_{\mu\nu}+e^{-\frac{2}{3}\phi}C_{\mu}C_{\nu}}&{e^{\frac{1}{3}\phi}C_{\mu}}\\{}&{}\\{e^{\frac{1}{3}\phi}C_{\nu}}&{e^{\frac{4}{3}\phi}}\\\end{array}\right).
{\calS}^{4}(x,\bar{y})=\frac12\left(6\sqrt{x^{0}}-6\sqrt{\bar{y}^{0}}+2\left[\frac{\bar{y}^{0}}{\sqrt{x^{0}}}-\frac{x^{0}}{\sqrt{\bar{y}^{0}}}\right]\right)\theta(\bar{y}^{0}-x^{0})
\gamma^{0}=\left(\begin{array}{cc}{\sigma_{3}}&{0}\\{0}&{\sigma_{3}}\\\end{array}\right)\quad\gamma^{1}=\left(\begin{array}{cc}{i\sigma_{2}}&{0}\\{0}&{i\sigma_{2}}\\\end{array}\right)\quad\gamma^{2}=\left(\begin{array}{cc}{-i\sigma_{1}}&{0}\\{0}&{i\sigma_{1}}\\\end{array}\right)\quad\gamma^{3}=\left(\begin{array}{cc}{0}&{i\sigma_{1}}\\{i\sigma_{1}}&{0}\\\end{array}\right)
\phi_{n}(\tilde{r}^{2})=2(-1)^{n}e^{-\frac{\tilde{r}^{2}}{\theta}}L_{n}\left(\frac{2\tilde{r}^{2}}{\theta}\right)\quad\mathrm{where}\quad\tilde{r}^{2}=\lambda^{2}(x^{1})^{2}+\frac{1}{\lambda^{2}}(x^{2})^{2}.
{\psi}_{\alpha}^{(a)}(x)=\frac{Z_{\alpha}^{(a)}}{\sqrt{({\gamma}^{0}{\gamma}^{+})_{{\alpha}{\alpha}}}}\mathrm{exp}[-i\sqrt{\pi}{\Lambda}_{{\alpha}r}^{(-)}(x)]{\sigma}_{\alpha}\mathrm{exp}[-i\sqrt{\pi}{\Lambda}_{{\alpha}r}^{(+)}(x)],
\langle\xi,\hat{\bboxp}\,|\,\hat{\bboxp}^{\prime},\xi^{\prime}\rangle=2\hat{p}^{0}\delta^{3}(\hat{\bboxp}-\hat{\bboxp}^{\prime})\,\delta_{\xi\xi^{\prime}}\,=2p^{0}m^{2}\delta^{3}(\bbox{p}-\bbox{p^{\prime}})\,\delta_{\xi\xi^{\prime}}\,.
I_{\mathrm{IIA,B}}=-{\frac{1}{2\kappa_{(10)}^{2}}}\intd^{10}x\sqrt{-G}\;\Biggl[e^{-2\Phi}\Bigl(R-4(d\Phi)^{2}+{\frac{1}{12}}(dB)^{2}\Bigr)+\sum{\frac{1}{2n!}}H^{(n)\;2}\Biggr]
S_{G}=\intd^{3}x\left[-\epsilon\frac{1}{2}\left(D_{\mu}Q^{A}D^{\mu}Q^{B}\eta_{AB}\right)-W_{G}(Q^{A})-\kappa\epsilon^{\mu\nu\rho}\mathrm{Tr}\left(\partial_{\mu}A_{\nu}A_{\rho}+\frac{2}{3}A_{\mu}A_{\nu}A_{\rho}\right)\right],
I=\intd^{4}x\epsilon^{\mu\nu\alpha\beta}\epsilon_{abcd}\bigg(c_{1}{^+}G_{\mu\nu}^{~~ab}{^+}G_{\alpha\beta}^{~~cd}+c_{2}{^-}G_{\mu\nu}^{~~ab}{^-}G_{\alpha\beta}^{~~cd}+c_{3}{^+}F_{\mu\nu}^{~~ab}{^+}G_{\alpha\beta}^{~~cd}+c_{4}{^-}F_{\mu\nu}^{~~ab}{^-}G_{\alpha\beta}^{~~cd}\bigg).
\Psi=\intd^{3}x\{\overline{{C}}\chi+\overline{{P}}N+\overline{{C}}^{i}\chi_{i}+\overline{{P}}^{i}N_{i}+\overline{{C}}_{(1)}\chi_{(1)}+\overline{{P}}_{(1)}N_{(1)}+\overline{{C}}_{1}^{1}\chi_{1}^{1}\},
f_{1}=1+{\frac{r_{1}^{2}}{r^{2}}},~~~~f_{5}=1+{\frac{r_{5}^{2}}{r^{2}}},~~~~f_{K}=1+{\frac{r_{K}^{2}}{r^{2}}},~~~~d=1-{\frac{r_{0}^{2}}{r^{2}}},
\Phi=(-1)^{j_{1}+j_{2}+j_{1}+k_{1}^{1}+k_{1}^{2}}\sqrt{\frac{(2I+1)^{3}(2k_{2}^{1}+1)^{2}(2k_{2}^{2}+1)(2k_{1}^{2}+1)(2r_{2}+1)(2j_{1}+1)(2j_{2}+1)}{(8\pi^{2})^{2}}}.
|B_{M}\rangle=|B\rangle_{\mathrm{gh}}\sum_{\lambda,\mu}|\lambda\rangle_{a}|{\hat{\mu}}\rangle_{{\tilde{a}}}|p=0\rangle\frac{\langle\mu,p=0|T_{M}|\lambda,p=0\rangle}{\langlep=0|p=0\rangle}~,
\eta^{2}\cdot\tau^{2}{\cal{G}}\;\;:=\;\;{\cal{Y}}_{[p_{1},...,p_{N}]}\circ\left({\frac{1}{4}}\sum_{i,j,k,l=1}^{N}\left(\eta^{(ij)}\eta^{(kl)}+2\eta^{(ik)}\eta^{(lj)}\right)\tau^{(ij)}\tau^{(kl)}\right){\cal{G}}
e^{3N\lambda_{0}^{\prime}}W^{\prime}\,\prod_{i,j>i}|\omega_{i}^{\prime}-\omega_{j}^{\prime}|^{2\beta_{ij}}=e^{3N\lambda_{0}}W\,\prod_{i,j>i}|\omega_{i}-\omega_{j}|^{2\beta_{ij}}\prod_{i=1}^{N}|c\omega_{i}+d|^{4}.
F_{(d)0}^{[1,0]}\left(p_{1}^{2},p_{2}^{2};p_{3}^{2}\right)=x_{1,8}\left(\Pi_{1}-\Pi_{2}\right)+\left[\left(x_{1,9}+x_{1,9}^{\prime}\frac{1}{\Pi_{2}}\right)\Pi_{1}-\left(x_{1,9}+x_{1,9}^{\prime}\frac{1}{\Pi_{1}}\right)\Pi_{2}\right]\Pi_{3}\,,
0=\sum_{n\neq0,1}\mathrm{e}^{in\sigma}\dot{z}_{n}+2\mathrm{e}^{i\sigma}p_{-1}+2\sum_{n=1}^{\infty}{\frac{1}{n}}(\mathrm{e}^{in\sigma}\bar{p}_{n}+\mathrm{e}^{-in\sigma}\bar{p}_{-n})\.
H=-\frac{e}{2}(\Pi^{2}-m^{2}+2\imathgF_{\mu\nu}\psi^{\mu}\psi^{\nu})+\imath(\Pi_{\mu}\psi^{\mu}+m\psi^{3})\chi-\imath(\varepsilon^{\mu\nu\lambda}\Pi_{\mu}\psi_{\nu}\psi_{\lambda}+\frac{\imath}{2}sm)\kappa\;,
\delta_{[\mu_{1}...\mu_{n}]}^{[\nu_{1}...\nu_{n}]}=\sum_{k=1}^{n}(-1)^{k+1}\delta_{\mu_{1}}^{\nu_{k}}\delta_{[\mu_{2}..\mu_{k}..\mu_{n}]}^{[\nu_{2}..\nu_{1}..\nu_{n}]}~~~.
[b_{Q}^{a}({\bf{k}}),\,\sum_{n=2}^{\infty}{\calA}_{n}\,]\,-\,g\,f^{a\beta\gamma}\intd{\bf{r}}\,e^{-i{\bf{k\cdotr}}}\,[a_{i}^{\beta}({\bf{r}}_{1})+x_{i}^{\beta}({\bf{r}})]\,[\sum_{n=2}^{\infty}{\calA}_{n-1},\,\Pi_{i}^{\gamma}({\bfr})]\approx0\;.
Z_{L}^{(m,n)}(\tau)=Z_{n}^{m}(\tau)^{4}Z_{0}^{0}(\tau)^{12}+Z_{n+1}^{m}(\tau)^{4}Z_{1}^{0}(\tau)^{12}+Z_{n}^{m+1}(\tau)^{4}Z_{0}^{1}(\tau)^{12},
\sum_{P_{2}^{0}}\tilde{T}_{m}\left(X/Y\right)T_{n-m}^{\mu}\left(Y,x\right)=\sum_{\Pi}\,\theta\left(x,x_{\pi_{1}},\ldots,x_{\pi_{n}}\right)C_{n}\left(x,x_{\pi_{1}},\ldots,x_{\pi_{n}}\right)
\gamma_{g,9}\gamma_{S_{1},9}\gamma_{g,9}^{-1}=\gamma_{S_{2},9}~,~~~\gamma_{g,9}\gamma_{S_{2},9}\gamma_{g,9}^{-1}=\gamma_{S_{1},9}~,~~~\gamma_{g,9}\gamma_{S_{3},9}\gamma_{g,9}^{-1}=-\gamma_{S_{3},9}~.
R_{\delta_{0}}^{j}\bigr(F_{\eta_{0}\alpha_{0}}^{\delta_{0}}F_{\beta_{0}\gamma_{0}}^{\eta_{0}}-R_{\alpha_{0}}^{i}F_{\beta_{0}\gamma_{0},i}^{\delta_{0}}+\mathrm{cyclic~perm}(\alpha_{0},\beta_{0},\gamma_{0})\bigr)=0,
M=i\left(\begin{array}{ccc}{\sqrt{2}\epsilonA_{1(\mathrm{tl})}^{\dagger}}&{(A_{2}^{\dagger})^{t}}&{\epsilonA_{3}^{\dagger}}\\{A_{2}^{\dagger}}&{0}&{-\frac{\mathrm{Tr}A_{1}^{\dagger}}{\sqrt{2}}I_{N_{c}-2}+\sqrt{2}A_{4}^{\dagger}}\\{(\epsilonA_{3}^{\dagger})^{t}}&{-\frac{\mathrm{Tr}A_{1}^{\dagger}}{\sqrt{2}}I_{N_{c}-2}+\sqrt{2}(A_{4}^{\dagger})^{t}}&{0}\\\end{array}\right),
\star\Phi_{(p)}=\frac{1}{p!(6-p)!}\,dx^{m_{1}}\wedge\ldotsdx^{m_{6-p}}\epsilon_{m_{1}\ldotsm_{6-p}}{}^{m_{6-p+1}\ldotsm_{6}}\Phi_{m_{6-p+1}\ldotsm_{6}}\,.
\begin{array}{c}{N_{H}-2N_{S}=2S+M_{C}+2\,\theta(p-1)\,M_{W}+}\\{-\left\lfloor\displaystyle\frac{1}{2}+\displaystyle\frac{S}{p+1}+\displaystyle\frac{\omega}{\pi}\right\rfloor-\left\lfloor\displaystyle\frac{1}{2}+\displaystyle\frac{S}{p+1}-\displaystyle\frac{\omega}{\pi}\right\rfloor.}\\\end{array}
K(\xi,\xi;\epsilon)=e^{2\chi(\xi)-\phi(\xi)}\left[\frac{e^{\phi}}{4\pi\epsilon}-\frac{1}{24\pi}\partial_{a}^{2}\phi+\frac{1}{4\pi}\left(\partial_{a}^{2}\chi-(\partial_{a}\chi)^{2}\right)+O(\epsilon)\right]
\begin{array}{ccc}{\deltax^{A}=\omega_{~B}^{A}x^{B}~~~}&{~~~\displaystyle{\delta\theta^{i}=\omega\theta^{i}}~~~}&{~~~\displaystyle{\delta\tilde{\theta}^{\tilde{\imath}}=\tilde{\omega}\tilde{\theta}^{\tilde{\imath}}}}\\\end{array}
x^{\mu}\,\quad\tilde{r}=\frac{\tilde{\ell}^{2}}{\alpha^{\prime}}r\,\quad\tilde{g}=g\Big(\frac{\alpha^{\prime}}{\tilde{\ell}^{2}}\Big)^{\frac{p-3}{2}}\,\quad\tilde{\ell}^{2}=\theta\alpha^{\prime}\,
V_{n}(p*)=\int_{\pi_{n}^{-1}(p*)}{\frac{\Omega^{d}}{d!}}=\sum_{\mathrm{\scriptsize{\it~\sum_{i=1}^sd_i=d~}}}<\prod_{i=1}^{s}\sigma_{d_{i}}>_{n}\prod_{i=1}^{s}{\frac{p_{i}^{2d_{i}}}{d_{i}!}}.
{\frac{M_{m}}{V_{5m}}}={\frac{l_{s}^{12}}{l_{m}^{12}}}{\frac{M}{V_{5}}}={\frac{k}{(2\pi)^{5}l_{m}^{6}}}+{\frac{l_{s}^{4}}{(2\pi)^{5}l_{m}^{12}}}\mu.
c={(4\pi)}^{4/3}\int_{0}^{\infty}du\sum_{l=0}^{\infty}\Big(\frac{u^{3l}{(-1)}^{l}l!}{32\pi^{2}(4l+2)!}-\frac{u^{3l+3/2}{(-1)}^{l}(2l+1)!!}{32\pi^{3/2}2^{l}(4l+4)!}\Big)
\intD\phi\{\frac{\delta}{\delta\phi_{p}}}\left({\frac{1}{2}}{\frac{\delta}{\delta\phi_{-p}}}+P_{\Lambda}^{-1}\phi_{p}-Q_{p}^{-1}\right)\e^{-S\left[\phi;\Lambda\right]+\int_{p}J_{p}Q_{\Lambda}^{-1}\phi_{-p}+f_{\Lambda}}=0,
\mathrm{d}\mu=e^{-8\pi^{2}/g^{2}}\,\prod_{i}\mathrm{d}\eta_{i}^{b}\,\frac{M_{\mathrm{PV}}}{\sqrt{2\pi}}\,\left\|\frac{\partial\Phi(\eta)}{\partial\eta_{i}^{b}}\right\|\;\prod_{k}\mathrm{d}\eta_{k}^{f}\,\frac{1}{\sqrt{M_{\mathrm{PV}}}}\,\left\|\frac{\partial\Phi(\eta)}{\partial\eta_{i}^{f}}\right\|^{-1}\,,
N^{{\calP}}~=~\sum_{m_{1}=-1/2}^{1/2}\cdots\sum_{m_{p}=-1/2}^{1/2}\left[~\delta_{(\sum_{n=1}^{p}m_{n}),0}~-~\frac12~\delta_{(\sum_{n=1}^{p}m_{n}),1}~-~\frac12~\delta_{(\sum_{n=1}^{p}m_{n}),-1}~\right]
\times\prod_{n>0}\frac{1}{[1-e^{-4\pin\frac{1}{\alphavT}}][1+e^{-4\pin\frac{1}{\alphaT}(1+\frac{1}{v})}-\frac{1+\frac{E^{2}}{\alpha\betav}}{1-\frac{E^{2}}{\alpha\betav}}(e^{-4\pin\frac{1}{\alphaT}}+e^{-4\pin\frac{1}{\alphavT}})]}.
V_{f}^{(1)}=\frac{1}{8\pi^{2}}\sum_{n=-\infty}^{\infty}(-1)^{n}\int_{0}^{+\infty}\frac{ds}{s^{3}}e^{-(m_{f}^{2}s+\beta^{2}n^{2}/4s)}(eHs)coth(eHs),
I=\sqrt{\frac{\pi|\omega^{2}|}{4g^{2}}}e^{\omega^{4}/4g^{2}}\left\{\mathrm{I}_{-1/4}\!\left(\!{\frac{\omega^{4}}{4g^{2}}}\right)-\epsilon(\omega^{2})\mathrm{I}_{1/4}\!\left(\!{\frac{\omega^{4}}{4g^{2}}}\right)\right\}\,
C_{2}^{\calM}=C_{0}^{{\calM}_{1}}C_{2}^{{\calM}_{2}}+C_{1/2}^{{\calM}_{1}}C_{3/2}^{{\calM}_{2}}+C_{1}^{{\calM}_{1}}C_{1}^{{\calM}_{2}}+C_{3/2}^{{\calM}_{1}}C_{1/2}^{{\calM}_{2}}+C_{2}^{{\calM}_{1}}C_{0}^{{\calM}_{2}}
\Gamma^{\mu}\Gamma^{\alpha\beta}=\eta^{\mu\alpha}\Gamma^{\beta}-\eta^{\mu\beta}\Gamma^{\alpha}-\frac{1}{7!}\epsilon^{\mu\alpha\beta\lambda_{1}\ldots\lambda_{7}}~\Gamma^{11}\Gamma^{\lambda_{1}}\ldots\Gamma^{\lambda_{7}}.
\Xi_{{m}_{{1}}\ldots{m}_{{5}}}^{{(n)}}\equivSp(v^{{T}}\tilde{\sigma}_{{m}_{{1}}{\bf...}{m}_{{5}}}v\sigma^{{(n)}})\equivv_{\alpha}^{{a}}(\tilde{\sigma}_{{m}_{{1}}{\bf...}{m}_{{5}}})^{\alpha\beta}v_{\beta}^{{b}}(\sigma^{{(n)}})_{{ab}}=0
\Psi^{T}=e^{2\pi\tau_{2}\nu(1-\nu)/2}(1+e^{-2\pi\tau_{2}\nu})\prod_{n=1}^{\infty}{\frac{(1+e^{-2\pi\tau_{2}(n-\nu)})(1+e^{-2\pi\tau_{2}(n+\nu)})}{(1-e^{-n2\pi\tau_{2}})^{2}}}\.
L_{B}=-{\frac{1}{4}}H^{2}{}_{\mu\nu}(B)+{\frac{m}{4!}}\epsilon^{\mu\nu\rho\omega}\,B_{\mu}\,\partial_{\,[\,\nu}A_{\rho\sigma\,]}-{\frac{1}{2\cdot3!}}F^{2}{}_{\mu\nu\rho}(A)
\mathcal{L}=\frac{_1}{^2}i\hbar(\Psi^{*}\partial_{t}\Psi-\partial_{t}\Psi^{*}\Psi)-\frac{\hbar^{2}}{2m}\Psi^{*}\overleftarrow{\nabla}\cdot\overrightarrow{\nabla}\Psi-V\left(\mathbf{x}\right)\Psi^{*}\Psi.
S=\frac{s}{2\pi}\intd\phidy\,{\mathrm{Tr}}\left[\fraci2\Pig^{-1}\partial_{y}g+\frac14\Pi^{2}+\frac14g^{-1}\partial_{\phi}gg^{-1}\partial_{\phi}g\right]+\frac{is}{12\pi}\int{\mathrm{Tr}}\,(G^{-1}dG)^{3}\,,
S=\frac{1}{4}A_{\mathrm{EH}}=\frac{1}{4}\left[(r^{2}C)^{\frac{\tilde{d}+1}{2}}D^{\frac{d-1}{2}}\right]_{r=r_{+}}=\frac{1}{4}\Omega_{\tilde{d}+1}r_{+}^{\tilde{d}+1}\left[1-\left(\frac{r_{-}}{r_{+}}\right)^{\tilde{d}}\right]^{\frac{\tilde{d}+2}{2\tilde{d}}}.
\Omega_{1}={\frac{B_{1}D_{1}}{C_{1}B_{2}}}+\theta_{1}{\frac{D_{2}}{C_{1}}}\\mathrm{and}\\tilde{\Omega}_{1}={\frac{B_{2}D_{1}}{C_{2}B_{1}}}+\theta_{1}\Gamma{\frac{D_{2}}{C_{2}}}
Z_{r,p}^{n}(q)=\sum_{\substack{n_{1},\dots,n_{p-2}\geq0\,N_{1}+\dots+N_{p-2}=n}}\frac{q^{N_{1}^{2}+\dots+N_{p-2}^{2}+N_{r}+\dots+N_{p-2}}}{(q)_{n_{1}}\dots(q)_{n_{p-2}}}
\!\!\!\!\!g(u,m;\rho)=\sum_{n_{1},n_{2}}{\frac{(-{\frac{1}{2}}+3\rho)_{n_{1}+n_{2}}(-1)^{n_{1}+n_{2}}}{(1+2\rho)_{n_{1}}(1+\rho)_{n_{1}-n_{2}}(-n_{1}+2n_{2})!}}\Bigl({\frac{\Lambda^{3}}{16mu}}\Bigr)^{n_{1}-\frac{1}{3}+2\rho}\Bigl({\frac{4m^{2}}{u}}\Bigr)^{n_{2}-\frac{1}{6}+\rho}.
K^{(i)}\left(\left.a+\bar{\varepsilon}_{\mu}\begin{array}{c}{a}\\{a^{\prime}}\\\end{array}\right|z\right)=f_{a}^{(i)}(z)\frac{[\bar{a}_{i}+\eta-z]}{[\bar{a}_{i}+\eta+z]}\frac{[\bar{a}_{\mu}+\eta+z]}{[\bar{a}_{\mu}+\eta-z]}\delta_{aa^{\prime}}.
L=\frac{m}{2}\epsilon^{\mu\nu\lambda}[F_{\mu\nu}^{a}A_{\alpha}^{a}-\frac{g}{3}f_{abc}A_{\mu}^{a}A_{\nu}^{b}A_{\alpha}^{c}]+D_{\mu}\phi_{a}^{\dagger}D^{\mu}\phi_{a}-V(\phi)
\sum_{\lambda}\epsilon^{\mu}(\lambda)\epsilon^{*\nu}(\lambda)=-g_{\perp}^{\mu\nu}+{\frac{1}{q^{+}}}\bigl(\eta^{\mu}q_{\perp}^{\nu}+\eta_{\perp}^{\mu}\bigr)+{\frac{{\bfq}_{\perp}^{2}}{(q^{+})^{2}}}\eta^{\mu}\eta^{\nu}\;,
\dot{\rho}_{\varphi0}+3H_{\mathrm{h}}(t)(\rho_{\varphi0}+p_{\varphi0})+\frac{\partial\overline{{V_{\chi}(\varphi_{\mathrm{h}},\tilde{\chi})}}}{\partial\varphi_{\mathrm{h}}}\dot{\varphi}_{\mathrm{h}}=0\,.
F_{a}\\left[J_{1}^{q_{i}^{1}}.J_{2}^{q_{i}^{2}}.J_{3}^{q_{i}^{3}}\right]=\omega^{q_{i}^{a}}\\left[J_{1}^{q_{i}^{1}}.J_{2}^{q_{i}^{2}}.J_{3}^{q_{i}^{3}}\right]\F_{a}.
{\calV}_{\alpha,\alpha^{\prime}}^{(n)}=\sum_{\beta_{1}\cdots\beta_{n}}w(\beta_{1},\alpha_{1},\beta_{2},\alpha_{1}^{\prime})w(\beta_{2},\alpha_{2},\beta_{3},\alpha_{2}^{\prime})\cdotsw(\beta_{n},\alpha_{n},\beta_{1},\alpha_{n}^{\prime})~,
V(z)=E[1-{\frac{z^{6-p}}{Q_{p}}}{\frac{1+{\frac{b^{6-p}}{2z^{6-p}}}}{(1+{\frac{b^{6-p}}{z^{6-p}}})^{2}}}]+{\frac{J_{\perp}^{2}z^{10-2p}}{2m_{f}Q_{p}^{2}}}{\frac{1}{(1+{\frac{b^{6-p}}{z^{6-p}}})^{2}}},
V_{1}(\mathrm{x)=\frac{1}{32}\mathrm{x^{6}-\frac{\omega}{16\mathrm{g}}\mathrm{x^{4}+(\frac{\omega^{2}}{32\mathrm{g^{2}}}-\frac{3j}{4}-\frac{3}{8})\mathrm{x^{2}-\frac{\omega}{4\mathrm{g}}(3j-\frac{1}{2})}}}}
\begin{array}{ll}{\Theta={\frac{\pi}{2}}(\mathrm{mod}2\pi),}&{0<F<{\frac{1}{2}}}\\{\Theta=-{\frac{\pi}{2}}[1-\mathrm{sgn}(m)](\mathrm{mod}2\pi),}&{F={\frac{1}{2}}}\\{\Theta=-{\frac{\pi}{2}}(\mathrm{mod}2\pi),}&{{\frac{1}{2}}<F<1}\\\end{array},
\sum_{n_{1},\dotsn_{6}}e^{-\pil[-s/2M_{s}^{2}+(n_{1}^{2}+\dots+n_{6}^{2})(RM_{s})^{2}/2+2n]}=e^{\pils/2M_{s}^{2}}(1+{\frac{8}{(RM_{s})^{6}l^{3}}})+\dots.
A_{j}^{i}\;=\;\left(\begin{array}{cc}{A_{1}^{1}}&{A_{2}^{1}}\\{A_{1}^{2}}&{A_{2}^{2}}\\\end{array}\right)\;=\;\left(\begin{array}{cc}{\frac{H+qA_{0}}{q+q^{-1}}}&{A_{+}}\\{A_{-}}&{\frac{H-q^{-1}A_{0}}{q+q^{-1}}}\\\end{array}\right)
{\calH}=\frac{1}{2}\left[\vec{E}_{\bot}^{2}+\vec{B}\,^{2}\right]+\frac{1}{2}{\calE}_{3}^{2}+\frac{1}{2L^{2}}e_{3}^{2}\left(\vec{x}_{\perp}\right)+\frac{1}{2}\vec{\eta}\,^{2}\left(\vec{x}_{\perp}\right)\.