harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{-\infty}^{\infty}\epsilon\left[\phi_{1},\phi_{2}\right]dz\geq\int_{-\infty}^{\infty}dz\,\frac{d}{dz}\left({\phi_{1}}-\frac{2}{3}\,\phi_{1}^{3}-2\phi_{2}^{2}\phi_{1}\right)=\frac{\sqrt2}{3},
\Phi=-\frac{4\pi\nu}{e}\int_{0}^{\infty}\frac{(2\nur^{2\nu-1})}{(r^{2\nu}-1)^{2}}dr=-\frac{4\pi\nu}{e}\int_{-1}^{\infty}\frac{dw}{w^{2}}\;\;\;(w=r^{2\nu}-1)
\int_{0}^{1}dx_{1}\int_{0}^{1-x_{1}-x_{2}-x_{3}-x_{4}}dx_{5}\left[\frac{2p_{0}x_{2}-q_{0}(1-x_{1}-2x_{2}-x_{3}-2x_{4}-x_{5})}{(1-x_{1}-x_{3}-x_{5})^{\frac{3}{2}}}\right]=\frac{4}{945}(2p_{0}+q_{0})
W_{\wedge}=\left\{\begin{array}{c}{L_{-1}\quadL_{0}\quadL_{1}}\\{J_{-2}\quadJ_{-1}\quadJ_{0}\quadJ_{1}\quadJ_{2}}\\{\Lambda_{-3}\quad\Lambda_{-2}\quad\Lambda_{-1}\quad\Lambda_{0}\quad\Lambda_{1}\quad\Lambda_{2}\quad\Lambda_{3}}\\{\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots}\\\end{array}\right\}
\left[g_{\mu\nu}(u)\right]=\left(\begin{array}{c|c}{1}&{u^{0}\vec{u}^{\mathrm{T}}}\\\hline{u^{0}\vec{u}}&{-I+\vec{u}\otimes\vec{u}^{\mathrm{T}}(u^{0})^{2}}\\\end{array}\right).
\wp(u)\equiv\wp(u|\{2\omega_{1},2\omega_{3}\})={\frac{1}{u^{2}}}+\sum_{m,\,n}{}^{\prime}\left[{\frac{1}{(u-\Omega_{m,\,n})^{2}}}-{\frac{1}{\Omega_{m,\,n}^{2}}}\right],
\sigma_{1,+}=\sigma_{1,1}=\frac{1}{2}\left(u_{1,1}+u_{1,2}\right)+i\sqrt{u_{1,3}-\left[\frac{1}{2}(u_{1,1}-u_{1,2})\right]^{2}}=\sigma_{1,3}^{*}=\sigma_{1,-}^{*}
\varphi_{1}={\frac{(1+\beta^{2})(1-3\beta^{2})}{\beta^{2}(1-\beta^{2})(1+3\beta^{2})}}\left[1-(1-x)^{\frac{2\beta^{2}}{1+\beta^{2}}}\right]+\left({\frac{2\beta^{2}}{(1+\beta^{2})(1-\beta^{2})}}\right)x,
{\calE_{C}}[\phi_{0}]=\Gamma^{(2)}[\phi_{0}]+\sum_{\ell=0}^{\infty}(2\ell+1)\left[\int_{0}^{\infty}\frac{dk}{\pi}E\frac{d}{dk}\left(\delta_{\ell}(k)-\delta_{\ell}^{(1)}(k)-\delta_{\ell}^{(2)}(k)\right)+\sum_{j}E_{j,\ell}\right]+(\omega-E_{0})
{\calD}_{Vv}^{\mathrm{sym}}\equiv{\calD}_{Vv}^{(l)}-r^{(1)}\bigl(4{\calD}_{e}^{(l+1)}+8l({\calN}_{V}^{(l+1)}-8(l+1)r^{(1)}\delta{\calN}_{V}^{(l+2)})\bigr).
\begin{aligned}{\big[W^{(8)},Q\big]}&{{}=\frac{\tilde{m}}{8}\,\GammaQ\,,}&{\big[W^{(5)},Q\big]}&{{}=\frac{\tilde{n}}{8}\,\GammaQ\,,}\\{\big[Z^{(2|1)},Q\big]}&{{}=C\,\GammaQ\,,}&{\big[Z^{(4|2)},Q\big]}&{{}=D\,\GammaQ\,,}\\\end{aligned}
\begin{array}{ccc}{\Psi_{I}}&{=}&{(w_{2})^{\frac{1}{2}}\Psi~~;}\\{F_{I}^{\mu\nu}}&{=}&{(w_{3})^{\frac{1}{2}}F^{\mu\nu}~~;}\\\end{array}\quad\begin{array}{ccc}{m_{I}}&{=}&{\frac{w_{0}}{w_{2}}m}\\{e_{I}}&{=}&{\frac{w_{1}}{w_{2}w_{3}^{\frac{1}{2}}}e}\\\end{array}
\{S^{p},T^{r}\}=\left(\sum_{k_{p}=0}^{r}...\sum_{k_{2}=0}^{k_{3}}\sum_{k_{1}=0}^{k_{2}}\alpha^{k_{1}}\alpha^{k_{2}}...\alpha^{k_{p}}\right)S^{p}T^{r}.
N_{\mathrm{out},t_{0}}(t_{1})=\sum_{i=1}^{N}(|\alpha_{p_{i},p_{z,i}}(\tau_{1},\tau_{0})|^{2}-|\beta_{p_{i},p_{z,i}}(\tau_{1},\tau_{0})|^{2})+N_{\mathrm{vac},t_{0}}(t_{1})
\begin{array}{l}{\alpha(K_{\alpha}^{})\,\Phi(-\alpha(q),v)\,A_{k}^{\alpha}(q,u)}\\{\mathrm{}+\,{\textstyle\frac{1}{2}}\,N_{\alpha,\theta\alpha}^{}\,\Phi(\alpha(q),u-v)\,\Bigl(A_{k}^{\alpha+\theta\alpha}(q,u)\,-\,A_{k}^{\alpha+\theta\alpha}(q,v)\Bigr)}\\{\mathrm{}+\,{\displaystyle\sum_{j=1}^{r}}\,\alpha_{j}^{}\,A_{j}^{\alpha}(q,u)A_{k}^{\theta\alpha}(q,v)~=~0~.}\\\end{array}
{\calL}_{\mathrm{RCS}}=|D_{0}\phi|^{2}+\kappaA_{0}B-\frac{\kappa}{2}\epsilon^{ij}A_{i}\dot{A}_{j}-|\vec{D}\phi|^{2}-\frac{1}{\kappa^{2}}|\phi|^{2}(|\phi|^{2}-v^{2})^{2}
\left<\Psi\right|{\frac{T_{ij}(\vec{y},\tau)}{N}}\left|\Psi\right>={\frac{i\xi_{mn}}{N^{2}}}\int{dtd^{3}\vec{x}f(t)\left[T_{ij}(\vec{y},\tau),T_{mn}(\vec{x},t)\right]}.
\epsilon_{0}c^{2}\intdt\int_{(\infty)}d^{3}\vec{x}\left\{-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}\left(\frac{\Phi_{0}}{2\pi\lambda}\right)^{2}\left[\mid(\nabla_{\mu}+i\frac{q}{\hbar}A_{\mu})\psi\mid^{2}-\frac{1}{2\xi^{2}}\left(\mid\psi\mid^{2}-1\right)^{2}\right]\right\},
W_{n}^{m}\approx\sqrt{\frac2{\pi\nu_{nm}}}\frac{|\eta|^{|n-m|}}{2^{\mu_{nm}}\mu_{nm}!|\xi|^{|n-m|+1}}\left[H_{\mu_{nm}}\left(\sqrt{\frac{|n-m|+1}{2|\xi|^{2}}}\right)\right]^{2}.
V(\sigma)=\frac{1}{2\lambda_{0}}\sigma^{2}-i\,\mathrm{tr}\\displaystyle\int_{0}^{\sigma}sds\frac{1}{L}\sum_{n=-\infty}^{\infty}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}\frac{1}{k^{2}-s^{2}}\,.
+\overline{{\nu}}_{e}(\alpha_{i}\sqrt{g^{2}+g_{0}^{2}}\gamma^{\mu}-\beta_{i}g\gamma^{\mu}\gamma^{5})\nu_{e}Z_{\mu}^{(i)}+\overline{{e}}(-\alpha_{i}\frac{g^{2}-g_{0}^{2}}{\sqrt{g^{2}+g_{0}^{2}}}\gamma^{\mu}+\beta_{i}g\gamma^{\mu}\gamma^{5})eZ_{\mu}^{(i)}].
g(r)=A(r)=\frac1{r^{2}}\left[2\sqrt{C}\left(C_{1}^{(0)}e^{2\sqrt{C}r}+C_{2}^{(0)}e^{-2\sqrt{C}r}\right)-\frac1{r}\left(C_{1}^{(0)}e^{2\sqrt{C}r}-C_{2}^{(0)}e^{-2\sqrt{C}r}\right)\right],\qquadk=0,
S^{(N)}\left[{\bfr}_{1},\dots{\bfr}_{N-1}\right]({\bfr}^{\prime\prime},{\bfr}^{\prime};t^{\prime\prime},t^{\prime})=\sum_{j=0}^{N-1}\left[\frac{M\left({\bfr}_{j+1}-{\bfr}_{j}\right)^{2}}{2\epsilon}-\epsilonV({\bfr}_{j},t_{j})\right]\;,
\langleR(\chi,\chi;-1)|U_{2}(\psi,w)|R(\chi,\chi;z)\rangle=(1+w)^{-2h_{\psi}}(1+z)^{-2\delta}\langleR(\chi,\chi;1)|U_{2}\left(\psi,{\frac{w}{w+1}}\right)|R\left(\chi,\chi;{\frac{z}{z+1}}\right)\rangle\,.
\Sigma(p)={\frac{i}{\theta}}\int_{0}^{1}dx\;\int{\frac{d^{3}q}{(2\pi)^{3}}}{\frac{\left(q_{\lambda}+xp_{\lambda}\right)\left[\left(p_{\rho}(1-x)-q_{\rho}\right)\;2\;g_{\lambda}^{\rho}-2MT^{3}\gamma_{\lambda}\right]}{\left[q^{2}+p^{2}x(1-x)-M^{2}x\right]^{2}}}
\begin{array}{l}{H_{1}=\hbar^{-1}x_{0}+H_{1}^{(0)}+\hbarH_{1}^{(1)}+{\calO}(\hbar^{2})\,,}\\{H_{2}=-\hbar^{-1}x_{0}+H_{2}^{(0)}+\hbarH_{2}^{(1)}+{\calO}(\hbar^{2})\,,}\\{H_{3}=\hbar^{-1}x_{+}+2\hbarH_{3}^{(1)}+{\calO}(\hbar^{2})\,,}\\{\bar{H}_{3}=\hbar^{-1}x_{-}+2\hbar\bar{H}_{3}^{(1)}+{\calO}(\hbar^{2})\,,}\\\end{array}
\dot{\Phi}^{2}\left[{\frac{1}{2}}{\frac{I}{c^{2}}}\dot{\Phi}^{2}-V+\hat{p}\sqrt{I}\right]{\frac{\dot{a}}{a}}+\dot{\Phi}\ddot{\Phi}\left[{\frac{1}{2}}{\frac{I}{c^{2}}}\dot{\Phi}^{2}+V+\hat{\rho}c^{2}\sqrt{I}\right]\leq0.
T_{\mu\nu}\left(q;p,p^{\prime}\right)=ie^{2}\frac{\left(q^{2}-m_{\pi}^{2}\right)}{f_{\pi}m_{\pi}^{2}}\intd^{4}xd^{4}ye^{ipx+ip^{\prime}y}\left\langle0\midT\left(\partial^{\lambda}A_{\lambda}^{3}\left(x\right)J_{\mu}\left(x\right)J_{\nu}\left(y\right)\right)\mid0\right\rangle.
\sqrt{-G}M^{3}W_{\mu\nu}^{5}\Bigr|_{y=0^{+}}=-\sqrt{-g^{0}}\frac{T_{\mu\nu}^{0}}{2}\,\qquad\sqrt{-G}M^{3}W_{\mu\nu}^{5}\Bigr|_{y=\pir_{c}^{-}}=\sqrt{-g^{\pi}}\frac{T_{\mu\nu}^{\pi}}{2}\.
\begin{aligned}{\Psi}&{{}=(-\frac{1}{2},0)\,,}&{\quad\theta}&{{}=(\frac{1}{2},0)\,,}&{\quad\partial}&{{}=(-1,0)\,,}&{\quad\gamma}&{{}=(\frac{1}{2},0)\,,}\\{\tilde{\Psi}}&{{}=(0,-\frac{1}{2})\,,}&{\quad\bar{\theta}}&{{}=(0,\frac{1}{2})\,,}&{\quad\bar{\partial}}&{{}=(0,-1)\,,}&{\quad\tilde{\gamma}}&{{}=(0,\frac{1}{2})\,}\\\end{aligned}
A_{k}^{a}(x^{0},{\bfx})\;\sim\frac{1}{\mid{\bfx}\mid^{1+\epsilon}}\;,\;\frac{\partial}{\partialx^{\ell}}A_{k}^{a}(x^{0},{\bfx})\;\sim\frac{1}{\mid{\bfx}\mid^{2+\epsilon}}\;,\;\epsilon>0.
\bar{\Gamma}_{~\mu\nu}^{\sigma}\rightarrow^{\prime}\bar{\Gamma}_{~\mu\nu}^{\sigma}=\bar{\Gamma}_{~\mu\nu}^{\sigma}+g^{\sigma\rho}\Lambda_{[\rho\mu\nu]}+g^{\sigma\rho}\underline{{T}}_{\rho\mu\nu}+M^{\sigma}g_{\mu\nu}+N_{\nu}\delta_{\mu}^{\sigma}+P_{\mu}g_{\nu}^{\sigma}
\begin{array}{ll}{det_{q}L(u)=}&{a(u+i\kappa/2)d(u-i\kappa/2)-b(u+i\kappa/2)c(u-i\kappa/2)}\\{\,}&{=Q_{1}u^{2}+Q_{2}u-Q_{3}+\kappa^{2}/4Q_{1}\quad,}\\\end{array}
C\left(F_{[\mu\nu]}F^{[\nu\rho]}\\mathrm{tr}(g^{-1}\partial_{\rho}g\g^{-1}\partial^{\mu}g)+\alphaF_{[\mu\nu]}F^{[\nu\mu]}\\mathrm{tr}(g^{-1}\partial_{\rho}g\g^{-1}\partial^{\rho}g)\right),
K_{\mathrm{np}}=d\left(\frac{S_{R}}{4\pi}\right)^{p/2}e^{-b\sqrt{S_{R}}}\left[1+{\calO}\left(\frac{1}{\piS_{R}},\frac{1}{\piT_{R}}\right)\right]+{\calO}(e^{-2\piS},e^{-2\piT}),
\pi\left(x^{0}+\tau,{\bfx}\right)=\pi\left(x^{0},{\bfx}\right)+\tau\sum_{{\bfx^{\prime}}}{\bfF}\left(\phi\left(x^{0}+\tau,{\bfx}\right)\right)_{{\bfx},{\bfx^{\prime}}}\phi\left(x^{0}+\tau,{\bfx^{\prime}}\right),
\alpha={\frac{b}{a}}\left(\begin{matrix}{n_{1}}&{m_{1}+p_{1}}&{m_{2}+p_{2}}&{m_{3}+p_{3}}\\{-m_{1}+p_{1}}&{n_{2}}&{-m_{3}+p_{4}}&{m_{2}+p_{5}}\\{-m_{2}+p_{2}}&{m_{3}+p_{4}}&{n_{3}}&{-m_{1}+p_{6}}\\{-m_{3}+p_{3}}&{-m_{2}+p_{5}}&{m_{1}+p_{6}}&{-n_{1}-n_{2}-n_{3}}\\\end{matrix}\right),
Z=\int{\calD}[f(\xi),g(\xi),X]\,\delta\left(g_{zz}^{f}\right)\delta\left(g_{\bar{z}\bar{z}}^{f}\right)\,\mathrm{det}\left(\frac{g_{zz}^{f}}{\deltaf}\right)\mathrm{det}\left(\frac{g_{\bar{z}\bar{z}}^{f}}{\deltaf}\right)\,e^{-S[g,X]}.
\left(\begin{array}{c}{\ddot{\hat{\Sigma}}_{n}^{t}}\\{\ddot{\hat{\Sigma}}_{n}^{r}}\\{\ddot{\hat{\Sigma}}_{n}^{\varphi}}\\\end{array}\right)+{\calV}\left(\begin{array}{c}{\hat{\Sigma}_{n}^{t}}\\{\hat{\Sigma}_{n}^{r}}\\{\hat{\Sigma}_{n}^{\varphi}}\\\end{array}\right)={\calG}^{-1}\left(\begin{array}{c}{\hat{U}_{n}^{t}}\\{\hat{U}_{n}^{r}}\\{\hat{U}_{n}^{\varphi}}\\\end{array}\right),
{\mathbfw}_{a}\Gamma\equiv-\sum_{\Phi}f_{abc}\Phi_{b}\frac{\delta\Gamma}{\delta\Phi_{c}}+\Gamma\frac{\stackrel{\leftarrow}{\delta}}{\delta\Psi}T_{a}\Psi-\overline{{\Psi}}T_{a}\frac{\stackrel{\rightarrow}{\delta}}{\delta\overline{{\Psi}}}\Gamma+\Gamma\frac{\stackrel{\leftarrow}{\delta}}{\deltaY}T_{a}Y-\overline{{Y}}T_{a}\frac{\stackrel{\rightarrow}{\delta}}{\delta\overline{{Y}}}\Gamma\;\;\;,
T(u)=\partial_{u}\partial_{u}\varphi-{\frac{1}{2}}\left(\partial_{u}\varphi\right)^{2}=\left\{J_{H}^{-1},u\right\}=\sum_{i=1}^{n-1}\left({\frac{1}{2(u-u_{i})^{2}}}+{\frac{c_{i}}{u-u_{i}}}\right).
\Omega^{\prime}=\left(\begin{array}{cc}{(G^{-1})^{rs}/4}&{B_{\alpha\gamma}G^{\gammar}/2}\\{B_{\alpha\gamma}(G^{-1})^{\gammas}/2}&{G_{\alpha\beta}-B_{\alpha\delta}(G^{-1})^{\delta\gamma}B_{\gamma\beta}}\\\end{array}\right)\.
q_{b}(a,r)\sim-\frac{(2\nu_{0}-n)r^{2\nu_{0}-n-2}f_{0}^{(q)}}{(2a)^{2\nu_{0}+1}\piS_{D}\sigma^{D-1}\Gamma^{2}(\nu_{0}+1)}\int_{ma}^{\infty}dz\frac{z^{2\nu_{0}+1}}{\sqrt{z^{2}-m^{2}a^{2}}}\frac{\bar{K}_{\nu_{0}}(z)}{\bar{I}_{\nu_{0}}(z)},\quadr\to0,
(H-E_{0})_{q,q^{\prime}}=-\frac{{\calP}}{(q-q^{\prime})^{2}}\;\psi_{0}(q)\psi_{0}^{\ast}(q^{\prime})+\;\delta(q-q^{\prime})\intdq^{\prime\prime}\frac{{\calP}}{(q-q^{\prime\prime})^{2}}\rho(q^{\prime\prime})
V_{p^{+},p^{-}}^{r,s}\left(z\right)=e^{iX^{-}p^{+}}\,\,e^{-\frac12X_{2}\left(1-2is\right)}\,F_{r,s}\left(e^{-X_{2}}p^{-}p^{+}\right)e^{iX^{+}p^{-}}
<\alpha|T_{1}|\gamma>=\frac{\left(2\pi\right)^{\nu}}{\left(2\pi\right)^{2\left(\nu-1\right)}}\frac{\delta\left(p-q_{3}-q_{4}\right)}{4\sqrt{{\omega}_{1}{\omega}_{2}{\omega}_{3}{\omega}_{4}}}\;\;\;(p=p_{1}+p_{2})
G\=\G_{0}e^{\frac{i}{2}\int\frac{d^{4}k}{(2\pi)^{4}}\\frac{1}{k^{2}+i\epsilon}\(g_{00}-\xi\frac{k_{0}^{2}}{k^{2}+i\epsilon})\\left|\widetildeJ^{0}(k)+J^{0}(k)\right|^{2}}\,
M_{1}^{(bm)}\simeq\kappa^{2/3}\left[4\pi(d-3)|\zeta^{\prime}(-2)|\right]^{1/3}+\frac{1}{3}\,\kappa^{4/3}\left[4\pi(d-3)|\zeta^{\prime}(-2)|\right]^{-1/3}(\betat)^{2}.
\left(\;\,\frac{\mathrm{d}^{2}}{\mathrm{d}r^{2}}+\frac{2}{r}\frac{\mathrm{d}}{\mathrm{d}r}-\mu_{0}^{2}\;\right)\phi\left(r\right)=-\frac{g^{2}N}{4\pi}\,\left|\,\psi\left(r\right)\right|^{2}\,,
\frac{\partialI_{i}}{\partialx^{\rho}}=A_{\mu}(c)\frac{\partialc^{\mu}}{\partial\tau}\vert_{\tau=\tau_{i+1}(x)}\cdot\frac{\partial\tau_{i+1}(x)}{\partialx^{\rho}}-A_{\mu}(c)\frac{\partialc^{\mu}}{\partial\tau}\vert_{\tau=\tau_{i}(x)}\cdot\frac{\partial\tau_{i}(x)}{\partialx^{\rho}}+D
\frac{d}{ds}\left[\frac{2}{\sqrt{4\pi}}\frac{\;\Gamma(s-1/2)}{\Gamma(s)}\zeta_{S_{4}}(s-1/2)\right]_{s=0}=\frac{d}{ds}\left[\frac{1}{3}\frac{L}{\sqrt{\pi}}\frac{\;\Gamma(s-1/2)}{\Gamma(s)}\left(\zeta(2s-4,3/2)-\frac{1}{4}\zeta(2s-2,3/2)\right)\right]_{s=0}=0,
\Psi_{E}^{n\leq\ell-1}\left(r,\theta\right)=\left(\begin{array}{l}{B_{n}^{+}J_{\ell+a-n}\left(kr\right)e^{in\theta}}\\{i\frac{k}{E-m}B_{n}^{+}J_{\ell+a-n-1}\left(kr\right)e^{i\left(n+1\right)\theta}}\\{B_{n}^{-}J_{\ell+a-n}\left(kr\right)e^{in\theta}}\\{i\frac{k}{E+m}B_{n}^{-}J_{\ell+a-n-1}\left(kr\right)e^{i\left(n+1\right)\theta}}\\\end{array}\right)
a_{n+2}(\vartheta+i\pi+i\varepsilon,\vartheta,\theta_{1},\theta_{2}....\theta_{n})\simeq\frac{1}{\varepsilon}\left[1-\prod_{i=1}^{n}S(\vartheta-\theta_{i})\right]a_{n}(\theta_{1},\theta_{2}....\theta_{n})
L_{SS}=-{\frac{1}{8\pi}}\intd^{2}\sigma\sqrt{-h}\bigg(h^{ab}\partial_{a}X^{\mu}\partial_{b}X_{\mu}+2i\bar{\psi}^{\mu}\gamma^{a}\partial_{a}\psi_{\mu}-i\bar{\chi}_{a}\gamma^{b}\gamma^{a}\psi^{\mu}\big(\partial_{b}X_{\mu}-{\frac{i}{4}}\bar{\chi}_{b}\psi_{\mu}\big)\bigg),
ds^{2}=\left[H(r)\right]^{-\frac{\widetilde{d}}{D-2}}\,dx^{\mu}\otimesdx^{\nu}\,\eta_{\mu\nu}+\left[H(r)\right]^{\frac{d}{D-2}}\,\left(dr^{2}+r^{2}ds_{S^{D-p-2}}^{2}\right)
\chi_{isos.}=\frac{<p_{g.s.}|M|p_{g.s.}>}{<p_{g.s.}|p_{g.s.}>}=\frac{<g.s.|M|g.s.>+\epsilon^{2}<p_{g.s.}^{1}|M|p_{g.s.}^{1}>+\epsilon^{4}<p_{g.s.}^{2}|M|p_{g.s.}^{2}>+\ldots}{<g.s.|g.s.>+\epsilon^{2}<p_{g.s.}^{1}|p_{g.s.}^{1}>+\epsilon^{4}<p_{g.s.}^{2}|p_{g.s.}^{2}>+\ldots}\quad.
\varphi^{a}=\Bigl(\delta_{b}^{a}-\frac{1}{2}\xi^{ae}\S_{ec}\delta_{j}^{c}G^{jk}\delta_{k}^{d}\xi_{db}\Bigr)\delta_{i}^{b}\,\phi^{i}-\Bigl(\xi^{ac}\S_{cb}\delta_{j}^{b}G^{ji}\Bigr)\,{\bar{\phi}}_{i}\;\;,
\epsilon_{\mathrm{Cas}}(x)=\frac{1}{2}\omega_{B}\phi_{B}^{2}(x)-m\int_{0}^{\Lambda}\frac{dk}{2\pi}\left(\frac{\omega}{k^{2}+\frac{1}{4}m^{2}}-\frac{1}{\omega}\right)\phi_{B}^{2}(x)-\frac{m}{2\pi}\left(\phi_{B}^{2}(x)+2\phi_{0}^{2}(x)\right)\\.
\int_{M}{\mathrm{Str}}_{\bf\mathrm{1}}{\calF}_{0}^{2}=\int_{M}{\mathrm{Str}}_{\bf\mathrm{1}}\Bigg(Q\Big({\cal{A}}_{0}Q{\cal{A}}_{0}+\frac{2}{3}{\calA}_{0}^{3}\Big)\Bigg),
S_{F}[\overline{{\psi}},\psi,A,h]\;=\;\sum_{f=1}^{N}\intd^{2}x\;\overline{{\psi}}^{(f)}(x)\gamma_{\mu}\Big(\partial_{\mu}-ieA_{\mu}(x)-i\sqrt{g}h_{\mu}(x)\Big)\psi^{(f)}(x)\;.
S={m}\,Tr\,\int\,d\tau\,\left({\frac{1}{2}}\,\dot{\Phi}^{i}\dot{\Phi}^{j}-{\frac{1}{4}}[\Phi^{i},\,\Phi^{j}]\,[\Phi^{i},\,\Phi^{j}]+i\Theta^{A}\dot{\Theta}^{A}-i\Theta^{A}\gamma_{AB}^{i}[\Phi^{i},\,\Theta^{B}]\right),
G_{gh}^{(f)}(z,z^{\prime})=-\frac{[1-(-1)^{2l_{1_{s}}}]}{2}\left[G_{gh}^{(f)(s)}(z,z^{\prime})+{\sum_{F_{s}}}^{\prime}P_{F_{s}}^{(a)}(z)\tilde{\chi}_{F_{s}}(z^{\prime})\right]
\Lambda_{magn}^{10-2F}M_{1}\tilde{M}_{2}^{2}M_{2}^{2F-3}\langleY\rangle^{3}\langleq^{*}\rangle^{2}\langle\bar{q}^{*}\rangle^{2}\langleY^{*}\rangle^{5}\frac{h^{2F+3}}{\mu^{4F}}.
\left(\Phi_{2}+2b^{2}e^{-2\Phi_{0}}\right)\left(1+16\beta\Phi_{2}e^{-2\Phi_{0}}\right)^{3}+32\betab^{4}e^{-6\Phi_{0}}\left(1+8\beta\Phi_{2}e^{-2\Phi_{0}}\right)=0.
\Psi_{r}[X_{r}^{i}(\sigma_{r})]\;=\;\mathrm{e}^{(\lambda_{r}^{\mathrm{\scriptscriptstyleT}}-\frac{c^{\mathrm{\scriptscriptstyleT}}}{12})\frac{\tau_{r}}{\alpha_{r}}}\int_{D_{r}}\!\!{\calD}^{g_{\scriptscriptstyle\mathrm{pl}}}X^{i}(w,\bar{w})\;\mathrm{e}^{-S[g_{\scriptscriptstyle\mathrm{pl}},X^{i}(w,\bar{w})]}\V_{r}^{\mathrm{\scriptscriptstyleT}}(z_{r},\bar{z}_{r})\.
\mathcal{H}=\intd^{\,2}\,x\,\left[\,\frac{1}{2}\,\left(\xi\,_{5}^{2}+\xi\,_{6}^{2}\right)+\frac{1}{2}\,\left(\partial_{1}\,\xi_{3}-\partial_{2}\,\xi_{2}\right)^{\,2}+\frac{1}{2}\,m^{2}\,\left(\xi\,_{2}^{2}+\xi\,_{3}^{2}\right)+m\left(\xi\,_{2}\,\xi\,_{6}-\xi\,_{3}\,\xi\,_{5}\right)\right];\tag{17}
L=\frac{1}{4\pi\alpha^{\prime}}\sum_{n}\left[\epsilon(\dot{X}_{n}^{i})^{2}-\frac{1}{\epsilon}\left(X_{n+1}^{i}-X_{n}^{i}\right)^{2}+4\pi\alpha^{\prime}B_{ij}\dot{X}_{n}^{i}\left(X_{n+1}^{j}-X_{n}^{j}\right)\right]
||u_{\underline{{m}}}^{~\underline{{a}}}||\equiv||u_{\underline{{m}}}^{~a}~~|~~u_{\underline{{m}}}^{i}||\equiv||u_{\underline{{m}}}^{\pm\pm}~~|~~u_{\underline{{m}}}^{i}||{}~~~~~~~~\in~~~~~~~~SO(1,D-1)
i_{X}\omega=\frac{1}{n!}{\displaystyle\sum_{s=1}^{n}(-1)^{s-1}X^{\mu_{s}}\omega_{\mu_{1}\ldots\mu_{s}\ldots\mu_{n}}d\sigma^{\mu_{1}}\ldotsd\sigma^{\mu_{s-1}}d\sigma^{\mu_{s+1}}\ldotsd\sigma^{\mu_{n}}},
W^{(\beta_{1},\beta_{2})}(\phi_{1},\phi_{2})=(-1)^{\beta_{1}}\sqrt{(\phi_{1}+(-1)^{\beta_{2}}\sigma)^{2}+\phi_{2}}[\frac{1}{3}(\phi_{1}^{2}+\phi_{2}^{2}-(-1)^{\beta_{2}}\sigma\phi_{1}+\sigma^{2})-1]\quad,
\begin{array}{l}{\delta_{b}x^{\mu}={b^{\mu}}_{\nu}n^{\nu},}\\{\delta_{b}A_{a}^{\mu}=-{b^{\mu}}_{\nu}\displaystyle\left(\varepsilon_{ab}\frac{g^{bc}}{\sqrt{-g}}{\Pi_{c}}^{\nu}-\xi_{a}n^{\nu}+i(\bar{\theta}\Gamma^{\nu}\partial_{a}\theta)\right).}\\\end{array}
W_{_C}[A]=\frac{e^{2}}{8\pi}\int\!\!dx\;A_{\mu}\left[\;a_{_C}g^{\mu\nu}-(\partial_{x}^{\mu}+\tilde{\partial}_{x}^{\mu})\frac{1}{\square}(\partial_{x}^{\nu}+\tilde{\partial}_{x}^{\nu})\right]A_{\nu}(x).
\Psi_{\bfC}\rightarrow\left(\begin{array}{cc}{e^{\frac{\mathrm{i}}{2}\mathrm{~{\bf~\sigma~\cdot}~}(\mathrm{~{\bf~\theta}~}-\mathrm{i}\mathrm{~{\bf~\phi}~})}}&{0}\\{0}&{e^{\frac{\mathrm{i}}{2}\mathrm{~{\bf~\sigma~\cdot}~}(\mathrm{~{\bf~\theta}~}+\mathrm{i}\mathrm{~{\bf~\phi}~})}}\\\end{array}\right)\cdot\Psi_{\bfC},
\mathrm{Re}\,\chi_{T}^{\mathrm{D}}(\omega)\approx-\frac{\hbarA}{4\pi^{2}}\left(\frac{k_{B}T}{\hbar}\right)^{5}\Omega^{4}e^{-\Omega/2}\int_{-1}^{1}ds\,s^{2}\sqrt{1-s^{2}}\,e^{-\Omegas^{2}/2}\int_{-\infty}^{\infty}dt\frac{\partial}{\partialt}\left[t\sqrt{1+t^{2}}(1+t^{2}-s^{2})\right]e^{-\Omegat^{2}/2}.
S_{0}=\intd^{4}x\{p_{i}\dot{A}_{i}-\frac{p_{i}^{2}}{2}-\frac{1}{4}(\partial_{i}A_{j}-\partial_{j}A_{i})^{2}+A_{0}\partial_{i}p_{i}+\lambda\tilde{\partial}_{i}A_{i}\}
{\calD}=\intdx\left\{j_{\mu}^{a}\frac{\delta}{\delta\gamma_{\mu}^{a}}+s\frac{\delta}{\delta\gamma}+b^{a}\frac{\delta}{\delta\gamma^{a}}+\bar{\eta}^{a}\frac{\delta}{\delta\omega^{a}}+\eta^{a}\left(\frac{1}{\alpha}\partial_{\nu}\frac{\delta}{\deltaj_{\nu}^{a}}-m\frac{\delta}{\deltab^{a}}\right)\right\},
\left|\begin{array}{cccc}{A_{11}}&{A_{12}v+A_{23}}&{A_{14}v^{2}+A_{15}v+A_{16}}&{0}\\{0}&{A_{11}}&{A_{12}v+A_{13}}&{A_{14}v^{2}+A_{15}v+A_{16}}\\{A_{21}}&{A_{22}v+A_{23}}&{A_{24}v^{2}+A_{25}v+A_{26}}&{0}\\{0}&{A_{21}}&{A_{22}v+A_{23}}&{A_{24}v^{2}+A_{25}v+A_{26}}\\\end{array}\right|=0
H_{c}=\intd^{2}x\left[({\vec{\nabla}}+i{\vec{A}})\phi^{*}\cdot({\vec{\nabla}}-i{\vec{A}})\phi+\pi_{\phi^{*}}\pi_{\phi}+A_{0}(j_{0}-\frac{\alpha}{2\pi}\epsilon_{ij}\partial_{i}A_{j})\right]
\begin{array}{lll}{\langleq_{i0}-x_{0}\rangle}&{=}&{0\,,}\\{}&{}&{}\\{\langle\dot{q}_{i0}\rangle}&{=}&{0\,,}\\{}&{}&{}\\{m\omega_{i}\omega_{j}\langle(q_{i0}-x_{0})(q_{j0}-x_{0})\rangle}&{=}&{k_{B}T\delta_{ij}\,,}\\{}&{}&{}\\{m\langle\dot{q}_{i0}\dot{q}_{j0}\rangle}&{=}&{k_{B}T\delta_{ij}\,,}\\{}&{}&{}\\{\langle\dot{q}_{i0}(q_{j0}-x_{0})\rangle}&{=}&{0\,,}\\\end{array}
(\frac{1}{4\pi^{2}})(q^{\mu}q^{\nu}-g^{\mu\nu}q^{2})\left\{\int_{0}^{1}d\alpha2\alpha(1-\alpha)ln\left[\frac{\Lambda^{2}}{-\alpha(1-\alpha)q^{2}+m^{2}}\right]-\frac{1}{3}\right\}.
e^{-\gamma\varphi(\tau,\sigma)}=e^{-\gamma\phi(\tau,\sigma)}\left(1+m^{2}\int\intdy\,d\bar{y}\,G(z,\bar{z};\,y,\bar{y})\,e^{2\gamma\phi(y)}\,e^{2\gamma\phi(\bar{y})}\right),
G_{1}^{I}(\lambda,l;V)=\sum_{m=0}^{\infty}(\tilde{A}_{1m}^{(l)}+\frac{G_{l-2}^{(l)}}{G_{l}^{(l)}}\tilde{A}_{0m}^{(l-2)})\frac{(1-V)^{m}}{m!}
\left(\Gamma^{\mu}(i\partial_{\mu}\,+\,eA_{\mu})\,-\,d\left(\begin{array}{cc}{0}&{\gamma^{\mu}\gamma^{\nu}\gamma_{5}}\\{\gamma^{\mu}\gamma^{\nu}\gamma_{5}}&{0}\\\end{array}\right)F_{\mu\nu}\,-\,m\right)\psi=0
\sim\int\frac{d^{3}p}{(2\pi)^{3}}D_{00}(p^{0}=0,\vec{p})e^{i\vec{p}\cdot\vec{R}}=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{e^{i\vec{p}\cdot\vec{R}}}{\vec{p}^{2}+m_{el}^{2}}=\frac{e^{-m_{el}R}}{4\piR}\;.
\intd\kappa\frac{e^{-i\kappa\tau}}{\frac{1}{2M}k^{2}-\kappa-i\epsilon}=2\pii\\mathrm{Res}\frac{e^{-i\kappa\tau}}{\frac{1}{2M}k^{2}-\kappa-i\epsilon}\times\mathrm{index\of\contour}=2\piie^{-i(\frac{k^{2}}{2M}-i\epsilon)\tau}\theta(\tau).
\left\{Q^{\alpha},Q^{\beta}\right\}=c\left(\Gamma^{a}{\calC}^{-1}\right)^{\alpha\beta}P_{a}+{\textstyle\frac{c_{2}}{2}}\left(\Gamma^{a_{1}a_{2}}{\calC}^{-1}\right)^{\alpha\beta}{\calZ}_{a_{1}a_{2}}^{(2)}+{\textstyle\frac{c_{5}}{5!}}\left(\Gamma^{a_{1}\cdotsa_{5}}{\calC}^{-1}\right)^{\alpha\beta}{\calZ}_{a_{1}\cdotsa_{5}}^{(5)}\,.
A_{0}=\left(\begin{array}{rrr}{-2}&{0}&{0}\\{0}&{1}&{1}\\{0}&{0}&{1}\\\end{array}\right),\;\;S=\left(\begin{array}{rrr}{-2}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{1}\\\end{array}\right),\;\;N=\left(\begin{array}{rrr}{0}&{0}&{0}\\{0}&{0}&{1}\\{0}&{0}&{0}\\\end{array}\right),\;\;
F^{(I\bar{I})}(\tau)\mathop{=}_{\epsilong^{2}\rightarrow0}^{R\rightarrow\infty}\theta(-\tau)\lambda\dot{q}_{0}^{(I)}(\tau;\tau_{I})-\theta(\tau)\lambda\dot{q}_{0}^{(\bar{I})}(\tau;\tau_{\bar{I}}),
\mathcal{H}=\intd^{\,2}\,x\,\left[\,\frac{1}{2}\,\left(\xi\,_{5}^{2}+\xi\,_{6}^{2}\right)+\frac{1}{2}\,\left(\partial_{1}\,\xi_{3}-\partial_{2}\,\xi_{2}\right)^{\,2}+\frac{1}{2}\,m^{2}\,\left(\xi\,_{2}^{2}+\xi\,_{3}^{2}\right)+m\left(\xi\,_{2}\,\xi\,_{6}-\xi\,_{3}\,\xi\,_{5}\right)\right];
U_{m}(v_{*}^{\alpha},X)(z):=\sum_{n=1}^{m}\sum_{\mu_{1},\ldots,\mu_{n}}\,\sum_{\substack{m_{1}\ge1,\ldots,m_{n}\ge1\,[.5mm]m_{1}+\cdots+m_{n}=m}}v_{\mu_{1},\ldots,\mu_{n}}^{\alpha}\prod_{j=1}^{n}\sqrt{-1}\,\partial^{m_{j}}X^{\mu_{j}}(z)\,,
ds^{2}=\alpha^{\prime}\Biggl[{\frac{U^{2}}{g_{\mathrm{YM}}{N^{1/2}}}}dx_{\parallel}^{2}+\frac{g_{\mathrm{YM}}{N^{1/2}}}{U^{2}}(dU^{2}+U^{2}d\Omega_{5}^{2})\Biggr]
\sum_{n_{1},n_{2}}(C(n_{1},n_{2})\bar{\chi}\partial_{\mu}\chiA_{\mu}(n_{1},n_{2})+c.c.)+\sum_{(n_{1},n_{2})\neq(0,0)}(mC(n_{1},n_{2})\chi\bar{\chi}\phi(n_{1},n_{2})+c.c.)
{\calL}=\frac{1}{2}\left(\partial_{\mu}\phi\partial^{\mu}\phi-\mu^{2}\phi^{2}\right)+\left(\partial_{\mu}B^{\mu}\right)^{2}+\beta\left(\varepsilon^{\mu\nu\lambda}\partial_{\nu}B_{\lambda}\right)^{2}+\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi+\alpha\phi\bar{\psi}\psi+\xi\partial_{\mu}B^{\mu}\bar{\psi}\psi,
{\frac{d\sigma}{d\varphi}}={\frac{p_{\perp}}{p}}\;|F(\varphi-\varphi_{p})|^{2}\mathrm{Tr}\;{\calR}(\vec{n}^{\prime})\;D(\varphi-\varphi_{p})\;{\calR}(\vec{n})\;D^{\dagger}(\varphi-\varphi_{p})
\Gamma_{\underline{\alpha}\underline{\beta}}^{a}=\left(\begin{array}{cc}{\gamma_{\alpha\beta}^{a}\delta_{q}^{r}}&{0}\\{0}&{\gamma^{a\alpha\beta}\delta_{q^{\prime}}^{r^{\prime}}}\\\end{array}\right)\,,\quada=0,1,...,p\,,
\begin{array}{lcl}{\lambda^{(0)}=\gamma_{1,6^{\prime}}\lambda^{(0)}\gamma_{1,6^{\prime}}^{-1}}&{,}&{\lambda^{(0)}=-\gamma_{\OmegaR,6^{\prime}}\lambda^{(0)^{T}}\gamma_{\OmegaR,6^{\prime}}^{-1}}\\{\lambda^{(3)}=e^{2\piiv_{i}}\gamma_{1,6^{\prime}}\lambda^{(3)}\gamma_{1,6^{\prime}}^{-1}}&{,}&{\lambda^{(3)}=-\gamma_{\OmegaR,6^{\prime}}\lambda^{(3)^{T}}\gamma_{\OmegaR,6^{\prime}}^{-1}}\\\end{array}
\zeta(s)=\frac{L_{x}\,L_{y}}{2\pi\,c^{2s}}\left[\left(\frac{\pi}{a}\right)^{2-2s}\frac{\zeta_{\mathrm{R}}(2s-2)}{s-1}+\frac{1}{2}\frac{\mu^{2-2s}}{s-1}\right].
{\calL}_{\rho}=-\frac{1}{4}F_{\mu\nu}^{a}F^{\mu\nua}-\frac{1}{4}(G_{\mu\nu}^{a}+f_{abc}F_{\mu\nu}^{b}\rho^{c})(G^{\mu\nua}+f_{abc}F^{\mu\nub}\rho^{c})+\frac{m}{2}\epsilon^{\alpha\beta\gamma}F_{\alpha\beta}^{a}\Phi_{\gamma}^{a}
\hat{\Omega}_{\alphaq,a}^{~~~~i}={\nabla}_{\alphaq}E_{a}^{~\underline{b}}u_{\underline{b}}^{~i}=2i\gamma_{qq^{\prime}}^{i}E_{a}^{\underline{\beta}}v_{\underline{\beta}}^{~\betaq^{\prime}}\epsilon_{\alpha\beta},
ds_{TdS}^{2}=-\left(\frac{2Gm}{r^{n-1}}-\frac{r^{2}}{l^{2}}\right)dt^{2}+\left(\frac{2Gm}{r^{n-1}}-\frac{r^{2}}{l^{2}}\right)^{-1}dr^{2}+r^{2}dx_{n}^{2},
\begin{array}{ll}{{[}W_{0}^{a},W_{0}^{b}]={f^{ab}}_{c}W_{0}^{c}}&{(a=1,\cdots,p^{2}-1)}\\{{[}W_{0}^{a},W_{k}^{b}]={f^{ab}}_{c}W_{k}^{c}}&{(k=1,2,\cdots,n-1)}\\{{[}C_{i},W_{0}^{a}]=[C_{i},W_{k}^{a}]=0}&{(i=2,3,\cdots,n)}\\\end{array}
\tilde{M}=\left(\begin{array}{cc}{-G^{-1}V\,G^{-1},}&{G^{-1}VG^{-1}B-G^{-1}U}\\{UG^{-1}-BG^{-1}VG^{-1},}&{V-UG^{-1}B+BG^{-1}VG^{-1}B-BG^{-1}U}\\\end{array}\right)\,.
\varepsilon^{rs}=\left(\begin{array}{ccc}{0}&{1}&{1}\\{1}&{0}&{1}\\{1}&{1}&{0}\\\end{array}\right)\,,\qquad\chi^{rs}=\left(\begin{array}{ccc}{0}&{1}&{-1}\\{-1}&{0}&{1}\\{1}&{-1}&{0}\\\end{array}\right)\,,\qquad\epsilon_{\alpha\beta}=\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)\,.
\langle\,\frac{\deltaS}{\delta\bar{c}_{x}^{a}}\,\bar{c}_{y}^{b}\,\rangle\,=\,\langle\,c_{y}^{b}\,\frac{\deltaS}{\deltac_{x}^{a}}\,\rangle\,=\,\delta^{ab}\,\delta_{xy}\;.
{\calM}_{b}^{vv}=\frac{ig^{4}}{2}[T^{a}T^{b}\otimesT_{a}T_{b}]\int\frac{d^{2}k}{(2\pi)^{2}}\left[\frac{H(p_{3},k)H^{*}(p,k)}{w_{k}+w_{p}}\right],
\sigma_{abs}^{\phi_{a}}=\frac{9\pi^{3}r_{1}^{6}r_{5}^{6}}{64(r_{1}^{2}+r_{5}^{2}+\sqrt{r_{1}^{4}+r_{5}^{4}-r_{1}^{2}r_{5}^{2}})^{2}}\omega(\omega^{2}+16\pi^{2}T_{L}^{2})(\omega^{2}+16\pi^{2}T_{R}^{2})\frac{e^{\frac{\omega}{T_{H}}}-1}{(e^{\frac{\omega}{T_{L}}}-1)(e^{\frac{\omega}{T_{R}}}-1)},
\begin{array}{l}{S=e^{-2\phi}=e^{-2\hat{\phi}}\sqrt{|G_{rs}|}=\sqrt{{\frac{H_{0}H_{1}}{H_{2}H_{3}}}}\;,}\\{T=\sqrt{|G_{rs}|}=\sqrt{{\frac{H_{0}H_{2}}{H_{1}H_{3}}}}\qquad,\qquadU=\sqrt{{\frac{H_{0}H_{3}}{H_{1}H_{2}}}}}\\\end{array}