harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\frac{\Gamma(\lambda)}{{4{\pi}^{\lambda+1}}{\Bigl({[x(\tau_{a})-x(\tau_{b})]}^{2}\Bigr)}^{\lambda}}}=\int_{0}^{\infty}d\bar{T}{(4\pi\bar{T})}^{-{\frac{D}{2}}}\mathrm{exp}\Biggl[-{\frac{{\Bigl(x(\tau_{a})-x(\tau_{b})\Bigr)}^{2}}{4\bar{T}}}\Biggr]\,
E_{M}\,=\,\frac{1}{2}\int_{0}^{\pi}d\sigma\,\left(\dot{{\bfx}}_{1}^{2}\,+\,{\bfx}_{1}^{2}\right)\,+\,\frac{1}{2}\int_{0}^{\pi}d\sigma\,\left(\dot{{\bfx}}_{2}^{2}\,+\,{\bfx}_{2}^{2}\,+\,{\bfx}_{2}^{2}\right).
[\delta_{h}^{(1)},\delta_{\chi}^{(1)}]_{A}^{\phantom{A}B}=\left(\begin{matrix}{0}&{h_{A}^{\phantom{A}D}\chi_{D}}\\{i(\chi^{\top}C)^{D}h_{D}^{\phantom{D}B}}&{0}\\\end{matrix}\right)\;,
{\calL}=\bar{\psi}^{(i)}\mathrm{i}\partial\!\!\!/\,\psi^{(i)}+\frac{g^{2}}{2}\left[(\bar{\psi}^{(i)}\psi^{(i)})^{2}-(\bar{\psi}^{(i)}\gamma_{5}\psi^{(i)})^{2}\right]\.
0=\sum_{n=0}^{\infty}\frac{\rho_{n}(s)}{t-n-\frac{1}{2}}-\sum_{n=0}^{\infty}\frac{r_{n}(t)}{s-n-\frac{1}{2}}\equiv\Psi_{2}(s,t),\\\\\\(s\sim-1,\t\sim-1).
Ht(\tau)=\bar{k}\sqrt{\frac{1-2\bar{k}^{2}}{1-\bar{k}^{2}}}\;\Pi\left(\frac{\bar{k}^{2}}{1-\bar{k}^{2}},\;\frac{\tau}{\sqrt{1-2\bar{k}^{2}}},\;\bar{k}\right).
H_{-1}=1+\sum_{i}{\frac{Q_{i}|\vec{z}-\vec{z}_{0}|^{4}}{[|\vec{x}-\vec{x}_{0\,i}|^{2}|\vec{z}-\vec{z}_{0}|^{2}+Q]^{4}}},\\\H_{3}={\frac{Q}{|\vec{z}-\vec{z}_{0}|^{4}}}.
\begin{array}{l}{\displaystyle\prod_{i=0}^{p-1}(A-q^{\pmi}B)=A^{p}-B^{p}~~~~(AB=BA)~,}\\{\displaystyle\prod_{i=0}^{p-1}(A+q^{\pmi}B)=A^{p}+B^{p}~~~~(AB=BA)~,}\\\end{array}
\left\langleL_{i_{1}}^{-}\ldotsL_{i_{m}}^{-},L_{j_{1}}^{+}\ldotsL_{j_{n}}^{+}\right\rangle=R_{i_{1}j_{n}}^{-1}\ldotsR_{i_{q}j_{p}}^{-1}\ldotsR_{i_{m}j_{1}}^{-1}\,
\bigg[\prod_{I=1}^{N}:e^{-i2\sqrt{\pi}\omega{(I)}U_{Ia}\Phi^{(I)}(x)}:_{M^{(I)}}\;e^{+i\frac{\theta}{N}}\;+\;\prod_{I=1}^{N}:e^{+i2\sqrt{\pi}\omega{(I)}U_{Ia}\Phi^{(I)}(x)}:_{M^{(I)}}\;e^{-i\frac{\theta}{N}}\bigg]\;.
\delta{\calL}=\frac{1}{2M}\epsilon^{\mu\nu\lambda}[(f_{\lambda}+\frac{1}{m_{+}}\epsilon_{\lambda\rho\omega}f^{\rho\omega})+(g_{\lambda}-\frac{1}{m_{-}}\epsilon_{\lambda\rho\omega}g^{\rho\omega})]\times[\epsilon_{\mu\sigma\beta}\partial_{\nu}\partial^{\sigma}\alpha^{\beta}]
\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}\gamma_{5}(\{(\eta^{2}+\Phi^{2}),\hat{F}_{\rho\sigma}\}+\hat{D}_{[\rho}\Phi\:\:\hat{D}_{\sigma]}\Phi)-(\{(\eta^{2}+\Phi^{2}),\hat{F}_{\mu\nu}\}+\hat{D}_{[\mu}\Phi\:\:\hat{D}_{\nu]}\Phi)=0.
\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{1}&{-\sqrt{2}\eta}&{\sqrt{2}\eta}\\{0}&{\sqrt{2}\eta}&{1-\eta^{2}}&{\eta^{2}}\\{0}&{\sqrt{2}\eta}&{-\eta^{2}}&{1+\eta^{2}}\\\end{matrix}\right).
{\calG}_{n}^{(0)}=\sum_{l=1}^{k-1}{\calG}_{k-l-1}^{(0)}{\calG}_{n+l-k-1}^{(0)}+\sum_{l=k+1}^{n}{\calG}_{l-k-1}^{(0)}{\calG}_{n+k-l-1}^{(0)},~~k=1,...,n,~
\Gamma_{j}=\left(\begin{array}{cccccc}{0}&{0}&{0}&{\ldots}&{0}&{1}\\{1}&{0}&{0}&{\ldots}&{0}&{0}\\{0}&{1}&{0}&{\ldots}&{0}&{0}\\{}&{}&{}&{\ldots}&{}&{}\\{0}&{0}&{0}&{\ldots}&{1}&{0}\\\end{array}\right).
V_{\{1,2\}}(r)=\frac{1}{r\,\mathrm{Im}\,\tau}\;\mathrm{Re}\left\{\tau\left(1+\bar{\tau}\right)\left(1-\frac{a_{D}^{*}\,(a+a_{D})}{|a_{D}|\,|a+a_{D}|}\right)\right\},
1=\int\,d\gamma\,\left(u_{0}-\left\langle\phi_{\mathrm{qu}}\left|\frac{\partialZ}{\partial\gamma}\right\rangle\right.\right)\delta\Big(\langle\phi_{\mathrm{qu}}|Z\rangle\Big)=\int\,d\gamma\,\left(u_{0}-\left\langle\phi_{\mathrm{qu}}\left|\frac{\partialZ}{\partial\gamma}\right\rangle\right.\right)\delta\Big(\xi_{0}u_{0}\Big)\.
d\bar{w}\,dw={\frac{[d(b+r)]^{2}}{8(b+r)}}+{\frac{[d(b-r)]^{2}}{8(b-r)}}+{\frac{b}{4(b^{2}-r^{2})}}\sum_{\nu=0}^{3}V_{\nu}^{2}-{\frac{r}{2(b^{2}-r^{2})}}(V_{0}V_{3})+d\bar{w}\,\Pi^{T}\,dw\,\,.
V_{eff}(\phi)|_{\mathrm{torus}}=-\sum_{I}(-)^{F_{I}}\,\,\frac{\Gamma(\frac{4+d}{2})}{32\pi^{\frac{12+d}{2}}}\,\,\prod_{i=1}^{d}R_{i}\,\,\sum_{\vec{n}\neq\vec{0}}\frac{e^{2\pii\sum_{i}n_{i}a_{i}^{I}(\phi)}}{\left[\sum_{i}n_{i}^{2}R_{i}^{2}\right]^{\frac{4+d}{2}}}
h_{\mu\nu}=g_{m}^{-1}\epsilon_{\mu\nu\rho\sigma}{\frac{1}{\partial^{2}}}\partial^{\rho}k^{\sigma}=g_{m}^{-1}{\frac{1}{2}}\epsilon_{\mu\nu\rho\sigma}{\frac{1}{\partial^{2}}}(\partial^{\rho}k^{\sigma}-\partial^{\sigma}k^{\rho}),
J(p_{1},p_{2})=\int\frac{d^{4}k_{1}d^{4}k_{2}}{(2\pi)^{8}}\frac{k_{1}^{2}p_{1}^{2}+k_{2}^{2}p_{2}^{2}-2k_{1}k_{2}p_{1}p_{2}}{\Omega(k,p)}
{\calN}_{\mathrm{CR}}=e^{-W}=e^{-NS_{PL}},\qquadS_{PL}=\frac{1}{96\pi}\intdx^{2}\sqrt{g^{(2)}}\,R^{(2)}\frac{1}{\nabla^{2}}R^{(2)}\:+a\intdx^{2}\sqrt{g^{(2)}}\,R^{(2)}
H^{(0)}(q,{\bfn})-m{\quad}\rightarrow{\quad}-\frac{1}{{2m}{q}^{2}}\frac{\partial}{{\partial}{q}}{q}^{2}\frac{\partial}{{\partial}{q}}+\frac{\bfL^{2}}{2mq^{2}},{\quad}{\bfP}^{(0)}(q,{\bfn})\rightarrow{\quad}-i\nabla_{\bfq}.
h_{\mu\nu}(t,x,r)=e^{i\omegat}e^{i\mux}\left[\begin{array}{ccc}{H_{tt}(r)}&{H_{tx}(r)}&{H_{tr}(r)}\\{H_{xt}(r)}&{H_{xx}(r)}&{H_{xr}(r)}\\{H_{rt}(r)}&{H_{rx}(r)}&{H_{rr}(r)}\\\end{array}\right].
\Pi_{o,np}^{(1)}=-2p^{2}\int_{0}^{1}dx\int_{0}^{x}dy\int_{0}^{y}dz\{5I_{2}(\nu)+[5p^{2}(1+y-x-z)(y-x-z)+4p^{2}+2m^{2}]I_{1}(\nu)\},
d({}^{*}\!F+\frac{\alpha}{2}A\wedgedA+\alphaA\wedge{}^{*}\!G+\alpha\Phi_{+}\,{}^{*}\!J_{m})={}^{*}\!J_{e}+\frac{\alpha}{2}(1-\omega){}^{*}\!J_{m}\wedge{}^{*}\!g\,.
Z=\frac{{\displaystyle\int}{\calD}{\bar{\psi}}{\calD}\psi\,\,\,{\displaystylee}^{iS({\bar{\psi}},\psi,A_{-},A_{+},A_{-}^{5},A_{+}^{5})}}{{\displaystyle\int}{\calD}{\bar{\psi}}{\calD}\psi\,\,\,{\displaystylee}^{iS({\bar{\psi}},\psi)}},
\tilde{\phi}_{V}^{(1)}=\left(\begin{array}{c}{x}\\{0}\\{0}\\\end{array}\right),\\tilde{\phi}_{V}^{(2)}=\left(\begin{array}{c}{0}\\{y}\\{0}\\\end{array}\right),\\tilde{\phi}_{V}^{(3)}=\left(\begin{array}{c}{0}\\{0}\\{z}\\\end{array}\right).
\nu{\frac{\theta_{ij}}{(z_{ij})^{2}}}\xi_{i}\epsilon_{j}(1-K_{ij}^{\mathrm{tot}}),\quad\nu{\frac{\theta_{ij}}{z_{ij}}}\xi_{i}\epsilon_{j}K_{ij}^{\mathrm{tot}}{\frac{\partial}{\partialz_{i}}},\quad\nu{\frac{1}{z_{ij}}}\xi_{i}\epsilon_{j}K_{ij}^{\mathrm{tot}}{\calD}_{i}^{0}\,.
\left(\frac{\partialT}{\partialt}\right)_{w,x^{i}}/\left(\frac{\partialr}{\partialt}\right)_{w,x^{i}}=-\left(\frac{\partialw}{\partialr}\right)_{T,x^{i}}/\left(\frac{\partialw}{\partialT}\right)_{r,x^{i}}=\mp\frac{\sqrt{f(r)+\dot{a}^{2}}}{f(r)\dot{a}(t)}.
\int_{0}^{2\pi}d\tau\left[\left(\frac{dx}{d\tau}\right)^{2}+\left(\frac{dy}{d\tau}\right)^{2}\right]=\int_{0}^{2\pi}d\tau\left(\frac{ds}{d\tau}\right)^{2}=\frac{\mathrm{L}^{2}}{2\pi}\mathrm{~}\tag{2.4}
S_{10}=\int\left(\frac{1}{4}\,\hat{R}-\frac{1}{2}\,\partial_{M}\hat{\phi}\,\partial^{M}\hat{\phi}-\frac{1}{12}\,e^{-2\hat{\phi}}\,\hat{H}_{MNP}\,\hat{H}^{MNP}\right)\sqrt{-\hat{g}}\,d^{10}\hat{x}.
0={\calM}^{4}-(\tilde{\gamma}\tilde{g}^{2}k_{q}+\frac1{k_{q}k_{T}})(q_{0}^{2}+\bar{q}_{0}^{2}){\calM}^{2}+\tilde{\gamma}\frac1{k_{T}}\tilde{g}^{2}(q_{0}^{2}+\bar{q}_{0}^{2})^{2}.
f_{0}^{n}=\frac{D_{0}^{n-1}}{D_{0}^{n}},\;f_{0}^{n}=\tau-\frac{\dot{D}_{0}^{n}}{D_{0}^{n}},\;f_{+}^{n}=\frac{D_{0}^{n+1}}{D_{0}^{n}}\:.
\begin{array}{cl}{}&{\left.\displaystyle\sum_{s=\pm1}e^{-s\frac{\pi\sqrt{-1}k}{r}}t_{k+s}^{k}(u_{2n-1}-u_{0})^{\varepsilon_{2n-1}}t_{k}^{k+s}(u_{2n}-u_{0})^{\varepsilon_{2n}}\right|_{u_{2n}=u_{2n-1}-1-\frac{\pi\sqrt{-1}}{\epsilon}}}\\{=}&{\varepsilon_{2n}e^{-\frac{\pi\sqrt{-1}}{2r}}\{k\}\delta_{\varepsilon_{2n-1}+\varepsilon_{2n},0}.}\\\end{array}
P(x,y)\;=\;\frac{1}{(2\pi)^{4}}\int_{-\infty}^{\infty}d\omega\int_{-\infty}^{\infty}dk\int_{0}^{\infty}\rho\:d\rho\int_{0}^{2\pi}d\varphi\;\hat{P}(\omega,k,\rho,\varphi)\;e^{i\omegat-ikr}\;.
{\frac{1}{p^{2}+\mu^{2}-i\varepsilon}}+{\frac{g^{2}}{M^{2}}}\int_{m_{0}^{2}}^{\infty}{\frac{dm^{2}}{\left(p^{2}+m^{2}-i\varepsilon\right)\left[\left(1-g^{2}{\frac{m^{2}}{M^{2}}}\right)^{2}+g^{4}\right]}}
ds^{2}=A^{2}(r)\left[-\left(1-{\frac{2m}{\rho}}\right)dt^{2}+{\frac{d\rho^{2}}{1-{\frac{2m}{\rho}}}}+\rho^{2}d\Omega_{2}^{2}\right]+B^{2}(r)dr^{2}+C^{2}(r)r^{2}d\Omega_{n-1}^{2}
h_{ij}\partial_{t}\varphi^{i}\partial_{t}\varphi^{j}={\frac{\left(1+{\frac{r^{2}}{\tilde{L}^{2}}}\right)^{2}\ell^{2}}{T_{p}^{2}V_{p}^{2}L^{4}\left({\frac{r^{p+1}}{\tilde{L}}}+{\frac{E}{T_{p}V_{p}}}\right)^{2}}}.
(G^{-1}dG)_{\mathrm{sym}}=U_{u}dq^{u}=\left(\begin{matrix}{0}&{P_{b}^{m}}&{U^{m}}&{0}\\{(P^{t})_{n}^{a}}&{0}&{0}&{V^{a}}\\{({U^{t}})_{n}}&{0}&{0}&{d\phi}\\{0}&{{V^{t}}_{b}}&{d\phi}&{0}\\\end{matrix}\right),
L_{int}=B_{k}H_{k}^{(ext)}=\frac{1}{2a}\varepsilon_{kmn}\left(F_{km}^{a}\phi^{a}-\frac{1}{ea^{2}}\varepsilon_{abc}\phi^{a}D_{k}\phi^{b}D_{m}\phi^{c}\right)H_{n}^{(ext)}.
\Pi_{VV}^{A}(p^{2})=\frac{\gamma^{2}}{16\pi^{2}}\frac{m_{V}^{4}}{v^{2}}\left(-\frac{10}{3}\frac{p^{2}}{m_{V}^{2}}+6-2\frac{m_{r}^{2}}{m_{V}^{2}}\right).
\left(\Psi\left|D^{3}\right|\Psi\right)=\left(\Psi\left|-D^{\mu}D_{\mu}\gamma^{\rho}D_{\rho}\right|\Psi\right)+\frac{1}{2}\left(\Psi\left|\gamma^{\mu\nu}F_{\mu\nu}\gamma^{\rho}D_{\rho}\right|\Psi\right).
\tilde{E}_{\mu\nu}=\Big(\frac{E}{\theta_{\mathrm{s}}E+1}\Big)_{\mu\nu}=\Big(\frac{g^{\mathrm{(s)}}(1-\theta_{\mathrm{s}}g^{\mathrm{(s)}})}{(1+\theta_{\mathrm{s}}g^{\mathrm{(s)}})(1-\theta_{\mathrm{s}}g^{\mathrm{(s)}})}\Big)_{\mu\nu}\.
\tilde{R}_{~\lambda\mu\nu}^{\sigma}(\tilde{\Gamma})\equiv\partial_{\mu}\tilde{\Gamma}_{~\lambda\nu}^{\sigma}-\partial_{\nu}\tilde{\Gamma}_{~\lambda\mu}^{\sigma}+\tilde{\Gamma}_{~\alpha\mu}^{\sigma}\tilde{\Gamma}_{~\lambda\nu}^{\alpha}-\tilde{\Gamma}_{~\alpha\nu}^{\sigma}\tilde{\Gamma}_{~\lambda\mu}^{\alpha}
Q=\intd^{2}x\left(\begin{array}{c}{\sqrt{2}(D_{0}\phi^{*}-iF^{*})\psi_{\uparrow}-i(\partial_{0}N+f_{0}-iG)\chi_{\uparrow}+\sqrt{2}(D_{-}\phi)^{*}\psi_{\downarrow}-i(\partial_{+}N+f_{+})\chi_{\downarrow}}\\{\sqrt{2}(D_{0}\phi^{*}+iF^{*})\psi_{\downarrow}-i(\partial_{0}N+f_{0}+iG)\chi_{\downarrow}-\sqrt{2}(D_{+}\phi)^{*}\psi_{\uparrow}+i(\partial_{-}N+f_{-})\chi_{\uparrow}}\\\end{array}\right)
\gamma_{n}(\tau)=-i\frac{l_{s}^{2}p^{+}}{2\mun}\left(\omega_{n}^{2}-\left(\frac{n}{l_{s}^{2}p^{+}}\right)^{2}\right)b_{n}e^{-i\nu_{n}\tau}\quad\textrm{for}\quadn\neq0
1+\Gamma=\left(\begin{matrix}{1+C}&{(1-C^{2})A^{-1}}\\{A}&{1-C}\\\end{matrix}\right)\,,\qquad1-\Gamma=\left(\begin{matrix}{1-C}&{-(1-C^{2})A^{-1}}\\{-A}&{1+C}\\\end{matrix}\right)\
(A\otimes1)\Phi_{\gamma}^{\alpha\beta}(v)=\sum_{IJij}<\stackrel{I}{e}_{i}{}^{*}\otimes\stackrel{J}{e}_{j}{}^{*},\Phi_{\gamma}^{\alpha\beta}(v)>(A\stackrel{I}{e}_{i})\otimes\stackrel{J}{e}_{j}
\left(\begin{array}{c}{Q^{N}}\\{Q^{R}}\\\end{array}\right)\to\Lambda\left(\begin{array}{c}{Q^{N}}\\{Q^{R}}\\\end{array}\right)=\left(\begin{array}{cc}{p}&{q}\\{r}&{s}\\\end{array}\right)\left(\begin{array}{c}{Q^{N}}\\{Q^{R}}\\\end{array}\right)
\frac{dg_{1}}{d\xi}+(3f-\frac{1}{f})g_{1}=-\frac{f_{1}+f_{2}}{f}\frac{dh}{d\xi}-6hf_{1}+\frac{9}{2}hf_{2}-\frac{hf_{2}}{2f^{2}}-\frac{h^{3}}{4f^{3}}-\frac{9h^{3}}{4f},
{\calH}_{0}=\eta_{0}^{2}(F_{ij}^{\alpha})^{2}+\frac{\tau_{1}^{2}}{2}(D_{i}\phi^{a})^{2}+\frac{\eta_{2}^{2}}{8}(D_{[i}\phi^{a}D_{j]}\phi^{b})^{2},
\left\langle\left[\sigma_{+}[Q^{W}]\right]^{q}\left[\sigma_{-}[Q^{W}]\right]^{n-q}\right\rangle_{Q^{\mu}}\;=\;\left\langle\left[\sigma_{+}[Q^{W}]\right]^{n-q}\left[\sigma_{-}[Q^{W}]\right]^{q}\right\rangle_{Q^{\mu}}\;.
\begin{array}{l}{\displaystyley^{0}=\frac{\xi^{n+1}}{\xi^{0}}~,~~~~~~(\xi^{0}\not=0)~,}\\{\displaystyley^{i}=\frac{\xi^{i}}{\xi^{0}}~,~~~~~~(i=1,~2,~\cdots,~n;~\xi^{0}\not=0)~,}\\\end{array}
i\int_{0}^{\infty}d\tau\int_{0}^{2\pi}d\theta\langlef|exp\{-\frac{i}{\hbar}\{\hbar\omega[\sum_{k=1}^{2}(b_{k}^{\dagger}b_{k}+\tilde{b}_{k}^{\dagger}\tilde{b}_{k})+2]-e^{2}\}\tau-i(b_{1}^{\dagger}b_{1}-b_{2}^{\dagger}b_{2}-\tilde{b}_{1}^{\dagger}\tilde{b}_{1}+\tilde{b}_{2}^{\dagger}\tilde{b}_{2})\theta\}|i\rangle
S=\frac{1}{2\pi}\int_{M}d^{2}x\sqrt{-g}\left[e^{-2\phi}\left(R+4(\nabla\phi)^{2}+4\lambda^{2}\right)-\frac{1}{2}(\nablaf)^{2}-\frac{\beta}{4}\left(f^{2}-\frac{\mu^{2}}{\beta}\right)^{2}e^{-2\phi}\right],
\tilde{V}_{\lambda}(r)=-\omega^{2}(f-1)+h\left[{\frac{3}{4r^{2}}}\left(1+{\frac{3r_{0}^{2}}{r^{2}}}\right)+{\frac{8R^{4}}{r^{2}(3r^{2}+R^{2})^{2}}}\right],
\left(\begin{matrix}{U^{1}}\\{U^{2}}\\\end{matrix}\right)\rightarrow(\Omega^{T})^{-1}\left(\begin{matrix}{U^{1}}\\{U^{2}}\\\end{matrix}\right)\;,\;\left(\begin{matrix}{C_{\mu\nu}}\\{B_{\mu\nu}}\\\end{matrix}\right)\rightarrow(\Omega^{T})^{-1}\left(\begin{matrix}{C_{\mu\nu}}\\{B_{\mu\nu}}\\\end{matrix}\right)\;.
\langle\mathrm{Tr}\,\lambda^{2}\rangle=16\pi^{2}m\langle\Phi^{2}\rangle=\pm16\pi^{2}\left[m\Lambda_{\mathrm{one-fl}}^{5}\right]^{1/2}=\pm16\pi^{2}\left[\frac{m\,e^{-8\pi^{2}/g^{2}}}{Zg^{4}}\,(M_{\mathrm{PV}})^{5}\right]^{1/2}\,.
\prod_{u=1}^{N}dv_{u}^{l}dv_{u}^{l*}=\prod_{k=1}^{l-1}d\xi_{k}^{l}d\xi_{k}^{l*}\cdot2^{N-l-1}(\xi_{l}^{l})^{2N-2l+1}\,d\xi_{l}^{l}\,d^{2N-2l+1}\hat{\Omega}_{l}\.
S_{6}\sim\intd^{6}x\sqrt{G_{6}}\left\{R_{6}-\left(\partial\phi\right)^{2}-\frac{e^{-2\phi}}{12}H^{2}-\frac{\alpha^{\prime}}{8}\sum_{\alpha}\left(v_{\alpha}e^{-\phi}+\tilde{v}_{\alpha}e^{\phi}\right)trF_{\alpha}^{2}\right\},
g_{ij,d}:=g_{ij,d;ji,d}=\frac{-d^{3}(\lambda_{i}-\lambda_{j})^{2}\prod_{k=1}^{N}\prod_{a=1}^{d-1}(a\lambda_{i}+(d-a)\lambda_{j}-d\lambda_{k})}{\prod_{a=1}^{Nd-1}(a\lambda_{i}+(Nd-a)\lambda_{j})}
2{\calS}^{AA^{\prime}}=[2a_{(i)(j)}-\delta_{(i)(j)}a_{(k)(k)}]{\bar{\pi}}_{(i)}^{A}\pi_{(j)}^{A^{\prime}}=2a_{(i)(j)}{\bar{\pi}}_{(i)}^{A}\pi_{(j)}^{A^{\prime}}-a_{(i)(i)}{\bar{\pi}}_{(j)}^{A}\pi_{(j)}^{A^{\prime}}.
i\sqrt{2}\partial_{+}^{x}G(x;y)+g^{2}\intdw^{-}G(x^{+},x^{-};x^{+},w^{-})|x^{-}-w^{-}|G(x^{+},w^{-};y^{+},y^{-})=i\delta(x^{-}-y^{-}).
\Lambda_{ijlm}(\hat{k})=\delta_{il}\delta_{jm}-2\hat{k}_{j}\hat{k}_{m}\delta_{il}+\frac{7}{8}\hat{k}_{i}\hat{k}_{j}\hat{k}_{l}\hat{k}_{m}-\frac{1}{8}\delta_{ij}\delta_{lm}+\frac{1}{8}\hat{k}_{l}\hat{k}_{m}\delta_{ij}+\frac{1}{8}\hat{k}_{i}\hat{k}_{j}\delta_{lm}
U_{[4]\times[N]}^{\prime}=-\left(\!\begin{array}{cccc}{U_{11}^{\prime}}&{U_{12}^{\prime}}&{\cdots}&{U_{1N}^{\prime}}\\{U_{21}^{\prime}}&{U_{22}^{\prime}}&{\cdots}&{U_{2N}^{\prime}}\\{U_{31}^{\prime}}&{U_{32}^{\prime}}&{\cdots}&{U_{3N}^{\prime}}\\{U_{41}^{\prime}}&{U_{42}^{\prime}}&{\cdots}&{U_{4N}^{\prime}}\\\end{array}\!\right),
\frac{\ddot{b}}{b}=\frac{\beta^{\prime\prime}}{\beta},~~~~\Biggl[\frac{d}{dt}\left(\frac{\dot{c}}{c}\right)+\frac{\dot{b}}{b}\frac{\dot{c}}{c}\Biggr]=\Biggl[\frac{d}{dx}\left(\frac{\kappa^{\prime}}{\kappa}\right)+\frac{\beta^{\prime}}{\beta}\frac{\kappa^{\prime}}{\kappa}\Biggr].
(x^{2}z_{1}^{-1})^{2n}k_{\varepsilon_{1}}^{\prime}(\zeta_{1})\overline{{G}}_{b}^{(n)}(\zeta_{1}^{-1},\tilde{\zeta})^{\varepsilon_{1}\,\tilde{\varepsilon}}=\overline{{G}}_{b}^{(n)}(x^{-2}\zeta_{1},\tilde{\zeta})^{\varepsilon_{1}\,\tilde{\varepsilon}},
\stackrel{(1)}{\cal{L}}={\frac{1}{6}}f_{bc}^{a}\^{*}C_{a}^{*}C^{b}C^{c}+f_{bc}^{a}3\^{*}A_{a}^{*}\wedgeA^{b}C^{c}+f_{abc}A^{a}\wedgeA^{b}\wedgeA^{c}\.
\frac{d\Lambda(t)}{dt}=\frac{d}{dt}\sum_{j=0}^{\infty}\frac{\left(\int_{0}^{t}\Omega(\eta)d\eta\right)^{j}}{j!}=\Omega(t)\sum_{j=0}^{\infty}\frac{\left(\int_{0}^{t}\Omega(\eta)d\eta\right)^{j}}{j!}
(\eta^{IJ})=\left(\begin{array}{cccccc}{1}&{}&{}&{}&{}&{}\\{}&{-1}&{}&{}&{}&{}\\{}&{}&{-1}&{}&{}&{}\\{}&{}&{}&{-1}&{}&{}\\{}&{}&{}&{}&{-1}&{}\\{}&{}&{}&{}&{}&{-{R}^{-2}}\\\end{array}\right)\,,
h_{nMN}=C_{\eta}^{n}\frac{g_{0MN}}{||g_{0MN}||}=\frac{1}{([n]_{q^{2}}!(1-q^{2})^{n})^{\frac{1}{2}}}C_{\eta}^{n}\frac{g_{nMN}}{||g_{0MN}||}
{G_{\hat{\mathcal{A}}\hat{T}}^{\mu}}=-g\alpha^{\prime}\left[{\frac{1}{Q^{2}}}\partial\cdot\hat{\mathcal{A}}\,{\partial^{\mu}}+\left(2+{\frac{1}{Q^{2}}}\right){\partial^{\mu}}\partial\cdot\hat{\mathcal{A}}\right]{{(\alpha^{\prime}\Box+1)}^{-1}}.
E_{m\bar{a}\bar{b}}\equivG_{m\bar{a}\bar{b}}-\mu\epsilon^{mnp}\partial_{n}h_{p\bar{a}\bar{b}}+\frac{\mu}{2}[\epsilon^{pna}\partial_{n}v_{b}+\epsilon^{pnb}\partial_{n}v_{a}-\frac{2}{3}\eta_{ab}\epsilon^{pmn}\partial_{m}v_{n}]=0,
\bar{M}=\sum_{l}(l+1)\partial^{(l)}\Phi^{A}\frac{\partial}{\partial(\partial^{(l)}\Phi^{A})}+\sum_{l}(l+2)\partial^{(l)}\Phi_{A}^{*}\frac{\partial}{\partial(\partial^{(l)}\Phi_{A}^{*})}.
\displaystyle{M=\left(\begin{array}{ccc}{w+\textstyle{\frac{1}{2}}\lambda+i\textstyle{\frac{1}{2}}\psi}&{-i\tilde{\mathrm{a}}}&{2\bar{\varepsilon}_{b}}\\{-i\mathrm{b}}&{\tilde{w}-{\textstyle\frac{1}{2}}\lambda+i\textstyle{\frac{1}{2}}\psi}&{2\rho_{b}}\\{2\bar{\rho}^{a}}&{2\varepsilon^{a}}&{t_{~b}^{a}+i\frac{2}{{\calN}}\psi\delta_{~b}^{a}}\\\end{array}\right)}\,,
\left[q^{\frac{L}{24}}\prod_{n=1}{(1-q^{n})}^{L}\,\cdot\,q^{\frac{L}{24}}\prod_{n=1}{(1+q^{n})}^{L}\right]^{-1}=\left[q^{\frac{2L}{24}}\prod_{n=1}{(1-q^{2n})}^{L}\right]^{-1}=\eta(2\tau)^{-L}
\vec{\lambda}_{j}^{(2N)}=\sum_{k=1}^{j}\vec{e}_{k},\quadj\leqN-2,\quad\vec{\lambda}_{N-1}^{(2N)}={\frac{1}{2}}\left(\sum_{1}^{N-1}\vec{e}_{k}-\vec{e}_{N}\right),\quad\vec{\lambda}_{N}^{(2N)}=\sum_{1}^{N}{\frac{\vec{e}_{k}}{2}}.
{H}_{int}^{1}=\bar{\psi}e\left(\gamma^{-}\bar{B}_{-}+\gamma^{i}\bar{B}_{i}\right)\ast\left[1+\frac{e}{2}\gamma^{+}\frac{1}{i\partial_{-}-e\bar{B}_{-}}\left(\gamma^{-}\bar{B}_{-}+\gamma^{j}\bar{B}_{j}\right)\right]\ast\psi.
u_{m}^{(i)}=\frac18(v_{\alphaA}^{+}\tilde{\sigma}_{m}^{\alpha\beta}v_{\beta\dot{A}}^{-})\gamma_{A\dot{A}}^{i}=-\frac18(v_{A}^{\alpha-}\sigma_{m\alpha\beta}v_{\dot{A}}^{\beta+})\gamma_{A\dot{A}}^{i}.
<\lambda,\mu\mid{\hat{\gamma}}_{(\tau,\nu,\alpha)}^{r}\mid\tau,\nu>={}^{*}\delta_{\lambda\tau}{}^{*}\delta_{\mu\nu}e_{(\tau,\nu,\alpha)}^{r},\quad<\tau,\nu\mid{\hat{\gamma}}_{r}^{(\tau,\nu,\alpha)}\mid\lambda,\mu>={}^{*}\delta_{\lambda\tau}{}^{*}\delta_{\mu\nu}e_{r}^{(\tau,\nu,\alpha)}.
\psi_{n,l_{max},m}(r\theta\phi)=\sqrt{{\frac{\omega^{n+{\frac{3}{2}}}n!}{\sqrt{\pi}(2n+1)!}}}\,2^{n+1}\;r^{n}e^{-{\frac{\omega}{2}}r^{2}}\,Y_{l_{max},m}(\theta\phi).
\Sigma_{2}=-g^{4}N^{2}\frac{\Gamma^{2}(\omega-1)}{32\pi^{2\omega}(2\omega-3)(2-\omega)}\intd\tau_{2}\,d\tau_{3}\,\frac{1}{\bigl[(\tau_{3}+\tau_{2})^{2}+L^{2}\bigr]^{2\omega-3}},
\theta_{\alpha\beta}={\frac{1}{2\kappa^{2}}}{\partial^{i}}\left[-{\partial_{i}}{h_{\alpha\beta}}+{\eta_{\alpha\beta}}\left({\partial_{i}}{h_{\gamma}^{\gamma}}+{\partial_{i}}{h_{j}^{j}}-{\partial_{j}}{h_{i}^{j}}\right)\right],
{\frac{\ddot{a}}{a}}={\frac{(p-1)(p-3)}{8}}\tilde{P}_{\phi}^{-2{\frac{p-1}{p-3}}}{\frac{\tilde{E}^{2}}{L_{p}^{2}}}\left({\frac{a}{L_{p}}}\right)^{2{\frac{p+1}{p-3}}},
<e^{ip\cdotX}>_{osc}\;\rightarrow\left[1-e^{-4\pi\tau}\right]^{-\frac{p_{B}^{(2)2}}{2\pi}}\left[1-e^{-4\pil^{\prime}}\right]^{-\frac{p_{B}^{(1)2}}{2\pi}}\;,
{\frac{\gamma}{2}}{\calL}={\frac{1}{(1-\vec{u}.\vec{v})}}\bigl(\partial_{+}\vec{u}.\partial_{-}\vec{v}+\partial_{-}\vec{u}.\partial_{+}\vec{v}\bigr)-\partial_{+}\phi\partial_{-}\phi-e^{2\phi}(1-2\vec{u}.\vec{v}),
{\calL}=-{\frac{1}{4g_{\mathrm{YM}}^{2}}}\mathrm{Tr}\Bigl[F^{\mu\nu}F_{\mu\nu}+2D_{\mu}\phi^{I}D^{\mu}\phi^{I}-2\sum_{I<J}[\phi^{I},\phi^{J}]^{2}\Bigr]
\Gamma_{A=0}=2V\int_{0}^{\infty}\frac{dt}{2t}\left[\mathrm{Tr}_{\mathrm{NS}}\left(\frac{1+(-1)^{\calF}}{2}\,q^{2L_{0}-1}\right)-\mathrm{Tr}_{\mathrm{R}}\left(\frac{1+(-1)^{\calF}}{2}\,q^{2L_{0}}\right)\right]
ds^{2}=\frac{d\xi^{0}d\xi^{0}}{1+\xi^{0}\xi^{0}}-(1+\xi^{0}\xi^{0})\left[(1+\rho^{2})^{-2}d\rho^{2}+(1+\rho^{2})^{-1}\rho^{2}d{\Omega_{(n-1)}^{2}}\right]~,
\left[\begin{array}{cc}{0}&{i\frac{m}{l}A_{l}^{\dagger}}\\{-i\frac{m}{l}A_{l}}&{0}\\\end{array}\right]\left[\begin{array}{c}{\xi}\\{\rho}\\\end{array}\right]=\omega\left[\begin{array}{c}{\xi}\\{\rho}\\\end{array}\right]
V_{\calN}^{-}(q)=\frac{C_{1}^{2}}{2}e^{2E_{0}q}+\frac12C_{1}\{2C_{3}-E_{0}(2{\calN}-1)\}e^{E_{0}q}+\frac12C_{2}\left(2C_{3}+E_{0}\right)e^{-E_{0}q}+\frac{C_{2}^{2}}{2}e^{-2E_{0}q}.
\int_{0}^{T}{dt}\omega_{\mu\nu}[\dot{x}]\to\int_{0}^{T}{dt\omega_{\mu\nu}}[\dot{x}^{cl}]+\frac{T}{2}\int_{0}^{T}{dt[\ddot{x}_{\mu}}(t)\dot{x}_{\nu}(t)-\ddot{x}_{\nu}(t)\dot{x}_{\mu}(t)],
\left(D^{k}\right)_{a}^{b}=\left(D^{k-1}\right)_{a}^{c}D_{c}^{b}=D_{a}^{c}\left(D^{k-1}\right)_{c}^{b},\qquad\left(D^{0}\right)_{a}^{b}=\delta_{a}^{b}
{\calL}_{MCS}=-\frac{1}{2}F_{\mu\nu}^{\dagger}F^{\mu\nu}+\frac{M}{2}\epsilon^{\mu\nu\rho}[\phi_{\mu}^{\dagger}\partial_{\nu}\phi_{\rho}+\phi_{\mu}(\partial_{\nu}\phi_{\rho})^{\dagger}].
\begin{aligned}{d\mathbf{g}_{H}\mathbf{g}_{H}^{-1}}&{{}=de^{\frac{1}{2}\phi^{i}H_{i}}e^{-\frac{1}{2}\phi^{i}H_{i}}}\\{}&{{}=\frac{1}{2}d\phi^{i}H_{i}}\\\end{aligned}
\{A,B\}^{\ast}=\{A,B\}-\int{d^{3}\sigma\{A,\zeta^{\mu}(\tau,\vec{\sigma})\}\{{\calH}_{\mu}^{\ast}(\tau,\vec{\sigma}),B\}}+\int{d^{3}\sigma\{A,{\calH}_{\mu}^{\ast}(\tau,\vec{\sigma})\}\{\zeta^{\mu}(\tau,\vec{\sigma}),B\}}.
I_{1}=\bar{M}^{\bar{g}}M_{\bar{g}}\,,\;\,I_{2}=\frac12\bar{B}^{\bar{g}f}B_{\bar{g}f}\,,\;\,I_{3}=\bar{M}_{\bar{g}}\bar{B}^{\bar{g}f}B_{\bar{k}f}M^{\bar{k}}\,,\;\,I_{4}=\frac12\bar{B}^{\bar{k}h}\bar{B}^{\bar{g}f}B_{\bar{g}h}B_{\bar{k}f}\,.
\hat{F}_{\alpha\beta}(q)=D_{\alpha}\left(\hat{A}_{\beta}^{I}+\hat{A}_{\beta}^{\pm}\right)-D_{\beta}\left(\hat{A}_{\alpha}^{I}+\hat{A}_{\alpha}^{\pm}\right)+i\,\left[\hat{A}_{\alpha}^{I}+\hat{A}_{\alpha}^{\pm},\,\hat{A}_{\beta}^{I}+\hat{A}_{\beta}^{\pm}\right],
\frac{d}{dt}\langleF(x)\rangle_{t}\sim\frac{1}{Z_{t}}\int_{D_{t}}dx\,\frac{\partial}{\partialx}e^{S(x)}F^{\prime}(x)=\left.\frac{1}{Z_{t}}e^{S(x)}F^{\prime}(x)\right|_{x_{-}(t)}^{x_{+}(t)}.
\hat{U}-h{\frac{\partial\hat{U}}{\partialh}}-\bar{h}{\frac{\partial\hat{U}}{\partial\bar{h}}}={\frac{1}{4}}{h^{1/2}<H>^{1/2}}+{\frac{1}{4}}{\bar{h}^{1/2}<\bar{H}>^{1/2}},
H^{2}=\left({\frac{\dot{a}}{a}}\right)^{2}={\frac{1}{3}}\rho-{\frac{k}{a^{2}}}={\frac{1}{3}}\left({\frac{1}{2}}\dot{\phi}^{2}+V(\phi)+\rho_{\alpha}\right)-{\frac{k}{a^{2}}}\.
ds_{\mathrm{induced}}^{2}=\left.-e^{2T(r,z)}dt^{2}+e^{2\left(A(r,z)+B(r,z)\right)}dr^{2}+e^{2\left(A(r,z)-B(r,z)\right)}r^{2}d\Omega_{2}^{2}\,\right|_{z=1}
\langle\mathrm{Tr}Z^{J}~\mathrm{Tr}\bar{Z}^{J}\rangle=JN^{J}\Bigg\{1+\sum_{h=1}^{\infty}{\frac{a_{h,4h}}{(4h)!}}{\frac{J^{4h}}{N^{2h}}}+\ldots\Bigg\}
{\calL}_{m}[B,\Lambda]:={\calL}_{m}[B^{\Lambda}]={\frac{1}{4\kappa}}(B_{\mu\nu}+\partial_{\mu}\Lambda_{\nu}-\partial_{\nu}\Lambda_{\mu})^{2}+{\frac{1}{12\eta^{2}}}(H_{\mu\nu\lambda})^{2}+{\frac{1}{4\gamma^{4}}}(\partial^{\lambda}H_{\lambda\mu\nu})^{2},
\left[{\frac{\hbar^{2}}{2}}\left(\int_{0}^{1}ds\,\sqrt{x^{\prime\,2}}\right)^{-1}\int_{0}^{1}ds\,\sqrt{x^{\prime\,2}}\,{\frac{\delta^{2}}{\delta\sigma^{\mu\nu}(s)\delta\sigma_{\mu\nu}(s)}}-m^{4}\right]\Psi[C]=0\,\qquadm^{2}\equiv1/2\pi\alpha^{\prime}\.
\mathrm{S}_{\mathrm{int}}=ie\int_{\Gamma^{*}}d\tau\omega_{\tau}^{M}A_{M}+i\mu\int_{\Gamma^{*}}d\tau\left(\lambda_{2}\dot{\theta}^{\alpha}{}_{i}D_{\alpha}{}^{i}W+\bar{\lambda}_{2}\dot{\bar{\theta}}_{\dot{\alpha}i}\bar{D}^{\dot{\alpha}i}\bar{W}\right).
H_{g-Q}=-\intd{\bfr}d{\bfx}\left[{\sfK}_{g}^{b}({\bfr})\left(4{\pi}|({\bfr}-{\bfx})|\right)^{-1}{\calK}_{0}^{b}({\bfx})+{\calK}_{0}^{b}({\bfx})\left(4{\pi}|({\bfr}-{\bfx})|\right)^{-1}{\sfK}_{g}^{b}({\bfr})\right]\,.
\vec{\alpha}^{(0)}=\sqrt{\frac{2}{V}}\left(\begin{matrix}{1}\\{0}\\{0}\\\end{matrix}\right),\;\vec{\alpha}^{(1)}=\sqrt{\frac{2}{\tau}}\left(\begin{matrix}{0}\\{1}\\{0}\\\end{matrix}\right),\;\vec{\alpha}^{(2)}=\sqrt{2\tau}\left(\begin{matrix}{0}\\{0}\\{1}\\\end{matrix}\right).
\begin{array}{rcl}{g_{1a}(t_{f},t^{\prime})}&{=}&{g_{2a}(t_{f},t^{\prime})}\\{g_{2a}(t_{0},t^{\prime})}&{=}&{g_{3a}(0,t^{\prime})}\\{g_{1a}(t_{0},t^{\prime})}&{=}&{e^{\beta\mu}g_{3a}(0,t^{\prime})}\\\end{array}\quad\begin{array}{rcl}{g_{a1}(t,t_{f})}&{=}&{g_{a2}(t,t_{f})}\\{g_{a2}(t,t_{0})}&{=}&{g_{a3}(t,0)}\\{g_{a1}(t,t_{0})}&{=}&{e^{-\beta\mu}g_{a3}(t,0)}\\\end{array}
\vec{\Pi}(t)=\left(\begin{array}{c}{{\frac{1}{6}}\kappa_{abc}t_{a}t_{b}t_{c}+\beta_{a}t_{a}+\gamma}\\{-{\frac{1}{2}}\kappa_{abc}t_{b}t_{c}+\alpha_{ab}t_{b}+\beta_{a}}\\{t_{a}}\\{1}\\\end{array}\right)
\begin{array}{ll}{R_{MN}=}&{\partial_{M}\phi\partial_{N}\phi+\frac{1}{4}e^{-2\phi}\left(H_{MN}^{2}-\frac{1}{6}H^{2}G_{MN}\right)}\\{}&{-\frac{\alpha^{\prime}}{4}\sum_{\alpha}\left(v_{\alpha}e^{-\phi}+\tilde{v}_{\alpha}e^{\phi}\right)tr\left(F_{\alpha\,MN}^{2}-\frac{1}{8}F_{\alpha}^{2}G_{MN}\right),}\\\end{array}
\Phi_{\alpha_{0}}^{*(1)}=N_{\alpha_{1}\alpha_{0}}n_{1}^{\alpha_{1}}+\cdots,\;\varphi_{\alpha_{1}}^{*(1)}=\left(-\right)^{\epsilon_{\alpha_{0}}\left(\epsilon\left(f_{\alpha_{1}}\right)+1\right)}N_{\alpha_{1}\alpha_{0}}\nu_{1}^{\alpha_{0}}+\cdots,
\left[\frac{d^{2}}{dt^{2}}+3H\frac{d}{dt}+\left[\frac{k^{2}}{a_{o}^{2}}+m^{2}(T_{o})+\frac{\lambda_{R}T_{c}^{2}}{24}\right]e^{-2Ht}-\frac{\lambda_{R}T_{c}^{2}}{24}\right]\varphi_{k}^{H}(t)=0
A_{\alpha}^{\,\,\,\,i}=\xi^{\beta}f_{\beta\alpha}^{\,\,\,\,\,\,\,i},\,\,\,\,\,\,\,\,\,\,\,\,A_{i}^{\,\,\alpha}=\xi^{\beta}f_{\betai}^{\,\,\,\,\,\alpha}.