harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{0}^{\infty}t^{s-1}e^{-{\frac{a}{t}}}e^{-\mut}dt=2({\frac{a}{\mu}})^{\frac{s}{2}}K_{s}((4a\mu)^{\frac{1}{2}})=\Gamma(s)\mu^{-s}+\Gamma(-s)({a})^{s}+{\calO}(\mu^{1-s},\mu)
\eta_{0}=\eta_{0}(p_{0},\vec{p}\,{}^{2}),\quad\eta_{i}=p_{i}\,\eta(p_{0},\vec{p}\,{}^{2}),\quad\eta_{4}=\sqrt{\kappa^{2}+\eta_{0}^{2}-\eta_{1}^{2}-\eta_{2}^{2}-\eta_{3}^{2}},
\begin{array}{rcl}{}&{}&{T_{i}(f_{i})=-t_{i}^{-1}e_{i}~,}\\{}&{}&{T_{i}(f_{j})=\displaystyle\sum_{m=0}^{-a_{ij}}(-1)^{m-a_{ij}}q_{i}^{m}\frac{f_{i}^{m}}{[m]_{q_{i}}!}f_{j}\frac{f_{i}^{-a_{ij}-m}}{[-a_{ij}-m]_{q_{i}}!}~,~~~\mathrm{if~}i\not=j~;}\\\end{array}
A(x,y)\;=\;-\sum_{n=0}^{\infty}\frac{(-q^{2})^{n}}{n!}\int_{-\frac{1}{2}}^{\frac{1}{2}}(\frac{1}{4}-\tau^{2})^{n}\:e^{-i\tauq(y-x)}\:d\tau\;T_{a}^{(n+1)}(x,y)\;e^{-i\frac{q}{2}\:(x+y)}\;\;\;.
\frac{\partialL}{\partial\tau}=A_{a\mu}\dot{x}^{\mu}\dot{I}^{a}+\phi_{a}\dot{I}^{a}=f^{abc}A_{a\mu}A_{b\nu}\dot{x}^{\mu}\dot{x}^{\nu}I^{c}+\phi_{a}\dot{I}^{a}=\phi_{a}\dot{I}^{a}.
\begin{array}{c}{{\bfJ}={\bfp}\times{\bfx}+{\bfp}_{+}\times{\bfe}_{+}+{\bfp}_{1}\times{\bfe}_{1}=}\\{={\bfp}\times{\bfx}+c(2\epsilon-q){\bfe}_{+}-2c{\bfe}_{-}.}\\\end{array}
G_{a}^{(1)}=\bigl\{T_{a},\,A^{(1)}\bigr\}_{(q,p)}+\bigl\{T_{a}^{(1)},\,A\bigr\}_{(q,p)}+\bigl\{T_{a}^{(2)},\,A^{(1)}\bigr\}_{(\eta)}\,.
[\phi(y)\,,\,\phi(y^{\prime})]=-{\frac{i\pi\beta^{2}}{2}}s\left(\frac{y-y^{\prime}}{R}\right)\quad,\quad[\bar{\phi}(\bar{y})\,,\,\bar{\phi}(\bar{y}^{\prime})]={\frac{i\pi\beta^{2}}{2}}s\left(\frac{\bar{y}-\bar{y}^{\prime}}{R}\right)\quad,
=\{\chi^{\mu\nu}\}+\left(\begin{array}{cccc}{((p^{2}-q^{2})a+2pqb)}&{(pa+qb)}&{(pb-qa)}&{((p^{2}-q^{2})a+2pqb)}\\{(pa+qb)}&{0}&{0}&{(pa+qb)}\\{(pb-qa)}&{0}&{0}&{(pb-qa)}\\{((p^{2}-q^{2})a+2pqb)}&{(pa+qb)}&{(pb-qa)}&{((p^{2}-q^{2})a+2pqb)}\\\end{array}\right).
\left(\begin{array}{cc}{M^{-1}}&{0}\\{0}&{M^{\dagger}}\\\end{array}\right)\left(\begin{array}{c}{\phi_{al}}\\{\bar{\chi}_{al}}\\\end{array}\right)=\left(\begin{array}{c}{\phi_{Mink}}\\{\bar{\chi}_{Mink}}\\\end{array}\right),
H_{|2}=\frac{1}{4l_{p}^{2}}\int_{\Sigma}d^{3}x\sqrt{g}G^{ijkl}\left[K^{-1\bot}\left(x,x\right)_{ijkl}+\left(\triangle_{2}\right)_{j}^{a}K^{\bot}\left(x,x\right)_{iakl}\right]
H_{0}^{\times}|\hat{\bboxp}j_{3}(wj)\rangle=E|\hat{\bboxp}j_{3}(wj)\rangle,\,\,\,\,\,\,\,E=w\sqrt{1+\hat{\bbox{p}}^{2}}=\sqrt{{\mathsfs}(1+\hat{\bboxp}^{2})}.
H=\omega_{1}\mathrm{a_{1}^{\dagger}a_{1}+\omega_{2}\mathrm{a_{2}^{\dagger}a_{2}+\mathrm{g(a_{1}^{\dagger})^{s}a_{2}^{r}+\bar{\mathrm{g}}a_{1}^{s}(a_{2}^{\dagger})^{r}}}}
\phi_{\alpha_{1}}(t)=\sqrt{\frac{\pi}{4H_{0}}}e^{-\frac{3}{2}H_{0}t}H_{\nu}^{(1)}(z),\phi_{\alpha_{2}}(t)=\sqrt{\frac{\pi}{4H_{0}}}e^{-\frac{3}{2}H_{0}t}H_{\nu}^{(2)}(z),
x=\frac{(z_{1}-z_{2})(z_{3}-z_{4})}{(z_{1}-z_{4})(z_{3}-z_{2})},{}~~~~~~I_{1}=\delta_{\epsilon_{1}\epsilon_{2}}\delta_{\epsilon_{3}\epsilon_{4}},{}~~~~~~I_{2}=\delta_{\epsilon_{1}\epsilon_{4}}\delta_{\epsilon_{2}\epsilon_{3}},{}~~~~~~\bar{m}_{1}+\bar{m}_{2}+\bar{m}_{3}+\bar{m}_{4}=0.
V_{i}\cdotV_{j}={\vec{U}}_{i}\cdot{\vec{U}}_{j}-\sum_{r}V_{i}^{r}V_{j}^{r}+\sum_{\ell:{\mathrm{right}}}T_{i}^{\ell}T_{j}^{\ell}-\sum_{\ell:{\mathrm{left}}}T_{i}^{\ell}T_{j}^{\ell}~.
\Phi=\left(\begin{matrix}{{\bf20}_{0}^{\prime}}&{{\bf10}_{-2}}&{\overline{{\bf10}}_{2}}\\{\overline{{\bf10}}_{2}}&{{\bf20}_{0}^{\prime}}&{{\bf1}_{4}}\\{{\bf10}_{-2}}&{{\bf1}_{-4}}&{{\bf20}_{0}^{\prime}}\\\end{matrix}\right)
J_{\mu\nu}[\xi|s]=\tilde{g}\epsilon_{\mu\nu\rho\sigma}[\bar{\tilde{\psi}}(\xi(s))\Omega_{\xi}(s,0)\gamma^{\rho}\tau_{i}\Omega_{\xi}^{-1}(s,0)\tilde{\psi}(\xi(s))]\tau^{i}\dot{\xi}^{\sigma}(s).
\frac{r!}{N^{2}}ru^{-2}[(\lambda_{\bf84}\phi+\mu_{\bf84})P_{Y_{(2r,2)}}+(\lambda_{\bf105}\phi+\mu_{\bf105})P_{Y_{(2r+2)}}+(\lambda_{\bf175}\phi+\mu_{\bf175})P_{Y_{(2r+1,1)}}]
\hat{e}_{\hat{\mu}}{}^{\hat{a}}=\left(\begin{array}{cr}{e^{-\frac{1}{6}\varphi}e_{\mu}{}^{a}}&{e^{\frac{1}{3}\varphi}L_{m}{}^{i}A^{m}{}_{\mu}}\\{}&{}\\{0}&{e^{\frac{1}{3}\varphi}L_{n}{}^{i}\,U^{n}{}_{m}}\\\end{array}\right)\,
\tilde{\varphi}_{\omega_{0},u_{0}}^{in}(v,p)=\int\!d\omega\e^{i\omegau_{0}}{\frac{e^{-(\omega-\omega_{0})^{2}/2\sigma^{2}}}{({2\pi})^{1/4}\sigma^{1/2}}}e^{-i\omegav}{\tilde{\psi}}_{\omega}^{in}(p)
\left(P_{(b)a}^{i},b_{i}^{a}\right),\;\left(P_{(b^{1})a}^{i},b_{i}^{(1)a}\right),\;\left(P_{(\bar{\eta})a}^{i},\bar{\eta}_{i}^{a}\right),\;\left(P_{(\bar{\eta}^{1})a}^{i},\bar{\eta}_{i}^{(1)a}\right),
\intd^{4}v\,V_{\rho}(y,v,0)=\frac{ig}{16\pi^{4}}\,(f^{2}-\frac{1}{2}g^{2})\,\gamma_{5}\gamma_{\sigma}\left(\frac{y_{\sigma}y_{\rho}}{y^{6}}-\frac{\delta_{\sigma\rho}}{4y^{4}}\right)\,.
\alpha_{0}=\frac13F^{\mu\nu\rho}\left(f_{abc}A_{\mu}^{a}A_{\nu}^{b}A_{\rho}^{c}+A_{\mu}^{a}F_{a\nu\rho}+A_{\rho}^{a}F_{a\mu\nu}+A_{\nu}^{a}F_{a\rho\mu}\right),
S_{0}^{\mathrm{A}}={\frac{\kappa}{4}}\intd^{3}x\,\epsilon^{\mu\nu\rho}\,A_{\mu}\,F_{\nu\rho}={\frac{\kappa}{2}}\intd^{3}x\,\epsilon^{\mu\nu\rho}\,A_{\mu}\,\partial_{\nu}\,A_{\rho}\,,
ds_{B}^{2}=g_{z{\bar{z}}}dzd{\bar{z}}+g_{z^{\prime}{\bar{z}^{\prime}}}dz^{\prime}d{\bar{z}^{\prime}}+g_{z{\bar{z}^{\prime}}}dzd{\bar{z}^{\prime}}+g_{z^{\prime}{\bar{z}}}dz^{\prime}d{\bar{z}}
N_{21}(x,y,a)=-\frac{1}{8ay}Re\left\{\sum_{k=0}^{\infty}\left(\frac{2\pi}{a}\right)^{k}\frac{(z-ia)^{k}}{k!}\zeta(-k)-\left(\frac{a}{2\pi}\right)(z-ia)^{-1}\right\}.
\left(\varepsilon_{\bar{f}}^{f}\right)_{\alpha}\to-\frac{i}{\sqrt{2}}\left[\delta_{1}^{f}\left(\delta_{\alpha}^{1}-i\delta_{\alpha}^{2}\right)\bar{\varepsilon}_{\bar{f}}-\delta_{2}^{f}\left(\delta_{\alpha}^{1}+i\delta_{\alpha}^{2}\right)\varepsilon_{\bar{f}}\right]\;.
\left(\begin{array}{cc}{0}&{-t\sum_{j}\mathrm{\Large~e^{i{\bf~k~\cdot~u}_{j}}~}}\\{-t\sum_{j}\mathrm{\Large~e^{i{\bf~k~\cdot~v}_{j}}~}}&{0}\\\end{array}\right)\left(\begin{array}{c}{c_{\bullet}}\\{c_{\circ}}\\\end{array}\right)=E({\bfk})\left(\begin{array}{c}{c_{\bullet}}\\{c_{\circ}}\\\end{array}\right)
N_{K}^{IJ}\Psi_{IJ}^{L}\Phi_{K}^{IJ}=\;N_{K}^{IJ}\;\;\delta_{K}^{L}\;id_{\stackrel{K}{V}}\;\;\;,\;\;\;\sum_{{K}}\Phi_{K}^{IJ}\Psi_{IJ}^{K}=id_{\stackrel{I}{V}\otimes\stackrel{J}{V}}.
{\calL}_{QCD}^{cov}(A_{\mu}^{a},\psi^{i,\alpha},G_{\mu\nu})=-{\frac{1}{4}}F_{\mu\nu}^{a}F_{\rho\sigma}^{a}\,{\frac{1}{2}}(G^{\mu\rho}\,G^{\nu\sigma}-G^{\mu\sigma}G^{\nu\rho})+{\bar{\psi}}_{i,\alpha}\gamma^{\mu}{\stackrel{\leftrightarrow}{\nabla}}_{\mu}\psi^{i,\alpha}
{\calL}_{\mathrm{bulk}}(t,y)={\frac{M_{3}^{2}}{2}}\sqrtg\\left(R-\frac{1}{2}(\partial\phi)^{2}-{\frac{3}{2}}\frac{e^{2\phi}{\calF}^{2}}{5!}\right)=-{\frac{3M_{3}^{2}\tilde{A}^{3}B}{\tilde{N}}}\left[\left({\frac{\dot{\tilde{A}}}{\tilde{A}}}\right)^{2}-{\frac{7}{4}}\left({\frac{\dot{D}}{D}}\right)^{2}\right].
W[\vec{V}^{R}]=\intd^{4}x{\calL}_{\mathrm{red}}=\intd^{4}x{\frac{1}{2}}\left(\dot{V}_{i}^{R}R_{ij}^{-1}\dot{V}_{j}^{R}-V_{i}^{R}\left(\vec{\partial}^{2}-M^{2}\right)R_{ij}^{-1}V_{j}^{R}\right)~.
\nabla^{2}G_{aa}(x,x)=-\frac{g^{2}}{N^{2}}\frac{f^{4}({\hat{\phi}})}{4}G_{aa}^{-1}(x,x)\left[1-\frac{1}{2}\frac{{\hbar}}{f({\hat{\phi}})}G_{aa}(x,x)\right].
G_{\mu\nu}[\xi|s]=4\pi{\tilde{g}}\epsilon_{\mu\nu\rho\sigma}\dot{\xi}^{\rho}(s)\Omega_{\xi}^{-1}(s,0)\{[{\bar{\psi}}(\xi(s))\gamma^{\sigma}\tau^{i}\psi(\xi(s)]\tau_{i}\}\Omega_{\xi}(s,0),
f_{2Q+1}(\varepsilon,M)=\frac{(-1)^{Q}\Gamma(-Q-1/2)}{2(4\pi)^{Q+1/2}}M^{D}+\ell^{-D}\sum_{n=0}^{Q}c_{n}(\varepsilon)(\ellM)^{2n}+O(\varepsilon)\,,\nonumber
\delta_{\xi}\left[\frac1{2\alpha}\intd^{4}x(\partial^{\mu}h_{\mu\nu})^{2}\right]=-\frac1{\alpha}\intd^{4}p(p^{\mu}h_{\mu\nu}(p))p^{\rho}(\hat{\xi_{;\rho}^{\nu}}+\hat{\xi_{\rho}^{;\nu}})(-p)
P(x,y)\;=\;\delta\:\int_{-\infty}^{\infty}\frac{dk^{0}}{2\pi}\int_{-\infty}^{\infty}\frac{dl^{0}}{2\pi}\:\frac{1}{V}\sum_{\vec{k},\vec{l}\inL^{3},\;\vec{k}=\vec{l}}\bar{t}_{m}(k)\:\bar{t}_{m}(l)\:e^{-ikx+ily}
\omega_{M}^{\bar{M}\bar{N}}=\frac{1}{2}h^{N\bar{M}}(\partial_{M}h_{N}^{\bar{N}}-\partial_{N}h_{M}^{\bar{N}})-\frac{1}{2}h^{N\bar{N}}(\partial_{M}h_{N}^{\bar{M}}-\partial_{N}h_{M}^{\bar{M}})-\frac{1}{2}h^{P\bar{M}}h^{Q\bar{N}}(\partial_{P}h_{Q\bar{R}}-\partial_{Q}h_{P\bar{R}})h_{M}^{\bar{R}}.
D_{\mu\nu}^{R}(q)J^{\mu}J^{\nu}=J_{0}{\frac{1}{\vec{q}^{2}+M^{2}}}J_{0}+J_{i}J_{j}\left(\delta_{ij}-{\frac{q_{i}q_{j}}{\vec{q}^{2}+M^{2}}}\right){\frac{1}{q^{2}-M^{2}}}~.
V_{\beta}^{1-loop}(\Phi_{0})={\frac{1}{2}}\Phi_{0}^{2}+\hat{\lambda}{\Phi_{0}}^{4}+{\frac{1}{8\pi}}(1+12\hat{\lambda}\Phi_{0}^{2})-{\frac{1}{12\pi}}(1-12\hat{\lambda}\Phi_{0}^{2})^{\frac{3}{2}}-{\frac{1}{24\pi}}+I_{1}^{\beta}(1+12\hat{\lambda}\Phi_{0}^{2})\,.
H_{D}=\int{d^{3}\sigma\Big[-A_{\tau}(\tau,\vec{\sigma})\Gamma^{\ast}(\tau,\vec{\sigma})+\lambda^{\mu}(\tau,\vec{\sigma}){\calH}_{\mu}^{\ast}(\tau,\vec{\sigma})+\mu_{\tau}(\tau,\vec{\sigma})\pi^{\tau}(\tau,\vec{\sigma})\Big]}
\frac{\partial}{\partialx^{\mu}}\psi^{q}=-i\,D_{x}(\hat{A})\left(D_{x}^{\dagger}(\hat{A})D_{x}(\hat{A})\right)^{-1}\sigma_{\mu}^{\dagger}\psi^{q}+\psi^{r}~R_{\mu}^{rq}(x),
P_{\tilde{p}}={\frac{\Omega_{p+2}}{\sqrt{2}\kappa_{D}}}(p+1)(r_{+}r_{-})^{\frac{p+1}{2}},\\\\{\calM}_{\tilde{p}}={\frac{\Omega_{p+2}}{2\kappa_{D}^{2}}}[(p+2)r_{+}^{p+1}-r_{-}^{p+1}].
\left\{\phi_{\varkappa}^{(1)},\phi^{(1)}\right\}=\left\{\phi_{\varkappa}^{(1)},T\right\}=\left.\left\{T,\phi_{3,n}^{(1)}\right\}\right|_{\phi=T=0}=\left.\left\{T,T\right\}\right|_{\phi=T=0}=0\;,\;\;\varkappa=1,2\,.
\Big[{\bfa}_{p}(\vec{k})\,,\,{\bfa}_{p^{\prime}}^{\dagger}({\vec{k}}^{\prime})\Big]\,=\,2\,(2\pi)^{n-1}w_{p}(\vec{k})\delta_{p\,p^{\prime}}\,\delta^{n-1}(\vec{k}-{\vec{k}}^{\prime})\,.
\sum_{A=2}^{m}\phi_{A}^{a}f_{tt}^{A}=-\sum_{A=2}^{m}[2\phi_{At}^{a}f_{t}^{A}+\sum_{B=2}^{m}(\phi_{AB}^{a}f_{t}^{B})f_{t}^{A}]-\phi_{tt}^{a},\;\;\;a=1,2,...,m-1
I=-\frac1{\pi}\intd^{2}x\sqrt{g}\Big[e^{-2\phi}(2\partial_{-}\partial_{+}\rho-4\omega\partial_{-}\phi\partial_{+}\phi+\lambda^{2}e^{2\rho})+\frac12\partial_{+}f_{i}\partial_{-}f_{i}-\kappa(\partial_{-}\rho\partial_{+}\rho+\epsilon\phi\partial_{-}\partial_{+}\rho)\Big]
T_{j}(s>0)=\left(\begin{array}{cc}{-i\rho_{j}/2s+O(1)}&{s^{(k-3)/2}p_{j}+O(s^{(k-1)/2})}\\{-s^{(k-3)/2}\bar{p}_{j}^{T}+O(s^{(k-1)/2})}&{T_{j}(0-)+O(s)}\\\end{array}\right),
\Theta_{a}^{I}|_{loops}=-\frac{i}{2}\,(\omega+\beta_{a}^{I})\,\langlet^{I}+\bar{t}^{\bar{I}}\rangle^{-1}~;{}~~~~~~~~\omega=-\frac{C}{8\pi^{2}}~,~~~\beta_{a}^{I}=\frac{b_{a}^{I}}{8\pi^{2}}.
R_{s}=\left(\begin{array}{cccccc}{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{K_{sts}}&{K_{stt}}&{0}\\{0}&{0}&{0}&{K_{sss}}&{K_{sst}}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}\\{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}\\\end{array}\right).
F(r)=-4\pi\psi_{n}\frac{r^{2}}{(1-{\phi_{c}^{\prime}}^{2})^{3/2}}\frac{d\psi_{n}}{dr}=-4\pie^{3}\frac{\psi_{n}}{r^{4}}\left(1+\frac{r^{4}}{e^{2}}\right)^{3/2}\frac{d\psi_{n}}{dr}.
(\gamma_{\mu\mu_{1}\mu_{2}\ldots\mu_{p}})^{(c)}=\delta_{C}^{p+1}\gamma_{\mu_{p}\ldots\mu_{2}\mu_{1}\mu}=\delta_{C}^{p+1}(-1)^{\frac12(p+1)p}\gamma_{\mu\mu_{1}\mu_{2}\ldots\mu_{p}},
S(a;b_{1},\ldots,b_{k-1})=\sum_{n=1}^{\infty}\frac{(\mathrm{sign}\,a)^{n}}{n^{|\,a|}}\prod_{i=1}^{k-1}\sum_{m_{i}=1}^{n-1}\frac{(\mathrm{sign}\,b_{i})^{m_{i}}}{m_{i}^{|\,b_{i}|}}\,.
2\dot{x}^{-}\ge(m_{a}^{2}y_{a}^{2}-\mu_{i}^{2}x_{i}^{2})+\dot{y_{a}}^{2}+\dot{x_{i}}^{2}\ge(-\mu_{i}^{2}x_{i}^{2})+\dot{y_{a}}^{2}+\dot{x_{i}}^{2}.
\begin{array}{rcl}{\tau^{0}}&{=}&{6w(w-1)(w-2)f}\\{\tau^{1}}&{=}&{6(w-1)(w-2)\nablaf}\\{\tau^{2}}&{=}&{3(w-2)(\nabla\circ\nablaf+w\Phi\circf)}\\{\tau^{3}}&{=}&{\nabla\circ\nabla\circ\nablaf+(3w-2)\Phi\circ\nablaf+w(\nabla\circ\Phi)f.}\\\end{array}
A_{2}\astA_{1}({\mit\Gamma},\bar{\mit\Gamma})=\int_{{\calL}_{1|2}}A_{2}({\mit\Gamma}_{1},\bar{\mit\Gamma})A_{1}({\mit\Gamma},\bar{\mit\Gamma}_{1})\frac{L_{s,b}({\mit\Gamma},\bar{\mit\Gamma}_{1})L_{s,b}({\mit\Gamma}_{1},\bar{\mit\Gamma})}{L_{s,b}({\mit\Gamma}_{1},\bar{\mit\Gamma}_{1})L_{s,b}({\mit\Gamma},\bar{\mit\Gamma})}d\mu({\mit\Gamma}_{1},\bar{\mit\Gamma}_{1})\,.
{\calH}=\pi^{\mu}{\dot{A}}_{\mu}+\pi^{<\mu\nu>}{\dot{B}}_{<\mu\nu>}+\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+\frac{m}{6}\epsilon^{\mu\nu\lambda\rho}H_{\mu\nu\lambda}A_{\rho}
T^{++}(x)=\mathrm{tr}\left[(\partial_{-}X^{I})^{2}+{\frac{1}{2}}\left(iu^{\alpha}\partial_{-}u^{\alpha}-i(\partial_{-}u^{\alpha})u^{\alpha}\right)\right],\qquadI=1\ldots8,\quad\alpha=1\ldots8
\frac{1}{2}\delta\Lambda_{I}=-\gamma^{\mu}\epsilon_{I}\partial_{\mu}\phi+\frac{1}{\sqrt2}\sigma^{\mu\nu}\left(e^{-\phi}F_{\mu\nu}\alpha_{IJ}-e^{\phi}\tilde{G}_{\mu\nu}\beta_{IJ}\right)^{-}\epsilon^{J}=0\.
g^{2}=\frac{1}{3}\frac{\partial^{4}V_{eff}}{\partialf_{0}^{4}}\bigg|_{(f_{0},g_{0})=(M^{\prime},0)}=\frac{1}{3}\frac{\partial^{4}V_{eff}}{\partialg_{0}^{4}}\bigg|_{(f_{0},g_{0})=(0,M^{\prime})}
\langleH_{k}\rangle={\frac{1}{2}}\left(\langle|{\widehat\Pi}_{k}|^{2}\rangle+k^{2}\langle|{\widehat\Psi}_{k}|^{2}\rangle\right)=k\left({\frac{S_{ex}}{S_{re}}}+{\frac{S_{re}}{S_{ex}}}\right),
\left(\omega_{\alpha}^{g_{1}g_{2},g_{3}}\right)\,\left(g_{3}^{*}\omega_{\alpha}^{g_{1},g_{2}}\right)\:=\:\left(\omega_{\alpha}^{g_{1},g_{2}g_{3}}\right)\,\left(\omega_{\alpha}^{g_{2},g_{3}}\right)
{\frac{\partial\widetilde{A}_{b}^{Mi}({\bfx})}{\partialA_{c}^{Mi}({\bfx})}}=\delta_{bc}+{\frac{\partial\,\deltaA_{b}^{Mi}({\bfx})}{\partialA_{c}^{Mi}({\bfx})}}=\delta_{bc}-gf_{bd}^{c}\delta\Lambda_{d}^{M}({\bfx}).
T_{q}^{+}=de_{q}^{+}-\frac12\omega^{(0)}e_{q}^{+}+\frac14\omega_{qp}e_{p}^{+},\T_{\dot{q}}^{-}=de_{\dot{q}}^{-}+\frac12\omega^{(0)}e_{\dot{q}}^{-}+\frac14\omega_{\dot{q}\dot{p}}e_{\dot{p}}^{-}.
\frac{2(d-2)!\,a_{3}\,}{d!\,a_{2}\,\alpha\,\mu}\,F+\frac{1}{4\,d^{2}\,a_{1}\,a_{2}\,\alpha}\,\widehat{G}=\left(h^{-1}+\mu\mathcal{F}\right)^{-1}\,\left[\mathrm{det}\,\left(h^{-1}+\mu\mathcal{F}\right)\right]^{\alpha}
D_{+}^{\alpha}\equiv\partial_{z}-\frac{i}{2}(A_{z}^{1}+\frac{1}{\sqrt{3}}A_{z}^{2}+\cdots+\frac{1}{\sqrt{(\alpha-1)(\alpha-2)/2}}A_{z}^{\alpha-2}+\frac{\alpha}{\sqrt{\alpha(\alpha-1)/2}}A_{z}^{\alpha-1}),
\begin{array}{ll}{[G_{n},G_{m}]=(n-m)G_{n+m},}&{n,m=0,\pm1,}\\{\rule{0mm}{0.7cm}\quadG_{\pm}^{\dagger}=G_{\mp},\quadG_{0}^{\dagger}=G_{0},}&{\mathrm{for}\quadsu(1,1),}\\{\rule{0mm}{0.5cm}\quadG_{\pm}^{\dagger}=-G_{\mp},\quadG_{0}^{\dagger}=G_{0},}&{\mathrm{for}\quadsu(2)}\\\end{array}
L_{+}=\frac{\ell^{2}}2\left[\frac32\left(\frac{f_{+}^{\prime\prime}}{f_{+}^{\prime}}\right)^{2}-\frac{f_{+}^{\prime\prime\prime}}{f_{+}^{\prime}}\right]\qquad,\qquadL_{-}=\frac{\ell^{2}}2\left[\frac32\left(\frac{f_{-}^{\prime\prime}}{f_{-}^{\prime}}\right)^{2}-\frac{f_{-}^{\prime\prime\prime}}{f_{-}^{\prime}}\right]\qquad.
(\zeta-b)\left[sM^{4}e^{-2\alpha\phi/M_{p}}+V_{1}(\upsilon+\chi)\right]+2V_{2}(\upsilon+\chi)+(\zeta+b)^{2}[F_{E}\overline{{E^{\prime}}}E^{\prime}+F_{U}\overline{{U^{\prime}}}U^{\prime}+F_{D}\overline{{D^{\prime}}}D^{\prime}]=0.
F[\Phi,\varphi]=\int_{-\infty}^{+\infty}dx~\Phi^{i}\;\left(\sqrt{1-{\varphi}^{2}}\frac{\stackrel{\leftrightarrow}{\partial}}{\partialx}\varphi^{i}+\varepsilon^{ijk}\varphi^{j}\frac\partial{\partialx}\varphi^{k}\right).
\partial_{\tau}\Phi_{\tau}\equiv\tilde{{\calG}}\Phi_{\tau}=\hbar^{2}Tr(D^{2}\Phi_{\tau})-\frac{\gamma^{2}}{2}Tr(R(\phi)DR(\phi)D\Phi_{\tau})-i({\calA}\phi+gG(\phi),D\Phi_{\tau})
F=-\frac{i}{6}\sum_{\alpha,\beta,\gamma}(D_{\alpha}\cdotD_{\alpha}\cdotD_{\gamma})t_{\alpha}t_{\beta}t_{\gamma}-\frac{\chi\zeta(3)}{2(2\pi)^{3}}\sum_{(d_{i})_{i=1,\dots,n}}n_{d_{1},\dots,d_{n}}{\calL}i_{3}(\Pi_{i=1}^{n}q_{i}^{d_{i}}),
T^{\mu\nu}={\frac{i}{2}}[{\bar{\psi}}\gamma^{\mu}\overrightarrow{\partial}{}^{\nu}\psi-{\bar{\psi}}\gamma^{\mu}\overleftarrow{\partial}{}^{\nu}\psi]+(\partial^{\mu}\phi)(\partial^{\nu}\phi)-g^{\mu\nu}{\cal{L}}\,
\mathrm{Re}\,\chi_{T}^{\mathrm{D}}(\omega)=-\frac{\hbarA}{\pi^{2}}\int\int_{R_{4}}d\omega_{\mathrm{in}}dk_{\|}k_{\|}\sqrt{k_{\|}^{2}-(\omega+\omega_{\mathrm{in}})^{2}}\sqrt{\omega_{\mathrm{in}}^{2}-k_{\|}^{2}}\;{\overline{n}}(-\omega_{\mathrm{in}})
I_{n+1}^{(\pm)}=\frac{1}{(\nu^{2}-\frac{1}{4})^{n}}\{8[W_{n}^{(\pm)}(1;k)+(\frac{dW_{n}^{(\pm)}}{d\alpha})_{\alpha=1}]+\int_{1}^{\infty}d\alphaW_{n+1}^{(\pm)}(\alpha;k)[\alpha^{\frac{\nu}{2}}+\alpha^{\frac{-\nu}{2}}]\},
H^{2}=\left(\frac{8\piG_{n+2}\sigma_{0}}{n}\right)^{2}e^{-a\phi_{0}}R^{2-2/N}-\frac{V_{0}N^{2}e^{-a\phi_{0}}R^{2-2/N}}{nN(N(n+2)-1)}+\frac{mN^{2}}{\sqrt{2nN(1-N)}R^{n+1/N}}.
d_{H}{\frac{\beta_{2}-2\beta_{3}}{\beta_{0}}}=2\left(1+\sqrt{1+{\frac{8}{Q^{2}}}}\right)\left[{\frac{1}{1+\sqrt{1-{\frac{4}{Q^{2}}}}}}-{\frac{1}{1+\sqrt{1-{\frac{2}{Q^{2}}}}}}\right]>0
|\Lambda,\Xi\rangle_{orb}=const\sum_{\nu,\nu^{\prime},\nu_{i}}(-1)^{\nu}(-1)^{\frac{s_{0}^{2}}{2}}|\lambda,\Xi+\nu\beta_{0}+\nu^{\prime}\beta_{\gamma}+\nu_{1}\beta^{(1)}+\dots+\nu_{r}\beta^{(r)}\rangle.
\frac{c_{-4}^{-\nu}}{c_{-2}^{-\nu}}=-\frac{h^{2}}{2^{5}}\bigg[1+O(h^{4})\bigg],\;\;\;\frac{c_{-4}^{-\nu}}{c_{0}^{-\nu}}=\frac{c_{-4}^{-\nu}}{c_{-2}^{-\nu}}\frac{c_{-2}^{-\nu}}{c_{0}^{-\nu}}=\frac{h^{4}}{3.2^{7}}\bigg[1+O(h^{4})\bigg]
\frac{A^{-1}2^{-1}i(2\delta_{1})^{-1/2}}{e^{\beta(\sqrt{A}-\mu+iB)}+1}\int_{-\pi}^{\pi}d\varphi\,e^{-\frac{i}{2}\varphi}=\frac{A^{-1}2i(2\delta_{1})^{-1/2}}{e^{\beta(\sqrt{A}-\mu+iB)}+1}
<r|z>=\left(\frac{2\tau_{2}}{k}\right)^{1/4}\,e^{-\frac{1}{2}|z|^{2}}\,e^{\frac{1}{2}(-iz)^{2}}\,\Theta\left[\begin{array}{c}{r/k}\\{0}\\\end{array}\right]\Big(-ik\sqrt{\frac{\tau_{2}}{\pik}}\,z\Big|k\tau\Big)\\\,
|\deltan\rangle=\sum_{l\neqn}{\frac{\langlel|V|n\rangle}{(E_{l}-E_{n})}}|l\rangle+\sum_{l,n\neqm}{\frac{\langlel|V|m\rangle\langlem|V|n\rangle}{(E_{l}-E_{m})(E_{m}-E_{n})}}|l\rangle+\cdots
A_{0}\left(x\right)=\frac{1}{\xi_{1}}\left(C_{1}e^{-x\xi_{1}}-C_{2}e^{x\xi_{1}}\right)+\frac{1}{\xi_{2}}\left(C_{3}e^{-x\xi_{2}}-C_{4}e^{x\xi_{2}}\right)+C_{7}\tag{25}
\tilde{S}^{-1}=\frac{1}{2c^{2}}\,\left(\begin{array}{cccc}{0}&{2-c^{2}}&{-1}&{1}\\{c^{2}-2}&{0}&{1}&{1}\\{1}&{-1}&{0}&{-1}\\{-1}&{-1}&{1}&{0}\\\end{array}\right)
D_{\mu\nu\rho\sigma}(x)\sim\frac{1}{\epsilon}(x^{2})^{\epsilon}\left[g_{\mu\rho}(x)g_{\nu\sigma}(x)+g_{\mu\sigma}(x)g_{\nu\rho}(x)-\frac{2}{D}\delta_{\mu\nu}\delta_{\rho\sigma}\right].
ds^{2}=-2dx^{+}dx^{--}\beta^{2}\sum_{k=1}^{n}a_{k}^{2}\left[(\tilde{x}^{2k-1})^{2}+(\tilde{x}^{2k})^{2}\right]+\sum_{i=1}^{2n}(d\tilde{x}^{i})^{2}+\sum_{i=2n+1}^{8}(dx^{i})^{2},
\begin{array}{ccccc}{\{L=(00011),M=7\}}&{+}&{\{L=(00111),M=2\}}&{=}&{\{L=(00001),M=2\}\nonumber}\\{(-5,-2,-11,10)}&{+}&{(6,3,14,-15)}&{=}&{(1,1,3,-5)}\\\end{array}
a_{2}=0,\quada_{3/2}=\frac{3\pi^{3/2}}{16Rc_{2}^{2}}\,(c_{1}-c_{2})^{2},\quad\frac{a_{5/2}}{(4\pi)^{3/2}}=\frac{857}{61440}\frac{(c_{1}-c_{2})^{2}}{R^{3}}.
V(\psi\otimes\bar{v},z)\;V(\chi\otimes\bar{w},\zeta)\;\left(\Omega\otimes\overline{{\Omega}}\right)=\sum_{\bar{w}^{\prime},\bar{v}^{\prime}}{R\;}_{\bar{v}^{\prime}\;\bar{w}^{\prime}}^{\bar{v}\;\bar{w}}\;\;V(\chi\otimes\bar{w}^{\prime},\zeta)\;V(\psi\otimes\bar{v}^{\prime},z)\;\left(\Omega\otimes\overline{{\Omega}}\right),
S=\frac{1}{16\pi}\,\int\,\,\left\{((-{\itR}+\,2\partial_{\mu}\Phi\partial^{\mu}\Phi)-\alpha^{\prime}e^{-2\Phi}(F_{a\mu\nu}\,F_{a}^{\mu\nu}-2\beta(\partial_{\mu}\Phi){K^{\mu}})\right\}\,\sqrt{-g}d^{4}x\;.
A_{L}^{\mu\nu}=-g^{\mu\nu}[{\frac{7\zeta(3)}{4\pi^{4}}}\Lambda^{2}+k^{2}\sum_{n=0}{\frac{(2^{2n-1}-1)n!B_{2n}\pi^{2n-2}{\sqrt\pi}}{2^{2n+4}(2n)!\Gamma(n+\frac{5}{2})}}\left({\frac{k^{2}}{\Lambda^{2}}}\right)^{n}].
\begin{aligned}{\phi^{m_{1}m_{2}p}{*\phi}_{n_{1}n_{2}n_{3}p}}&{{}=\phi^{[m_{1}}{}_{[n_{1}n_{2}}\delta_{n_{3}]}^{m_{2}]},}\\{\phi^{mp_{1}p_{2}}{*\phi}_{n_{1}n_{2}p_{1}p_{2}}}&{{}=4\phi^{m}{}_{n_{1}n_{2}}.}\\\end{aligned}
\left[e^{-t\bar{H}_{s}^{2}}\right]_{\mathrm{diag}}={\frac{1}{(4\pit)^{3/2}}}~~~,{}~~~\left[e^{-t\bar{H}_{d}^{2}}\right]_{\mathrm{diag}}={\frac{r_{d}}{(4\pit)^{3/2}}}\left[1+\frac12\kappa^{2}t\right]~~~.
I_{n}(\mu)=-\left(\frac{\mu}{2}\right)^{4-D}\frac{M}{(4\pi)^{(D-1)/2}}\frac{\Gamma(\frac{2n+D-1}{2})}{\Gamma(\frac{D-1}{2})}(-ME)^{(2n+D-3)/2}\Gamma\left(\frac{3-D-2n}{2}\right)
\left(\frac{\lambda_{j}+\frac{i}{2}}{\lambda_{j}-\frac{i}{2}}\frac{\lambda_{j}+i}{\lambda_{j}-i}\right)^{N}=-\prod_{k=1}^{M}\frac{\lambda_{j}-\lambda_{k}+i}{\lambda_{j}-\lambda_{k}-i},\qquadj=1\dotsM.
{\textstyle\frac{1}{4}}\nabla_{a}k_{(I)}^{b}\Gamma^{a}{}_{b}={\textstyle\frac{1}{4}}f_{ia}{}^{b}\Gamma_{\mathrm{Adj}}(u^{-1})^{i}{}_{I}\gamma^{a}{}_{b}=-\Gamma_{\mathrm{Adj}}(u^{-1})^{i}{}_{I}\Gamma_{s}(M_{i})\,,
\beta_{w}\equiv\oint_{r_{min}}^{r_{max}}\,d\tau=\oint_{r_{min}}^{r_{max}}\,{\frac{dr}{\sqrt{\Big(1-{\frac{2m}{r}}+{\frac{q^{2}}{r^{2}}}\Big)\Big(1-{\frac{2m}{r}}+{\frac{q^{2}}{r^{2}}}-(2\pi\sigmar)^{2}\Big)}}}.
\sum_{a_{k}=1}^{L_{k}}\left(\bar{\phi}_{a_{k}}^{(\alpha_{k})}\right)^{2}=\frac{2}{\lambda}\left[\mu^{2}-\left(\frac{\alpha_{1}}{R}\right)^{2}\right]\delta_{k,1}\qquad\mathrm{for~k=1,2,\cdots,M~.}
f={\frac{1}{2}}\alpha^{\prime\,{\frac{1}{2}}}g_{YM}^{-2}\left[\sqrt{m^{2}+{\frac{\alpha^{\prime\,-1}g_{YM}^{6}N}{U^{7}}}\left(P_{U}^{2}+{\frac{\vec{L}^{2}}{U^{2}}}\right)}+m\right],\\\\g={\frac{\vec{L}^{2}}{U^{2}}}.
\mathcal{L}=-\overline{{\Psi}}(x)\left[\Gamma_{\mu}\left(\partial_{\mu}-ieA_{\mu}-gA_{\mu}^{B}I^{B}\right)+m\right]\Psi(x)-\frac14\mathcal{F}_{\mu\nu}^{2}-\frac14\left(F_{\mu\nu}^{B}\right)^{2},
{\calL}[v_{\mu},j_{\mu}^{5},\theta]=-\,\frac{1}{4}\,\left(\partial_{[\mu}v_{\nu]}\right)^{2}-\,e\,\epsilon^{\mu\nu}\,v_{\mu}\,j_{\nu}^{5}+\frac{1}{2}\,j_{\mu}^{5}\,(\partial^{\mu}\theta)-\frac{1}{2\alpha_{j}}\,j_{5}^{\mu}j_{\mu}^{5}\,.
I_{n}^{p}\equiv\frac{\mu_{0}^{(p+1)/2}}{n}\,\int_{\Sigma}d^{p+1}\sigma\,e(\,\sigma\,)\,\left[\,{\frac{(\mathrm{det}\gamma_{mn})^{n/2}}{e(\,\sigma\,)^{n}}}+n-1\,\right]
\hat{\tilde{\tilde{q}}}^{ab}(x)\Psi_{\gamma}(A)={\frac{\delta}{\deltaA_{a}^{i}}}{\frac{\delta}{\deltaA_{b}^{i}}}\Psi_{\gamma}(A)=\dot{\gamma}^{a}(x)\dot{\gamma}^{b}(x)\Psi_{\gamma}(A)
{\calL}=\intd^{2}\thetad^{2}\bar{\theta}\;\Phi^{\dagger}e^{V_{a}T^{a}}\Phi+\frac{1}{4g^{2}}\left(\intd^{2}\theta\;\mathrm{Tr}\;(W^{\alpha}W_{\alpha})+\intd^{2}\theta\;{\calW}(\Phi)+\mathrm{h.c.}\right)
{\calL}=\frac{9}{\alpha}\intd^{2}\!\thetad^{2}\!{\bar{\theta}}\left(SS^{\dagger}\right)^{\frac{1}{3}}+\left\{\intd^{2}\theta\,S\left[\mathrm{ln}\left(\frac{S}{\Lambda^{3}}\right)^{N_{c}}-N_{c}\right]+\mathrm{h.c.}\right\}\,
G_{\beta}(\vec{x},\tau)={\frac{1}{\beta}}\sum_{n}\int{\frac{d^{3}p}{(2\pi)^{3}}}{\frac{e^{-i\omega_{n}\tau+i\vec{p}\cdot\vec{x}}}{E_{p}^{2}+\omega_{n}^{2}}}\;\;.
Tr\gamma_{1,3,(0,\pm1,0)}-\frac{1}{3}Tr[\gamma_{1,{\tilde{7}}_{3},(0)}\gamma_{W4,{\tilde{7}}_{3}}^{\pm1}]-\frac{1}{3}Tr\gamma_{1,7_{4},(\pm1)}+\frac{1}{3}Tr[\gamma_{1,7_{5},(0)}\gamma_{W4,7_{5}}^{\pm1}]=0
\frac{1}{2\pii}\int_{\Re\,s=s_{0}}ds\,\Gamma(s)\left(\frac{\pi}{2}NT_{II}\beta^{2}\right)^{-s}Z_{\mathrm{g}}|_{\bf0}^{\bf0}|(2s)\left\{\intd\tau_{WP}\Omega_{2}^{s}I_{\mathrm{g}}(\tau)\right\}_{(\mathrm{Reg})}.
{\calM}_{\mathrm{d}}(\theta)=\frac{ig^{4}}{m\Theta^{2}}[\varepsilon^{cab}T_{c}\otimesT_{b}T_{a}]\int\frac{d^{2}k}{(2\pi)^{2}}\frac{[{\bfq}\wedge{\bfk}-{\bfp}_{1}\wedge{\bfp}_{3}]({\bfq}\wedge{\bfk})}{{\bfq}^{2}({\bfk}-{\bfp}_{1})^{2}({\bfk}-{\bfp}_{3})^{2}}
\epsilon\propto4\frac{v^{2}}{r^{2}}+\frac{2}{3}\left(\frac{w^{\prime}}{r}-\frac{w}{r^{2}}\right)+4\frac{v^{2}w^{2}}{r^{4}}+\frac{2}{r^{2}}\left(v^{2}-1\right).
{\calS}_{0}=\intd^{4}x\left[i\Psi_{L}^{\dagger}\left(D_{0}+\vec{\sigma}\cdot\vec{D}\right)\Psi_{L}+i\Psi_{R}^{\dagger}\left(D_{0}-\vec{\sigma}\cdot\vec{D}\right)\Psi_{R}\right].
A(\lambda_{i},\varepsilon_{i},p_{i})_{G,\Theta}=Tr\bigg(\lambda_{1}\lambda_{2}\dots\lambda_{k}\bigg)\intd\tau_{i}^{\prime}\langle\prod_{i=1}^{k}\varepsilon_{i}\cdot{\frac{dX}{d\tau}}exp\bigg(ip_{i}\cdotX\bigg)(\tau_{i}^{\prime})\rangle_{G,\Theta}.
\int_{0}^{\infty}P_{r}^{\bfV}(y)P_{s}^{\bfV}(y)\,y^{-1}e^{-y}\,dy\,\propto\int_{0}^{\infty}L_{r-1}^{(1)}(y)L_{s-1}^{(1)}(y)y\,e^{-y}\,dy\,\propto\delta_{r\,s},
\int\frac{d^{d}p}{(2\pi)^{d}}\,\frac{p\cdot(p+k)}{k^{2}\,p^{2}\,(k+p)^{2}}=-\frac{1}{2}\,\frac{1}{(4\pi)^{d/2}}\,B\left(\frac{d}{2}-1\,,\,\frac{d}{2}-1\right)\,\Gamma\left(2-\frac{d}{2}\right)\,(k^{2})^{d/2-2}~~.
I=\frac{1}{2\kappa^{2}}\intd^{11}x\sqrt{-g}\left(R-\frac{1}{2\cdot4!}(F_{(4)})^{2}\right)-\frac{1}{6}\frac{1}{2\kappa^{2}}\intC_{(3)}\wedgeF_{(4)}\wedgeF_{(4)},
u_{r}={u^{cl}}_{r}\pm2\Lambda^{N_{c}-\frac{1}{2}}\Big[\delta_{N_{c},r}(\sqrt{\lambdaM+m_{Q}}-\frac{\lambdaM}{2\sqrt{\lambdaM+m_{Q}}})+\delta_{N_{c}-1,r}\frac{\lambda}{2\sqrt{\lambdaM+m_{Q}}}\Big].
S=\intd^{3}x\left[\frac{1}{4}\left(F_{\mu\nu}^{a}\right)^{2}+\frac12\left(D_{\mu}\Phi^{a}\right)^{2}+\frac{\lambda}{4}\left(\left(\Phi^{a}\right)^{2}-\eta^{2}\right)^{2}-i\bar{\psi}\left(\vec{\gamma}\vec{D}+h\frac{\tau^{a}}{2}\Phi^{a}\right)\psi\right].
{\calE}=(S+{\bar{S}})^{2}\left|-\frac{1}{(S+{\bar{S}})}+\frac{{\calB}^{\prime}}{\calB}\right|^{2}+\frac{1}{3}\left|3+(T+{\bar{T}})\frac{H^{\prime}(T)}{H(T)}\right|^{2}-3,