harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[\beta_{e}^{x},\beta_{e^{\prime}}^{x}\right]=\varepsilon\left(e\right)\kappa_{\left\verte\right\vert}\delta_{e+e^{\prime}},\;\;\left[\beta_{o}^{p},\beta_{o^{\prime}}^{p}\right]=\varepsilon\left(o\right)\kappa_{\left\verto\right\vert}\delta_{o+o^{\prime}},\;\;\left[\beta_{e}^{x},\beta_{o}^{p}\right]=0,
\tilde{F}^{\prime}(A)=\frac{2(\tilde{\epsilon}-\tilde{\delta})^{2}-(\tilde{\epsilon}+\tilde{\delta})^{2}}{16}=\frac{1}{8}(\tilde{\epsilon}-\tilde{\delta})^{2}-\frac{1}{2A}(\tilde{\epsilon}-\tilde{\delta}).
\Phi=\,{\frac{-1}{2i\pi}}\,G_{1}^{0}G_{2}^{0}G_{3}^{0}\,\beta_{1}\beta_{2}\beta_{3}\,\left[\,V_{12}^{0}\,\psi_{12}\,+\,V_{23}^{0}\,\psi_{23}\,+\,V_{31}^{0}\,\psi_{31}\,\right]
{|F_{\it3}|^{2}}/{|F_{\it5}|^{2}}\sim\frac{m^{2}R^{2}}{g^{2}r^{2}}\biggl/\frac{1}{g^{2}Z^{1/2}r^{2}}\sim\frac{m^{2}gN\alpha^{2}}{r^{2}}\sim\frac{gN}{n^{2}}\,
C\:\alpha_{1}^{1-p}\:\beta_{1}^{1-q}\left(\prod_{j=2}^{p}\alpha_{1}^{a_{j}-\alpha_{j}}\:\alpha_{j}^{-a_{j}}\right)\left(\prod_{k=2}^{q}\beta_{1}^{b_{k}-\beta_{k}}\:\beta_{k}^{-b_{k}}\right)\;.
T_{ij}\equiv\frac{\varphi^{\frac{2}{p+1}-1}}{\sqrt{-\gamma}}\left(\frac{\deltaS}{\delta\gamma^{ij}}+\frac{\varphi}{2}\frac{\deltaS}{\delta\varphi}\right)=\frac{1}{p+1}\gamma^{kl}g_{kl}\gamma_{ij}-g_{ij}.
{\calT}_{MN}=-\LambdaG_{MN}+\delta_{M}^{\mu}\delta_{N}^{\nu}\left[T_{\mu\nu}^{\mathrm{mat}}+T_{\mu\nu}^{\psi}+e^{-2\psi}T_{\mu\nu}^{\mathrm{em}}-\sigmag_{\mu\nu}\right]{\frac{\sqrt{-g}}{\sqrt{-G}}}\delta(y),
H_{\alpha\beta\gamma}^{5}=\ell^{-1}\epsilon_{\alpha\beta\gamma}\;\H_{abc}^{5}=\ell^{-1}\epsilon_{abc}\;\H_{\muab}^{5}=-\ell^{-1}K_{I}^{m}A_{\mu}^{I}e_{m}^{c}\epsilon_{abc}\.
\int[dK^{4}][dK_{,\theta}^{4}]\bar{W}[\bar{K},\bar{L},t]\left|_{L=0}=W\left[K_{,\theta}^{\alpha\mu};K_{,\theta}^{\alpha5};-t_{,\theta}^{\alpha};K^{\alpha\mu};K^{\alpha5};t^{\alpha}\right]\right.
V\equiv\int\,d\tau\,\int_{L}^{r}\gamma\,\beta\,\alpha^{2n}\,dr\,\mathrm{Vol(M)}=(\sqrt{c}L)^{2n+2}\,\times\frac{4\pi(n+1)}{\lambda}\mathrm{Vol(M)}\,\int_{1}^{\rho}(\rho^{2}-1)^{n}\,d\rho.
e^{-iH_{+}^{\times}t}|[j,{\mathsfs}],b_{\mathrm{rest}}^{-}\rangle\,=e^{-i{\sqrt{\mathsfs}}\,t}|[j,{\mathsfs}],b_{\mathrm{rest}}^{-}\rangle\,,\,\,\,\mathrm{for}\,\,\,t\geq0\,\,\,\mathrm{only},
\Psi^{(I)}(t)~=~\sum_{\{\alpha\}}~\prod_{abc}^{\{\alpha\}}~\bar{\psi}_{a}^{(+)}(t)~\psi_{b}^{(+)}(t)~\varphi_{c}^{(+)}(t)~\equiv~\sum_{\{\alpha\}}\,\Psi_{\{\alpha\}}^{(+)(I)}(x_{i},t)
\frac{1}{4}Tr(D^{\mu}\PhiD_{\mu}\Phi)=\frac{1}{2}(\partial_{\mu}\phi_{1})^{2}+\frac{1}{2}|D_{\mu}^{3}\phi_{2}|^{2}+\frac{9}{2}e^{2}f^{2}[(A_{\mu}^{1})^{2}+(A_{\mu}^{2})^{2}]+...
\mathbf{M}\,[\phi(\alpha,s)]\,=\,\sum_{k}\frac{b_{k}}{s-\lambda_{k}}\,+\,\int_{1}^{\infty}\phi(\tau)\tau^{s-1}\,d\tau\,+\,\alpha\,\int_{1}^{\infty}\phi(\tau)\tau^{h-s-1}\,d\tau.\,[7pt]
H|\psi\rangle=-i\xi{\widetilde\xi}|\Psi\rangle=\frac{1}{cos^{2}(\theta)-sin^{2}(\theta)e^{2i\phi}}(\sqrt{2}sin(\theta)cos(\theta)e^{i\phi}\xi-(cos^{2}(\theta)+sin^{2}(\theta)e^{2i\phi})/2)|\Psi\rangle,
L=\sum_{a=1}^{N}\biggl({\frac{m}{2}}{\dot{\bfx}}_{a}^{2}+e{\dot{\bfx}_{a}}\cdot{\bfA}({\bfx}_{a},t)-eA_{0}({\bfx}_{a},t)\biggr)+\theta\intd^{2}x\biggl({\frac{1}{2}}\epsilon^{ij}{\dot{A}}_{i}A_{j}-A_{0}B\biggr)\,.
S=\intd^{4}x\sqrt{g}\Bigl\{\frac{R}{2\kappa^{2}}-\frac{1}{24\kappa^{2}}e^{-2\tilde{\phi}}\tilde{H}^{2}-\frac{1}{4\kappa^{2}}\partial_{\mu}\tilde{\phi}\partial^{\mu}\tilde{\phi}-\frac{1}{16\kappa^{2}}e^{-\tilde{\phi}}\tilde{F}^{2}\Bigr\}
\times\left\{\left(\sum_{\mu=1}^{M}(e^{i\omega_{\mu}}|z_{\mu+}^{+}|^{2}+e^{-i\omega_{\mu}}|z_{\mu-}^{+}|^{2})+|z_{2M+1}^{+}|^{2}\right)\left(\sum_{\mu=1}^{N}(e^{i\omega_{\nu}}|z_{\nu+}^{-}|^{2}+e^{-i\omega_{\nu}}|z_{\nu-}^{-}|^{2})+|z_{2M+1}^{-}|^{2}\right)\right\}^{n}\\\.
\langle{\calO}_{W_{2}}^{n_{2}}{\calO}_{W_{3}}^{n_{3}}\cdots{\calO}_{W_{D}}^{n_{D}}\rangle=d^{n_{2}}\langle{\calO}_{W_{3}}^{n_{3}}\cdots{\calO}_{W_{D}}^{n_{D}}\rangle
K_{zz}K_{\bar{z}\bar{z}}={\frac{4}{H^{2}}}{\frac{\partial\bar{\omega}\bar{\partial}\omega}{(1+\omega\bar{\omega})^{2}}}{\frac{\partial\omega\bar{\partial}\bar{\omega}}{(1+\omega\bar{\omega})^{2}}}=2g_{z\bar{z}}\mathrm{e}^{\phi_{2}},
S_{\nu}={\frac{\intd^{\nu^{2}}\Phie^{-\mathrm{Tr}U(\Phi)}\mathrm{Tr}\left[\left({\frac{\partial}{\partial\Phi}}\right)U^{\prime}(\Phi)\right]}{\intd^{\nu^{2}}\Phie^{-\mathrm{Tr}U(\Phi)}}}
\begin{array}{c}{I_{ca}=-\frac{1}{2}\intd^{4}x\sqrt{-g}\[u_{n}{\calF}^{\alphamn}({\calL}_{\alpha\beta}F_{pm}^{\beta}+\({\calL}^{T}{\calM}{\calL})_{\alpha\beta}{\calF}_{mp}^{\beta})g^{pq}u_{q}}\\{+g^{mn}g^{pq}{\Lambda}_{mp}^{\alpha}({\calL}_{\alpha\beta}F_{nq}^{\beta}+\({\calL}^{T}{\calM}{\calL})_{\alpha\beta}{\calF}_{qn}^{\beta})],}\\\end{array}
T_{a}^{(n)}(x,y)\;\asymp\;\frac{1}{16\pi^{3}}\sum_{l=n-1}^{\infty}\frac{(-1)^{l}}{l!\:(l+1-n)!}\:\frac{a^{l+1-n}\:\xi^{2(l+n-1)}}{4^{l}}\:(\Phi(l+1)+\Phi(l))\;\;,\;\;\;\;n\geq1\;.
\frac{i}{2}\mathrm{Tr}{{\bar{\psi}}{\Gamma^{\mu}}(U_{\mu}\psiU_{\mu}^{\dagger}-U_{\mu}^{\dagger}\psiU_{\mu})}=\frac{i}{2}\psi_{a}^{\dagger}\sum_{\mu}\Gamma^{0}(\Gamma^{\mu}u_{\mu}-\Gamma^{\mu}u_{\mu}^{\dagger})\psi_{a}.
|g.s.>=\frac{1}{\sqrt{\cal{N}}}(|\psi_{8}>+\alpha|\psi_{6}^{(1)}>+\beta|\psi_{6}^{(2)}>+\gamma|\psi_{4}^{(1)}>+\delta|\psi_{4}^{(2)}>+\epsilon|\psi_{4}^{(3)}>+\zeta|\psi_{2}>)
{\calL}_{ij}^{R}=\delta_{ij}+R\left(p_{i}\delta_{ij}+m(1-\delta_{ij})\frac{\sigma(q_{ij}+z)}{\sigma(q_{ij})\sigma(z)}\right)+{\calO}(R^{2})=\delta_{ij}+R{\calL}_{ij}^{CM}+{\calO}(R^{2})
\Pi_{\mathrm{static}}^{\mu\nu}=u^{\mu}u^{\nu}\,\Pi_{\mathrm{L}}^{\mathrm{static}}+\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{\tilde{p}^{2}}\,\tilde{\Pi}_{\mathrm{T}}^{\mathrm{static}}
\sigma_{3}=\left[\begin{array}{cc}{-q}&{0}\\{0}&{q^{-1}}\\\end{array}\right]\;,\qquad\sigma_{-}=\left[2\right]^{1/2}\left[\begin{array}{cc}{0}&{0}\\{-q^{1/2}}&{0}\\\end{array}\right]\;,
R(u_{1},u_{2},u_{3})_{i_{1}i_{2}i_{3}}^{j_{1}j_{2}j_{3}}\stackrel{\rho}{=}R((\omegau_{2})^{-1},u_{1},(\omegau_{3})^{-1})_{-j_{2}i_{1}-i_{3}}^{-i_{2}j_{1}-j_{3}}.
\delta\hat{\omega}{}_{\hat{\imath}\hat{\jmath}}^{(2)}=-{\textstyle\frac{1}{2\pi\alpha^{\prime}}}\hat{\chi}_{\hat{\imath}\hat{\jmath}}-{\textstyle\frac{m}{2}}(2\pi\alpha^{\prime})\left({\textstyle\frac{1}{2\pi\alpha^{\prime}}}2\hat{\lambda}+\partial\hat{\rho}^{(0)}\right)_{[\hat{\imath}}\hat{b}_{\hat{\jmath}]}+2\partial_{[\hat{\imath}}\hat{\rho}^{(1)}{}_{\hat{\jmath}]}\,.
S\left|p_{1},....,p_{n}\right\rangle_{\alpha_{1}....\alpha_{n}}^{in}=\left|p_{1},....,p_{n}\right\rangle_{\alpha_{1}^{\prime}....\alpha_{n}^{\prime}}^{in}S_{\alpha_{1}....\alpha_{n}}^{\alpha_{1}^{\prime}....\alpha_{n}^{\prime}}(p_{1},....p_{n})
y={\frac{\bar{E}^{1/2}}{c}}\Big({\frac{3\bar{\Theta}}{4\tilde{\hbar}}}\Big)^{3/2}c_{_\mathrm{S}}^{1/2}\Big({\frac{c^{2}}{{\Gamma}^{2}}}+{\frac{(c_{_\mathrm{S}}^{\,2}-c^{2})\bar{E}^{2}}{c^{4}\mu^{2}}}\Big)^{-1}\,
\partial_{I}^{\,I}f_{L_{1}\ldotsL_{r}}^{K_{1}\ldotsK_{q}}(u)=(\delta_{I}^{K_{1}}+\ldots+\delta_{I}^{K_{q}}-\delta_{L_{1}}^{I}-\ldots-\delta_{L_{r}}^{I})f_{L_{1}\ldotsL_{r}}^{K_{1}\ldotsK_{q}}(u)
Z_{\;\;j_{1}\ldotsj_{p-k-1}}^{i_{1}\ldotsi_{p-k}}=\frac1{\left(p-k-1\right)!}\partial_{}^{\left[i_{1}\right.}\delta_{\;\;j_{1}}^{i_{2}}\ldots\delta_{\;\;j_{p-k-1}}^{\left.i_{p-k}\right]},\;k=1,\ldots,p-1,
\vertw\vert=det{w^{a}}_{\mu}=det({e^{a}}_{\mu}+{t^{a}}_{\mu}),\quad{t^{a}}_{\mu}={\frac{\kappa}{2i}}\sum_{j=1}^{10}(\bar{\psi}^{j}\gamma^{a}\partial_{\mu}{\psi}^{j}-\partial_{\mu}{\bar{\psi}^{j}}\gamma^{a}{\psi}^{j}),
\begin{array}{c}{\left[k_{0}^{[+]}\right]^{2}=-E+i\,\varepsilon-i\,Z=\kappa^{2}=(s-i\,t)^{2},}\\{\left[k_{1}^{[+]}\right]^{2}=-E-i\,\varepsilon-i\,Z=\lambda^{2}=(p-i\,q)^{2},}\\{\left[k_{0}^{[-]}\right]^{2}=-E+i\,\varepsilon+i\,Z=\left[\lambda^{*}\right]^{2},}\\{\left[k_{1}^{[-]}\right]^{2}=-E-i\,\varepsilon+i\,Z=\left[\kappa^{*}\right]^{2}\.}\\\end{array}
{\bfC}_{[\alpha_{1}],[\alpha_{2}],(\alpha_{3})}^{\epsilon}=G_{R}(\alpha_{1})\epsilon{\bfC}_{[Q-\alpha_{1}],[\alpha_{2}],(\alpha_{3})}^{\epsilon}=G_{NS}(\alpha_{3}){\bfC}_{[\alpha_{1}],[\alpha_{2}],(Q-\alpha_{3})}^{\epsilon},
|\hat{F}(t,\star;\rho,W,\rho^{\prime},W^{\prime},\bullet,\bullet^{\prime})|\leqC_{(\star,\bullet,\bullet^{\prime})}(1+|t|)^{\frac{N}{2}}[(1+\rho)(1+|W|)(1+\rho^{\prime})(1+|W^{\prime}|)]^{N}\notag
Z(4)\propto\int_{0}^{\infty}x^{d-1}z^{d-2}dxdz\int_{-\infty}^{\infty}dy\prod_{i=1}^{3}dx_{i}\left[\frac{}{}(x_{3}^{2}+\cdots+x_{d}^{2})x^{2}z^{2}\right]^{\frac{k}{2}}e^{-S_{B}},
\phi_{a}(x_{0},\psi_{0},\bar{\psi}_{0},\tau_{a})=\int_{x_{0},\psi_{0},\bar{\psi}_{0}}{\calD}x_{d}^{I}{\calD}\psi_{\alpha}{\calD}\bar{\psi}_{\dot{\alpha}}V_{a}e^{-S(x_{d}^{I},\psi_{\alpha},\bar{\psi}_{\dot{\alpha}})}
\sigma{\frac{\stackrel{\leftarrow}{\partial}\tilde{b}_{[l-1]0}}{\partial(\phi^{A})^{0(k)}}}\sigma^{AB}(\phi_{B}^{*})^{0(k)}=\tilde{d}{\frac{\stackrel{\leftarrow}{\partial}\tilde{b}_{[l-2]0}}{\partial(\phi^{A})^{0(k)}}}\sigma^{AB}(\phi_{B}^{*})^{0(k)}
\tilde{\varphi}(z,\bar{z},u)=\tilde{\varphi}_{0}(z,\bar{z})+y(z,\bar{z},u)=\tilde{\varphi}_{0}(z,\bar{z})+u\sum_{n=0}^{\infty}\frac{1}{n!}\Lambda_{n}(z,\bar{z})u^{n}\,.
N_{R}=\sum_{n=1}^{\infty}n(a_{n}^{i}{}^{\dagger}a_{n}^{i}+s_{n}^{a}{}^{\dagger}s_{n}^{a}),~~~~~N_{L}=\sum_{n=1}^{\infty}n(\tilde{a}_{n}^{i}{}^{\dagger}\tilde{a}_{n}^{i}+\tilde{s}_{n}^{a}{}^{\dagger}\tilde{s}_{n}^{a}),
f_{0}^{-1}\partial_{\mu}H^{\mu}(x)=-if_{0}^{-1}\partial_{\mu}\Omega^{\mu}=-i{\frac{g^{2}}{32\pi^{2}f_{0}}}F_{\mu\nu}^{a}\,\tilde{F}^{a\mu\nu}(x)\equiv-i\rho(x)\.
R(a_{1},a_{2})=2\pi\sqrt{a_{2}}\sum_{n=-\infty}^{\infty}\frac{1}{\sqrt{a_{2}n^{2}+c^{2}}\left(e^{2\pi\sqrt{(a_{2}n^{2}+c^{2})/a_{1}}}-1\right)}-f(a_{1})\,.
K_{x\mu\nu}^{R\rho\sigma}G_{\alpha\beta,\rho\sigma}^{(2)}(x,y)=-(1-P^{\dagger})\left(\frac{1}{2}(\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha})\delta^{N}(x-y)\right)
\hat{X}^{\Lambda}\to\hat{U}_{\Sigma}^{\Lambda}\hat{X}^{\Sigma},\\\\\\\hat{F}_{\Lambda}\to[(\hat{U}^{T})^{-1}]_{\Lambda}^{\\Sigma}\hat{F}_{\Sigma}+[(\hat{U}^{T})^{-1}{\calC}]_{\Lambda\Sigma}\hat{X}^{\Sigma},
{F}^{\alpha^{\prime}\beta^{\prime}}=\partial^{\alpha(\alpha^{\prime}}{\calA}_{\alpha}^{\beta^{\prime})}+{\calA}^{\alpha(\alpha^{\prime}}{\calA}_{\alpha}^{\beta^{\prime})}=0\,\qquad\nabla^{\alpha\alpha^{\prime}}\chi_{\alpha}=0\,
{\calL}(w)=\left(\begin{array}{ccccc}{-p_{1}}&{e^{{\frac{1}{2}}(q_{2}-q_{1})}}&{0}&{}&{-{\frac{1}{w}}e^{{\frac{1}{2}}(q_{n}-q_{1})}}\\{e^{{\frac{1}{2}}(q_{2}-q_{1})}}&{-p_{2}}&{e^{{\frac{1}{2}}(q_{3}-q_{2})}}&{\ldots}&{0}\\{0}&{e^{{\frac{1}{2}}(q_{3}-q_{2})}}&{-p_{3}}&{}&{0}\\{}&{}&{\ldots}&{}&{}\\{-{w}e^{{\frac{1}{2}}(q_{n}-q_{1})}}&{0}&{0}&{}&{-p_{n}}\\\end{array}\right)
[D,Q^{\pmi}]=-\frac{1}{2}Q^{\pmi}\,,\quad[D,Q_{i}^{\pm}]=-\frac{1}{2}Q_{i}^{\pm}\,,\quad[D,S^{\pmi}]=\frac{1}{2}S^{\pmi}\,,\quad[D,S_{i}^{\pm}]=\frac{1}{2}S_{i}^{\pm}\,,
H_{I}=\int_{x}\{{\calV}_{G}(\varphi)+\varphi(\mu^{2}-{\frac{\lambda}{3}}\varphi^{2}):(\phi_{x}-\varphi):+{\frac{\lambda}{3!}}\varphi:(\phi_{x}-\varphi)^{3}:+{\frac{\lambda}{4!}}:(\phi_{x}-\varphi)^{4}:\},
\theta^{ij}=-(2\pi\alpha^{\prime})^{2}\left(\frac{1}{g+2\pi\alpha^{\prime}B}B\frac{1}{g-2\pi\alpha^{\prime}B}\right)^{ij}~,\quadG^{ij}=\left(\frac{1}{g+2\pi\alpha^{\prime}B}g\frac{1}{g-2\pi\alpha^{\prime}B}\right)^{ij}~.
\begin{array}{ll}{\tilde{T}_{(0)tt}=U\delta(x)\delta(y)+\frac{(J^{2}+\nuS^{2})}{4\pi}\nabla^{2}\left(ln\frac{r}{r_{0}}\right)^{2},}\\{\tilde{T}_{(0)zz}=-T\delta(x)\delta(y)+\frac{(J^{2}-\nuS^{2})}{4\pi}\nabla^{2}\left(ln\frac{r}{r_{0}}\right)^{2},}\\{\tilde{T}_{(0)ij}=(J^{2}+\rhoS^{2})\delta_{ij}\delta(x)\delta(y)-\frac{(J^{2}+\nuS^{2})}{2\pi}\partial_{i}\partial_{j}ln(r/r_{0}),}\\\end{array}
V=\frac{g^{2}}{2}\left(-2|\Phi|^{2}+\xi^{2}+|Q_{A}|^{2}\right)^{2}+g_{A}^{2}|\Phi|^{2}|Q_{A}|^{2}+\frac{1}{4}|g_{A}Q_{A}^{2}|^{2}\,.
\{\tilde{H_{i}}(x)\,,\,\tilde{H_{j}}(x^{\prime})\}=\tilde{H_{j}}(x)\frac{\partial}{\partialx^{i}}\delta(x,x^{\prime})-\tilde{H_{i}}(x^{\prime})\frac{\partial}{\partialx^{\primej}}\delta(x,x^{\prime})
\begin{array}{rl}{\mathrm{graviton}:}&{(3,3)\nonumber}\\{\mathrm{gravitino}:}&{2(3,2)\\mathrm{or}\2(2,3)\nonumber}\\{\mathrm{self\dual\tensor}:}&{(3,1)\\mathrm{or}\(1,3)\nonumber}\\{\mathrm{vector}:}&{(2,2)\nonumber}\\{\mathrm{spinor}:}&{2(2,1)\\mathrm{or}\2(1,2)\nonumber}\\{\mathrm{scalar}:}&{(1,1).}\\\end{array}
\langle\rho(k)\rho(-k)\rangle=\left({\frac{e}{2\pi}}\right)^{2}\left(k^{2}-{\frac{k^{4}}{M^{2}+k^{2}}}\right)=\left({\frac{e}{2\pi}}\right)^{2}{\frac{M^{2}k^{2}}{M^{2}+k^{2}}}\.
T_{1}=igf_{abc}\bigl\{:\frac{1}{2}A_{\mua}A_{\nub}F_{c}^{\nu\mu}:-:A_{\mua}u_{b}\partial^{\mu}\tilde{u}_{c}:\bigr\}+ig\tilde{f}_{abc}:\partial_{\nu}A_{a}^{\nu}u_{b}\tilde{u}_{c}:\quad,
\theta_{i}\to\theta_{Bi}+\delta\dot{b}_{i},\quad\theta\to\theta_{B}+\frac13\sum_{i}\delta\dot{b}_{i},\quad\sum_{i}\theta_{i}^{2}\to\sum_{i}\theta_{Bi}^{2}+2\sum_{i}\theta_{Bi}\,\delta\dot{b}_{i},\quadV\toV_{B}+V_{B}\sum_{i}\deltab_{i}.
\stackrel{(0)}{\omega}_{1}=-W_{ab}\left(H_{0}^{a}\eta^{(2)b}+C_{i}^{(2)a}A^{bi}\right)-M_{bc}^{a}\left(B_{a}^{0i}\eta^{(2)b}A_{i}^{c}+\frac{1}{2}\eta_{a}^{(2)ij}A_{i}^{b}A_{j}^{c}\right).
\intd^{4}x\frac{\left(x^{2}\right)^{n}}{\left(x^{2}+\rho^{2}\right)^{m}}=\pi^{2}\left(\rho^{2}\right)^{n-m+2}\frac{{\mit\Gamma}(n+2){\mit\Gamma}(m-n-2)}{{\mit\Gamma}(m)}\,
\Pi_{k}^{out}=\frac{2^{1-n_{1}}}{\pi}A_{\pi}\Gamma\left(1-n_{1}\right)a_{1}^{m}e^{l\phi_{1}/2}k^{-1}k_{1}^{1/2}\left(\frack{k_{1}}\right)^{n_{1}}-i\frac{2^{n_{1}-1}}\piB_{\pi}\Gamma\left(n_{1}-1\right)a_{1}^{m}e^{l\phi_{1}/2}k_{1}^{-1/2}\left(k_{1}k\right)^{-n_{1}+1}\eta^{2-2n_{1}}
\begin{array}{l}{D_{u_{1},u_{j}}\Phi^{(b-1)}\left(u_{1},\ldots,u_{j-1},u_{j+1},\ldots,u_{b},g,h\right)=}\\{\displaystyle{\qquad{\frac{u_{1}\Phi^{(b-1)}(u_{1},\ldots,u_{j-1},u_{j+1},\ldots,u_{b},g,h)-u_{j}\Phi^{(b-1)}(u_{2},\ldots,u_{b},g,h)}{u_{1}-u_{j}}}}\,}\\\end{array}
\frac{\mathrm{d}^{2}\rho_{n}(t)}{\mathrm{d}t^{2}}=-\frac{(\gamma_{1}-\gamma_{-1})^{2}}{\gamma_{1}\gamma_{-1}}\frac{1}{(2\kappa)^{2}}\big(\mathrm{e}^{-\rho_{n-2}(t)}-2\mathrm{e}^{-\rho_{n}(t)}+\mathrm{e}^{-\rho_{n+2}(t)}\big).
V=\frac{1}{2{b}_{11}}\left(\tilde{m}^{\dagger}\tilde{m}-m^{\dagger}m\right)^{2}+\frac{2}{{b}_{11}}\left|a_{D}\right|^{2}\left(mm^{\dagger}+\tilde{m}\tilde{m}^{\dagger}\right)+\frac{{b}_{01}}{{b}_{11}}D_{0}\left(mm^{\dagger}-\tilde{m}^{\dagger}\tilde{m}\right)
{\calV}_{{\mu_{1}}{\nu_{1}}\,{\mu_{2}}{\mu_{3}}}^{ghost}(k_{1},k_{2},k_{3})=\frac\kappa2\left[\eta_{{\mu_{2}}{\mu_{3}}}({k_{2}}_{\mu_{1}}{k_{3}}_{\nu_{1}}+{k_{3}}_{\mu_{1}}{k_{2}}_{\nu_{1}})-(\eta_{{\mu_{1}}{\mu_{2}}}{k_{1}}_{\nu_{1}}+\eta_{{\nu_{1}}{\mu_{2}}}{k_{1}}_{\mu_{1}}){k_{2}}_{\mu_{3}}\right].
2\int_{0}^{1}dy\frac{\deltab(y,t)}{b_{0}}=\frac{m_{B}^{2}}{24k^{2}}{\frac{e^{2kb_{0}(2+\epsilon)}}{k^{2}v_{0}^{2}b_{0}\epsilon^{2}}}(\rho_{4}(T)-3P_{4}(T))\approx\frac{m_{B}^{2}}{18k^{2}m_{r}^{2}}{\frac{(\rho_{4}(T)-3P_{4}(T))}{kb_{0}M_{Pl}^{2}e^{-2kb_{0}}}}.
V_{\mu\nu}^{i}=\partial_{\mu}V_{\nu}^{i}-\partial_{\nu}V_{\mu}^{i}\,\,\,\mathrm{and}\,\,\,{}F_{\mu\nu}^{i}=V_{\mu}^{k}\partial_{k}V_{\nu}^{i}-V_{\nu}^{k}\partial_{k}V_{\mu}^{i}\,\,.
\intd\xi_{0}\mathrm{Tr}\left(\left(\hat{A}(\xi_{0},\xi)\right)^{\dagger}\star\left({\frac{\partial}{\partial\xi_{0}}}-{\frac{\theta^{\prime}}{2}}\bar{v}{\frac{\partial}{\partialq_{o}}}\right)\hat{A}(\xi_{0},\xi)\right)=\langle\Psi|\hat{c}_{0}|\Psi\rangle\,,
\left[\begin{matrix}{\varphi_{1k_{-}}}\\{\varphi_{2k_{-}}}\\\end{matrix}\right]=\sqrt{{\frac{\kappa}{2}}}\left[\begin{matrix}{e^{ik_{-}x^{-}}e^{i\pi/2}}\\{e^{-ik_{-}x^{-}}e^{-i\pi/2}}\\\end{matrix}\right].
S_{2}=\frac{1}{2}\alpha_{\mu}^{a}\left[{-(D^{2})^{ab}\delta_{\mu\nu}-g(J\cdotF)_{\mu\nu}^{ab}}\right]\alpha_{\nu}^{b}+\bar{c}^{a}\left[{(D^{2})^{ab}}\right]c^{b}.
\left\langle\Phi_{D}^{2}\right\rangle_{\phi}=\left\langle\Phi_{ND}^{2}\right\rangle_{\phi}=\left\langle\Phi_{D}^{2}\Phi_{ND}^{2}\right\rangle_{\phi}=\left\langle\Phi_{D}^{2}\Phi_{D}^{2}\right\rangle_{\phi}\equiv0.
\mathcal{S}_{\Omega}(A)=\frac1\xi\intd^{3}x_{1}....d^{3}x_{n}F_{\mu_{1}\nu_{1}}(x_{1})....F_{\mu_{n}\nu_{n}}(x_{n})\Omega^{\mu_{1}\nu_{1}...\mu_{n}\nu_{n}}(x_{1},...,x_{n})\;,
{\frac{\dot{\varrho}_{m}}{\varrho_{m}}}=-3{\frac{\dot{a}}{a}}(1+w)-2{\frac{\dot{c}}{c}}{\frac{\varrho_{c}}{\varrho_{m}}}+2{\frac{\dot{c}}{c}}{\frac{\varrho+\varrho_{\delta\sigma}}{\varrho_{m}}}-{\frac{\varrho}{\varrho_{m}}}{\frac{\dot{G}_{4}}{G_{4}}}-{\frac{3Cc\dot{c}}{4\piG_{4}a_{0}^{4}\varrho_{m}}}.
H_{bog}=\sum_{{\bf{k}}}\epsilon_{{\bf{k}}}d_{(1/2){\bf{k}}}^{\dagger}({\bf{k}})d_{(1/2){\bf{k}}}({\bf{k}})+\sum_{{\bf{q}}\neq0}\frac{v_{{\bf{q}}}}{2}[{\sqrt{N_{0}}}d_{-{\bf{q}}/2}(-{\bf{q}})+d_{{\bf{q}}/2}^{\dagger}({\bf{q}}){\sqrt{N_{0}}}][{\sqrt{N_{0}}}d_{{\bf{q}}/2}({\bf{q}})+d_{-{\bf{q}}/2}^{\dagger}(-{\bf{q}}){\sqrt{N_{0}}}]
{\calH}={\frac{1}{2}}\rhov_{i}v_{i}~,~~\{v_{i}(x),\rho(y)\}={\frac{\partial}{\partialx^{i}}}\delta(x-y)~,~~\{v_{i}(x),v_{j}(y)\}=-{\frac{\partial_{i}v_{j}-\partial_{j}v_{i}}{\rho}}\delta(x-y).
g(z)=\,\left(\begin{array}{cr}{~~\,\psi(z)~~~}&{\,\chi(z)~~~~~~~~}\\{\beta^{-1}\psi\,^{\prime}(z)+\rho^{-1}J_{2}(z)\psi(z)}&{~\beta^{-1}\chi\,^{\prime}(z)+\rho^{-1}J_{2}(z)\chi(z)}\\\end{array}\right),
\prod_{k=0}^{\infty}\left(\frac{2k+1}{2\pi}\right)^{26/2}=\prod_{k=0}^{\infty}\left[\left(\frac{2k}{\pi}\cdot\frac{2k+1}{\pi}\right)\left(\frac{4\pi}{2k}\cdot\frac{2k+1}{\pi}\right)\right]^{26/4}.
J(s)=\sum_{p}\sum_{n_{2}}\int_{0}^{\infty}\frac{dt}{t}t^{s}\int_{-\infty}^{\infty}dxe^{-2\piipx-[(x-\tilde{t}_{1})^{2}+\frac{1}{|\tau|^{2}}\tilde{n}_{2}^{2}]t}
P^{\mu}P_{\mu}=M^{2}(1+{\frac{J^{2}}{L^{2}}}{\frac{a_{1}a_{4}}{({\frac{a_{1}a_{2}}{2}}+a_{3})}})~~,~~S^{\mu\nu}S_{\mu\nu}=2J^{2}~~,~~S^{\mu\nu}P_{\nu}=0.
{\calR}^{(i)}=\sum_{n_{1},\dotsn_{4}\geq0}p_{n_{1},\dots,n_{4}}^{(i)}\left({\frac{s}{M_{s}^{2}}},{\frac{t}{M_{s}^{2}}}\right)e^{-2\pi\tau_{2}(n_{1}\eta_{1}+n_{2}\eta_{2}+n_{3}\eta_{3}+n_{4}\eta_{4})}\,
(-1)^{F}=(-1)^{1+\sum_{j=2}^{n}\sum_{r>0}(b_{-r}^{-j}b_{r,j}+b_{-r}^{j}b_{r,-j})+\sum_{j=n+1}^{5}\sum_{n>0}(b_{-n}^{-j}b_{n,j}+b_{-n}^{j}b_{n,-j})}\cdot\!\!\!\prod_{k=n+1}^{5}(b_{0}^{k}b_{0}^{-k}-b_{0}^{-k}b_{0}^{k})
\{J^{\mu\nu}\}=\left(\begin{array}{cccc}{0}&{-\frac{1}{B}}&{\frac{1}{2}}&{0}\\{\frac{1}{B}}&{0}&{0}&{\frac{1}{2}}\\{-\frac{1}{2}}&{0}&{0}&{-\frac{B}{4}}\\{0}&{-\frac{1}{2}}&{\frac{B}{4}}&{0}\\\end{array}\right).
\frac{d\varepsilon}{dt}=q{\vec{E}\cdot\vec{v}}\left(1+\frac{q^{2}|\vec{E}|^{2}}{A^{2}m^{2}}\right)+\frac{q^{2}c^{3}}{A^{2}m{\frac{d\tilde{\tau}}{dt}}}\left[\frac{1}{2}\frac{d|\vec{E}|^{2}}{dt}+\frac{1}{c^{2}}\left(\vec{v}\cdot\frac{\partial\vec{E}}{\partial{t}}\right)\vec{E}\cdot\vec{v}+v^{j}E_{i}\partial_{i}{E_{j}}\right]\,{.}
S=\int_{M}\varepsilon^{\mu\nu\alpha\beta}\left(c_{1}\,^{+}G_{\mu\nu}^{ab}\,{^+G}_{\alpha\beta}^{cd}+c_{2}\,{^-G}_{\mu\nu}^{ab}\,{^-G}_{\alpha\beta}^{cd}+c_{3}\,^{+}{\calR}_{\mu\nu}^{ab}\,{^+G}_{\alpha\beta}^{cd}+c_{4}\,{^-{\calR}}_{\mu\nu}^{ab}\,{^-G}_{\alpha\beta}^{cd})\epsilon_{abcd}\right),
(f_{I}^{\Lambda},\bar{f}_{\bar{I}}^{\Lambda})=(f_{i}^{\Lambda}P_{I}^{i},\bar{f}_{\bar{\imath}}^{\Lambda}\bar{P}_{\bar{I}}^{\bar{\imath}}),\quad\quadP_{i}^{I}\bar{P}_{\bar{\jmath}}^{\bar{J}}\eta_{I\bar{J}}=g_{i\bar{\jmath}}
|\psi_{j,{\mathsfs}_{R}}^{G}(t)^{-}\rangle\equive^{-iH^{\times}t}|[j,{\mathsfs}_{R}],b_{\mathrm{rest}}^{-}\rangle=e^{-iM_{R}t}e^{-\frac{\Gamma_{R}t}{2}}|[j,{\mathsfs}_{R}],b_{\mathrm{rest}}^{-}\rangle\,,\quadt\geq0\,.
S={\frac{1}{8\pi}}\intdxdy\left[\sum_{i=1}^{2}(\partial_{x}\phi_{i})^{2}+(\partial_{y}\phi_{i})^{2}-\Lambda^{\prime}\left(e^{i{\frac{\beta}{\sqrt{2}}}(\phi_{1}+\sqrt{3}\phi_{2})}+e^{i{\frac{\beta}{\sqrt{2}}}(\phi_{1}-\sqrt{3}\phi_{2})}+e^{-i\sqrt{2}\beta\phi_{1}}\right)\right]
\nu^{AF}=-\frac{g}{8i\pi^{2}}\epsilon_{jkl}\intd^{3}\Sigma_{j}\,\mathrm{tr}\,\tau_{3}\,\bar{\omega}\Omega_{\mathrm{D}}\partial_{k}\Omega_{\mathrm{D}}^{\dagger}\bar{\omega}^{\dagger}\,\partial_{l}a_{0}.
{\calL}(x)=\overline{{\Psi}}(x)\left(i\gamma^{\mu}\partial_{\mu}-e\gamma^{\mu}A_{\mu}(x)\right)\Psi(x)-\frac{1}{4}F^{\mu\nu}(x)F_{\mu\nu}(x)-\frac{\lambda}{2}\left(\partial_{\mu}A^{\mu}(x)\right)^{2}\;,
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{array}{ll}{}&{g\longrightarrowhgh\\\,\\\g\epsilonSL(2,R);\\\f\longrightarrowh^{c}fh^{c}\\\,\\\f\epsilonU(1)}\\{}&{{\bf{A}}\longrightarrowh({\bf{A}}+\partial)h^{-1}\\\,\\\\overline{{\bf{A}}}\rightarrowh^{-1}(\overline{{\bf{A}}}-\overline{{\partial}})h}\\\end{array}
{\calL}=\frac{1}{2}\left\{\partial_{m}\phi^{\prime}\partial^{m}\phi^{\prime}+\imath\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-\left({\calW}^{2}+{\calW}^{\prime}{\calW}^{\prime\prime\prime}\right)\phi^{2}-{\calW}^{\prime\prime}\bar{\psi}\psi\right\}.
B_{4}=2{u}^{A}\hat{{u}}^{\dot{B}}P_{A\dot{B}}-{\frac{1}{a}}{u}^{A}{u}^{B}Z_{AB}-{\frac{1}{a}}{\bar{u}}^{\dot{A}}{\bar{u}}^{\dot{B}}\bar{Z}_{\dot{A}\dot{B}}=0,
V=1+{\frac{m_{1}}{|\vec{r}_{1}|}}+{\frac{m_{2}}{|\vec{r}_{2}|}}\,\qquad\vec{\partial}V={\frac{m_{1}\vec{r}_{1}}{|\vec{r}_{1}|^{3}}}+{\frac{m_{2}\vec{r}_{2}}{|\vec{r}_{2}|^{3}}}\,
\quade_{A}^{~M}=\left(\begin{array}{cc}{e^{-1}(1+{\frac{ef\tilde{f}}{\tilde{e}}})}&{{\frac{if}{\tilde{e}}}}\\{{\frac{-i\tilde{f}}{\tilde{e}}}}&{\tilde{e}^{-1}(1+{\frac{e\tilde{f}f}{\tilde{e}}})}\\\end{array}\right)\,,
\sigma_{ab}(x)=\intdy^{-}|x^{-}-y^{-}|\big[\psi_{a}(x^{+},y^{-})\psi_{b}^{\dagger}(x^{+},y^{-})-{\frac{1}{N}}\delta_{ab}\psi_{c}(x^{+},y^{-})\psi_{c}^{\dagger}(x^{+},y^{-})\big].
X_{(3t_{c})}={\frac{1}{3(2\mathrm{sin}{\frac{3g_{s}}{2}})^{2}}}(e^{{\frac{3}{2}}i\lambda_{1}}-e^{-{\frac{3}{2}}i\lambda_{1}})(e^{{\frac{3}{2}}i\lambda_{2}}-e^{-{\frac{3}{2}}i\lambda_{2}}).
\delta\chi^{(\alpha)i_{(\beta)}}=-{\textstyle{\frac{1}{\sqrt{2}}}}\gamma^{\mu}\partial_{\mu}\phi^{(\alpha\beta)}\epsilon^{ij}\epsilon_{j_{(\beta)}}-g\Sigma_{\alpha\beta\gamma}\Omega_{\gamma\delta}e^{\lambda_{\delta}/2}\epsilon_{ij}\epsilon^{j_{(\beta)}}+\Omega_{\alpha\delta}e^{-\lambda_{\delta}/2}F_{\mu\nu}^{(\delta)}\gamma^{\mu\nu}\epsilon^{i_{(\beta)}},
\frac{\partialI_{i}}{\partialx_{\rho}}=A_{\mu}(c)\frac{\partialc^{\mu}}{\partialx^{\rho}}\vert_{c=c(x,x_{0}),\tau_{i}(x))}^{c=c(x,x_{0},\tau_{i+1}(x))}-\int_{\tau_{i}}^{\tau_{i+1}}F_{\nu\mu}\frac{\partialc^{\mu}}{\partialx^{\rho}}\cdot\frac{\partialc^{\nu}}{\partial\tau}d\tau
\mathcal{K}_{\alpha\beta}(p,t)=i\alpha\sum_{n=1}^{\infty}(\alpha\betap)^{n}\sum_{d|n}\frac{\left(i^{d}-(-i)^{d}\right)}{4d}\left(\beta^{\frac{n+d}{2d}}+\beta^{-\frac{n+d}{2d}}\right)K\left(\frac{nt}{d^{2}}\right)
\begin{array}{rcl}{E_{n}^{(1)}=\hbarg\omega\left(2n+\frac{9}{2}\right)\\\}&{,}&{\\\d_{n}^{(1)}=\frac{1}{120}(n+1)(n+2)(n+3)(n+4)(n+5)\\\,}\\{E_{n}^{(2)}=\hbarg\omega\left(2n+\frac{15}{2}\right)\\\}&{,}&{\\\d_{n}^{(2)}=\frac{1}{120}(n+1)(n+2)(n+3)(n+4)(n+5)\\\.}\\\end{array}
Q_{\mathrm{I}}=\frac{1}{\sqrt{2}}(e^{i\pi/4}Q_{2}+e^{-i\pi/4}\overline{{Q}}_{\dot{2}}),\qquadQ_{\mathrm{II}}=\frac{1}{\sqrt{2}}(e^{-i\pi/4}Q_{1}+e^{i\pi/4}\overline{{Q}}_{\dot{1}}),
\widehat{c_{\infty}}(p)\;=\;\frac{\bigl(1-\widehat{\chi}(p)\bigr)\;\widehat{\chi}(p)}{(p^{2})^{1-\frac{\epsilon}{2}}}\;=\;\bigl(1-\widehat{\chi}(p)\bigr)\;\widehat{C_{\infty}}(p).
(-iG_{k}(t,t))\approx\frac{1}{2\omega_{<}(k)}\left[1+\frac{\mu^{2}+m^{2}}{\mu^{2}-\frac{\lambda}{2}\phi_{i}^{2}-k^{2}}[cosh(2W_{k}t)-1]\right]coth[\beta_{i}\omega_{<}(k)/2]
C=\int_{x}[-{\frac{1}{2}}f_{xx}+{\frac{1}{2}}\int_{y}J_{x}h_{xy}^{-1}J_{y}+{\frac{1}{2}}I_{0}(M^{2})-{\frac{M^{2}}{4}}I_{1}(M^{2})+J_{x}\Phi]\;.
\Psi_{n}=\Psi_{n+1}=\Psi,\;\;\;\tilde{\Psi}_{n+\frac{1}{3}}=\tilde{\Psi}_{n+\frac{4}{3}}=\tilde{\Psi},\;\;\;\hat{\Psi}_{n+\frac{2}{3}}=\hat{\Psi}_{n+\frac{5}{3}}=\hat{\Psi},
T^{t}=\left(\begin{array}{cc}{A}&{C}\\{0}&{B}\\\end{array}\right)\quad,\quad\quad\quad(T^{t})^{-1}=\left(\begin{array}{cc}{A^{-1}}&{-A^{-1}CB^{-1}}\\{0}&{B^{-1}}\\\end{array}\right)
G(t-t^{\prime},\tau)=\frac{1}{a^{D}}\int_{1/\Lambda^{2}}^{\infty}ds\frac{e^{-\frac{t-t^{\prime}}{4s}}}{(4\pis)^{\frac{1}{2}}}\int_{0}^{\infty}\phi_{r}(\tau)|C(r)|^{2}e^{-\frac{s}{a^{2}}(r^{2}+\rho_{D}^{2})}dr\,,
{\frac{8\pi^{2}}{g^{2}N_{c}}}={\frac{9}{4}}{\frac{u}{(1-u)}}\left[\mathrm{Li}_{2}(u)-\left(f_{1}u+f_{2}{\frac{u^{2}}{2}}\right)-{\frac{1}{9}}\left(h_{1}(1)u+h_{2}(1){\frac{u^{2}}{2}}\right)\right],
dJ^{+}=\Bigl(-e\Bigr)\times\Bigl(e^{-2\pi\lambda(k^{+},k_{\bot})}\Bigr)\times\Bigl(2\Bigr)\times\Biggl({\frac{dp^{+}}{2\pi}}\Biggr)\times\Biggl({\frac{d^{2}p_{\bot}}{(2\pi)^{2}}}\Biggr)\;.
{\bf\Phi}_{\pm}^{(n_{\rho},m)}={\bf\Phi}_{\pm}^{(n_{\rho},m)}(\rho,k)=\left(\begin{array}{c}{\phi_{1\pm}^{(n_{\rho},m)}(\rho,k)}\\{\phi_{2\pm}^{(n_{\rho},m)}(\rho,k)}\\\end{array}\right)