harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
M_{\mathrm{p}}=\sqrt{{\frac{3}{4\pi}}}\left({\frac{\widetilde{M}_{\mathrm{p}}^{2}}{\sqrt{\lambda}}}\right)\widetilde{M}_{\mathrm{p}}\,,~~\Lambda={\frac{4\pi}{\widetilde{M}_{\mathrm{p}}^{3}}}\left[\widetilde{\Lambda}+\left({\frac{4\pi}{3\widetilde{M}_{\mathrm{p}}^{3}}}\right)\lambda^{2}\right]\,.
{\frac{2J+1}{\pi}}\int{\frac{d\xi^{*}d\xi}{\left(1+\vert\xi\vert^{2}\right)^{2}}}\vert\xi\rangle\langle\xi\vert\equiv\intd\mu\left(\xi,\xi^{*}\right)\vert\xi\rangle\langle\xi\vert={\bf1}_{J}\,
\zeta_{2}=\frac{2^{2s-2}\mu^{-1}\Gamma(s+\frac{1}{2})\Gamma(-s)}{\pi\Gamma(s)\Gamma(\frac12-s)}\intdk\\tilde{S}(k)\tilde{S}(-k)\left(\frac{k^{2}}{\mu^{2}}\right)^{-s-\frac{1}{2}}\,.
p_{L}\sim\frac{n_{I}^{{\bf\alpha}_{i}}X^{I}}{X^{0}}=n_{1}^{{\bf\alpha}_{i}}+n_{2}^{{\bf\alpha}_{i}}TU+m_{1}^{{\bf\alpha}_{i}}T+m_{2}^{{\bf\alpha}_{i}}U=0\,
L_{Rel}^{(k)}=\\frac{1}{2}\left[\{g}_{pq}\d_{\tau}Y^{p}d_{\tau}Y^{q}+\{g}_{pq}\i\bar{\alpha}^{p}\gamma^{0}D_{\tau}\alpha^{q}+\\frac{1}{12}R_{pqrs}(\bar{\alpha}^{p}\alpha^{r})(\bar{\alpha}^{q}\alpha^{s})\right]
\sigma^{\mu\mu_{2}\dots\mu_{p+1}}=\int_{\partial\Sigma}\,y^{\mu}\,dy^{\mu_{2}}\wedge\dots\wedgedy^{\mu_{p+1}}\equiv\int_{\partial\Sigma}d\sigma^{\mu\mu_{2}\dots\mu_{p+1}}(\,\vec{s}\,)\,\qquadp\ge1
G_{ab}^{\mu\nu}(k)=\frac{-i\delta^{ab}}{k^{2}}\left\{g^{\mu\nu}-\frac{k^{\mu}n^{\nu}+k^{\nu}n^{\mu}}{k\cdotn}+\frac{k^{2}n^{\mu}n^{\nu}}{(k\cdotn)^{2}}\right\},
{}_{\mathrm{NS}}^{(0)T}\langle\mathrm{D}p_{\psi},\eta|=\langle\widetildeM|\langleL|\left(\widehatC{\widehat{\gamma}}^{6}\dots{\widehat{\gamma}}^{5+s}\frac{1-\mathrm{i}\eta\widehat\gamma}{1-\mathrm{i}\eta}\right)_{LM}
\tilde{f}(\omega,\kappa,\bar{\kappa},0)=\frac{1}{2\pi^{2}}\int_{-\infty}^{\infty}d\nu\sum_{n=-\infty}^{\infty}{\frac{\vert\kappa^{2}\vert^{-1+i\nu}\left({\frac{\kappa}{\vert\kappa\vert}}\right)^{n}\vert\bar{\kappa}^{2}\vert^{-1-i\nu}\left({\frac{\bar{\kappa}}{\vert\bar{\kappa}\vert}}\right)^{-n}}{\omega-\frac{g^{2}C_{2}}{8\pi^{2}}\Omega(\omega,\nu,n)}}
\begin{array}{cc}{a=\frac{1}{2}\left(\begin{array}{cc}{-1}&{\mp1}\\{-1}&{\pm1}\\\end{array}\right)}&{b=\frac{1}{2}\left(\begin{array}{cc}{-1}&{\pm1}\\{1}&{\pm1}\\\end{array}\right)}\\{}&{}\\{c=\frac{1}{2}\left(\begin{array}{cc}{-1}&{\pm1}\\{1}&{\pm1}\\\end{array}\right)}&{d=\frac{1}{2}\left(\begin{array}{cc}{-1}&{\mp1}\\{-1}&{\pm1}\\\end{array}\right)}\\\end{array},
\omega=i\mathrm{~}(\frac{|q|}{m})\sqrt{\frac{(k_{f}+q/2)^{2}-(k_{f}-q/2)^{2}exp(-(\frac{2\mathrm{~}\pi\mathrm{~}q}{m})(\frac{1}{v_{q}}))}{1-exp(-(\frac{2\mathrm{~}\pi\mathrm{~}q}{m})(\frac{1}{v_{q}}))}}
\left\langleT_{ik}\right\rangle_{SUB}=\frac{D(\xi_{c}-\xi)}{2^{D-1}\pi^{D/2}\Gamma(D/2+1)}\,\mathrm{p.v.}\,\int_{0}^{\infty}dt\,t^{D}e^{-2t\left|x\right|}\frac{\beta_{1}t+1}{\beta_{1}t-1}\,\,\mathrm{diag}(1,0,-1,\cdots,-1).
\mu^{\prime}=\alpha^{2}\left(e^{2\gamma\psi}N(K^{\prime})^{2}+\frac{1}{2}Nx^{2}(H^{\prime})^{2}e^{-4\gamma\psi}+\frac{1}{2x^{2}}(K^{2}-1)^{2}e^{2\gamma\psi}+K^{2}H^{2}e^{-4\gamma\psi}\right.
\delta\left(\begin{matrix}{g_{ij}}\\{\pi^{ij}}\\\end{matrix}\right)=\{\left(\begin{matrix}{g_{ij}}\\{\pi^{ij}}\\\end{matrix}\right),T_{\mu}\}F^{\mu},\:\:\deltaN^{\mu}=\dot{F}^{\mu}-U_{\beta\gamma}^{\mu}N^{\beta}F^{\gamma}.
A_{\mu}^{a}\mathbf{t}^{a}\mapstoA_{\mu}^{a}\mathbf{t}^{a}+\frac{1}{g}\left(D_{\mu}\mathbf{\alpha}^{a}\right)\mathbf{t}^{a}+i\alpha^{a}A_{\mu}^{b}f^{abc}\mathbf{t}^{c}
{\Phi}^{(I)}=-\frac{s\,e^{2}F(1-F)}{2\pi|m|}\left[\frac{1}{6}(F-\frac{1}{2})+\frac{1}{4\pi}\int_{1}^{\infty}\frac{dv}{v{\sqrt{v-1}}}\frac{A{v}^{F}-A^{-1}v^{1-F}}{A{v}^{F}+2+A^{-1}v^{1-F}}\right].
{\calL}_{mass}^{(3/2)}=\frac{M}{2}\left(\bar{\Psi}_{\mu}-\partial_{\mu}\bar{\lambda}\right)\gamma^{\mu\nu}\left(\Psi_{\nu}-\partial_{\mu}\lambda\right)=\frac{M}{2}\left(\bar{\Psi}_{\mu}\gamma^{\mu\nu}\Psi_{\nu}-2\bar{\Psi}_{\mu}\gamma^{\mu\nu}\partial_{\mu}\lambda\right)\,.
|B>_{(\frac{1}{2},\frac{1}{2})}=\sum_{k_{p}}\left((\tilde{A}_{\mu\nu}(k_{p})\psi_{-1/2}^{\mu}\psi_{-1/2}^{\nu}+\tilde{B}(k_{p})(\bar{\beta}_{-\frac{1}{2}}\gamma_{-\frac{1}{2}}-\beta_{-\frac{1}{2}}\bar{\gamma}_{-\frac{1}{2}})\right)|\Omega,k>\,
\frac{1}{4\,e^{2}}~\left\{\int_{N}d^{4}x\,F_{\mu\nu}(x)\,F^{\mu\nu}(x)+\,2\int_{N}d^{4}x\,\left(\partial_{\mu}\varphi\right)(x)\left(\partial^{\mu}\varphi\right)(x)\right\}~.
b_{1}={\frac{a_{1}}{2}}[1+{\frac{({\bar{u}}{\bar{z}}^{\prime}{\bar{u}})}{(ux^{\prime}{\bar{u}})}}],\;\;{\bar{b}_{1}}={\frac{a_{1}}{2}}[1+{\frac{(u{z}^{\prime}u)}{(ux^{\prime}\bar{u})}}].
\displaystyle\alpha=\omega\,\frac{p+1}{p}+\pi\frac{p-1}{2p}\left(\left\lfloor\frac{1}{2}+\frac{S}{p+1}+\frac{\omega}{\pi}\right\rfloor-\left\lfloor\frac{1}{2}+\frac{S}{p+1}-\frac{\omega}{\pi}\right\rfloor\right)
\mathrm{div\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q^{2}(q+p)^{2}\vec{p}^{2}(\vec{q}+\vec{p})^{2}}=\frac{-4\Gamma(2-\omega-\sigma)}{p^{2}\vec{p}^{2}(4\pi)^{2}}\left|{\begin{array}{l}{{\scriptstyle\sigma\to(1/2)^{+}}}\\{{\scriptstyle\omega\to(3/2)^{+}}}\\\end{array}}~,\right.}
\sqrt{J^{\prime}}\prod_{i=1}^{N}d^{2}\omega_{i}^{\prime}d\lambda_{0}^{\prime}\prod_{j=1}^{N-1}d\alpha_{j}=\sqrt{J}\prod_{i=1}^{N}d^{2}\omega_{i}d\lambda_{0}\prod_{j=1}^{N-1}d\alpha_{j}.
\begin{array}{c}{\displaystyle{z_{0}^{-1}\delta\left(\frac{z_{1}-z_{2}}{z_{0}}\right)Y_{M}(u,z_{1})Y_{M}(v,z_{2})-z_{0}^{-1}\delta\left(\frac{z_{2}-z_{1}}{-z_{0}}\right)Y_{M}(v,z_{2})Y_{M}(u,z_{1})}}\\{\displaystyle{=z_{2}^{-1}\left(\frac{z_{1}-z_{0}}{z_{2}}\right)^{-r}\delta\left(\frac{z_{1}-z_{0}}{z_{2}}\right)Y_{M}(Y(u,z_{0})v,z_{2})}}\\\end{array}
(ds)^{2}=-\left(-\mu^{2}+\frac{\tilde{x}^{2}}{x_{+}^{2}}\right)d^{2}{\tilde{t}}+\left(-\mu^{2}+\frac{{\tilde{x}^{2}}}{x_{+}^{2}}\right)^{-1}d^{2}{\tilde{x}},
e^{2\nu}=e^{-2\lambda}=k+\frac{r^{2}}{2\tilde{\alpha}}\left(1\mp\sqrt{1+\frac{64\piG\tilde{\alpha}M}{(d-2)\Sigma_{k}r^{d-1}}-\frac{4\tilde{\alpha}}{l^{2}}}\right),
R_{q}^{v^{\prime}v^{\prime\prime}}=\displaystyle\int\tilde{\psi}_{v^{\prime}}^{\ast}M\tilde{\psi}_{v^{\prime\prime}}dx=\displaystyleM_{0}\delta_{v^{\prime}v^{\prime\prime}}+M_{1}\frac{N_{v^{\prime}}N_{v^{\prime\prime}}}{\beta}\intXH_{v^{\prime}}(X)H_{v^{\prime\prime}}(X)e^{-X^{2}}dx~.
\tilde{S}_{SU(3)}(\theta,\sigma_{21})=\left(\begin{array}{ccc}{-1}&{(\sigma_{12},2)}&{-(\sigma_{12}/2,1)}\\{-(\sigma_{21},2)}&{-1}&{-(\sigma_{21}/2,3)}\\{-(\sigma_{21}/2,3)}&{-(\sigma_{12}/2,1)}&{-1}\\\end{array}\right)\,.
\langleV_{{\bfa}_{+}}(z)V_{{\bfa}_{1}}(z_{1})V_{{\bfa}_{2}}(z_{2})V_{{\bfa}_{3}}(z_{3})\rangle=G_{{\bfa}_{+}{\bfa}_{1}{\bfa}_{2}{\bfa}_{3}}(z,z_{1},z_{2},z_{3})\,,
T^{F}(z)={\frac{2\sum_{i,j=1}^{h}\omega_{i}^{\prime\prime}A_{ij}\overline{\omega}_{j}-3\left(\sum_{i,j=1}^{h}\omega_{i}^{\prime}A_{ij}\overline{\omega}_{j}\right)^{2}}{2\left(\sum_{i,j=1}^{h}\omega_{i}A_{ij}\overline{\omega}_{j}\right)^{2}}}.
\psi^{\prime}(\tau,\vec{x})=\mathrm{~e}^{ie\chi(\tau,\vec{x})}\psi(\tau,\vec{x})~~,~\bar{\psi}^{\prime}(\tau,\vec{x})=\bar{\psi}(\tau,\vec{x})\mathrm{~e}^{-ie\chi(\tau,\vec{x})}
\left(\begin{array}{lll}{j_{1}}&{j_{2}}&{I}\\{\sigma_{1}}&{\sigma_{2}}&{m}\\\end{array}\right)\left(\begin{array}{lll}{j_{3}}&{j_{4}}&{I}\\{\sigma_{3}}&{\sigma_{4}}&{m^{\prime}}\\\end{array}\right)C_{m^{\prime}m}C_{m_{I}^{\prime}m_{I}}\delta_{II^{\prime}}\frac{\mathrm{e}^{-M_{I}|x_{2}-x_{1}|}}{\left(\frac{2\pi}{M}|x_{2}-x_{1}|\right)^{3/2}}.
X^{m}=\left(\begin{array}{cccc}{x_{1}^{m}\otimes1_{N\timesN}}&{0}&{\dots}&{0}\\{0}&{x_{1}^{m}\otimes1_{N\timesN}}&{\dots}&{0}\\{\dots}&{\dots}&{\dots}&{\dots}\\{0}&{\dots}&{0}&{x_{k}^{m}\otimes1_{N\timesN}}\\\end{array}\right),\m=2p+1,\dots,26,\,N\rightarrow\infty\.
\sum_{n=-\infty}^{\infty}\intd\tau\,d^{d-2}x^{i}\left(\epsilon^{2}\partial_{\tau}\phi_{n}^{*}\partial_{\tau}\phi_{n}^{\vphantomx}+\frac{2in}{R}\phi_{n}^{*}\partial_{\tau}\phi_{n}^{\vphantomx}-\partial_{i}\phi_{n}^{*}\partial_{i}\phi_{n}^{\vphantomx}-M^{2}\phi^{*}\phi\right)\.
E_{0}=\frac{\left|\mu_{eff}^{2}\right|^{1/2}}{16\pi^{2}r}\sum_{n=1}^{\infty}\frac{1}{n^{3}}\left\{\eta^{3}\left[\frac{H_{1}\left(\eta\right)}{2\eta}-\frac{H_{2}\left(\eta\right)}{\eta^{2}}-\frac{H_{3}\left(\eta\right)}{2\eta}+\frac{\eta}{60\pi}\right]-\etaN_{2}\left(\eta\right)+\eta^{2}N_{3}\left(\eta\right)\right\}_{\lambda=1}.
\langleL_{\mu\nu}^{i}[\xi|s]L_{\mu^{\prime}\nu^{\prime}}^{i^{\prime}}[\xi^{\prime}|s^{\prime}]\rangle={\frac{i\,a_{\xi^{\prime}}(s^{\prime})}{2}}\delta^{ii^{\prime}}\delta(s-s^{\prime})\,\Box_{\xi^{\prime}}^{-1}(s^{\prime})\prod_{\bar{s}=0}^{2\pi}\delta^{4}(\xi(\bar{s})-\xi^{\prime}(\bar{s}))\,(g_{\mu\mu^{\prime}}g_{\nu\nu^{\prime}}-g_{\mu\nu^{\prime}}g_{\nu\mu^{\prime}}).
M_{scalar\,\pm}^{2}~~=~~M_{\pm}^{2}~-~{\frac{3}{4}}\,\alpha\gamma\,\mu\,\widetilde{m}_{\lambda}\Bigg(1~\pm~\sqrt{1+x}~\Bigg)\Bigg(2~\pm~{\frac{1}{\sqrt{1+x}}}~\Bigg),
\mathrm{tr}\left[QP^{,\rho}\right]=0.\qquad\mathrm{tr}\left[QP^{,\rho}P^{,\sigma}\right]=\frac{1}{k^{2}}P^{\rho\sigma}.\qquad\mathrm{tr}\left[QP^{,\rho\sigma}\right]=\frac{2}{k^{2}}P^{\rho\sigma}
\widetilde{M}_{\lambda}(j_{p},j_{q})=-e{\frac{1}{2\sqrt{2\omega_{k}E_{q}E_{p}}}}e^{-i{\frac{\pi}{2}}(|l|-|n|)}\delta(E_{p}+E_{q}-\omega_{k})\delta(p_{3}+q_{3}-k_{3})\;m_{\lambda}
\eta=-\frac{b_{0}\frac{g^{2}}{(4\pi)^{2}}+b_{1}\,\frac{g^{4}}{(4\pi)^{4}}+b_{2}\,\frac{g^{6}}{(4\pi)^{6}}+\dots}{1+d_{1}\,\frac{g^{2}}{(4\pi)^{2}}+d_{2}\,\frac{g^{4}}{(4\pi)^{4}}+d_{3}\,\frac{g^{6}}{(4\pi)^{6}}+\dots},
P_{\nu}^{\mu}(z)\approx\frac{2^{\nu}\Gamma(\nu+1/2)}{\pi^{1/2}\Gamma(\nu-\mu+1)}z^{\nu}+\frac{\Gamma(-\nu-1/2)}{2^{\nu+1}\pi^{1/2}\Gamma(-\nu-\mu)}z^{-\nu-1}\:,\nonumber
\delta_{(\pm)n}\gamma_{(\pm)lm}=-i\sum_{j=1}^{|n|-1}\sqrt{j|n+j|}\alpha_{(\pm)lj}^{-1}\left[\alpha_{(\pm)|n+j|m}^{*}-\sum_{q=1}^{\infty}\beta_{(\pm)|n+j|q}^{*}\gamma_{(\pm)qm}\right],\quad\mathrm{for}\n<0
\psi^{\theta}\{p^{\mu}\}\equiv\left[\begin{array}{c}{\left(\xi\,\Theta_{[\,j]}\right)\,\phi_{L}^{\ast}(p^{\mu})}\\{\left(\xi\,\Theta_{[\,j]}\right)^{\ast}\,\phi_{R}^{\ast}(p^{\mu})}\\\end{array}\right]\quad,
\chi_{\mathcal{E}}^{(h_{1})}\chi_{\mathcal{E}}^{(h_{2})}=\chi_{\mathcal{E}}^{(h_{1}+h_{2})}-\chi_{\mathcal{E}}^{(h_{1}+h_{2}+1)}+\cdots\pm\chi_{\mathcal{E}}^{(h_{1}-h_{2})}.
Z_{F}^{\eta}(q,\alpha)\=\\mathrm{Tr}[\eta^{F}q^{H\pi/l}]\=\q^{\alpha^{2}-{\frac{1}{2}}\alpha+{\frac{1}{24}}}\,(1+\etaq^{\alpha})\prod_{n=1}^{\infty}(1+\etaq^{{\frac{n}{2}}+\alpha})(1+\etaq^{{\frac{n}{2}}-\alpha}),
\frac{d}{d\tau}mu^{\mu}=eF_{IN}^{\mu\nu}(x)u_{\nu}+\frac{2}{3}\frac{e^{2}}{4\pi}(\ddot{u}^{\mu}-(u\cdot\ddot{u})u^{\mu}),\:\:u^{\mu}=\frac{\dot{x}^{\mu}}{\sqrt{\dot{x}^{2}}}.
(\bar{\nabla}_{\mu}+\partial_{\mu}\bar{\Phi})({\calF}^{\mu\nu}-\bar{F}_{\alpha}^{~~\nu}h^{\alpha\mu}-\bar{F}_{~~\beta}^{\mu}h^{\beta\nu})+\bar{F}^{\mu\nu}(\delta\Gamma_{\sigma\mu}^{\sigma}(h)+(\partial_{\mu}\phi))=0,
S=\intd^{4}x\int_{0}^{L}dy\sqrt{-G}\left(M^{3}F+{\scrL}\right)+\intd^{4}x\sqrt{-g^{0}}L^{0}\Bigr|_{y=0}+\intd^{4}x\sqrt{-g^{L}}L^{L}\Bigr|_{y=L}\,
\begin{aligned}{\Gamma^{i}}&{{}=\gamma^{i}\otimes1\,,}&{\Gamma^{y}}&{{}=-i\gamma^{*}\otimes1\,,}\\{\Gamma^{a}}&{{}=\gamma^{*}\gamma^{a}\otimes\sigma^{1}\,,}&{\Gamma^{\underline{{1}}}}&{{}=-1\otimes\sigma^{2}\,,}\\\end{aligned}
I_{mn}^{\prime}=\frac{1}{A^{4}}(\int\frac{d^{d}k}{{(2\pi)}^{d}}\frac{k_{m}k_{n}}{(k^{2}+a_{3})({(k+p)}^{2}+a_{2})({(k+p)}^{2}+a_{3})}+fin)
r\widetilde{F}\left(\beta\right)\approx-\frac{\pi^{4}}{45}\left(\frac{r}{\beta}\right)^{4}-\frac{1}{240}+\frac{\mu_{eff}^{2}}{2}\left(\frac{r}{\beta}\right)^{2}+\frac{\mu_{eff}^{2}}{48}.
V^{1}(x)=-\Phi^{(0)}{\frac{x^{2}}{(x^{1})^{2}+(x^{2})^{2}}},\quadV^{2}(x)=\Phi^{(0)}{\frac{x^{1}}{(x^{1})^{2}+(x^{2})^{2}}},\quadV^{\nu}(x)=0,\,\,\,\nu=\overline{{3,d}},
A=\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{-\bar{q}\mathcal{D}c^{\ast}}\\{c}&{\mathcal{D}a^{\ast}}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{-qc^{\ast}\mathcal{D}}\\{c}&{a^{\ast}\mathcal{D}}\\\end{array}\right)\,,
\mathrm{Res}\left[I_{C}(k,q),q_{0}=|\vec{q}+\vec{k}|-k_{0}\right]=-\frac{1}{2|\vec{k}||\vec{q}+\vec{k}|}\left.\left(4q\cdotu-4k\cdotu\frac{k\cdotq}{k^{2}}-k\cdotu\right)\right|_{q_{0}=|\vec{q}+\vec{k}|-k_{0}}
\varphi(x_{4},r)=\pi+\pi{\frac{(r^{2}-x_{4}^{2})^{3}+3(r^{4}-x_{4}^{4}-x_{4}^{2})+r^{2}-1}{[(r^{2}-x_{4}^{2})^{2}+2(r^{2}+x_{4}^{2})+1]^{\frac{3}{2}}}}.
H_{D}^{fer}=\frac{1}{\sqrt{2}}\intd^{3}x\xi^{\dagger}(\vec{x})\left(\frac{1}{i\partial_{-}-eV_{-}}\ast\xi\right)(\vec{x})+\frac12\frac{e^{2}}{2L}\intd^{2}x_{\perp}\Gamma^{i}(x_{\perp})\left({\calG}_{(\perp)}[{\calM}^{2}]\ast\Gamma^{i}\right)(x_{\perp})
E_{mnl}=(-1)^{l}\left\{\begin{array}{ll}{\displaystyle\tilde{E}_{m}\sqrt{1+\frac{4\pi^{2}(n+\tau)^{2}}{L_{3}^{2}{\tilde{E}_{m}}^{2}}}\,,\,}&{m\neq0}\\{\displaystyle-\frac{2\pi}{L_{3}}(n+\tau)\,,\,}&{m=0}\\\end{array}\right\}\,,
{\frac{e^{2}}{6}}\,c_{A}\,\sum_{k}{\frac{1}{\omega_{k}^{3}}}\intd^{3}x\;E^{2}\,={\frac{e^{2}}{6}}{\frac{c_{A}}{2\pi^{2}}}\,log\left({\frac{\Lambda}{\mu}}\right)\,\intd^{3}x\;E^{2}.
\langleT(z,\bar{z})T(0,0)\rangle=\frac{m^{2}|z|^{2}}{8z^{4}}\left(4K_{1}^{2}(m|z|)\!+\!m^{2}|z|^{2}K_{0}^{2}(m|z|)\!+\!4m|z|K_{1}(m|z|)K_{0}(m|z|)\right),\nonumber
D_{c}^{(n)}(\eta,{\eta}^{\prime})=\frac{1}{|{\eta}-{\eta}^{\prime}|^{n-2}}-\frac{1}{|\frac{a_{{\eta}^{\prime}}}{a_{B}}{\eta}-\frac{a_{B}}{a_{{\eta}^{\prime}}}{\eta}^{\prime}|^{n-2}}
P_{l}^{m}(x)=\frac{(l+m)!}{2^{m}\,l!}\,(1-x^{2})^{\fracm2}\,P_{l-m}^{(m,\,m)}(x),\qquad\mathrm{and}\qquadP_{l}^{m}(x)=(1-x^{2})^{\fracm2}\,\frac{d^{m}}{dx^{m}}P_{l}(x).
\int_{0}^{\infty}dr~r~J_{0}(pr)~\frac{e^{-a\sqrt{r^{2}+M^{2}}}}{\sqrt{r^{2}+M^{2}}}~=~\frac{e^{-M\sqrt{p^{2}+a^{2}}}}{\sqrt{p^{2}+a^{2}}}~~~.
\vec{\calF}\otimes\varepsilon^{1}\wedge\varepsilon^{2}=\left[\frac1c\left(\boldsymbol{j}_{e}\times\boldsymbol{B}-\boldsymbol{j}_{m}\times{\boldsymbol{E}}\right)+\rho_{e}{\boldsymbol{E}}+\rho_{m}\boldsymbol{B}\right]\otimes\varepsilon^{1}\wedge\varepsilon^{2}=\frac1c*\wedge\left(\omega,{\calJ}\right)+
B\,=\,\frac{1}{2}dz^{M}dz^{N}B_{NM}=\frac{1}{2}E^{A}{E_{A}}^{M}\,E^{B}\,{E_{B}}^{N}\,B_{NM}\,=\,(-)^{b(m+a)}\frac{1}{2}E^{A}E^{B}\,{E_{A}}^{M}{E_{B}}^{N}B_{NM},
\phi_{\triangle\omega}(t,\rho,z)={\frac{1}{2\pi}}\int_{0}^{\infty}d\omega\intd^{2}k~u_{\omega,k}(t,\rho)e^{-ik_{j}z^{j}}\tilde{\phi}_{\triangle\omega}(\omega)\tilde{\varphi}(k)~~~.
\Phi(r)=m(r)\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{-\frac{1}{2}}&{0}\\{0}&{0}&{-\frac{1}{2}}\\\end{array}\right)+q(r)\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{\frac{3}{2}}&{0}\\{0}&{0}&{-\frac{3}{2}}\\\end{array}\right)
\dot{r}=p_{r}\\,\\\dot{p}_{r}=\frac{p_{\theta}^{2}}{r^{3}}-\omega^{2}\,r\\;\\,dot{\theta}=\frac{p_{\theta}}{r^{2}}+g\xi\\,\\\dot{p}_{\theta}=0\\;\\\dot{z}=p_{z}+\xi\\,\\\dot{p}_{z}=0\\,
V=\left(\begin{array}{l}{X(u)=\frac{2}{\pi}\int_{u_{0}}^{u}\,\sqrt{{\frac{2}{1+t}}}\,K\left({\frac{2}{1+t}}\right)\,dt\,=\,{\frac{8}{\pi}}\,\sqrt{\frac{1+u}{2}}\,E\left({\frac{2}{1+u}}\right)\,+\,\mathrm{const}}\\{F_{A}(u)=\frac{2}{\pi}i\,\int_{u_{0}}^{u}\,K\left({\frac{1-t}{2}}\right)\,dt\,=-\frac{4i}{\pi}\,(1-u)\,B\left(\frac{1-u}{2}\right)\,+\,\mathrm{const.}}\\\end{array}\right)
V_{ij}={\frac{N(x^{\prime})}{N(x)N_{1}(x^{\prime})}}\sum_{i^{\prime}j^{\prime}}c_{i^{\prime}i}(R_{\perp})K_{i^{\prime}j^{\prime}}c_{j^{\prime}j}(R_{\perp}^{\prime}),\qquadN_{1}(x^{\prime})={\frac{\alpha\pi}{m^{2}}}x^{\prime}(1-x^{\prime}).
S_{(f)\sigma}(z,z^{\prime})=G_{(\sigma)}(z,z^{\prime})-\sum_{r=1}^{n}\sum_{N_{r}=\mu_{r},\nu_{r}}\int_{C_{b}^{(r)}}G_{(\sigma)}(z,z^{\prime\prime})\frac{d\theta^{\prime\prime}dz^{\prime\prime}}{2\pii}\tilde{Y}_{\sigma,N_{r}}^{(1)}(t^{\prime\prime})\Psi_{\sigma,N_{r}}^{(0)}(z^{\prime})
\left<\Psi\right|{\frac{T_{ij}(\vec{y},\tau)}{N}}\left|\Psi\right>=-\xi_{mn}{\frac{2}{N^{2}}}\int{dtf(t)\epsilon(\tau-t)Im\intd^{3}\vec{x}\left<T_{ij}(\vec{y},\tau)T_{mn}(\vec{x},t)\right>}.
I_{YM,\theta}={\frac{1}{8\pi}}\int_{X}d^{4}x\sqrt{g}\bigg({\frac{4\pi}{g_{YM}^{2}}}\mathrm{tr}[F_{\mu\nu}F^{\mu\nu}]+{\frac{i\theta}{4\pi}}\varepsilon_{\mu\nu\rho\sigma}\mathrm{tr}[F^{\mu\nu}F^{\rho\sigma}]\bigg),
G_{T}W_{\mathrm{eff};a}^{\ast}\frac{\pi_{\lambda}}{\pi_{a}}\left(1-\alpha_{B}\right)=G_{S}W_{\mathrm{eff};\lambda}^{\ast}\left(1-\alpha_{B}\right)+\beta_{B}X_{B}+M^{2}\big|\frac{\pi_{a}}{\Lambda}\big|^{2},
{\calH}_{0}=\frac{1}{2}\sum_{a=1,2}\left\{\frac{(\vec{\nabla}\vec{E}^{a})^{2}}{(g\langle\chi\rangle)^{2}}+(\vec{E}^{a})^{2}+(\vec{\nabla}\times\vec{A}^{\,a})^{2}+(g\langle\chi\rangle)^{2}(\vec{A}^{\,a})^{2}\right\}+\frac{1}{2}(\vec{E}^{3})^{2}+\frac{1}{2}(\vec{\nabla}\times\vec{A}^{\,3})^{2}\.
{\calL}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\theta}{4}\epsilon^{\mu\nu\alpha}F_{\mu\nu}A_{\alpha}+\frac{i}{2}\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-\frac{i}{2}(\partial_{\mu}\bar{\psi})\gamma^{\mu}\psi-m\bar{\psi}\psi+e\bar{\psi}\gamma^{\mu}A_{\mu}\psi,
L_{eff,present}^{(Yuk,modified)}=-\frac{\upsilon}{M_{p}}\left(\frac{m_{1}}{m_{2}}\right)^{2}(\beta-\frac{1}{2}\alpha)\left[6h\left(\frac{m_{1}}{m_{2}}\right)^{4}+\tilde{h}\right]\overline{{\Psi^{\prime}}}\Psi^{\prime}\phi.
<jj|e^{\tilde{\mu}^{*}J_{+1}^{-}}\;e^{\tilde{\mu}J_{-1}^{+}}|jj>=1+\sum_{n=1}^{\infty}\frac{(\tilde{\mu}^{*}\tilde{\mu})^{n}}{n!}\prod_{l=1}^{n}(2j+k-1+l)
g_{D+d}=||g_{MN}||=\left(\begin{array}{ccc}{g_{1}}&{g_{1}K}&{g_{1}{\calA}}\\{K^{T}g_{1}}&{g_{2}+K^{T}g_{1}K}&{g_{2}{\calB}+K^{T}g_{1}{\calA}}\\{{\calA}^{T}g_{1}}&{{\calB}^{T}g_{2}+{\calA}^{T}g_{1}K}&{e^{-\psi/2}g_{D}+{\calA}^{T}g_{1}{\calA}+{\calB}^{T}g_{2}{\calB}}\\\end{array}\right),
{\hat{R}_{\mu\nu}=R_{\mu\nu}+\frac{1}{2}\nabla^{2}\left(\sum_{a}{A_{a}}\right)g_{\mu\nu}-\frac{3}{2}\sum_{a}A_{,\mu}^{a}A_{,\nu}^{a}-\frac{1}{2}\sum_{a\neqb}A_{,\mu}^{a}A_{,\nu}^{b},}
\tilde{\Psi}_{l\zetaj}^{{\vec{k}}{\vec{m}}}\left(x_{\alpha}+L_{1}\right)=\left(e^{-i\pi/\kappa}\right)^{N_{A}-1}e^{-i\left(\beta/\kappa+2\pim_{1}/\kappa\right)}\tilde{\Psi}_{l\zeta,j+1}^{{\vec{k}}{\vec{m}}}\left(x_{\alpha}\right)
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\}_{(y)}\,.
\hat{\delta}\Theta=-{\frac{1}{2}}\epsilon\left[\mathrm{det}(e^{-1})\partial_{m}\phi\gamma^{m}\gamma_{1^{\prime}}-2\gamma^{01}\left\{v^{\tilde{m}}\gamma_{\tilde{m}}+\mathrm{det}(e^{-1})v_{\tilde{m}}\gamma^{\tilde{m}}\right\}\right].
\Pi_{\nu}^{\mu}={\frac{\rho}{12}}\left[\begin{array}{ccc}{-(\rho+2\delta\rho)}&{}&{2a(\rho+P)(v+B)_{|j}}\\{}&{}&{}\\{-2a^{-1}(\rho+P)v^{|i}}&{}&{\left\{2P+\rho+2(1+P/\rho)\delta\rho+2\deltaP\right\}\delta_{j}^{i}-(1+3P/\rho)\delta\pi_{j}^{i}}\\\end{array}\right]\,.
\Pi_{4}=\frac{g^{4}}{4}\int\frac{d^{3}p}{(2\pi)^{3}}\,\frac{N(\omega)}{2\omega}\left[\frac{1}{\hbar^{2}k^{2}+2\hbark\cdotp}+\frac{1}{\hbar^{2}k^{2}-2\hbark\cdotp}\right]\,.
D_{++}^{+}(x)=\frac{2}{(2{\pi})^{3}}\int\frac{dk_{1}dk_{2}}{k_{\bot}^{2}}\mathrm{exp}[-i(k_{1}x^{1}+k_{2}x^{2})]\{{\partial}_{+}J(x^{-},x^{+})-i{\epsilon}(x^{+}){\partial}_{+}I(x^{-},x^{+})\}
G\approx\frac{\lambda^{2}}{8\pi^{2}}\bigg(1-\frac{\mu_{T}^{2}}{m_{T}^{2}}\bigg)\approx\frac{\lambda^{2}}{8\pi^{2}}\bigg(1-\frac{2^{3+\lambda^{2}/2\pi}}{\lambda^{4}}\bigg(\frac{m^{2}}{T^{2}}\bigg)^{2-\lambda^{2}/4\pi}\bigg)
\mathrm{I=\frac{\pi^{\omega+\sigma}\Gamma(2-\omega-\sigma)}{(2\pi)^{2(\omega+\sigma)}}\int_{0}^{1}dxx^{-\sigma}\int_{0}^{x}dG\int_{G}^{1-x+G}dy(H_{0}^{(1)}+H_{0}^{(2)})^{\omega+\sigma-4}+Finite~Terms~,}
\lbraceA,B\rbrace{}^{*}=\lbraceA,B\rbrace-i[\lbraceA,\pi_{\theta\,i}+{\frac{i}{2}}\theta_{i}^{*}\rbrace\lbrace\pi_{\theta^{*}\,i}+{\frac{i}{2}}\theta_{i},B\rbrace+\lbraceA,\pi_{\theta^{*}\,i}+{\frac{i}{2}}\theta_{i}\rbrace\lbrace\pi_{\theta\,i}+{\frac{i}{2}}\theta_{i}^{*},B\rbrace],
[\textstyle{\frac{1}{2}}s_{~b}^{a}t_{1}^{b}{}_{a},\textstyle{\frac{1}{2}}s_{~d}^{c}t_{2}^{d}{}_{c}]=\textstyle{\frac{1}{2}}s_{~b}^{a}[t_{1},t_{2}]_{~a}^{b}\,.
{\calH}_{0}^{B}=\frac{1}{2}[\pi_{1}^{2}+(\partial_{1}\phi_{1})^{2}]-\frac{m}{\pi}\mucos(2\sqrt{\pi}\phi_{1})+\frac{1}{2}[\pi_{2}^{2}+(\partial_{1}\phi_{2})^{2}]+\frac{1}{2}[\pi_{3}^{2}+(\partial_{1}\phi_{3})^{2}].
Q_{\mathrm{I}}=\frac{1}{\sqrt{2}}(e^{i\pi/4}Q_{2}+e^{-i\pi/4}\bar{Q}_{\dot{2}}),\,\,Q_{\mathrm{II}}=\frac{1}{\sqrt{2}}(e^{-i\pi/4}Q_{1}+e^{i\pi/4}\bar{Q}_{\dot{1}}),\,\,Q_{\mathrm{III}}=\frac{1}{\sqrt{2}}(e^{i\pi/4}Q_{1}+e^{-i\pi/4}\bar{Q}_{\dot{1}}).
S[B_{m},A_{m},V_{mn}]=\intd^{5}x\left\{{\textstyle\frac{1}{2\cdot3!}}\mathbb{H}_{m}\mathbb{H}_{m}-{\textstyle\frac{1}{4}}F_{m}F_{m}-{\textstyle\frac{1}{4}}{\calF}_{m}{\calF}_{m}\right\}\,,
\partial_{\mu}(ee_{{f}}^{\mu}u_{{m}}^{{\{f\}}})+{\frac{2}{c(\alpha^{\prime})^{{1/2}}}}\epsilon^{\mu\nu\rho}e_{\mu{f}}u^{{m\{f\}}}\partial_{\nu}\theta\Gamma_{{mn}}C^{-{1}}\partial_{\rho}\theta=0
R_{i}^{~j}\equivdA_{i}^{~j}+A_{i}^{~k}A_{k}^{~j}=u_{i}^{~\underline{c}}R_{\underline{c}}^{~\underline{d}}u_{\underline{d}}^{~j}-\hat{\Omega}_{i}^{~a}\hat{\Omega}_{a}^{~j}.
+R^{\lambda\nu\mu\rho}F_{\lambda\nu}F_{\mu\rho}-2R_{\mu\nu}F^{\rho\mu}F_{\rho}~^{\nu}-2(F_{\rho}~^{\nu}F^{\rho\mu})_{;\nu\mu}+(F_{\nu\rho}F^{\nu\rho})^{;\mu}~_{\mu}~~~,
{\ttT}^{\mathrm{lor}}=(T,\{l_{s}^{2}=a^{2},l_{t}^{2}=-a^{2}\})\;\;\stackrel{W}{\mapsto}\;\;{\ttT}^{\mathrm{eu}}=(T,\{l_{s}^{2}=a^{2},l_{t}^{2}=a^{2}\}).
{\calH}_{int}(x)=-e\overline{{\psi}}^{(0)}(x)\beta^{\mu}A_{\mu}^{(0)}(x)[I+\frac{e}{m}(I-\beta_{0}^{2})\beta^{\nu}A_{\nu}^{(0)}(x)]\psi^{(0)}(x)\,\,,
{\calL}=-\frac{1}{4}F_{\mu\nu}\left({1+e^{2}a\left[{1-12\alpha^{2}g^{2}\partial^{2}}\right]}\right)F^{\mu\nu}\,+\frac{\chi}{2}\varepsilon_{\mu\nu\lambda}A^{\mu}\partial^{\nu}A^{\lambda}\,.
[J_{k}^{\,j}(p),J_{m}^{\,l}(p^{\prime})]=pN_{f}\delta_{m}^{\,j}\delta_{k}^{\,l}\delta(p+p^{\prime})+\delta_{k}^{\,l}J_{m}^{\,j}(p+p^{\prime})-\delta_{m}^{\,j}J_{k}^{\,l}(k+k^{\prime}).
\hat{\partial}_{z^{a}}\hat{f}=\theta_{a\bar{a}}[\hat{\bar{z}}^{\bar{a}},\hat{f}]\qquad\mathrm{and}\qquad\hat{\partial}_{\bar{z}^{\bar{a}}}\hat{f}=\theta_{\bar{a}a}[\hat{z}^{a},\hat{f}].
\mathcal{S}_{\mathrm{MCS}}(A)=\frac12\intd^{3}x\,\varepsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}+\frac1{4m}\intd^{3}x\,F_{\mu\nu}F^{\mu\nu}\;=\mathcal{S}_{\mathrm{CS}}(\widehat{A})\;,
+{\calC}^{1\mu}S_{\mu\nu}\bar{\calP}_{2}^{\nu}+\frac{1}{M^{2}}{\calC}^{2\mu}P_{\mu}P^{\nu}\bar{\calP}_{1\nu}+\frac{1}{M^{2}}{\calC}^{2\mu}P_{\mu}S^{\nu\sigma}P_{\sigma}\bar{\calP}_{2\nu}-{\calC}^{\prime}P^{\prime}\bar{\calP}^{\prime}.
\left\{\begin{array}{c}{1,e_{1},e_{2},e_{3},}\\{1|e_{1},e_{1}|e_{1},e_{2}|e_{1},e_{3}|e_{1},}\\{1|e_{2},e_{1}|e_{2},e_{2}|e_{2},e_{3}|e_{2},}\\{1|e_{3},e_{1}|e_{3},e_{2}|e_{3},e_{3}|e_{3}}\\\end{array}\right\}.
\{F,G\}=-\intd^{2}\theta\,\Bigr\{\theta^{2}\gamma_{B}^{A}\frac{\partial^{2}\deltaF}{\partial\theta^{2}\delta\Phi^{A}(\theta)}\,\frac{\deltaG}{\delta\overline{{\Phi}}_{B}(\theta)}+(-1)^{\epsilon(F)\epsilon(G)}(F\leftrightarrowG)\Bigr\}.\nonumber
{\calJ}_{i}^{0}\equiv-\frac\partial{\partialx}\Phi^{i}-2\varepsilon^{ijk}\Phi^{j}\Pi_{k}=-\sqrt{1-{\varphi}^{2}}\;\varpi_{i}-\varepsilon^{ijk}\varphi^{j}\varpi_{k}\equivJ_{i}^{0}.
\tilde{A}_{p}\;=\;\textstyle{\frac{1}{p!}}A_{\mu_{1}\ldots\mu_{p}}\left(X\right)\partial_{i_{1}}X^{\mu_{1}}\ldots\partial_{i_{p}}X^{\mu_{p}}\;d\xi^{i_{1}}\wedge\ldots\wedged\xi^{i_{p}}\;.
Q_{b}^{(n)}(w|\zeta)=\sum_{\varepsilon_{j}=\pm\atop\varepsilon_{1}+\cdots+\varepsilon_{2n}=0}v_{\varepsilon_{1}}\otimes\cdots\otimesv_{\varepsilon_{2n}}Q_{b}^{(n)}(w|\zeta)^{\varepsilon},
{}\dot{P}_{\mu}(x)=\{P_{\mu}(x),{\bfH}_{T}\}=\int_{M}d^{3}x^{\prime}\bar{N}^{\nu}(x^{\prime})\{P_{\mu}(x),Q_{\nu}^{\sigma}(x^{\prime})\}H_{\sigma}^{G}(x^{\prime}),
\begin{array}{l}{sA_{\mu}^{a}=-\left(\partial_{\mu}c^{a}+gf^{abc}A_{\mu}^{b}c^{c}\right),}\\{sc^{a}=\frac12gf^{abc}c^{b}c^{c},}\\{s\phi^{a}=gf^{abc}c^{b}\phi^{c},}\\{s\overline{{c}}^{a}=b^{a},\,\,sb^{a}=0\;.}\\\end{array}
a_{2}=\int_{M_{\beta}}\left({\frac{1}{180}}R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho}-{\frac{1}{180}}R_{\mu\nu}R^{\mu\nu}-\frac16(\frac15-\xi)\BoxR+\frac12(\frac16-\xi)^{2}R^{2}\right)~~~,...,
\hat{E}^{\hat{\underline{{\alpha}}}}=(E^{\underline{{\alpha}}},\bar{E}^{\underline{{\alpha}}})\equiv(E^{\underline{{\alpha}}1}+iE^{\underline{{\alpha}}2},E^{\underline{{\alpha}}1}-iE^{\underline{{\alpha}}2}),\qquad\underline{{\alpha}}=1,...,16
\left(\begin{array}{c}{\Gamma^{(4)}(q,P)}\\{\Gamma^{(5)}(q,P)}\\\end{array}\right)^{AS}=\left(\begin{array}{c}{\gamma_{4}}\\{\gamma_{5}}\\\end{array}\right)f+\left(\begin{array}{c}{\gamma_{\mu4}}\\{\gamma_{\mu5}}\\\end{array}\right)(q^{\mu}g+P^{\mu}h)+\epsilon_{\mu\nu\rho}P^{\mu}q^{\nu}\left(\begin{array}{c}{\gamma^{\rho5}}\\{-\gamma^{\rho4}}\\\end{array}\right)k,
L\left(m^{2}(\beta),\beta\right)=Q(D)m(\beta)^{D-4}\int_{0}^{\infty}dt\;e^{-t}\;t^{-\frac{D}{2}+1}\left(\Theta_{3}(\pi,e^{-\frac{m^{2}\beta^{2}}{4t}})-1\right),