harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Tr\rho_{r}^{2}=\langle\Sigma_{p}\Sigma_{p^{\prime}}\psi_{\xi}(p)~\psi_{\xi}^{*}(p^{\prime})~\psi_{\xi^{\prime}}(p^{\prime})~\psi_{\xi^{\prime}}^{*}(p)\rangle_{\xi,\xi^{\prime}}
{\calL}_{4}=-\,\frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}+\frac{m^{2}}{4}\,F_{\mu\nu}\Bigl(1+\frac{\partial_{t}^{2}}{\nabla^{2}}+\frac{\partial_{t}^{4}}{\nabla^{4}}\Bigr)\,\frac{1}{\nabla^{2}}\,F^{\mu\nu}-\frac{1}{2\alpha}\,\Bigl(\partial_{\mu}A^{\mu}\Bigr)^{2}\,.
I(k)=\frac{(2\pi)^{2}}{(2\pi)^{4}}\intdp_{\parallel}p_{\parallel}\intdp_{\perp}p_{\perp}\frac{J_{0}(\Thetap_{\parallel}k_{\parallel})J_{0}(\Thetap_{\perp}k_{\perp})}{p_{\parallel}^{2}+p_{\perp}^{2}}.
\langlec|\mathrm{e}^{-\pi\ellH}|c\rangle_{\mathrm{R-R}}\sim\sum_{m,k}\left[1-(-1)^{m}\right]\left[1+(-1)^{k}\right]\mathrm{e}^{-{\frac{\pi\alpha^{\prime}\ell}{2R^{2}}}(k-\gammam)^{2}}
{\calZ}\left({\calJ},\bar{{\calJ}},{\calK}\right)=\int\left[{\calD}A\right]\left[{\calD}\varphi\right]\left[{\calD}\bar{\varphi}\right]e^{i\intdx\left({\calL}+{\calJ}\varphi+\bar{{\calJ}}\bar{\varphi}+{\calK}A\right)}
{\mathcalJ}^{\mu}\;=\;\sum_{i=1\atops=\uparrow,\downarrow}^{N}{\mathcalJ}^{(i,s)\mu}~,~~\widehat{{\mathcalJ}}_{\ell/r}^{\mu}\;=\;\sum_{i=1\atops=\uparrow,\downarrow}^{N}\widehat{{\mathcalJ}}_{\ell,r}^{(i,s)\mu}~.
\sigma^{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,,\quad\sigma^{2}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\,,\quad\sigma^{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\,.
F_{\mu\nu\lambda\rho}=\partial_{\mu}A_{\nu\lambda\rho}-\partial_{\nu}A_{\mu\lambda\rho}+\partial_{\lambda}A_{\rho\mu\nu}-\partial_{\rho}A_{\lambda\mu\nu}\equiv\partial_{\left[\mu\right.}A_{\left.\nu\lambda\rho\right]}.
[\frac{1+\sigma_{3}}{2},\frac{\sigma_{1}\pmi\sigma_{2}}{2}]=\pm\frac{\sigma_{1}\pmi\sigma_{2}}{2}~~,~~[\frac{1-\sigma_{3}}{2},\frac{\sigma_{1}\pmi\sigma_{2}}{2}]=\mp\frac{\sigma_{1}\pmi\sigma_{2}}{2}
\begin{array}{l}{{}^{A}(\lambda\Phi+\mu\Psi)^{(n)}=\lambda{}^{A}\Phi^{(n)}+\mu{}^{A}\Psi^{(n)},\quad\forall\lambda,\mu\inC,}\\{({}^{A}\Phi,{}^{A}\Psi)=\S_{n=0}^{\infty}({}^{A}\Phi^{(n)},{}^{A}\Psi^{(n)}).}\\\end{array}
R(u,\lambda)=\sum_{i=1}^{n}E_{i,i}\otimesE_{i,i}+\sum_{i\neqj}\alpha(u,\lambda_{ij})E_{i,i}\otimesE_{j,j}+\sum_{i\neqj}\beta(u,\lambda_{ij})E_{i,j}\otimesE_{j,i}
\int_{0}^{r}\frac{u_{0}^{2}}{(I_{0}+\lambda)^{2}}W^{2}(r^{\prime})dr^{\prime}=\left.\frac{-W^{2}}{I_{0}+\lambda}\right|_{0}^{r}+\int_{0}^{r}\frac{2WW^{\prime}}{(I_{0}+\lambda)}dr^{\prime}.
D_{\;\;\beta_{2}}^{\alpha_{2}}\Phi_{\alpha_{2}}^{*}=\delta\left(Z_{\;\;\beta_{2}}^{\alpha_{1}}\eta_{\alpha_{1}}^{*}+\frac12C_{\beta_{2}}^{\alpha_{0}\beta_{0}}\Phi_{\alpha_{0}}^{*}\Phi_{\beta_{0}}^{*}\right).
V_{i}\cdot{\calN}_{\overline{{\alphaV}}}\equiv{\vec{U}}_{i}\cdot{\vec{P}}_{\overline{{\alphaV}}}-\sum_{r}V_{i}^{r}N_{\overline{{\alphaV}}}^{r}+\sum_{\ell:{\mathrm{right}}}T_{i}^{\ell}J_{\overline{{\alphaV}}}^{\ell}-\sum_{\ell:{\mathrm{left}}}T_{i}^{\ell}J_{\overline{{\alphaV}}}^{\ell}~.
(\dot{\Gamma}_{i}^{a}\delta\tilde{E}^{ia}-\dot{\tilde{E}^{ia}}\delta\Gamma_{i}^{a})-\partial_{i}(i\hat{\lambda}^{a}\delta\tilde{E}^{ia})=0\\mathrm{mod}\left(\dot{\tilde{E}^{ia}}+i\varepsilon^{abc}\lambda^{b}\tilde{E}^{ic}=0\right),
\left[{\frac{d^{4}}{dy^{4}}}+{\frac{2}{y}}{\frac{d^{3}}{dy^{3}}}-\left(1+{\frac{l(l+1)}{y^{2}}}\right){\frac{d^{2}}{dy^{2}}}-{\frac{2}{y}}{\frac{d}{dy}}\right]c_{l}(y)=0.
{\calL}=-\frac{1}{4}F_{\mu\nu}\,F^{\mu\nu}-\frac{3\gamma}{16}\,\Biggl[\,(F_{\mu\nu}F^{\mu\nu})^{2}-2\,(F_{\mu\lambda}F_{\nu\rho}F^{\mu\nu}F^{\lambda\rho})\,\biggr]\,.
\rho_{o}(N)\simeq\frac{c\;(ap)^{\frac{D-p}{2(p+1)}}}{\sqrt{2\pi(p+1)}}\;N^{-\frac{D+1}{2(p+1)}}\;e^{AN^{\frac{p}{p+1}}}\;,\;\;\;A=\frac{p+1}{p}\;(ap)^{\frac{1}{p+1}}
\psi_{0}^{(-)}\proptoe^{-\int_{x_{0}}^{x}W(y,a_{0})dy}~,~\psi_{n}^{(-)}(x,a_{0})=\left[-\frac{d}{dx}+W(x,a_{0})\right]\psi_{n-1}^{(-)}(x,a_{1})~~(n=1,2,3,...)~.
\int_{R^{8+1}}{\hat{C}}_{{\hat{\mu}}_{1}{\hat{\mu}}_{2}{\hat{\mu}}_{3}}D{\hat{X}}^{{\hat{\mu}}_{1}}D{\hat{X}}^{{\hat{\mu}}_{2}}D{\hat{X}}^{{\hat{\mu}}_{3}}\wedge\mathrm{Tr}({\hat{F}}\wedge{\hat{F}}\wedge{\hat{F}})\,.
\mathrm{Tr}_{s}\left(q^{L_{0}/\hbar}e^{i2\piT_{0}^{3}/\hbar}\right)=\frac{q^{\hbars(s+1)/k}\sum_{n=-\infty}^{\infty}(-1)^{2s+2nk/\hbar}(2s+1+2nk/\hbar)q^{{n^{2}k/\hbar}+(2s+1)n}}{\Pi_{m=1}^{\infty}(1-q^{m})^{3}}.
\psi_{1}=\left(\begin{array}{c}{|0\rangle\!\rangle\langle\!\langle0|}\\{|0\rangle\!\rangle\langle\!\langle1|}\\{\vdots}\\{|0\rangle\!\rangle\langle\!\langlek-1|}\\\end{array}\right)\,,\quad\psi_{2}=\left(\begin{array}{c}{0}\\{0}\\{\vdots}\\{0}\\\end{array}\right)\,,\quad\xi=S_{k}\,,
{\calL}_{NBI}=(2\pi\alpha^{\prime})^{-2}\mathrm{STr}\left[{\calI}-\sqrt{\left({\calI}+\frac{1}{4}(2\pi\alpha^{\prime})^{2}F^{2}\right)^{2}}\right]=-\frac{1}{4}\mathrm{Tr}(F^{2}),
a_{n}^{\prime}(\omega)=\sum_{p=0}^{n}(-1)^{p}{\frac{\mu^{p}}{p!}}\bar{a}_{n}(\omega),~~~b_{n}^{\prime}(\omega)=\sum_{p=0}^{n}(-1)^{p}{\frac{\mu^{p}}{p!}}\bar{b}_{n}(\omega).
J_{4}=Tr~({\calZ}^{\dagger}{\calZ})^{2}-{\frac{1}{4}}(Tr~{\calZ}^{\dagger}{\calZ})^{2}+{\frac{1}{96}}(\epsilon_{ABCDEFGH}{\calZ}^{AB}{\calZ}^{CD}{\calZ}^{EF}{\calZ}^{GH}+c.c)
\gamma^{-{\frac{1}{2}}}\delta_{\\i^{\prime}j^{\prime}}^{ij}(\Omega-\Omega^{\prime})=\sum_{p^{\prime}=7i/2}^{+i\infty}W_{(p^{\prime})\i^{\prime}j^{\prime}}^{\ij}(\mu(\Omega,\Omega^{\prime})).
P(l)=\frac{4\pi^{2}}{l!^{2}(l+1)!^{2}}\left(\frac{\lambda\omegar_{0}}{4}\right)^{2l+2}\left|\frac{N_{1}}{N_{0}}\right|^{2}=P_{0}(l)\left|\frac{N_{1}}{N_{0}}\right|^{2}\,.
\left(\begin{array}{cc}{\deltaM}&{\deltaB}\\{\delta\bar{B}}&{\deltaN}\\\end{array}\right)=i\left(\begin{array}{cc}{[M,m]+b\bar{B}-\bar{B}b}&{Mb-Bn+bN-Mb}\\{-\bar{B}m+n\bar{B}-N\bar{b}+\bar{b}M}&{[n,N]+\bar{b}B-\bar{B}b}\\\end{array}\right).
S=\int_{t_{1}}^{t_{2}}dt\int\theta(P)\left[\rho\left(\frac{{\bfv}^{2}}{2}-\Phi({\bfx})-\varepsilon(\rho,s)\right)-\rho\frac{D\phi}{Dt}-\eta\frac{Ds}{Dt}-\beta\frac{D\alpha}{Dt}-\tauK\right].
\Sigma_{A}(0)~\simeq~e^{2}(4|eH|)\int\frac{d^{4}\hat{q}}{(2\pi)^{4}}\frac{\mathrm{e}^{-\hat{q}_{\perp}^{2}}}{\hat{q}^{2}}\frac{\Sigma_{A}(q_{\parallel})}{2|eH|\hat{q}_{\parallel}^{2}+\Sigma_{A}^{2}(q_{\parallel})}.
\sqrt{\phi(k)}\doteq\frac{\sqrt{-1}}{2\pi}\frac{l(\partial(\rho^{2}(k)))}{\zeta(k)}d\zeta(k)\underset{DT}{=}\frac{\sqrt{-1}}{2\pi}\frac{\left(\frac{\sqrt{3}}{3}a\right)q(k)}{\zeta(k)}d\zeta(k),
\hat{H}\leftrightarrow\beta\varepsilon(\vec{\pi})+e\Phi-\mu_{B}\frac{mc^{2}}{\varepsilon({\bfp})}\beta\vec{\Sigma}{\bfB}-\mu_{B}\frac{mc^{3}}{\varepsilon({\bfp})(\varepsilon({\bfp})+mc^{2})}\left[{\bfp}\times\vec{\Sigma}\right]{\bfE}
S={\frac{1}{4}}\int_{0}^{T}d\tau\dot{x}^{2}(\tau)+{\frac{1}{4L}}\int_{0}^{T}d\sigma_{1}\int_{0}^{T}d\sigma_{2}B(\sigma_{1},\sigma_{2}|\tau_{1},\tau_{2})x(\sigma_{1})\cdotx(\sigma_{2}),
\tilde{\Pi}^{\underline{{m}}}=d\tilde{X}^{\underline{{m}}}-id\tilde{\Theta}^{1}{\sigma}^{\underline{{m}}}\tilde{\Theta}^{1}-id\tilde{\Theta}^{2}\tilde{\sigma}^{\underline{{m}}}\tilde{\Theta}^{2}=d\tau\tilde{\Pi}_{\tau}^{\underline{{m}}}
\left(2q-1\right)\cdot\left(CS\Gamma_{[m_{1}}\Gamma_{\left|l\right|}\Gamma_{m_{2}\ldotsm_{2q-1}]}\right)_{\left(sym\right)}=-\left(2q-1\right)\cdot\left(CS\Gamma_{lm_{1}\ldotsm_{2q-1}}\right)\quad.
\left[(hr^{3}{\frac{d}{dr}})^{2}+(r^{2}+R^{2})^{2}(r^{2}+r_{K}^{2})\omega^{2}-{\frac{8r^{4}R^{4}}{(r^{2}+R^{2})^{2}}}h\right]\nu(r)=0\.
H_{c}=\int[\frac{1}{2}(\pi_{i}^{a})^{2}+\frac{1}{2}m^{2}(A_{i}^{a})^{2}+\frac{1}{4}(F_{ij}^{a})^{2}+\frac{m^{2}}{2}(A_{0}^{a})^{2}-A_{0}^{a}T_{2}^{a}]
R(t)=\frac{t}{2m_{0}^{2}}+\dots\,;\;s_{2n}=\frac{(2n-1)!!}{(n+1)!}+\dots\,;\;\rho\left(\frac{x}{m_{0}}\right)=\frac{m_{0}}{2\pi}\sqrt{2-x^{2}}+\dots
e^{-F}=\int\prod_{k}D\mu[M_{k}]e^{-N^{2}S[M]}=\int\prod_{k}D\mu[M_{k}]e^{-N^{2}(S[M]-S_{0}[M])}e^{-N^{2}S_{0}[M]}=e^{-F_{0}}\Big<e^{-N^{2}(S-S_{0})}\Big>_{S_{0}}\;.
{\calL}=-\kappa\dot{B}\left(a\mp\frac{1}{\kappa}B^{\prime}\right)+i\psi^{\ast}\dot{\psi}-A_{0}(\kappaB^{\prime}+\rho)-\frac{1}{2m}|(\partial_{x}-ia)\psi|^{2}-\frac{1}{2m}B^{2}\rho
S={\frac{1}{16\piG_{D+d+1}}}\intd\bar{t}\intd^{D+d}x\left(\prod_{K}e^{\bar{\alpha}_{K}}\right)\sum_{I}\left[2\ddot{\bar{\alpha}}_{I}+\dot{\bar{\alpha}}_{I}^{2}+\dot{\bar{\alpha}}_{I}\sum_{J}\dot{\bar{\alpha}}_{J}\right]
ds_{H}^{2}=ds_{6}^{2}-H_{s}\tilde{H}_{s}dx^{i}dx^{i}-\frac{H_{s}}{\tilde{H}_{s}}\left(dz+{A}_{i}^{(1)}dx^{i}\right)^{2}\quad,\quade^{2\phi}=H_{s}\,.
{\bfV}_{\mu}=V_{\mu}^{a}(z)\frac{\partial}{\partialz^{a}}+{\bar{V}}_{\mu}^{\bar{a}}(\bar{z})\frac{\partial}{\partial\bar{z}^{a}},\quad[{\bfV}_{\mu},{\bfV}_{\nu}]=C_{\mu\nu}^{\lambda}{\bfV}_{\lambda},
\widetilde\chi^{(3)}\left(e_{i},e_{j}\right)=\chi_{F}^{(3)}\left(e_{i},e_{j}\right)+\sum_{k=1}^{3h-3}a_{k}\oint_{\calC}\Omega^{k+1-h_{0}}\left[e_{i},e_{j}\right],\qquadh_{0}\equiv{\frac{3}{2}}h.
{\frac{\partialg_{\alpha\bar{\beta}}}{\partialw^{\gamma}}}={\frac{\partialg_{\gamma\bar{\beta}}}{\partialw^{\alpha}}},\quad\mathrm{or}\quad{\frac{\partialg_{\alpha\bar{\beta}}}{\partial\bar{w}^{\gamma}}}={\frac{\partialg_{\alpha\bar{\gamma}}}{\partial\bar{w}^{\beta}}}.
\frac{da^{\prime}}{d\ell}=1\,,\quada^{\prime}\vert\,{}_{\ell=0}=a\,;\quad\frac{du^{\prime}}{d\ell}=\alpha(t,x,a^{\prime},\nu)\,e^{-a^{\prime}u^{\prime}/\nu}-\frac{u^{\prime}}{a^{\prime}}\,,\quadu^{\prime}\vert\,{}_{\ell=0}=u\,.
\cdot\left\{(2n-1)(4n-1){\calG}_{n}+2{\calG}_{0}\sum_{k=0}^{n-1}{\calG}_{n-k}{\calG}_{k+1}c(k,n)-2\sum_{j=0}^{n-1}\sum_{k=0}^{j+1}{\calG}_{n-j}{\calG}_{j+1-k}{\calG}_{k}d(j,k,n)\right\},
(-\Theta)^{2}(-2\Theta)(-2\Theta-1)\prod_{j=0}^{7}(-8\Theta-j)\prod_{j=0}^{11}(-12\Theta-j)\tilde{\Pi}(t^{\prime})=e^{-t^{\prime}}\prod_{j=0}^{23}(24\Theta-j)\tilde{\Pi}(t^{\prime}).
\langle\bar{W}^{4}|W^{4}|W^{4}|\bar{W}^{4}\rangle|_{N\to\infty}^{\scriptsize\mathrm{one-loop}}\propto\frac{\Phi(s,t)}{x_{13}^{8}x_{42}^{8}}\frac{t}{s}\left(1+\frac{1}{s}+\frac{1}{s^{2}}\right)\;.
\lambda^{k}\frac{\delta}{\deltaJ_{1}}\dots\frac{\delta}{\deltaJ_{4}}\langle\boldsymbol\beta,\psi_{k}\rangle|_{J=0,m=0}=(k-1){\mathrm{R}es}({\tilde{G}}_{k}^{(4)}(p_{1},\dots,p_{4})).
J_{\mathrm{asym}}=\kappa\int_{R_{*}^{2}}d^{2}x\,\,\partial_{j}\,\left\{{\frac{1}{2}}x^{j}(\bar{A}^{k}\bar{A}_{k})-\bar{A}_{j}(x^{k}\bar{A}_{k})\right\}-n\pi\rho_{e}|\vec{q}|^{2}.
\delta_{(k)}|A\rangle=\sum_{j=0}^{k-1}(-)^{j}\!\!\!\!\sum_{l_{1},\ldots,l_{j+1}=1}^{k}\delta_{kl_{j+1}}\prod_{i=1}^{j}\frac{\varepsilon^{l_{i}l_{i+1}}{\bfa}_{l_{i+1}l_{i}}^{}}{k+1-l_{i}}D_{l_{1}}|\Lambda_{k}\rangle\,.
\rho^{2}(n^{2}\rho^{2}+8sin^{2}nF)F^{\prime\prime}+2n^{2}\rho^{3}F^{\prime}+\frac{1}{n}sin2nF(4n^{2}\rho^{2}F^{2}-4sin^{2}nF-n^{2}\rho^{2})=0
\gamma_{\scriptscriptstyle{L,L}}=\Phi/2\pi-L+\frac{a^{L}\,b_{\scriptscriptstyle{L-1,L}}}{f_{\scriptscriptstyle{L,L}}(a)}\,(1+\mathrm{O}(ka)^{2})+\mathrm{O}(ka)^{2}\,.
V_{i}\cdotP_{\overline{{\alphaV}}}+{\frac{\delta_{i1}}{m_{1}}}D\equivV_{i}\cdotP_{\overline{{\alphaV}}}+\sum_{r}T_{i}^{r}N^{r}=s_{i}+\sum_{j}k_{ij}\alpha_{j}~({\mathrm{mod}}~1)~,
\begin{array}{rclcl}{\xi^{\mu}\,\gamma_{\mu}\,\epsilon_{aI}}&{=}&{\mathrm{i}\,\varepsilon_{ab}\,\epsilon^{bI}}&{;}&{i=1,\dots,n_{max}}\\{\epsilon_{aI}}&{=}&{0}&{;}&{i>n_{max}}\\\end{array}
\theta_{ab}=\left(\begin{array}{ccccc}{0}&{\theta_{1}}&{0}&{0}&{0}\\{-\theta_{1}}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{\theta_{2}}&{0}\\{0}&{0}&{-\theta_{2}}&{0}&{0}\\{0}&{0}&{0}&{0}&{\theta_{ij}}\\\end{array}\right),\i,j=5,\dots,2p\.
\langle\Psi_{1}^{\primeI_{1}}(z_{1})\Psi_{2}^{\primeI_{2}}(z_{2})\cdots\Psi_{n}^{\primeI_{n}}(z_{n})\rangle=\langle\Psi_{1}^{I_{1}}(z_{1})\Psi_{2}^{I_{2}}(z_{2})\cdots\Psi_{n}^{I_{n}}(z_{n})\rangle
\begin{array}{ll}{\lambda_{\theta}=-ip_{n}\sigma^{n}\bar{\psi},}&{\qquad\lambda_{\bar{\theta}}=i\psi\sigma^{n}p_{n},}\\{\lambda_{\rho}=-2ip_{n}\sigma^{n}\lambda_{\bar{\theta}}\approx0,}&{\qquad\lambda_{\bar{\rho}}=2i\lambda_{\theta}\sigma^{n}p_{n}\approx0.}\\\end{array}
\sum_{\epsilon_{l}\leq0}u_{\epsilon_{l}l}\left(r\right)u_{\epsilon_{l}l}\left(r^{\prime}\right)+\int_{0}^{\infty}\frac{\mathrm{d}k}{2\pi}\,u_{kl}\left(r\right)u_{kl}\left(r^{\prime}\right)=\delta\left(r-r^{\prime}\right)
({\frac{\delta\tilde{\Sigma}}{\deltaA}},{\frac{\delta\tilde{\Sigma}}{\deltaK}})+({\frac{\delta\tilde{\Sigma}}{\deltac}},{\frac{\delta\tilde{\Sigma}}{\deltaL}})+{\frac{\delta\tilde{\Sigma}}{\deltaV}}\cdot{\frac{\delta\tilde{\Sigma}}{\deltaM}}=0
H_{nn}^{\{{\bfR}\}}({\bfR}^{-1}{\bff})=(-1)^{n}n!\sum_{s=0}^{n}(R_{12}-\sqrt{R_{11}R_{22}})^{s}(R_{12}+\sqrt{R_{11}R_{22}})^{n-s}L_{s}^{-\frac12}(x_{1})L_{n-s}^{-\frac12}(x_{2}).
\{\varphi^{6},\varphi^{2}\bar{\Psi}\Psi,n_{A}={\calB}_{\Sigma}(\hat{A}^{\natural}A),n_{\Psi}={\calB}_{\Sigma}(\bar{\Psi}^{\natural}\Psi+\bar{\Psi}\Psi^{\natural}),n_{\varphi}={\calB}_{\Sigma}(\varphi^{\natural}\varphi)\}.
\omega_{WP}=i\overline{\partial}{\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}}}.
L_{m}^{c}(\mathrm{~\frac{1}{2}~}\epsilon_{ab}L_{m}^{b}L_{m}^{a}+m^{2})=\mathrm{~\frac{1}{2}~}m^{2}(\sigma^{\alpha})_{~d}^{c}L_{m}^{d}L_{\alpha},\qquad[L_{\alpha},\mathrm{~\frac{1}{2}~}\epsilon_{ab}L_{m}^{b}L_{m}^{a}+m^{2}]=0.
\begin{array}{l}{\left[D_{\bar{\theta}}\;-\;H_{\bar{\theta}}^{z}\partial_{z}\;+\;\frac{1}{2}(D_{\theta}H_{\bar{\theta}}^{z})D_{\theta}\right]\Xi\;=\;\frac{j}{2}(\partial_{z}H_{\bar{\theta}}^{z})\Xi.}\\\end{array}
c_{IR}={\frac{1}{16}}\left(7\,N_{c}^{2}-2-9{\frac{N_{c}^{4}}{N_{f}^{2}}}\right),~~~~a_{IR}={\frac{3}{16}}\left(2\,N_{c}^{2}-1-3{\frac{N_{c}^{4}}{N_{f}^{2}}}\right).
\theta_{(3)}^{(1,0)}(\tau)=\sum_{(m_{1},m_{2})\in{\bfZ}^{2}}q^{\frac{1}{9}+m_{1}+3(m_{1}^{2}+m_{2}^{2}-m_{1}m_{2})}=:q^{\frac{1}{9}}{\tilde{\theta}_{(3)}^{0}}(\tau),
={\frac{2(d-1)}{(d-2)!}}\sum_{k=0}^{\infty}C_{k}(z)\sigma^{k}\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}\left(n+\gammam+{\frac{d-1}{2}}\right)^{-2z-2k}\sum_{p=0}^{d-2}n^{p}w_{p}~~~,
\frac{d^{2}\phi}{dx^{2}}-\frac{(1+x^{2})}{x(1-x^{2})}\frac{d\phi}{dx}-\frac{l(l+4)}{4x^{2}(1-x^{2})}\phi+\frac{(\lambda^{2}+\rho_{0}^{2}/x^{2})}{4x(1-x^{2})^{2}}\phi=0\,.
e^{i\delta_{n}}=\frac{\Gamma(\frac{ir_{n}^{2}}{2v}+\frac{1}{2}-\frac{ic}{v})\Gamma(\frac{ir_{n}^{2}}{2v}+\frac{1}{2}+\frac{ic}{v})\Gamma^{6}(\frac{ir_{n}^{2}}{2v}+\frac{1}{2})(r_{n}^{2}-iv)}{\Gamma^{2}(\frac{ir_{n}^{2}}{2v}+1-\frac{ic}{2v})\Gamma^{2}(\frac{ir_{n}^{2}}{2v}+1+\frac{ic}{2v})\Gamma^{2}(\frac{ir_{n}^{2}}{2v}-\frac{ic}{2v})\Gamma^{2}(\frac{ir_{n}^{2}}{2v}+\frac{ic}{2v})(r_{n}^{2}+iv)}
+2(n+1)^{2}G_{2n}c_{-k+2-2n}^{(3)}+(n+1)^{3}G_{3n}c_{-k-3n}^{(3)}+c_{-k+2}^{(1)}+c_{-k-n}^{(1)}G_{n}+c_{-k}^{(0)}=0
R_{f}^{(1)}(z,z^{\prime};\hat{m};l_{1},l_{2})\to\frac{1}{z-z^{\prime}}+\frac{\hat{W}_{1}(z;\hat{m};l_{1},l_{2})}{z^{\prime}-u}+\frac{\hat{W}_{2}(z;\hat{m};l_{1},l_{2})}{(z^{\prime}-u)^{2}}\,.
\chi_{\boldsymbol{\epsilon},\boldsymbol{\alpha},\delta;p}(\boldsymbol{g},\boldsymbol{a},e):=\alphaa(p)+\betab(p)+\gammac(p)+(2\delta{-}\alpha)e(p)+\mathrm{\small~\displaystyle~\sum_{j=1}^8~}\epsilon^{j}g^{j}(p)\,,
L=\frac{1}{2}\intd^{3}x\;\mathrm{tr}\left\{{\bfE}^{2}-{\bfB}^{2}+(D_{0}a)^{2}-({\bfD}a)^{2}+(D_{0}b)^{2}-({\bfD}b)^{2}-\bigl(-i[a,b]\bigr)^{2}\right\},
f_{N}^{\prime}\left[\phi(x)\right]={\frac{1}{Z_{0}}}\sum_{n=0}^{N}{\frac{\left(-1\right)^{n}}{n!}}e^{-S_{0}}\left({\frac{\lambda}{4}}\int\mathrm{d^{d}}x\phi^{4}\right)^{n}\mathrm{Ch}(\Omega_{N},\left\{\phi(x)\right\})
\begin{array}{c}{Z_{N}(+\infty)=N\pi+\chi_{\infty}\left(N_{H}-2N_{S}-M_{C}-2\theta(p-1)M_{W}\right)+}\\{+\pi\,\textrm{sign}(p-1)\,M_{W}+\displaystyle\intdx\,G(x)\,{\calQ}_{N}(+\infty)+\alpha.}\\\end{array}
\begin{array}{rcl}{|z_{1}|}&{=}&{\frac{1}{4}|q_{1}+q_{2}+q_{3}+q_{4}|\,,}\\{}&{}&{}\\{|z_{2}|}&{=}&{\frac{1}{4}|q_{1}-q_{2}+q_{3}-q_{4}|\,,}\\{}&{}&{}\\{|z_{3}|}&{=}&{\frac{1}{4}|q_{1}+q_{2}-q_{3}-q_{4}|\,,}\\{}&{}&{}\\{|z_{4}|}&{=}&{\frac{1}{4}|q_{1}-q_{2}-q_{3}+q_{4}|\,.}\\{}&{}&{}\\\end{array}
{}_{\mathrm{R}}^{(0)T}\langle\mathrm{D}p_{\psi},\eta|=\langle\widetildeB|\langleA|\left(\bar{C}\bar{\gamma}^{0}...\bar{\gamma}^{r}\frac{1+\mathrm{i}\eta\bar{\gamma}}{1-\mathrm{i}\eta}\right)_{AB}
S(\bar{\psi},\psi,\varphi)=\intd^{4}x\left(\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-\sum_{n=1}^{4}\lambda_{n}\varphi^{n}+\bar{\psi}(i\not{\partial}-M-g\varphi)\psi\right)+counterterms,
G_{E}(z_{E})=\int\frac{d^{3}k_{E}}{(2\pi)^{3}}\frac{e^{-ik_{E}\cdotz_{E}}}{k_{E}^{2}+\mu^{2}}=\frac{\mu}{{(2\pi)}^{\frac{3}{2}}(\muz)^{\frac{1}{2}}}K_{\frac{1}{2}}(\muz)\;,
\Gamma_{inv}=\int\left\{-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}(D_{\mu}\underline{\varphi})(D^{\mu}\underline{\varphi})-\frac{1}{8}\frac{m_{H}^{2}}{m^{2}}e^{2}(\varphi_{1}^{2}+2\fracme\varphi_{1}+\varphi_{2}^{2})^{2}\right\}
\Pi_{m}^{r}=\partial_{m}X^{\mu}e_{\mu}^{~r}+i\bar{\theta}\Gamma^{r}\partial_{m}\theta+i\partial_{m}X^{\mu}(\bar{\theta}\Gamma^{r}\Omega_{\mu}\theta)+\frac{i}{2}\partial_{m}X^{\mu}e_{\mu}^{~r}(\bar{\theta}\Gamma^{11}\Omega_{11}\theta)+{\calO}(\theta^{4})~,
X_{\alpha\beta}(x,y)=\left(\begin{array}{cccccc}{1}&{0}&{0}&{0}&{0}&{0}\\{0}&{\delta^{ij}}&{0}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{\delta^{ij}}&{0}&{0}\\{0}&{0}&{0}&{0}&{I_{3\times3}}&{0}\\{0}&{0}&{0}&{0}&{0}&{\frac{1}{m}I_{3\times3}}\\\end{array}\right)\delta(x-y),
\Phi_{\alpha}(x)=\sum_{a=0}^{N^{2}-1}\phi_{\alpha}^{a}(x)T^{a}~~,~~A_{\mu}(x)=\sum_{a=0}^{N^{2}-1}A_{\mu}^{a}(x)T^{a}~~,~~\Psi(x)=\sum_{a=0}^{N^{2}-1}\psi^{a}(x)T^{a}
x^{2}\hat{G}\simeq\frac{(2\!-\!\xi)}{3\pi^{2}N}\int_{0}^{x}dy\;\hat{G}^{3}(y)+\frac{x^{2}}{\pi^{2}N}\int_{x}^{\infty}dy\;\frac{\hat{G}^{3}(y)}{y^{2}}+\frac{x^{4}\xi}{3\pi^{2}N}\int_{x}^{\infty}dy\;\frac{\hat{G}^{3}(y)}{y^{4}}.
\hat{f}_{\mu}-\partial_{\mu}\hat{\phi}=\epsilon_{\mu\alpha\beta}\partial^{\alpha}\hat{A}^{\beta}+\epsilon_{\mu\alpha\beta}\partial^{\alpha}\hat{a}^{\beta}-\hat{h}_{\mu}+(\epsilon_{\nu\alpha\beta}\partial^{\alpha}\hat{A}^{\beta}-\hat{f}^{\nu}+\partial^{\nu}\hat{\phi})\frac{\delta\hat{h}_{\nu}}{\delta\hat{f}^{\mu}}+O(\theta^{2}),
J_{\nu_{1}\nu_{2}\nu_{3}}^{(d)}=\frac{1}{\pi^{d}}\int\!\!\int{d^{d}k_{1}~d^{d}k_{2}}~P_{k_{1},m_{1}}^{\nu_{1}}P_{k_{1}-k_{2},m_{2}}^{\nu_{2}}P_{k_{2},m_{3}}^{\nu_{3}}.
2^{N}\prod_{a=1}^{N}\sum_{n_{a}=0}^{\infty}\frac{(\omega\beta^{(a)})^{2n_{a}}}{(2n_{a})!}\prod_{b=1}^{N}\sum_{q_{b}=0}^{2n_{b}}{\binom{2n_{b}}{q_{b}}}\left\langle\prod_{c=1}^{N}\left[\chi_{+}^{(c)}[Q^{0,S}]\right]^{q_{c}}\left[\chi_{-}^{(c)}[Q^{0,S}]\right]^{2n_{c}-q_{c}}\right\rangle_{Q^{0,S}}\;.
{\calL}_{0}=(\partial^{\mu}+ig_{0}A^{\mu})\phi\,(\partial_{\mu}-ig_{0}A_{\mu}){\bar{\phi}}-m_{0}^{2}\phi\,{\bar{\phi}}-\frac{\lambda_{0}}{4}(\phi\,{\bar{\phi}})^{2}-\frac{1}{4\Omega_{D-2}}\,F_{\mu\nu}F^{\mu\nu}.
\chi_{\gamma}\left(2\right)=(n-2)!\\sum_{i\in{\calI}}\frac{D(p_{1},\ldots,p_{i-1},p_{i}-2,p_{i+1},\ldots,p_{m})}{p_{1}!\cdotsp_{i-1}!\(p_{i}-2)!\p_{i+1}!\\cdotsp_{m}!},
\left.\begin{array}{lll}{\gamma_{1,3,(m,0,\pm1)}}&{=}&{-\alpha{\bf1}_{1}}\\{\gamma_{1,{\tilde{7}}_{3},(m)}}&{=}&{-\alpha{\bf1}_{1}}\\{\gamma_{1,7_{4},(0)}}&{=}&{-\alpha{\bf1}_{2}}\\{\gamma_{1,7_{5},(\pm1)}}&{=}&{{\bf0}}\\\end{array}\right.
e_{n_{+},n_{-}}^{\alpha}(z)=\prod_{r=0}^{n_{+}-1}\left(1-e^{4\pii\alpha\alpha_{+}}q_{+}^{r}\right)\prod_{r^{\prime}=0}^{n_{--}1}\left(1-e^{4\pii\alpha\alpha_{-}}q_{-}^{r^{\prime}}\right)\tilde{e}_{n_{+},n_{-}}^{\alpha}(z)~,
F(z_{12})^{-1}=\left(\begin{array}{cc}{i{\calX}_{12}^{-1}}&{-2i{\calX}_{12}^{-1}\theta_{12}^{j}}\\{-2i\bar{\theta}_{12i}{\calX}_{12}^{-1}}&{V_{i}^{~j}(z_{12})}\\\end{array}\right)
\Theta_{\;j}^{i}=\varepsilon_{0}\vec{E}_{i}\vec{E}_{j}+\frac{1}{\mu_{0}}\vec{B}_{i}\vec{B}_{j}+[{\calH}_{p}-\frac{\varepsilon_{0}}{2}\vec{E}_{k}^{2}-\frac{1}{2\mu_{0}}\vec{B}_{k}^{2}]\,\delta_{j}^{i}.
S=\frac{1}{16\pi}\int\left\{-R+2\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}e^{4\phi}{\partial_{\mu}}\kappa\partial^{\mu}\kappa-e^{-2\phi}F_{\mu\nu}F^{\mu\nu}-\kappaF_{\mu\nu}{\tilde{F}}^{\mu\nu}\right\}\sqrt{-g}d^{4}x,
T(z)\timesV_{p^{+}p^{-}}^{r,s}(w)\rightarrow\frac{-r+\frac1k\left(\frac14+s^{2}\right)}{(z-w)^{2}}V_{p^{+}p^{-}}^{r,s}(w)+\frac1{z-w}\partial_{w}V_{p^{+}p^{-}}^{r,s}(w)
\begin{array}{ccccccccccccc}{0\!\!\!}&{\!\stackrel{}{\longrightarrow}\!\!\!}&{\!\Omega^{0}\!\!\!}&{\!\stackrel{d_{0}}{\longrightarrow}\!\!\!}&{\cdots\!\!\!}&{\!}&{\stackrel{d_{k-3}}{\longrightarrow}}&{\!\Omega^{k-2}\!\!\!}&{\!\stackrel{d_{k-2}}{\longrightarrow}\!\!\!}&{\!\Omega_{k-1}^{\prime}\!\!\!}&{\!\stackrel{d_{k-1}^{*}d_{k-1}}{\longrightarrow}\!\!\!}&{\!0,\!\!\!}&{}\\\end{array}
g_{{\bf{k}},{\bf{q}}}({\bf{x}})=-e^{-i{\bf{q.x}}}(\frac{1}{2\mathrm{~}N\mathrm{~}\epsilon_{{\bf{q}}}})\Lambda_{{\bf{k}}}(-{\bf{q}})\omega_{{\bf{k}}}({\bf{q}})+i\mathrm{~}U_{{\bf{q}}}({\bf{x}})\Lambda_{{\bf{k}}}(-{\bf{q}})=-f_{{\bf{k}},{\bf{q}}}^{*}({\bf{x}})
S\left(A\right)=\mathcal{D}^{-1}\left(\begin{array}{cc}{d}&{-p^{-2}b}\\{-p^{2}c}&{a}\\\end{array}\right)=\left(\begin{array}{cc}{d}&{-q^{-2}b}\\{-q^{2}c}&{a}\\\end{array}\right)\mathcal{D}^{-1}\,,
{M_{c}=\left(\begin{matrix}{0}&{1}&{0}&{0}&{\cdots}&{0}\\{0}&{0}&{1}&{0}&{\cdots}&{0}\\{0}&{0}&{0}&{1}&{\cdots}&{0}\\{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}\\{1}&{0}&{0}&{0}&{\cdots}&{0}\\\end{matrix}\right),}
E_{\mathrm{NP}}^{(\pm)}(\epsilon,N_{\pm})=\alpha^{2}\frac{(-1)^{N_{\pm}+1}}{N_{\pm}!}\left(-\frac{2}{g^{2}}\right)^{\pm\epsilon+2N_{\pm}}\Gamma\left(\mp\epsilon-N_{\pm}\right)+O(\alpha^{4}).
\Pi_{ik}^{+}\left(L_{n}^{1/2}(x)\right)_{kl}{\calL}_{n}(x)\Pi_{lj}^{+}=-(x+i/2)(x-3i/2)\rho_{i}A_{n}(x)\rho_{j}^{+}(\rho^{+}\rho+1)^{-1}
{\calV}={\textstyle\frac{1}{4}}e^{-\varphi}\,\left[2{\calM}^{nq}f_{mn}{}^{p}f_{pq}{}^{m}+{\calM}^{mq}{\calM}^{nr}{\calM}_{ps}f_{mn}{}^{p}f_{qr}{}^{s}\right]\,.
\hat{I}=-\frac{1}{4}\intd^{4}x\;[F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}\theta^{\alpha\beta}F_{\alpha\beta}F^{\mu\nu}F_{\mu\nu}+2\theta^{\alpha\beta}F_{\alpha\mu}F_{\beta\nu}F^{\mu\nu}]+O(\theta^{2})\;,
E\left(\mathcal{P}_{\left\{n_{i}+m_{i}\theta_{i}\right\}},\Pi_{k_{i}}\right)=\left(\prod_{i=1}^{s}\left(n_{i}+m_{i}\theta_{i}\right)\right)\left(\prod_{i=s+1}^{l}\frac{k_{i}}{p_{i}}\right).
\lambda=\frac{\tilde{p}_{0}-\sqrt{{\mathrm{\bf~p}^{2}}+m^{2}}}{m},\quad\dot{\mathrm{\bf~x}}=\frac{{\calP}}{m}\frac{\mathrm{\bf~p}}{\sqrt{{\mathrm{\bf~p}^{2}}+m^{2}}}{.}
+i\{gg^{\mu\nu}\delta(x-y)+{\frac{1}{e^{2}}}N^{\mu\beta}(x)N^{\nu\lambda}(y)({\frac{a}{a^{2}\partial^{2}+4\alpha^{2}}}g_{\beta\lambda}\delta(x-y)+{\frac{2\alpha}{\partial^{2}(a^{2}\partial^{2}+4\alpha^{2})}}\epsilon_{\beta\lambda\sigma}\partial^{\sigma}\delta(x-y))\},
\frac{\partial\Psi^{0}}{\partial\Pi^{i}}=-\frac{1}{2}\frac{R_{vac}^{2}\Pi^{*i}}{\left(\sqrt{\sum_{s=1}^{N}|\Pi^{s}|^{2}+R_{vac}^{2}}\right)^{3}},\frac{\partial\Psi^{0}}{\partial\Pi^{*k}}=-\frac{1}{2}\frac{R_{vac}^{2}\Pi^{k}}{\left(\sqrt{\sum_{s=1}^{N}|\Pi^{s}|^{2}+R_{vac}^{2}}\right)^{3}}
a=\frac{1}{4\pi}\sum_{j,m}\left(2j+1\right)\left(\begin{array}{c}{a_{+}\left(j,m;r\right)D_{+1,m}^{j}\left(\theta,\varphi\right)}\\{a_{0}\left(j,m;r\right)D_{0,m}^{j}\left(\theta,\varphi\right)}\\{a_{\tilde{o}}\left(j,m;r\right)D_{\tilde{o},m}^{j}\left(\theta,\varphi\right)}\\{a_{-}\left(j,m;r\right)D_{-1,m}^{j}\left(\theta,\varphi\right)}\\\end{array}\right)
\bar{\lambda}\xi=\bar{\xi}\lambda\;\qquad\bar{\lambda}\gamma^{a^{\prime}}\xi=\bar{\xi}\gamma^{a^{\prime}}\lambda\;\qquad\bar{\lambda}\gamma^{a^{\prime}b^{\prime}}\gamma_{ab}\xi=\bar{\xi}\gamma^{a^{\prime}b^{\prime}}\gamma_{ab}\lambda\.
\gamma_{\lambda}:=\mu\frac\partial{\partial\mu}\lambda_{\mathrm{R}}=2\lambda_{\mathrm{R}}\left[\frac{13}6-\frac{\lambda_{\mathrm{R}}}{2}+\lambda_{\mathrm{R}}\xi_{\mathrm{R}}(1-\xi_{\mathrm{R}})\right]\frac{C_{2}(G)}{(4\pi)^{2}}g_{\mathrm{R}}^{2}.
|V_{3}\rangle=\mathrm{exp}\left[-b_{0}^{A}(\tilde{N}_{0}^{AB}|E^{-1}|c^{B+})-(b^{A+}|E\tilde{N}^{AB}E^{-1}|c^{B+})\right]|+\rangle_{1}|+\rangle_{2}|+\rangle_{3}.
\frac{1}{z}F(z)+z^{l+1}F(\frac{1}{z})+\frac{\psi(1)(-1)^{l}}{2(l+1)}\frac{1-z^{l+1}}{1-z}=\frac{\lambda-1}{2}(-1)^{l}(1-z)^{l}F^{\prime}(\frac{1}{1-z}).
\left\langle{{\xi}\left\vert{\xi^{\prime}}\right\rangle}\right.={\frac{\left({1-{\vert\xi\vert}^{2}}\right)^{K}\left({1-{\vert\xi^{\prime}\vert}^{2}}\right)^{K}}{\left({1-\xi^{*}\xi^{\prime}}\right)^{2K}}}\,
\sigma_{ik}^{\prime}=\eta\Bigl({\frac{\partialv_{i}}{\partialx_{k}}}+{\frac{\partialv_{k}}{\partialx_{i}}}-{\frac{2}{3}}\delta_{ik}{\frac{\partialv_{l}}{\partialx_{l}}}\Bigr)+\zeta\delta_{ik}{\frac{\partialv_{l}}{\partialx_{l}}}\,
u_{1}^{\dot{a}}Q^{\dot{a}+}u_{2}^{a}Q^{b-}\midB\rangle=u_{1}^{\dot{a}}\gamma_{\dot{a}b}^{I}u_{2}^{b}\oint(\partialX^{I}+M^{IJ}\bar{\partial}X^{J})\midB\rangle