harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{\hat{g}_{\mathrm{R}}^{2}}:=\frac{1}{g^{2}}+\frac{7}{3}\frac{1}{16\pi^{2}}\,t\quad\Rightarrow\quad\hat{\beta}_{g^{2}}:=\partial_{t}\hat{g}_{\mathrm{R}}^{2}=-\frac{7}{3}\frac{1}{16\pi^{2}}\,\hat{g}_{\mathrm{R}}^{4}.
{\calL}_{gf}=-\frac{1}{2}\sqrt{-g}(1+a_{2}^{2})\left(\nabla^{\rho}h_{\rho}^{\mu}-\frac{1}{2}\nabla^{\mu}h_{\rho}^{\rho}\right)\left(\nabla^{\sigma}h_{\mu\sigma}-\frac{1}{2}\nabla_{\mu}h_{\sigma}^{\sigma}\right)\.
S_{5}^{\mathrm{\tiny{A}}}=\intd^{5}x\left[\frac{1}{4g^{2}}\intd^{2}\theta\;W^{\alpha}W_{\alpha}+\mathrm{h.c.}+\intd^{4}\theta\frac{1}{g^{2}}\left((\sqrt{2}\partial_{5}V-\bar{\phi})(\sqrt{2}\partial_{5}V-\phi)-\partial_{5}V\partial_{5}V\right)\right]
\psi=\left[\begin{matrix}{0}\\{0}\\{q^{4}-iq^{3}}\\{q^{2}-iq^{1}}\\\end{matrix}\right]=\left[\begin{matrix}{0}\\{0}\\{M^{1}}\\{M^{2}}\\\end{matrix}\right],\quad\quad\gamma_{i}=\left(\begin{matrix}{0}&{i\sigma_{i}}\\{-i\sigma_{i}}&{0}\\\end{matrix}\right),\quad\gamma_{4}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right),
R(r)\equivR_{nl}(r;\delta_{1},\delta_{2})=C_{nl}(\delta_{1},\delta_{2})(\varepsilonr)^{l+\frac{\delta_{1}+\delta_{2}}{2}}\mathrm{e}^{-\frac{\varepsilon}{2}r}{_1F}_{1}(-n+l+1,2l+\delta_{1}+\delta_{2}+2;\varepsilonr),
{}_{1}^{S}\!\langlei|\rho_{1}|j\rangle_{1}^{S}={}_{1}^{S}\!\langlei|\left(\sum_{k,l}|k\rangle_{1}^{S}{}_{2}\!\langle\tilde{k}|\tilde{l}\rangle_{2}{}_{1}^{S}\!\langlel|\right)|j\rangle_{1}^{S}={}_{2}\!\langle\tilde{i}|\tilde{j}\rangle_{2}\propto\delta_{ij}
\hat{Q}_{1}^{\prime}(\bar{z})=\hat{Q}_{1}(\bar{z})-(\frac{1}{2}+\frac{1}{3\sqrt{3}})p-\frac{2}{3\sqrt{3}}q,\\,hatQ_{2}^{\prime}(\bar{z})=\hat{Q}_{2}(\bar{z})-(\frac{1}{2}-\frac{1}{3\sqrt{3}})p+\frac{2}{3\sqrt{3}}q.
\widetilde{f}({\bfs})=\frac{n_{D}}{(2\pi)^{{D}/2}}\intd^{{D}}ue^{-i{\bfs}\cdot{\bfu}}f({\bfu})=\frac{n_{D}}{s^{{D}/2-1}}\,\int_{0}^{\infty}f(u)J_{{D}/2-1}(su)u^{{D}/2}\,du\;.
\frac{e}{\midx_{i}\mid}=\frac{e}{[(q_{i}^{2}-\theta^{ij}q_{i}Q_{j}+O(\theta^{2})]^{\frac{1}{2}}}\approx\frac{e}{\midq_{i}\mid}(1-\frac{\theta^{ij}q_{i}Q_{j}}{2q_{i}^{2}}+O(\theta^{2})).
E_{(2)}^{(0)}=-\frac12\int_{M}^{\infty}\frac{d\kappa}{\pi}\sqrt{\kappa^{2}-M^{2}}\left[\frac{m}{\kappa^{2}-m^{2}/4}+\frac{2m}{\kappa^{2}-m^{2}}-\frac{3m}{\kappa^{2}}\right]\,.
C_{W_{L}W_{L}W_{L}W_{L}}\propto\mathrm{{Tr}}(\{T^{a},T^{b}\}\{T^{c},T^{d}\})\frac{N_{c}^{2}}{N_{f}}=\left(d^{abe}d^{cde}+\frac{1}{2N_{f}}\delta^{ab}\delta^{cd}\right)\frac{N_{c}^{2}}{N_{f}}.
\begin{array}{l}{z_{0}=i\left((\alpha_{1}-\alpha_{2})T+{\frac{1}{3}}\right)^{1/2}e^{-i(\alpha_{1}+\alpha_{2})\sigma}}\\{z_{1}=i\left((-\alpha_{1}-2\alpha_{2})T-{\frac{1}{3}}\right)^{1/2}e^{i\alpha_{1}\sigma}}\\{z_{2}=\pmi\left((2\alpha_{1}+\alpha_{2})T-{\frac{1}{3}}\right)^{1/2}e^{i\alpha_{2}\sigma}}\\\end{array}
M_{r=\infty}=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{1}&{1}&{0}&{0}\\{0}&{5}&{1}&{0}\\{0}&{0}&{1}&{1}\\\end{matrix}\right),\qquadM_{c}=\left(\begin{matrix}{1}&{-5}&{-1}&{-1}\\{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\\end{matrix}\right),\qquadM_{g}=\left(\begin{matrix}{-4}&{-10}&{-2}&{-1}\\{1}&{1}&{0}&{0}\\{0}&{5}&{1}&{0}\\{0}&{0}&{1}&{1}\\\end{matrix}\right)
{\calZ}(G,\lambdaA,N)=\sum_{\nu\in\Sigma_{G}}\frac{(-1)^{\tilde{t}}}{|S_{\nu}|}e^{-\frac{(n+\tilde{n})\lambdaA}{2}}\frac{(\lambdaA)^{(i+t+\tilde{t}+h)}}{i!t!\tilde{t}!h!}N^{(n+\tilde{n})(2-2G)-2(t+\tilde{t}+h)-i}\left[1+{\calO}\left({\frac{1}{N}}\right)\right],
\theta^{\rho\sigma}{\calR}_{\mu\nu\rho\sigma}^{(1)}=\frac{1}{2}\theta^{\rho\sigma}\left[2{\calR}_{\mu\rho}^{\\i}{\calR}_{\nu\sigmai}-\omega_{\rho}^{\\i}\left(\partial_{\sigma}{\calR}_{\mu\nui}+D_{\sigma}{\calR}_{\mu\nui}\right)\right]{\bf1},
\tilde{\tau}=\tau-\eta(\tau)\\,\\\delta_{\eta}x^{\mu}=\tilde{x}^{\mu}(\tau)-x^{\mu}(\tau)=\eta\dot{x}^{\mu}(\tau)\\,\\\delta_{\eta}e(\tau)=\frac{d}{d\tau}\left(\eta(\tau)e(\tau)\right)\,
\begin{array}{rcl}{\hat{D}_{i}e_{j}^{a}}&{=}&{\Gamma_{ij}^{k}e_{k}^{a}+N_{ij}e^{a}}\\{\hat{D}_{i}e^{a}}&{=}&{-{N_{i}}^{k}e_{k}^{a}.}\\\end{array}
B\Psi(\alpha_{a_{0}},\ldots,\alpha_{a_{n}})=\frac{1}{n+1}\Bigl(\Psi(1,\alpha_{a_{0}},\ldots,\alpha_{a_{n}})+(-1)^{n}\Psi(\alpha_{a_{0}},\ldots,\alpha_{a_{n}},1)\Bigr)\pmcycl.
P^{\mu}={\frac{\partialL}{\partialu_{\mu}}}=L^{-1}[M^{2}u^{\mu}+{\frac{MJ}{2}}\epsilon^{\mu\nu\lambda}\sigma_{\nu\lambda}]~~,~~S^{\mu\nu}={\frac{\partialL}{\partial\sigma_{\mu\nu}}}={\frac{L^{-1}}{2}}[J^{2}\sigma^{\mu\nu}+{MJ}\epsilon^{\mu\nu\lambda}u_{\lambda}].
\delta_{\lambda}\tilde{F}=i[\lambda,\tilde{F}]_{M}\Longrightarrow\quad\delta_{\lambda}(\frac1{(\tilde{D}^{2})^{2}}\star\tilde{F})=i[\lambda,\frac1{(\tilde{D}^{2})^{2}}\star\tilde{F}]_{M},
M_{d}^{d-2}=M_{d+2}^{d}\,\int\psi_{1}(z)^{2}\,dz\,d\theta=e^{u_{0}}u_{0}^{(d+1)/2d}\Gamma\left(\frac{d-1}{2d}\right)\pi\gamma^{2}M_{d+2}^{d}\.
\Psi_{(3,3,2)}=\prod_{i<j}(z_{i}^{(1)}-z_{j}^{(1)})^{3}\prod_{i<j}(z_{i}^{(2)}-z_{j}^{(2)})^{3}\prod_{i,j}(z_{i}^{(1)}-z_{j}^{(2)})^{2}\;\Psi_{0}.
{\bfM}=\frac{1}{AD-BC}\left(\begin{array}{crc}{\frac{1}{2}({A^{2}}+{B^{2}}+{C^{2}}+{D^{2}})}&{AB+CD}&{\frac{1}{2}(-{A^{2}}+{B^{2}}-{C^{2}}+{D^{2}})}\\{AC+BD}&{AD+BC}&{-AC+BD}\\{\frac{1}{2}(-{A^{2}}-{B^{2}}+{C^{2}}+{D^{2}})}&{-AB+CD}&{\frac{1}{2}({A^{2}}-{B^{2}}-{C^{2}}+{D^{2}})}\\\end{array}\right)\,\,\,.
{\calE}\equivM=\int_{-\infty}^{\infty}dx\left[\frac{1}{2}{\dot{\phi}}^{2}+\frac{1}{2}{{\phi}^{\prime}}^{2}+\frac{\lambda}{4}\left(\phi^{2}-\frac{m^{2}}{\lambda}\right)^{2}+\varepsilon\frac{m^{3}}{\sqrt\lambda}\phi\right]=\frac{2{\sqrt2}m^{3}}{3\lambda}
ds_{4}^{2}=-(H_{0}H_{1}H_{2}H_{3})^{-{\frac{1}{2}}}(1-{\frac{r_{0}}{r}})dt^{2}+(H_{0}H_{1}H_{2}H_{3})^{\frac{1}{2}}[{\frac{1}{1-{\frac{r_{0}}{r}}}}dr^{2}+r^{2}d\Omega_{2}^{2}]
x_{2}-x_{1}=\left(\begin{matrix}{0}\\{-1/2}\\\end{matrix}\right),\;\;\;x_{3}-x_{2}=\left(\begin{matrix}{-1/2}\\{1/2}\\\end{matrix}\right),\;\;\;x_{1}-x_{3}=\left(\begin{matrix}{1/2}\\{0}\\\end{matrix}\right).
\epsilon:={\frac{\calE}{\delta^{2}H_{c}^{2}}}={\frac{2\pi}{\lambda}}\left\{[r\varphi^{\prime}\varphi]_{0}^{\infty}+\int_{0}^{\infty}dr\;r\left[h^{2}+{\frac{\lambda}{2}}(1-\varphi^{4})\right]\right\},
{\calR}={\frac{p}{\tilde{L}^{2}}}\left[(p+1)(\tilde{\ell}^{2}-\tilde{E}^{2})\left({\frac{\tilde{L}}{a}}\right)^{2(p+1)}+(p-1)\tilde{\ell}^{2}\left({\frac{\tilde{L}}{a}}\right)^{2p}\right].
F_{1}(0)=Z+\int\frac{{\mathrmd}^{2}{\vec{k}}_{\perp}{\mathrmd}x}{16\pi^{3}}\Big[\psi_{+\frac{1}{2}}^{\uparrow\*}(x,{\vec{k}}_{\perp})\psi_{+\frac{1}{2}}^{\uparrow}(x,{\vec{k}}_{\perp})+\psi_{-\frac{1}{2}}^{\uparrow\*}(x,{\vec{k}}_{\perp})\psi_{-\frac{1}{2}}^{\uparrow}(x,{\vec{k}}_{\perp})\Big]\,
\Gamma_{3}=\frac{\lambda^{3}}{8\left(2\pi\right)^{5}}\int\frac{\mathrm{d}^{5}Q}{|{\bfQ}|}N\left(|{\bfQ}|\right)\left\{\frac{k_{1}^{2}\,k_{2}\cdotk_{3}}{\left(Q\cdotk_{1}\right)^{2}Q\cdotk_{2}Q\cdotk_{3}}+permutations\right\},
H_{T}=\intd^{3}x\Bigg[\frac{g^{2}}{2}\left(\Pi_{ai}-\frac{\theta}{8\pi^{2}}B_{ai}\right)^{2}+\frac{1}{2g^{2}}B_{ai}^{2}-A_{a0}D_{i}\Pi_{ai}+\lambda_{a}\Pi_{a}\Bigg]~,
\lbrack(\bar{P}+\bar{K})_{\bar{l}},\chi^{i\alpha}\rbrack=\lbrack(\bar{P}+\bar{K})_{\bar{l}},\bar{\chi}_{i}^{\alpha}\rbrack=0\qquad\qquad\{\chi^{i\alpha},\bar{\chi}_{k\beta}\}=\delta_{k}^{i}\delta_{\beta}^{\alpha}
S=\,{\frac{8\pi^{2}}{g^{2}}}+2\pi^{2}\rho^{2}(|v_{+}|^{2}+|v_{-}|^{2})-{\frac{i}{\sqrt{2}}}\left(\begin{array}{llll}{\bar{v}_{+}}&{}\\{}&{\bar{v}_{-}}\\\end{array}\right)_{f}^{\,\,\dot{\beta}}\mu_{\dot{\beta}}\,({\calK}_{f}+\tilde{\calK}_{f})\,
K_{\rho}(\rhoz)\sim\sqrt{\frac{\pi}{2\rho}}{\frac{e^{-\rho\eta}}{(1+z^{2})^{\frac{1}{4}}}}\left[1+\sum_{k=1}^{\infty}(-1)^{k}{\frac{u_{k}(t)}{\rho^{k}}}\right],
u^{-m\ell/2}\tilde{\calH}_{r}u^{m\ell/2}=-2\omega\left[u{\frac{d^{2}}{du^{2}}}+\left(m\left({\ell+g\brace\ell+{\frac{1}{2}}(g_{e}+g_{o})}\right)+1-u\right){\frac{d}{du}}-{\frac{m\ell}{2}}\right].
\psi(x)=(2\pi)^{-3/2}\!\!\!\int\!\!d^{3}p\left(b_{s}({\mathbf{p}})u_{s}({\mathbf{p}})e^{-ipx}\!+\!{d_{s}({\mathbf{p}})}^{\dagger}v_{s}({\mathbf{p}})e^{ipx}\right).
P=P_{0}(l)\left(\frac{2}{\lambda}\right)^{2l+3}\frac{(l+1)!^{2}}{\pi|C_{l}|^{2}}=P_{0}(l)\left(\frac{2\piT}{\omega}\right)^{2l+3}\frac{(l+1)!^{2}}{\pi|C_{l}|^{2}}\,,
S_{\psi}^{(2)}=\intd\taud\sigma\sum_{n}~\theta_{0}^{\mu_{1}}\cdots\theta_{0}^{\mu_{n}}~\theta_{0}^{p+1}\cdots\theta_{0}^{9}~\frac{1}{n!}A_{\mu_{1},\cdots,\mu_{p}}^{(n)}\left(X(\tau,\sigma)\right).
{\calL}_{m}(g)=\frac{1}{2^{m}}\sqrt{-g}\;\;\delta_{c_{1}d_{1}\cdotsc_{m}d_{m}}^{a_{1}b_{1}\cdotsa_{m}b_{m}}\;\;R_{a_{1}b_{1}}{}^{c_{1}d_{1}}\cdotsR_{a_{m}b_{m}}{}^{c_{m}d_{m}}
S_{\mathrm{total}}=S_{\mathrm{EM}}+S_{\mathrm{M}}=\frac{1}{16\piG}\intd^{4}x\sqrt{-g_{(4)}}\left[R_{(4)}-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}\right]+S_{\mathrm{M}},
D_{\mu\nu}(x-x^{\prime})=\int\frac{d^{4}q}{(2\pi)^{4}}\frac{\mathrm{e}^{iq\cdot(x-x^{\prime})}}{q^{2}-i\epsilon}\left(g_{\mu\nu}-(1-\xi)\frac{q_{\mu}q_{\nu}}{q^{2}}\right),
W^{\mu\nu}\equiv\frac{\partial^{2}L}{\partial\dot{x}_{\mu}\partial\dot{x}_{\nu}}=-\frac{m}{\sqrt{\dot{x}^{2}}}\left(g^{\mu\nu}-\frac{\dot{x}^{\mu}\dot{x}^{\nu}}{\dot{x}^{2}}\right)=-m\left(g^{\mu\nu}-\dot{x}^{\mu}\dot{x}^{\nu}\right)\;,
{\cal{F}}\left\{P^{\alpha}\right\}(x)=2^{2\alpha+\frac{\nu}{2}}\frac{\Gamma\left(\alpha+\frac{\nu}{2}\right)}{\Gamma\left(-\alpha\right)}sin\pi\left(\alpha+\frac{\nu}{2}\right)Q_{-}^{-\alpha-\frac{\nu}{2}}
X_{a}(R^{c}{}_{b\mu\nu})-X_{b}(R^{c}{}_{a\mu\nu})+f^{c}{}_{ad}R^{d}{}_{b\mu\nu}+f^{d}{}_{ba}R^{c}{}_{d\mu\nu}-f^{c}{}_{bd}R^{d}{}_{a\mu\nu}=0.
a_{2}=f\sum_{i=1}^{i_{3}}{\calU}_{i}^{(2,2)}\;I_{i}^{(2)}+f_{;N}\sum_{i=1}^{i_{2}}{\calU}_{i}^{(2,3/2)}\;I_{i}^{(3/2)}+f_{;NN}\sum_{i=1}^{i_{1}}{\calU}_{i}^{(2,1)}\;I_{i}^{(1)}.
H_{\vec{p},L}^{(1M)}=m_{ML}\;(b_{\vec{p},h=-1}^{(0)+}b_{-\vec{p},h=-1}^{(0)+}+b_{-\vec{p},h=-1}^{(0)}b_{\vec{p},h=-1}^{(0)}+d_{\vec{p},h=1}^{(0)+}d_{-\vec{p},h=1}^{(0)+}+d_{-\vec{p},h=1}^{(0)}d_{\vec{p},h=1}^{(0)}).
S=\int_{M_{6}}[\frac{1}{2}P^{\mu\nu}\partial_{5}B_{\mu\nu}-\frac{1}{4}({}^{*}H^{ij}+\frac{1}{2}P^{ij})({}^{*}H_{ij}+\frac{1}{2}P_{ij})-2^{*}H^{0i}{}^{*}H_{0i}]d^{6}x,
\left[r_{\dot{J}}^{\dot{I}},f_{\dot{L}}^{\dot{K}}\right]=\sum_{\dot{K}_{1}\dot{K}_{2}=\dot{K}}\delta_{\dot{J}}^{\dot{K}_{2}}f_{\dot{L}}^{\dot{K}_{1}\dot{I}}-\sum_{\dot{L}_{1}\dot{L}_{2}=\dot{L}}\delta_{\dot{L}_{2}}^{\dot{I}}f_{\dot{L}_{1}\dot{J}}^{\dot{K}}.
[\hat{\phi}^{a}(x),\hat{\pi}^{b}(y)]=i\delta^{ab}\delta(x-y)\qquad,\qquad\{\hat{\psi}_{\alpha}^{a}(x),\hat{\psi}_{\beta}^{b}(y)\}=\delta^{ab}\delta_{\alpha\beta}\delta(x-y)\quad,
\left(\begin{array}{c}{x^{1}}\\{x^{2}}\\\end{array}\right)\longrightarrow\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)\left(\begin{array}{c}{x^{1}}\\{x^{2}}\\\end{array}\right)\quad\mathrm{with}\quadad-bc=1.
I=(\frac{K_{R}^{2}s}{4M})\frac{(v_{i}-u_{i})^{2}}{1-C}+\frac{Ms}{2}+\frac{K_{R}(v_{i}-u_{i})^{2}}{4(1-C)}(\frac{K_{R}s}{M}-S)]+(s\leftrightarrows^{\prime}).
{\calH}=-\frac{1}{4}\int_{-\infty}^{\infty}d\tau:[P^{2}+{X^{\prime}}^{2}-\frac{\sqrt{\pi}}{\betav^{2}}(PX^{\prime}P+\frac{1}{3}{X^{\prime}}^{3})-\frac{1}{2\beta\sqrt{\pi}}(\frac{v^{\prime\prime}}{3v^{3}}-\frac{(v^{\prime})^{2}}{2v^{4}})X^{\prime}]:,
4g_{4}\lambda^{3}+6\nu_{3}c_{1}\lambda^{2}+2(g_{2}+2\widetilde{g}_{2}c_{2}+\nu_{1}c_{1}^{2})\lambda+2(\muc_{1}+\nu_{1}c_{1}c_{2}+2\nu_{2}c_{1}^{3}+\nu_{3}c_{3})=2P\int_{-2a}^{2b}d\tau\,\frac{u(\tau)}{\lambda-\tau},
{\calW}^{(3)}={\frac{C_{F}C_{A}(LT)^{2}(2T)^{4-4\omega}}{\pi^{2\omega}}}e^{-2i\pi\omega}[I_{1}(\beta^{2})+I_{2}(\beta^{2})+I_{3}(\beta^{2})+I_{4}(\beta^{2})]
S=-\frac{1}{2\pi}\intd^{2}\sigma\left[\partial_{\alpha}X^{\mu}\partial^{\alpha}X_{\mu}-i\;\bar{\psi}^{\mu,j}\;\rho^{\alpha}\;\partial_{\alpha}\;\psi_{\mu,j}+i\;\bar{\phi}^{\mu,k}\;\rho^{\alpha}\;\partial_{\alpha}\;\phi_{\mu,k}\right]
\mathrm{(B)}~~:~~\frac{c_{0}}{d_{0}}=-\bigg[\frac{(\phi_{0}^{\prime})^{2}}{K^{\prime\prime}K^{\prime}K}\bigg]_{z=z_{c}^{-}}-\int_{0}^{z_{c}}dz\bigg(\frac{\phi_{0}^{\prime}}{K^{1/2}K^{\prime}}\bigg)^{2}.
M(s,t,u)=\sum_{i=0}^{\infty}{\alpha}_{i}^{(u)}(u){\nu}_{u}^{i}+\sum_{R_{s}}^{}\frac{\rho^{(us)}(u)}{(\sigma-2M_{R}^{2})-u+{\nu}_{u}}+\sum_{R_{t}}^{}\frac{\rho^{(ut)}(u)}{(\sigma-2M_{R}^{2})-u-{\nu}_{u}}\.
\zeta_{1,2}=\left(\frac{K}{2}\pmZ\right)^{2}=-\rho^{2}+(w\pm{\frac{1}{2}})^{2}t,\\zeta_{1,2}^{\prime}=\left(\frac{K}{2}\pmZ^{\prime}\right)^{2}=-{\rho^{\prime}}^{2}+(w^{\prime}\pm{\frac{1}{2}})^{2}t\\\tag{3.3}
T_{0}=[{\bfq}\wedge{\bfk}]\frac{w_{p}^{2}}{w_{k}}+[{\bfp}_{1}\wedge{\bfp}_{3}]\frac{w_{p}^{2}+w_{k}^{2}}{2w_{k}}+\frac{L}{\pi}[{\bfp}_{1}\wedge{\bfp}_{3}]
\partial_{\mu}X^{a^{\prime}}\partial_{\nu}X^{a^{\prime}}{\calH}^{*\mu\nu}\,\qquad\partial_{\mu}X^{a^{\prime}}\partial_{\nu}X^{a^{\prime}}\bar{\lambda}\gamma^{\mu}\partial^{\nu}\lambda\,\qquad{\calH}^{*\mu\nu}{\calH}_{\mu\nu}(\partialX^{a^{\prime}})^{2}
R_{\mu\nu\sigma}^{\lambda}(\Gamma)\equiv\Gamma_{\mu\nu,\sigma}^{\lambda}-\Gamma_{\mu\sigma,\nu}^{\lambda}+\Gamma_{\alpha\sigma}^{\lambda}\Gamma_{\mu\nu}^{\alpha}-\Gamma_{\alpha\nu}^{\lambda}\Gamma_{\mu\sigma}^{\alpha}
V(r)={\frac{1}{4\alphar}}[n_{1}n_{1}^{\prime}\mathrm{Tr}(Y^{2})(1-e^{-\mu_{0}r})+n_{3}n_{3}^{\prime}\mathrm{Tr}(\lambda_{i}\lambda_{j})(1-e^{-\mu_{3}r})+n_{8}n_{8}^{\prime}\mathrm{Tr}(T_{a}T_{b})(1-e^{-\mu_{8}r})]\
{\calH}={\frac{u}{\calR}}\sqrt{\left({\frac{u}{\calR}}\right)^{2}{P_{u}}^{2}+\vec{L}^{2}+\alpha^{\prime}\,m^{2}\,L^{2}g_{s}^{\frac{-2}{D-3}}e^{-2{\frac{D-4}{D-3}}\phi}}-qA_{t}\,,
I^{(6)}({\bf{q}}_{1},\ldots,{\bf{q}}_{6})=K_{D}\sum_{\{I_{1},I_{2}\},I_{3}}^{45\;\mathrm{terms}}\Big\langleC(\hat{\bf{q}}^{\prime}+{\bf{Q}}_{1})C(\hat{\bf{q}}^{\prime}-{\bf{Q}}_{2})\Big\rangle_{\hat{\bf{q}}^{\prime}}\;.
\frac{d^{2}\varphi_{k}^{H}}{dt^{2}}+3\frac{\dot{a}}{a}\frac{d\varphi_{k}^{H}}{dt}+\left[\frac{\vec{k}^{2}}{a^{2}}+{\cal{V}}^{(2)}(\phi(t))\right]\varphi_{k}^{H}=0
\rho^{\prime\prime}+\frac{2m}{\hbar}\left({\frac{mv^{2}}{2\hbar}}+{\lambdav\rho_{0}^{2}}\right)\rho-{\frac{m^{2}v^{2}}{\hbar^{2}}}{\frac{\rho_{0}^{4}}{\rho^{3}}}-{\frac{2mv\lambda}{\hbar}}\rho^{3}=0
\dot{\varrho}_{m}+3{\frac{\dot{a}}{a}}\left(\varrho_{m}+{\frac{\varrho_{m}}{c^{2}}}\right)=-\dot{\varrho}_{\Lambda}-\varrho{\frac{\dot{G}_{4}}{G_{4}}}+{\frac{3kc\dot{c}}{4\piG_{4}a^{2}}}.
\mathop{\calE}_{k}^{\underline{{a}}}(\vec{x},t)=-{\frac{\lambda}{2}}\sum_{\alpha}m_{\alpha}\varepsilon_{kj}\,\dot{x}_{\alpha}^{j}\cdot\xi_{\alpha}^{\underline{{a}}}\cdot\delta^{(2)}(\vec{x}-\vec{x}_{\alpha}(t)),
T_{\mu\nu}^{VV}|_{AT}=T_{\mu\nu}^{AA}|_{AT}=\frac{2i\pi^{2}}{3(2\pi)^{4}}\left\{s_{\mu}(s+p)_{\nu}+g_{\mu\nu}s\cdot(s+p)+s_{\nu}(s+p)_{\mu}\right\}.
[K,\mathcal{O}_{+}^{in}(\Omega)]=-2\overline{{c}}\mathcal{O}_{+}^{out}(\Omega)\,\left[K,[K,\mathcal{O}_{+}^{in}(\Omega)]\right]=4c\overline{{c}}\mathcal{O}_{+}^{in}(\Omega)\.
\gamma^{2}=\frac{2GM}{b}+\frac{2G}{b}\sqrt{M^{2}-\frac{8J^{2}\alpha^{2}}{9}}\quad;\quad\frac{\omega^{2}}{\alpha^{2}}=\frac{4GM}{b}-\frac{4G}{b}\sqrt{M^{2}-\frac{8J^{2}\alpha^{2}}{9}}
C^{\Lambda,\Lambda_{0}}(x-y)=\frac{1}{{(2\pi)}^{d}}\int_{\mathbf{R}^{d}}dk\frac{e^{ik(x-y)}}{k^{2}+m^{2}}\left(e^{-\frac{k^{2}+m^{2}}{{\Lambda_{0}}^{2}}}-e^{-\frac{k^{2}+m^{2}}{{\Lambda}^{2}}}\right).
\widetilde{V}(r)=\frac{1}{3\ell^{2}(r^{2}+N^{2})^{2}}[3r^{6}+(\ell^{2}-15N^{2})r^{4}-3N^{2}(2\ell^{2}-15N^{2})r^{2}-6mr\ell^{2}-3N^{4}(\ell^{2}-15N^{2})]
\begin{array}{l}{R^{(1)}X_{1}X_{2}=\kappa_{1}X_{2}X_{1}\quad,}\\{\,}\\{Z_{1}^{\dagger}R^{(2)}X_{2}=X_{2}Z_{1}^{\dagger}\quad,}\\{\,}\\{Z_{2}^{\dagger}R^{(3)}X_{1}=X_{1}Z_{2}^{\dagger}\quad,}\\{\,}\\{\kappa_{2}Z_{1}^{\dagger}Z_{2}^{\dagger}=Z_{2}^{\dagger}Z_{1}^{\dagger}R^{(4)}\quad,}\\\end{array}
\Lambda(\alpha,0)=\int\frac{d^{D}k}{(2\pi)^{D}}\frac{1}{\left[k^{2}-m_{0}^{2}\right]^{\alpha}}=\frac{i}{(4\pi)^{D/2}}\left(-m_{0}^{2}\right)^{2-\alpha}m_{0}^{2\epsilon}\frac{\Gamma(\alpha-2-\epsilon)}{\Gamma(\alpha)}
\phi_{1\mu}\equivp_{\mu}-{\frac{m\dot{n}_{\mu}}{\sqrt{\dot{n}^{2}}}}\approx0;\,\nonumber\phi_{2\mu}\equivp_{\mu}^{(n)}-{\frac{m}{\sqrt{\dot{n}^{2}}}}({\dot{x}}_{\mu}-{\frac{(\dot{x}\dot{n}){\dot{n}}_{\mu}}{{\dot{n}}^{2}}})\approx0;\phi_{3}\equivp^{(\lambda)}\approx0.
S_{2}^{(0)}={\frac{1}{2}}x^{i}\Lambda_{ij}x^{j}+y_{\alphaa}^{*}D_{2\beta}^{\alpha}c^{\betaa}+\bar{y}_{\alpha}D_{4\beta}^{\alpha}B^{\beta}+\varepsilon^{ab}c_{\alphaa|b}^{*}D_{2\gamma}^{-1\alpha}D_{4\beta}^{\gamma}B^{\beta}.
S_{WZNW}(f,l)=-{\frac{l}{4\pi}}\int[\mathrm{Tr}|f^{-1}\mathrm{d}f|^{2}~+~{\frac{i}{3}}\mathrm{d}^{-1}\mathrm{Tr}(f^{-1}\mathrm{d}f)^{3}].
(\omega^{2}-{\omega^{\prime}}^{2})(\int_{0}^{\theta}\!\phi_{\omega}^{E}(\sigma)\:\phi_{\omega^{\prime}}^{E}(\sigma)\,d\sigma+\int_{\theta}^{2\pi}\!\phi_{\omega}^{L}(\sigma)\:\phi_{\omega^{\prime}}^{L}(\sigma)\,d\sigma)=0
Z_{pert}(m^{2}=1,\lambda)=\frac{4}{3\lambda}\int_{0}^{\infty}db\,e^{-\frac{4b}{3\lambda}}B(b)=\sqrt{\frac{2}{\pi}}\frac{1}{\sqrt{\lambda}}e^{\frac{1}{\lambda}}K_{\frac{1}{4}}(\frac{1}{\lambda}).
\beta_{o}^{b}:={\frac{1}{{\theta^{\prime}}}}q_{|o|}-{\frac{i}{2}}\epsilon(o)x_{|o|},\qquad\beta_{o}^{c}:={\frac{1}{2}}y_{|o|}-{\frac{i}{{\theta^{\prime}}}}\epsilon(o)p_{|o|}\,.
V_{1}^{\mu}\equivp^{\mu}-\left[\frac{(i+\omega)p^{-}}{(1+i+l+\omega)(n\!\cdot\!n^{*})}\right]\;n^{\mu}-\left[\frac{(i+\omega)p^{+}}{(1+i+j+\omega)(n\!\cdot\!n^{*})}\right]\;n^{*\mu}\:.
\psi_{1}=\left(-\right)^{1+\epsilon\left(\Phi^{B}\right)\left(\epsilon\left(\Phi^{A}\right)+1\right)}\frac{\delta^{2L}\psi\left[\Phi^{A}\right]}{\delta\Phi^{A}\delta\Phi^{B}}\mu_{3}^{A}\mu_{2}^{B}+
\tau_{A}=\left\|\begin{array}{ccc}{cosh(\tau)}&{0}&{sinh(\tau)}\\{0}&{1}&{0}\\{sinh(\tau)}&{0}&{cosh(\tau)}\\\end{array}\right\|\\quad{\bfa}_{\bfT}=\left\|\begin{array}{ccc}{1+{\bfa}^{2}/2}&{-{\bfa}^{T}}&{{\bfa}^{2}/2}\\{-{\bfa}}&{1}&{-{\bfa}}\\{-{\bfa}^{2}/2}&{{\bfa}^{T}}&{1-{\bfa}^{2}/2}\\\end{array}\right\|\
G(x)=\prod_{i=1}^{n_{f}}(\sqrt{x}-im_{i})=\sum_{k=0}^{n_{c}}(-1)^{k}s_{2k}(m)x^{n_{c}-k}-i\sqrt{x}\sum_{k=0}^{n_{c}-1}(-1)^{k}s_{2k+1}(m)x^{n_{c}-k-1}.
J^{\mu\nu}=Eg^{\mu\nu},\\\j_{1}^{\mu}={\frac{\nabla^{\mu}\phi}{(\nabla\phi)^{2}}},\\\j_{2}^{\mu}=j_{\calR}^{\mu}+V{\frac{\nabla^{\mu}\phi}{(\nabla\phi)^{2}}},\\\j_{3}^{\mu}=Ej_{1}^{\mu},
G\left(\tau_{2}+\epsilon_{2},\tau_{1}-\epsilon_{1};\tau_{2},\tau_{1}\right)=\frac12|\tau_{1}-\tau_{2}-\epsilon_{1}-\epsilon_{2}|-\frac12\left(\tau_{1}-\tau_{2}-\epsilon_{1}-\epsilon_{2}\right)-\frac{\epsilon_{1}\epsilon_{2}}{\tau_{1}-\tau_{2}}
V_{0}(x)=\frac{2\lambda}{H^{2}}\psi_{0}^{2}(0)\int_{\gamma}^{\infty}\frac{d\gamma^{\prime}}{{\gamma^{\prime}}^{2}}{\cal{S}}_{0}(z)\sim-\frac{\lambda\psi_{0}^{2}(0)}{4\pi}\frac{1}{R}.
\tilde{F}^{\prime}(A)=\frac{f(A^{2}x)^{2}}{2A^{2}}-\frac{f(A^{2}x)}{A^{2}}=-\sum_{n=1}^{\infty}\frac{n^{n-3}}{(n-1)!}A^{2n-2}x^{n}.
\begin{array}{c}{\left(D(\beta,R)-i\mu\gamma^{0}\right)K_{bc}(t,\vec{x};t^{\prime},\vec{x^{\prime}})=\delta(\vec{x}-\vec{x^{\prime}})\delta(t-t^{\prime})}\\{BK_{bc}(t,\vec{x};t^{\prime},\vec{x^{\prime}})=0,{\\mathrm{for\}}|\vec{x}|=1,}\\{K_{bc}(1,\vec{x};t^{\prime},\vec{x^{\prime}})=-K_{bc}(0,\vec{x};t^{\prime},\vec{x^{\prime}}).}\\\end{array}
M^{2}\phi_{\mu}^{\dagger}\phi^{\mu}\rightarrow\frac{M}{2}\varepsilon^{\mu\nu\rho}\phi_{\mu}^{\dagger}\partial_{\nu}\phi_{\rho}+\frac{M}{2}\varepsilon^{\mu\nu\rho}\phi_{\mu}(\partial_{\nu}\phi_{\rho})^{\dagger}
G_{10}(x_{1}-x_{1}^{\prime})=e^{-ix_{1}^{0}h_{0}}\frac{1}{2}\Big(\epsilon(x_{1}^{0}-x_{1}^{\prime0})+\frac{h_{0}}{\sqrt{h_{0}^{2}}}\Big)\delta^{3}(\bf{x}_{1}-\bf{x}_{1}^{\prime})\gamma_{0}e^{ix_{1}^{\prime0}h_{0}^{\prime}},
\int_{0}^{R}\mathrm{d}r\,\left[u_{1\epsilon\kappa}^{2}\left(r\right)+u_{2\epsilon\kappa}^{2}\left(r\right)\right]=u_{2\epsilon\kappa}\left(R\right)\partial_{\epsilon}u_{1\epsilon\kappa}\left(R\right)-u_{1\epsilon\kappa}\left(R\right)\partial_{\epsilon}u_{2\epsilon\kappa}\left(R\right)\,.
[\alpha_{k}^{(\sigma)},\alpha_{k^{\prime}}^{(\sigma^{\prime})\,\dagger}]=[\beta_{k}^{(\sigma)},\beta_{k^{\prime}}^{(\sigma^{\prime})\,\dagger}]=\delta_{\sigma\sigma^{\prime}}\,\delta(k-k^{\prime})\,{,}\quad\sigma,\sigma^{\prime}=\pm\,{,}
I^{(0)}(x,x^{\prime})=\epsilon(x_{0}-x_{0}^{\prime})\delta({\bfx-x^{\prime}})\frac{e^{i\frac{3\pi}{4}}e^{ig\Lambda}}{4\sqrt{\pi}}\int_{e^{i\pi}\infty}^{\infty}t^{-\frac{3}{2}}e^{-i(x_{0}-x_{0}^{\prime})^{2}t}dt.
T^{00}={\frac{2cosh^{2}(\gamma)}{g^{2}}}\left[-C^{2}sech^{2}(Cr)\Big(1-Cr\;tanh(Cr)\Big)^{2}+{\frac{(C^{2}r^{2}sech^{2}(Cr)+1)^{2}}{2r^{2}}}\right]
\left|\Phi\right\rangle=\left\{\phi_{0},\phi_{1}^{(\mu_{1})}\left(p_{1}\right),\phi_{2}^{(\mu_{1})(\mu_{2})}\left(p_{1,}p_{2}\right),...,\phi_{n}^{(\mu_{1})...(\mu_{n})}\left(p_{1},...,p_{n}\right),...\right\}\;\,,
P\Gamma_{KZ}{[A,B,C]}=<\!0|\alpha_{1}^{p}A_{pq}\bar{\alpha}_{1}^{q}\left(-\frac{1}{2}G^{il}C_{kl}\alpha_{-1}^{k}B_{ij}\bar{\alpha}_{-1}^{j}|0\!>\;\;-\;\;\frac{1}{2}\alpha_{-1}^{i}B_{ij}G^{jl}C_{lk}\bar{\alpha}_{-1}^{k}|0\!>\right).
\sum_{i,j,k,l=1}^{h}a_{ij}^{p}A_{ik}A_{jl}\overline{a}_{kl}^{q}=\sum_{i,j,k,l,m,n=1}^{h}b_{ijm}^{p}A_{ik}A_{jl}A_{mn}\overline{b}_{kln}^{q}.
\intd^{4}x\left[-\frac{1}{2}G_{\mu}G^{\mu}+\frac{1}{8\alpha}(\varepsilon^{\mu\nu\sigma\lambda}\partial_{\nu}B_{\sigma\lambda})^{2}\right]=\nonumber\,\frac{1}{4}\intd^{4}xB_{\mu\nu}L_{(\alpha)}^{\mu\nu\sigma\lambda}B_{\sigma\lambda},
m^{2}=g_{\infty}^{2}\biggl[\frac{\vec{Q}_{R}^{2}}{g_{\infty}^{4\gamma}}+2(N_{R}-{\textstyle{\frac{1}{2}}})\biggr]=g_{\infty}^{2}\biggl[\frac{\vec{Q}_{L}^{2}}{g_{\infty}^{4\gamma}}+2(N_{L}-1)\biggr]
|\Gamma_{AA}^{(1)}|=g^{2}\frac{C_{2}}{32}\intd^{3}p\sqrt{p^{2}}\Bigg\{\Big(g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\Big)\Big(1+\frac{\alpha^{2}}{4p^{2}}\Big)+\frac{i\alpha}{p^{2}}\epsilon_{\mu\lambda\nu}p_{\lambda}\Bigg\}A_{\mu}^{a}(p)A_{\nu}^{a}(-p)\;.
\zeta(s)=2\left(\frac{L\tau_{2}}{2\pi}\right)^{2s}\!\!\sum_{Z^{2}}\left[\left(n_{1}+H_{1}+\frac{1}{2}-\tau_{1}\left(n_{2}+H_{2}+\frac{1}{2}\right)\right)^{2}\!\!\!+\tau_{2}^{2}\left(n_{2}+H_{2}+\frac{1}{2}\right)^{2}\right]^{-s}\!\!\!\!,
C(\omega,\omega^{\prime})=\delta_{{\omega+\omega^{\prime}},0}f(\omega,\omega^{\prime},\theta)-4\sum_{\omega_{1},\omega_{2}>0}^{N}{\omega_{1}}^{2}{\omega_{2}}^{2}C_{\omega_{1},\omega_{2}}^{-\omega^{\prime}}\:C_{\omega_{1},\omega_{2}}^{\omega}+\sum_{\omega_{1}>0}^{N}{\omega_{1}}^{2}\:(\omega-\omega_{1})^{2}\:C_{\omega_{1},\omega-\omega_{1}}^{-\omega^{\prime}}
\begin{array}{c}{\displaystyle\displaystyle\sum_{j=1}^{M}\phi(\Theta-\vartheta_{j},1/2)=\displaystyle\int_{asymp}^{\Theta}dx\,W(x)}\\{\displaystyle\sum_{j=1}^{M}\phi(\Theta+\vartheta_{j},1/2)=-\displaystyle\sum_{j=1}^{M}\phi(-\Theta-\vartheta_{j},1/2)=-\displaystyle\int_{asymp}^{-\Theta}dx\,W(x).}\\\end{array}
L_{V,A}=-{\frac{1}{2(p+1)!}}\,\left(\,\partial_{\,[\,\mu_{1}}V_{\mu_{2}\dots\mu_{p+1}\,]}+m\,A_{\mu_{1}\dots\mu_{p+1}}\,\right)^{2}-{\frac{1}{2(p+2)!}}F^{2}{}_{\mu_{1}\dots\mu_{p+2}}(A)
\frac{M_{i}}{k^{2}-M_{i}^{2}+i0}\,\bar{D}^{2}\delta^{4}(\theta_{1}-\theta_{2})\quad\mathrm{and}\quad\frac{M_{i}}{k^{2}-M_{i}^{2}+i0}\,D^{2}\delta^{4}(\theta_{1}-\theta_{2})
\Gamma_{I}^{\phi_{P}^{0}}(-i\Omega_{m},\stackrel{\rightharpoonup}{p})=-G/(-\Omega_{m}^{2}-{\stackrel{\rightharpoonup}{p}}^{2})\sum_{Q}g_{QQ}K_{Q}^{T}(-i\Omega_{m},\stackrel{\rightharpoonup}{p}).
W[Y]=\int\frac{d^{4}x}{32\pi^{2}}\int_{0}^{\infty}\frac{dT}{T^{3}}\rho(T,\Lambda^{2})e^{-TK}\mathrm{tr}\left(e^{-TM\tau_{3}}[1-f_{1}(T,B)+f_{2}(T,B)-\tilde{f}_{3}(T,B)+\ldots]\right),
\left(\begin{array}{c}{U_{1}}\\{U_{2}}\\\end{array}\right)\rightarrow\left(\begin{matrix}{\alpha}&{\beta}\\{\gamma}&{\delta}\\\end{matrix}\right)\left(\begin{array}{c}{U_{1}}\\{U_{2}}\\\end{array}\right)\quad;\quad\left(\begin{matrix}{\alpha}&{\beta}\\{\gamma}&{\delta}\\\end{matrix}\right)\,\in\,SU(1,1)
\begin{array}{rcccl}{\psi}&{\stackrel{\itT}{\longrightarrow}}&{\eta}&{\stackrel{\itT}{\longrightarrow}}&{\psi}\\{\chi_{i}}&{\stackrel{\itT}{\longrightarrow}}&{\psi_{i}}&{\stackrel{\itT}{\longrightarrow}}&{\chi_{i}\.}\\\end{array}
[D_{tree}(\phi;k)]_{ab}={\frac{\delta_{a1}\delta_{b1}}{k^{2}+m^{2}+{\frac{\lambda_{\Phi}}{2}}\phi^{2}}}+{\frac{\delta_{a2}\delta_{b2}}{k^{2}+\omega^{2}+{\frac{\lambda_{\Phi\Psi}}{2}}\phi^{2}}}~.
\Omega_{B}=\frac{\kappa^{2}\rho_{B}}{3H_{0}^{4}},\;\;\Omega_{M}=\frac{\kappa^{2}\rho_{M0}}{3H_{0}^{2}},\;\;\Omega_{\phi}=\frac{\kappa^{2}\rho_{\phi0}}{3H_{0}^{2}}.
[f_{ij}^{(1)}]^{-1}(x,y)=\frac{1}{m^{2}}\left(\begin{array}{cccc}{0}&{-\partial_{x}^{1}}&{-\partial_{x}^{2}}&{1}\\{-\partial_{x}^{1}}&{0}&{-m}&{0}\\{-\partial_{x}^{2}}&{m}&{0}&{0}\\{-1}&{0}&{0}&{0}\\\end{array}\right)\delta^{2}(x-y).
{\calL}={\frac{1}{2\kappa^{2}}}\sqrt{-g}\left[R-{\frac{1}{2\cdot4!}}{\calA}_{\mu}^{ijkl}{\calA}_{ijkl}^{\mu}-{\frac{1}{4}}(F_{\mu\nu\,IJ}^{+}(2S^{IJ,KL}-\delta_{KL}^{IJ})F^{+\,\mu\nu}{}_{KL}+\mathrm{h.c.})-V\right],