harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
x_{m}^{j}=\frac{1}{2\pii}\oint_{0}\xi^{j}\frac{d\zeta}{\zeta^{m+1}}=\frac{1}{2\pii}\oint_{0}\tilde{\xi}^{j}\frac{d\zeta}{\zeta^{m+2j-1}}-\frac{1}{2\pii}\oint_{0}\frac{\partialH}{\partial\alpha^{j}}\frac{d\zeta}{\zeta^{m+1}}
T={\frac{1}{2\piM}}\quad{\frac{\sqrt{(M^{2}-|z_{1}|^{2})\;(M^{2}-|z_{2}|^{2})}}{\left[\sqrt{M^{2}-|z_{1}|^{2}}+\sqrt{M^{2}-|z_{2}|^{2}}\right]^{2}}}\.
\begin{array}{lcc}{}&{SU(2)_{L}}&{SU(2)_{R}}\\{E}&{\gamma_{y}\gamma_{x}}&{\gamma_{\bar{x}}\gamma_{y}}\\{H\;\;\;}&{\left[\gamma_{x},\gamma_{\bar{x}}\right]+\left[\gamma_{y},\gamma_{\bar{y}}\right]\;\;\;}&{\left[\gamma_{\bar{x}},\gamma_{x}\right]-\left[\gamma_{\bar{y}},\gamma_{y}\right]}\\{F}&{\gamma_{\bar{x}}\gamma_{\bar{y}}}&{\gamma_{\bar{y}}\gamma_{x}}\\\end{array}
\begin{array}{ll}{\nabla\cdot\mathbf{E}^{\prime}=\rho_{e}^{\prime},}&{\nabla\times\mathbf{B}^{\prime}=\mathbf{j}_{e}^{\prime}+\frac{\partial\mathbf{E^{\prime}}}{\partialt},}\\{\nabla\cdot\mathbf{B}^{\prime}=\rho_{m}^{\prime}=0,}&{\nabla\times\mathbf{E}^{\prime}=-\frac{\partial\mathbf{B^{\prime}}}{\partialt}.}\\\end{array}
(G_{{\calL}}^{T}G_{{\calR}})^{ij}=H_{L}^{ik}(-l^{k\underline{{m}}}r_{\underline{{m}}}^{l}+{\frac{2(r^{(++)\underline{{n}}}l_{\underline{{n}}}^{k})(l^{(--)\underline{{n^{\prime}}}}r_{\underline{{n^{\prime}}}}^{l})}{(l^{(--)\underline{{n^{\prime\prime}}}}r_{\underline{{n^{\prime\prime}}}}^{(++)})}})H_{R}^{jl}~,
\delta^{(2)}S_{\phi}=-\frac{3}{8\kappa_{5}^{2}}\int\sqrt{-g_{0}}\left(\frac{{\calL}_{\phi}}{3}h_{ab}h^{ab}+(\partial\delta\phi)^{2}+\frac{1}{2}\frac{\partial^{2}U}{\partial\phi^{2}}\delta\phi^{2}+2h^{ab}\partial_{a}\delta\phi\partial_{b}\phi\right)
\begin{array}{rl}{x^{25}(\tau,\sigma)=\sqrt{2\alpha^{\prime}}\left[\right.}&{q^{25}+\alpha_{0}^{25}(\tau-\sigma)+\bar{\alpha}_{0}^{25}(\tau+\sigma)}\\{}&{}\\{}&{\left.+\frac{1}{2}i{\sum_{n}}^{\prime}\frac{1}{n}\left(\alpha_{n}^{25}e^{-2in(\tau-\sigma)}+\bar{\alpha}_{n}^{25}e^{-2in(\tau+\sigma)}\right)\right]\.}\\\end{array}
\left((D^{2}({\Phi}^{(0)}))^{ab}+g\epsilon^{adc}{\tilde{A}}_{i}^{d(0)}D_{i}^{cb}({\Phi}^{(0)})\right)\tilde{\sigma}^{b}=-g\epsilon^{abc}{\tilde{A}}_{i}^{a(0)}E_{i}^{c}~.
W^{I}=W_{L}^{I}+W_{R}^{I}\,W_{L}^{I}=\int_{0}^{\pi/2}d\sigma\partial_{\sigma}X^{I}(\sigma)\,W_{R}^{I}=\int_{\pi/2}^{\pi}d\sigma\partial_{\sigma}X^{I}(\sigma)\,
I_{N}(s)=\beta_{N}\sum_{n=0}^{\fracN2-1}c_{2n+1}^{N}\left({\frac{1}{2}}n!(a^{2}b)^{n+1-s}{\frac{\Gamma(s-n-1)}{\Gamma(s)}}-2H_{n}(s;a\sqrtb)\right),
\nabla_{\alpha}v\nabla_{\mu}v\:m_{\beta}m_{\nu}R^{\alpha\beta\mu\nu}\biggl|_{{\calI}^{+}}=-\frac{8}{r}\:\frac{\partial^{2}}{{\partialu}^{2}}\:{\bfC}+O\left(\frac{1}{r^{2}}\right)
\omega\lfloor\eta={\frac{1}{p!}}\omega_{j_{1}\ldotsj_{p}}\eta_{i}g^{j_{p}i}dx^{j_{1}}\cdotsdx^{j_{p-1}}-{\frac{1}{p!}}\omega_{j_{1}\ldotsj_{p}}\eta_{i}g^{j_{p-1}i}dx^{j_{1}}\cdotsdx^{j_{p-2}}dx^{j_{p}}+\cdots.
\begin{array}{lll}{J}&{\rightarrow}&{J-{\frac{im\pi}{2}}}\\{J_{1}}&{\rightarrow}&{J_{1}+{\frac{im\pi}{4}}}\\{\varepsilon}&{\rightarrow}&{\varepsilon}\\{A}&{\rightarrow}&{A+{\frac{im\pi}{4}}}\\\end{array}
i{\frac{\partialg}{\partiala^{(j)(l)}}}a_{(k)(l)}-i{\frac{\partialg}{\partiala^{(l)(k)}}}a_{(l)(j)}-2i{\frac{\partialg}{\partial{\bar{m}}^{(k)(l)}}}{\bar{m}}_{(l)(j)}+2i{\frac{\partialg}{\partialm^{(j)(l)}}}m_{(k)(l)}.
M_{\mathrm{int}}^{2}(x_{1},x_{2},s_{1},s_{2})\equiv\frac{s_{1}^{2}+r_{n}}{x_{1}}+\frac{s_{2}^{2}+r_{n}}{x_{2}}+\frac{(s_{1}+s_{2})^{2}+r_{n}}{1-x_{1}-x_{2}}.\nonumber
|(\bar{R}\kappa_{o}R)_{ee^{\prime}}|=\left|{\frac{16\,i^{e+e^{\prime}}\left(ee^{\prime}\right)^{2}}{\pi^{2}}}\sum_{o=1}^{\infty}{\frac{1}{o(e^{2}-o^{2})(e^{\prime2}-o^{2})}}\right|\quad<\infty\,.
G^{(2)}(x_{1},x_{2})=\frac{2^{9}\pi^{3}}{e^{2}}M_{W}^{3}\mathrm{exp}(-S_{\mathrm{cl}}+i\sigma)\int\mathrm{d}^{3}x\epsilon^{\alpha^{\prime}\beta^{\prime}}S_{\mathrm{F}\alpha\alpha^{\prime}}(x_{1}-x)S_{\mathrm{F}\beta\beta^{\prime}}(x_{2}-x)\.
\int_{-\infty}^{0}dx_{1}dx_{2}\int_{-\infty}^{\infty}dt_{1}dt_{2}G_{N}(t,x;t_{1},x_{1})~G_{N}(t^{\prime},x^{\prime};t_{1},x_{1})~G_{N}(t_{1},x_{1};t_{2},x_{2})
\|c\|\equiv\|c_{jl}\|\equiv\left(\begin{array}{c}{{\;1\;\;\;\;c_{12}\;\;\;c_{13}\;\;\;\ldots\;\;c_{1N}}}\\{{c_{12}\;\;\;\;1\;\;\;\;c_{23}\;\;\;\ldots\;\;c_{2N}}}\\{{c_{13}\;\;\;c_{23}\;\;\;\;1\;\;\;\;\ldots\;\;c_{3N}}}\\{{\ldots\ldots\ldots\ldots\ldots\ldots\ldots}}\\{{c_{1N}\;\;c_{2N}\;\;c_{3N}\;\;\ldots\;\;\;1\;\;}}\\\end{array}\right)\;\;,
{\calH}^{2}=\frac{\rho_{M}^{2}}{36}+\frac{1}{18}U_{B}\rho_{M}+\frac{U_{B}^{2}}{36}-\frac{2}{3}{\calQ}-\frac{1}{9}{\calE}-\frac{k}{a^{2}}+\frac{{\calA}}{a^{4}}\;.
\frac{\partial^{2}}{{\partialu}^{2}}\:{\bfC}=-\:\frac{\partial}{\partialu}\:\left\{\:\frac{\partial}{\partialu}\;\frac{r}{2}\:m_{\beta}m_{\nu}\:\frac{1}{\Box}\:\Bigl(R^{\nu\beta}+O[R_{..}^{2}]\Bigr)\right\}\Biggl|_{{\calI}^{+}}\;.
16\,e^{4\sqrt{2}c}\,\phi_{1}^{2}\,(\phi_{1}^{2}+\phi_{2}^{2})+(1-e^{4\sqrt{2}c})^{2}\,\phi_{2}^{4}\,(2\phi_{1}-\phi_{2}^{2}+1)(2\phi_{1}+\phi_{2}^{2}-1)=0
{\frac{9\mathrm{Vol}(S^{3})a_{H}^{2}}{32\pi^{2}G_{4}^{2}LM(1+c_{1})}}\le\gamma\le{\frac{9\mathrm{Vol}(S^{3})a_{H}^{2}}{32\pi^{2}G_{4}^{2}LM(1-c_{1})}}.
A_{BPS}=4\pi^{2}[(Q_{1}^{(1)}Q_{1}^{(2)}Q)(1-\textstyle{\frac{1}{2}}(L_{1}-L_{2})^{2})]^{\frac{1}{2}}=4\pi^{2}[Q_{1}^{(1)}Q_{1}^{(2)}Q-\textstyle{\frac{1}{4}}J^{2}]^{\frac{1}{2}}.
\left(\Psi_{{l}}^{\prime\prime}-\frac{l(l+1)}{r^{2}}\Psi_{{l}}\right)+\left(\frac{r_{0}}{r}\right)^{4}\left(\Psi_{{l}}^{\prime\prime}-\frac6r\Psi_{{l}}^{\prime}+\frac{6}{r^{2}}\Psi_{{l}}\right)=0\,
\begin{array}{c}{L=\left[1+\frac12h\right]i\bar{\psi}\gamma^{a}\left[\eta_{a}^{\mu}+h_{a}^{\mu}\right]\left[\partial_{\mu}+\Omega_{\mu}\right]\psi++\left[1+\frac12h\right]m\bar{\psi}\psi}\\\end{array}
M_{1}^{-1}={\frac{1}{g_{1}}}\tilde{\bfR}\,\quadM_{3}M_{1}^{-1}={\frac{1}{\sqrt{2}g_{1}}}\biggl(l\mu_{q}^{\ast}+{\frac{i\hbar}{l}}\mu_{p}^{\ast}\biggr)I_{2}\equivg_{4}{\bfI_{2}}\\.
Z_{1}(x)=\sum_{u,v\in\mathbbZ}e^{iux/R}q^{\frac{u^{2}}{4R^{2}}+v^{2}R^{2}}=\sum_{u,v\in\mathbbZ}e^{ix(\alpha_{u,v}+\bar{\alpha}_{u,v})}\,q^{h_{u,v}+\bar{h}_{u,v}}
g=\frac12\sqrt{\fracrl}\left(\begin{array}{cc}{1+\frac{it}{2l}}&{\frac{it}{2l}}\\{-\frac{it}{2l}}&{1-\frac{it}{2l}}\\\end{array}\right)\left(\begin{array}{cc}{1+\frac{l}{r}}&{1-\frac{l}{r}}\\{1-\frac{l}{r}}&{1+\frac{l}{r}}\\\end{array}\right)\,.
{\calL}_{\mathrm{b}}~=~-\frac{1}{4l^{2}}\partial_{\mu}l\partial^{\mu}l+\frac{1}{4l^{2}}h_{\mu}h^{\mu}-G_{i\bar{\jmath}}\partial_{\mu}{z^{i}}\partial^{\mu}{\bar{z}^{\bar{\jmath}}}-\frac{i}{2}(G_{lj}\partial_{\mu}{z^{j}}-G_{l\bar{\jmath}}\partial_{\mu}{\bar{z}^{\bar{\jmath}}})h^{\mu},
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\left.-3gA_{\mu51}\frac{1}{\theta}\displaystyle\sum_{A=1}^{4}\sigma_{A}\bar{\psi}_{AiCjDk}\gamma^{\mu}\psi_{AiCjDk}\right)\quad
I_{X2}\equiv\int\frac{d^{4}q}{(2\pi)^{4}}\,\Bigg\{\frac{2q_{\mu}(q_{\mu}+k_{\mu})}{\Big((k+q)^{2}+M^{2}\Big)^{2}\Big(q^{2}+M^{2}\Big)^{2}}-\frac{4M^{2}}{\Big((k+q)^{2}+M^{2}\Big)\Big(q^{2}+M^{2}\Big)^{3}}\Bigg\}.\qquad
m_{n}\frac{g_{\bar{z}j}(z_{n})(\dot{z}_{n}^{j}+N^{j})}{\sqrt{N^{2}-g_{ij}(\dot{z}_{n}^{i}+N^{i})(\dot{z}_{n}^{j}+N^{j})}}\rightarrowP_{n\bar{z}}.
\begin{array}{l}{\langle(-D_{FP})^{-1}(x,y)\rangle=\int_{0}^{\infty}d\tau\langleK_{\tau}(x,y)\rangle}\\{=\xi^{-1-\gamma}\int_{0}^{\infty}d\tau\tau^{-\frac{2}{1+\gamma}}F\left(\tau^{-\frac{1}{2(1+\gamma)}}\left(y-x\right)\right)\simeq\xi^{-1-\gamma}\vertx-y\vert^{-2+2\gamma}}\\\end{array}
\frac1{k_{T}}\sum_{j=1}^{p}q_{i_{j}}^{2}-\frac{\tilde{g}^{2}}{2\tilde{N}_{c}}\{(\tilde{N}_{c}-p)\sum_{j=1}^{p}(q_{i_{j}}^{2}-\bar{q}_{i_{j}}^{2})-p\sum_{k\neqi_{1},\cdots,i_{p}}(q_{k}^{2}-\bar{q}_{k}^{2})\}+pm_{\bar{q}}^{2}=0.
\{\partial_{T}\partial_{\bar{T}}+\frac{2}{(T-{\bar{T}})^{2}}\}{\calI}=-\frac{4}{(T-{\bar{T}})^{2}}\intd^{2}\tau{\bar{F}}({\bar{\tau}})\partial_{\tau}(\partial_{\bar{\tau}}^{2}+\frac{i}{\tau_{2}}\partial_{\bar{\tau}})(\tau_{2}\sum_{P_{L},P_{R}}q^{P_{L}^{2}/2}{\bar{q}}^{P_{R}^{2}/2}).
S=Tr\intd^{4}x\left\{-\textstyle{\frac{1}{4}}F_{\mu\nu}(Q)^{2}-\textstyle{\frac{1}{2}}\left[D^{\mu}(B)(Q_{\mu}-B_{\mu})\right]^{2}-D_{\mu}(B)b\cdotD^{\mu}(Q)c\right\}
\left(L^{\frac{3}{2}}\right)_{+}=\left(LL^{\frac{1}{2}}\right)_{+}=\partial^{3}+{\frac{1}{4}}u\partial+{\frac{1}{8}}u_{x}=\partial^{3}+{\frac{1}{8}}(\partialu+u\partial)={\frac{1}{4}}B
\bar{\calR}\left|J_{1}\,M_{1}\,z_{1}\right\rangle\otimes\left|J_{2}\,M_{2}\,z_{2}\right\rangle=\!\!\sum_{M_{1}^{\,\prime}\,M_{2}^{\,\prime}}\left(\bar{\calR}^{J_{1}\,z_{1}\,\omega_{1}\,,\,J_{2}\,z_{2}\,\omega_{2}}\right)_{M_{1}\,M_{2}}^{M_{1}^{\,\prime}\,M_{2}^{\,\prime}}\left|J_{1}\,M_{1}^{\,\prime}\,z_{1}\right\rangle\otimes\left|J_{2}\,M_{2}^{\,\prime}\,z_{2}\right\rangle
G^{[F]}=-dx^{0}\otimesdx^{0}+dx^{1}\otimesdx^{1}+g_{i^{\prime}j^{\prime}}^{[F]}(x,y)dx^{i^{\prime}}\otimesdx^{j^{\prime}}+g_{i^{\prime}j^{\prime}}^{[F]}(x,y)\deltay^{i^{\prime}}\otimes\deltay^{j^{\prime}}
\left\{i\left[V_{a}^{\mu}\gamma^{a}\left(\partial_{\mu}-i\tilde{e}\tilde{A}_{\mu}\right)+\gamma^{a}C_{ab}^{b}+\frac{1}{4}\omega_{\mu}^{cd}(\gamma^{a}\sigma_{cd}+\sigma_{cd}\gamma^{a})V_{a}^{\mu}\right]+\frac{1}{2}\gamma^{a}V_{a}^{\mu}\sigma_{,\mu}+U^{\prime}(\overline{{\Psi}}\Psi)\right\}\Psi=0
\int[d\varphi^{I}]\,\epsilon^{a}\,\lbrace\,\,{\frac{\partial}{\partial\chi^{a}}}\Big(S_{1}\,+\,\hbarM_{1}\Big)\,+\,J^{r}{\overline{R}}_{a}^{r}\,\rbrace\,exp{\frac{i}{\hbar}}\Big(S_{_\Sigma}+\hbarM_{1}\,+\,J^{A}\,\phi_{A}\Big)\,=\,0
{\frac{d^{2}x^{i}}{d\tilde{\lambda}^{2}}}=-{\frac{F}{\sigma+\varrho}}\left[{\frac{1}{a_{0}^{2}}}\sum_{j}\left({\frac{dx^{j}}{d\tilde{\lambda}}}\right)^{2}+{\frac{2}{a_{0}^{4}}}\right]{\frac{dx^{i}}{d\tilde{\lambda}}}.
\Omega_{+2}^{+2i}+\Omega_{-2}^{-2i}+4i\left(\psi_{+2q}^{1+}\gamma_{q\dot{q}}^{i}\psi_{-2\dot{q}}^{1-}-\psi_{-2q}^{1+}\gamma_{q\dot{q}}^{i}\psi_{+2\dot{q}}^{1-}+\psi_{+2q}^{2-}\gamma_{q\dot{q}}^{i}\psi_{-2\dot{q}}^{2+}-\psi_{-2q}^{2-}\gamma_{q\dot{q}}^{i}\psi_{+2\dot{q}}^{2+}\right)=0.
g=\left(\begin{array}{ccccc}{-1}&{0}&{0}&{\dots}&{0}\\{0}&{(g_{ij})}&{\dots}&{0}&{0}\\{0}&{\dots}&{1}&{\dots}&{0}\\{0}&{\dots}&{\dots}&{\ddots}&{0}\\{0}&{0}&{\dots}&{\dots}&{1}\\\end{array}\right)
\xi=\left[1+\left({\frac{(2\pi)^{p-2}U_{0}^{3-p}}{2c_{p+4}\lambda_{p}}}\right)^{2}\right]^{1/2}-{\frac{(2\pi)^{p-2}U_{0}^{3-p}}{2c_{p+4}\lambda_{p}}}\,.
\Psi\left(x\right)=\left(\begin{array}{c}{\varphi_{+}}\\{\varphi_{-}}\\{\chi_{+}}\\{\chi_{-}}\\\end{array}\right)=\left(\begin{array}{c}{a_{1}\midn-1\rangle}\\{a_{0}\midn\rangle}\\{a_{\widetilde{0}}\midn-1\rangle}\\{a_{2}\midn\rangle}\\\end{array}\right)
\phi(x)=m\left({\frac{\int\mathrm{d^{d}}x\phi^{4}}{4n}}\right)^{1/2}\varphi(mx)=m^{\mathrm{d}/2-1}\left({\frac{4n}{\int\mathrm{d^{d}}u\varphi^{4}(u)}}\right)^{1/2}\varphi(mx),
P_{R}^{\prime}=({\rho}\frac{\tilde{m}}{2}\;\;\;+(G_{\bot}\;\;-\;\;B_{\bot}+\frac{1}{4}A_{\bot}^{t}C_{\bot}A_{\bot}){\tilde{n}}-\;\;\frac{1}{2}A_{\bot}^{t}C_{\bot}{\tilde{l}},0),
\Gamma\lbrackA\rbrack=\int_{0}^{\infty}{\frac{dT}{T}}\,\mathrm{e}^{-m^{2}T}\int{\calD}x\,\mathrm{exp}\left[-\int_{0}^{T}\!\!\!d\tau\left({\frac{1}{4}}{\dot{x}}^{2}+ieA_{\mu}\dot{x}^{\mu}\right)\right].
\omega_{0}\equiv-{{\textstyle\frac{1}{2}}}\left({\displaystyle\sum_{d>1}\omega_{d}}\right)\+\F_{1}/2\,\quad\omega_{1}\equiv-{{\textstyle\frac{1}{2}}}\left({\displaystyle\sum_{d>1}\omega_{d}}\right)\-\F_{1}/2\.
\langle\bar{\lambda}^{a}\lambda^{a}\rangle\\approx\4\,(N_{c}^{2}-1)\int{\frac{d^{4}k}{(2\pi)^{4}}}\,{\frac{\widetilde\Sigma_{\lambda}(k^{2})}{k^{2}+\widetilde\Sigma_{\lambda}^{2}(k^{2})}}\,{\frac{1}{A_{\lambda}(k^{2})}}\\sim\(N_{c}^{2}-1)\,{\frac{\Lambda_{1}^{3}}{4\pi^{2}}}\.
{\cal{G}}(E):=\frac{i}{\hbar}\intdT\;e^{\frac{i}{\hbar}(E+i\epsilon)T}{\cal{K}}=\frac{i}{\hbar}\int_{0}^{\infty}dT\;e^{\frac{i}{\hbar}(E+i\epsilon-{\cal{H}})T}=\frac{1}{({\cal{H}}-E-i\epsilon)},
G_{N}^{M}=\frac{6}{l^{2}}\delta_{N}^{M}+\kappa^{2}\frac{\sqrt{-g_{brane}}}{\sqrt{-g}}T_{N}^{M}=\frac{6}{l^{2}}\delta_{N}^{M}+e^{-\beta}\kappa^{2}T_{N}^{M},\:\:(M,N=y,t,x^{i}).
q_{SUB}=\frac{2^{2-D}\pi^{-D/2}}{a^{D}\Gamma(D/2-1)}\sum_{n=-\infty}^{+\infty}\int_{ma}^{\infty}dz\,z^{3}\left(z^{2}-a^{2}m^{2}\right)^{D/2-2}\frac{\bar{I}_{n}(z)}{\bar{K}_{n}(z)}
L(A_{\mu},\phi,\theta)={\frac{-d}{F(\phi)}}G_{\mu\nu\rho\sigma}G^{\mu\nu\rho\sigma}-{\frac{\theta}{4}}\epsilon_{\mu\nu\rho\sigma}G^{\mu\nu\rho\sigma}-{\frac{1}{2}}\partial^{\mu}\phi\partial_{\mu}\phi-{\frac{1}{2}}m^{2}\phi^{2},
L_{eff}=L_{eff}^{GR}-\frac{1}{2}q_{~\mu\nu}^{a}H_{a~~b}^{~\mu\nu~\alpha\beta}q_{~\alpha\beta}^{b}-\frac{1}{2}\lambda_{~\mu}^{a}X_{a~b}^{~\mu~\nu}\lambda_{~\nu}^{b}-\lambda_{~\sigma}^{b}Z_{b~a}^{~\sigma~\mu\nu}q_{~\mu\nu}^{a}
H=-\frac{1}{2|\gamma|^{1/4}}\partial_{i}|\gamma|^{1/2}\partial_{i}\frac{1}{|\gamma|^{1/4}}-\frac{1}{2|g|^{1/4}|\gamma|^{1/4}}\hat{\partial}_{\mu}\lambda^{\mu\nu}|\gamma|^{1/2}\hat{\partial}_{\nu}\frac{|g|^{1/4}}{|\gamma|^{1/4}}+V(y).
\bigg[\prod_{I=1}^{N}:e^{-i2\sqrt{\pi}\omega{(I)}U_{Ia_{j}}\Phi^{(I)}(x_{j})}:_{M^{(I)}}e^{+i\frac{\theta}{N}}\;+\prod_{I=1}^{N}:e^{+i2\sqrt{\pi}\omega{(I)}U_{Ia_{j}}\Phi^{(I)}(x_{j})}:_{M^{(I)}}e^{-i\frac{\theta}{N}}\bigg]\;\Bigg\rangle_{K^{\mu}}\;.
\bar{\delta}_{\epsilon}H_{\mu}^{a}=\partial^{\nu}\eta_{\mu\nu}^{a}+g\left(\frac{\deltaW_{bc}}{\delta\varphi_{a}}A^{b\nu}\epsilon_{\mu\nu}^{c}-\frac{\deltaW_{bc}}{\delta\varphi_{a}}H_{\mu}^{c}\epsilon^{b}+\frac{\delta^{2}W_{bc}}{\delta\varphi_{a}\delta\varphi_{d}}B_{d\mu\nu}A^{c\nu}\epsilon^{b}\right),
j_{\sigma_{l}}^{i}(\xi)=\left(\left.\frac{\partialg^{\eta_{k}}(g^{\prime},g)}{\partialg^{\sigma_{l}}}\right|_{g^{\prime}=e}\left.\frac{\partialg_{\rho_{k}}^{\prime\prime}(g^{\prime},g)}{\partialg^{\eta_{k}}}\right|_{g^{\prime}=g^{-1}}\right)\Lambda^{\rho_{k}i}
D_{++}^{CPV}(x)=-{\frac{i}{(2\pi)^{2}}}\inte^{ikx}d^{2}k{\frac{\partial}{\partialk_{-}}}\left[\textrm{P}\left({\frac{1}{k_{-}}}\right)\right]=-{\frac{i}{2}}|x^{-}|\delta(x^{+}).
\frac{\partial^{2}A_{\varphi}}{\partial\rho^{2}}+\frac{1}{\rho}\frac{\partialA_{\varphi}}{\partial\rho}-\frac{A_{\varphi}}{\rho^{2}}-\gammaC_{2}\bigg(A_{\varphi}+\frac{1}{\rho^{2}}\int_{0}^{\rho}A_{\varphi}(\rho^{\prime})\rho^{\prime}d\rho^{\prime}\bigg)=m_{CS}\gammae\lambdaK_{1}(\lambda\rho).
C(l)_{n,n^{\prime}}^{m,m^{\prime}}=\left(\begin{array}{cc}{n}\\{l}\\\end{array}\right)\left[\begin{array}{c}{m^{\prime}+n^{\prime}}\\\end{array}\right]_{l}-\left(\begin{array}{cc}{n^{\prime}}\\{l}\\\end{array}\right)\left[\begin{array}{c}{m+n}\\\end{array}\right]_{l}
+\frac{n_{k-1}}{2\rho_{k}}(2n_{k-1}-1)\left(\frac{\rho_{k}}{\rho_{k-1}}\right)^{n_{k-1}}\frac{\Gamma\left(\frac{2n_{k-1}-1}{2n_{k}}\right)}{\Gamma\left(\frac{1}{2n_{k}}\right)}.
\frac{\partial^{r}\tilde{\phi}^{I}}{\partialz_{\;\:b_{1}}^{i_{\mu_{1}}a_{1}}\partialz_{\;\:b_{2}}^{i_{\mu_{2}}a_{2}}\ldots\partialz_{\;\:b_{r}}^{i_{\mu_{r}}a_{r}}}=P_{a_{2}}^{b_{1}}(I(\mu_{1},\mu_{2}))P_{a_{3}}^{b_{2}}(I(\mu_{2},\mu{3}))\ldotsP_{a_{1}}^{b_{r}}(I(\mu_{r},\mu_{1}))
y^{2}=x^{3}+\sum_{k=-4}^{4}\sum_{l=0}^{8}f_{kl}z^{\primel}z^{4+k}p^{8-l-qk}x+\sum_{k=-6}^{6}\sum_{l=0}^{12}g_{kl}z^{\primel}z^{6+k}p^{12-l-qk}
|\psi_{l}>=exp\{-i\int_{0}^{t}d\tauE(\tau)\}(Pexp\int_{0}^{t}d\tauiA(\tau))_{lm}|\psi_{m}^{(0)}>=exp\{-i\int_{0}^{t}d\tauE(\tau)\}(U(t))_{lm}|\psi_{m}^{(0)}>.
\mathbb{M}^{2}=p_{i\alpha}p_{i\alpha}+\frac{1}{M!}g_{\alpha\alpha_{1}\ldots\alpha_{M}}g_{\alpha\beta_{1}\ldots\beta_{M}}x_{i_{1}\alpha_{1}\ldots}x_{i_{M}\alpha_{M}}x_{i_{1}\beta_{1}\ldots}x_{i_{M}\beta_{M}}\,
{\calU}_{\mu\nu\rho\sigma\delta\tau}(q_{1},q_{2},q_{3})=-\frac{1}{2}\biggl[q_{2(\mu}q_{3\nu)}\biggl(2\eta_{\rho(\delta}\eta_{\tau)\sigma}-\eta_{\rho\sigma}\eta_{\delta\tau}\biggr)+q_{1(\rho}q_{3\sigma)}\biggl(2\eta_{\mu(\delta}\eta_{\tau)\nu}-\eta_{\mu\nu}\eta_{\delta\tau}\biggr)+...\biggr],
<\mathrm{Tr}\varphi^{2n}\mathrm{Tr}\varphi^{4}>_{c}=a^{2n+4}{\frac{12}{n+2}}{\frac{(2n)!}{(n!)^{2}}}=a^{2n+4}<\mathrm{Tr}\varphi^{2n}\mathrm{Tr}\varphi^{4}>_{c0}
Q(\sigma,\sigma^{\prime},\tau)=(\rho\rho^{\prime})^{-\frac{d-3}{2}}R(\mu\rho,\mu\rho^{\prime},\frac{\tau}{\rho\rho^{\prime}})\equiv(\rho\rho^{\prime})^{-\frac{d-3}{2}}\sum_{l=0}^{\infty}R_{l}\,(\mu\rho,\mu\rho^{\prime})(\frac{\tau}{\rho\rho^{\prime}})^{l},
\begin{array}{rcl}{{\calL}_{\mathrm{g.f.}}}&{=}&{\displaystyle-\frac{1}{\xi}\mathrm{Tr}(ff),}\\{f}&{=}&{\partial^{\mu}G_{\mu}+ig\xi(\phi^{\dagger}t^{A}\phi_{0}-\phi_{0}^{\dagger}t^{A}\phi)t^{A}.}\\\end{array}
K_{\rho}^{\prime}(\rhoz)\sim-\sqrt{\frac{\pi}{2\rho}}e^{-\rho\eta}{\frac{(1+z^{2})^{\frac{1}{4}}}{z}}\left[1+\sum_{k=1}^{\infty}(-1)^{k}{\frac{v_{k}(t)}{\rho^{k}}}\right],
\begin{array}{c}{{\calH}=g^{a{\bar{b}}}(\pi_{a}{\bar{\pi}}_{b}+{U}_{a}{\bar{U}}_{\bar{b}})-iU_{a;b}\eta_{1}^{a}\eta_{2}^{b}+i{\bar{U}}_{\bar{a};\bar{b}}{\bar{\eta}}_{1}^{\bar{a}}{\bar{\eta}}_{2}^{\bar{b}}-}\\{-R_{a\bar{b}c\bar{d}}\eta_{1}^{a}\bar{\eta}_{1}^{b}\eta_{2}^{a}\bar{\eta}_{2}^{d},}\\\end{array}
Y_{3,0}^{(1,1)}=J\lrcorner\,Y_{3}-\frac{1}{3}(J^{2}\lrcorner\,Y_{3})J-\frac{i}{8}(\Omega\lrcorner\,Y_{3})\lrcorner\,\overline{{\Omega}}+\frac{i}{8}(\overline{{\Omega}}\lrcorner\,Y_{3})\lrcorner\,\Omega,
p^{3}\,\dot{m}^{\prime\prime\prime}+6p^{2}\,\dot{m}^{\prime\prime}+\left[6p+\frac{8}{N\pi^{2}}\left(p+\xi\right)\right]\dot{m}^{\prime}+\frac{16}{N\pi^{2}}\dot{m}+\frac{8}{N\pi^{2}}m^{\prime}=0.
\langle{\bfx}^{\prime\prime},T\vert{\bfx}^{\prime},0\rangle=\intd{\bfx}_{N}\dots\intd{\bfx}_{1}\langle{\bfx}^{\prime\prime},T\vert{\bfx}_{N},t_{N}\rangle\langle{\bfx}_{N},t_{N}\vert{\bfx}_{N-1},t_{N-1}\rangle\dots\langle{\bfx}_{1},t_{1}\vert{\bfx}^{\prime},0\rangle\,.
\stackrel{\left(J\right)}{\omega}_{1}=\tilde{\omega}_{J}\left(\left[F_{ij}^{a}\right],\left[\pi_{a}^{\mu}\right],\left[y^{\alpha_{0}}\right],\left[\mathcal{P}_{1a}\right],\left[\mathcal{P}_{2a}\right]\right)e^{J+1}\left(\eta_{2}^{a}\right),
\left[D_{0}\right]_{\mu\nu}(x,y)=\int\frac{d^{4}p}{(2\pi)^{4}}\,e^{-ip(x-y)}\,\left[\delta_{\mu\nu}+(\alpha-1)\frac{p_{\mu}p_{\nu}}{p^{2}}\right]\,\frac{1}{p^{2}}.
\betaP=\rho\pm{\frac{1}{4}}\lambda^{2}\rho^{2}+{\frac{1+3\nu^{2}}{36}}\lambda^{4}\rho^{3}\mp{\frac{\nu^{2}}{16}}\lambda^{6}\rho^{4}-{\frac{1-100\nu^{2}+5\nu^{4}}{3600}}\lambda^{8}\rho^{5}+\cdots.
\left(\begin{array}{cc}{L_{_{1}}^{(L)\;2}+L_{_{2}}^{(L)\;2}+\alpha^{2}L_{_{3}}^{(L)\;2}+\frac{3}{4}\alpha^{2}+\alpha^{2}L_{_{3}}^{(L)}}&{\alpha(L_{_{1}}^{(L)}-iL_{_{2}}^{(L)})}\\{\alpha(L_{_{1}}^{(L)}+iL_{_{2}}^{(L)})}&{L_{_{1}}^{(L)\;2}+L_{_{2}}^{(L)\;2}+\alpha^{2}L_{_{3}}^{(L)\;2}+\frac{3}{4}\alpha^{2}-\alpha^{2}L_{_{3}}^{(L)}}\\\end{array}\right),
\zeta_{0}^{(p)}(\nu)={\frac{2}{(4\pi)^{D/2}\Gamma(\nu)}}\sum_{n=0}^{\infty}\sum_{m=0}^{[n/2]}{\frac{\Gamma(m+1/2)\breve{a}_{2m,n}}{\sqrt{\pi}(2\nu+2(n-m)-D)}}.
\begin{array}{rl}{\dot{\phi}\=}&{{\displaystyle-\,\left({\frac{6|\xi|}{1+6|\xi|}}\right){\frac{H\phi}{3}}\(n-2)\,}}\\{H\=}&{{\displaystyle\left({\frac{\lambda}{6n|\xi|}}\right)^{1/2}\phi^{n-1}}\.}\\\end{array}
G=e^{-6A-B}{\frac{\epsilon_{\\nu\alpha\beta}^{\mu}}{6}}\partial_{\mu}D\theta^{\nu}\wedge\theta^{\alpha}\wedge\theta^{\beta}\wedge\theta^{11}+e^{-6A-C}{\frac{\epsilon_{\mu\nu\alpha\beta}}{24}}\partial_{11}D\theta^{\mu}\wedge\dots\wedge\theta^{\beta}\.
{\bf\Phi}=\left[\begin{array}{c}{\,\,\varphi^{+}(x)\,}\\{\,\,\varphi^{-}(x)\,}\\\end{array}\right],\,\,\,{\bf\Phi}^{T}=\Big[\,\varphi^{+}(x)\,\,\,\varphi^{-}(x)\,\Big].
S_{\mathrm{bound}}=-{\frac{3\piM_{\mathrm{P}}^{2}}{2}}\int_{0}^{\infty}drR(r)(1-R^{\prime}(r))={\frac{3\pi^{2}M_{\mathrm{P}}^{2}R^{2}(0)}{8}}\cdot\left(-{\frac{2}{\pi}}\right)\,
\langle\Psi_{1}(x_{1})\cdots\Psi_{n}(x_{n})\rangle=\displaystyle{\frac{F(\hat{X}_{1(i)}{\cdot\hat{X}_{1(j)}})}{\displaystyle{\prod_{l<m}(x_{lm}^{2})^{-\frac{1}{(n-1)(n-2)}(\eta_{1}+\cdots+\eta_{n})+\frac{1}{n-2}(\eta_{l}+\eta_{m})}}}}\,,
\psi=e^{-i\omegat}\cdot\left\{F_{l\pm1,l}\Psi_{\frac{l\pm1}{2}\frac{l}{2}mm^{\prime}}^{(l)}+G_{l\pm1,l}\Psi_{\frac{l\pm1}{2}\frac{l}{2}mm^{\prime}}^{(l\pm1)}\right\}
A^{0}(x)=(2\pi)^{-3/2}\!\!\!\int\!\!\frac{d^{3}k}{k^{0}\sqrt{2}}i\left(a^{0}({\mathbf{k}})e^{-ikx}+a^{0}({\mathbf{k}})^{\dagger}e^{ikx}\right).\nonumber\,
T^{00}={\frac{2cosh^{2}(\gamma)}{r^{2}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]
\frac{\partial}{\partialu_{j}^{(n)}}\Gamma=-(\Theta\lambda^{n}E_{jj}{\Theta}^{-1})_{-}\Gamma+\Theta({\bfu},\lambda)\lambda^{n}E_{jj}D({\bfu},\lambda)=(\Theta\lambda^{n}E_{jj}{\Theta}^{-1})_{+}\Gamma
\begin{array}{ll}{H=\sum_{i=2}^{M}}&{(A_{x}s_{i}^{x}s_{j}^{x}+A_{y}s_{i}^{y}s_{j}^{y}-Bs_{i}^{z})-B_{1}s_{1}^{z}}\\{}&{+A_{x}s_{1}^{x}s_{M}^{x}+A_{y}s_{1}^{y}s_{M}^{y}+MB/2}\\\end{array}
V_{\mathrm{W_{B}}}(r)=\frac{NT_{0}}{L}\sqrt{\Bigl(1+\frac{r^{2}}{L^{2}}\Bigr)\Bigl(1+\frac{L^{2}r^{6}}{16(N^{2}-1)}\Bigr)}-\frac{NT_{0}}{4\sqrt{N^{2}-1}}\frac{r^{4}}{L}\,,
\frac{\alpha}{2\pi^{2}}\int_{0}^{1}\frac{dx}{1-x}\intd^{2}q_{\perp}\frac{\bbox{q}_{\perp}^{2}+(2-x)^{2}M^{2}}{\bbox{q}_{\perp}^{2}+x^{2}M^{2}+(1-x)\mu^{2}}\,,
\left(\begin{array}{c}{L}\\{M-u(x){\bf1}_{B}}\\\end{array}\right)=-\left(\begin{array}{cccc}{-1}&{}\\{}&{\rho}\\\end{array}\right)\frac{1-k}{\sqrt{2}}\left(\begin{array}{c}{L}\\{M-x{\bf1}_{3}}\\\end{array}\right)\frac{1+k}{\sqrt{2}}\,\rho^{-1}
{\mathcalD}_{0}^{-1}=(h-\widehat{/\!\!\!\partial}^{2})^{-1}P_{L}(a^{\dagger}-\widehat{/\!\!\!\partial})+(\tilde{h}-\widehat{/\!\!\!\partial}^{2})^{-1}P_{R}(a-\widehat{/\!\!\!\partial})\;.
L_{(\xi>0)}=-\sqrt{-g}\left[{\frac{1}{2}}\xi\hat{\phi}^{2}R+{\frac{1}{2}}\partial_{\mu}\hat{\phi}\partial^{\mu}\hat{\phi}+{\frac{1}{2}}(1+6\xi)f\hat{\phi}^{2}(fS_{\mu}S^{\mu}+D_{\mu}S^{\mu})+{\frac{1}{4}}H_{\mu\nu}H^{\mu\nu}+V(\hat{\phi})\right].
\gamma_{\alpha\beta}^{s}\gamma_{\alpha^{\prime}\beta^{\prime}}^{st}+\gamma_{\alpha^{\prime}\beta}^{s}\gamma_{\alpha\beta^{\prime}}^{st}+(\beta\leftrightarrow\beta^{\prime})=2\left(\delta_{\alpha\alpha^{\prime}}\gamma_{\beta\beta^{\prime}}^{t}-\delta_{\beta\beta^{\prime}}\gamma_{\alpha\alpha^{\prime}}^{t}\right)
\frac{\delta^{2}\mathcal{A}}{\deltax_{\mu}(s)\deltax_{\mu}(s^{\prime})}\propto\frac{1}{|s-s^{\prime}|^{4}}(1-D)(1-\dot{\phi}^{2})+\frac{1}{|s-s^{\prime}|^{2}}\left(\frac{1-D}{12}\dot{\phi}\stackrel{{\,...}}{\phi}+\frac{3-D}{4}\ddot{\phi}^{2}\right).
L=\frac{\kappa}{4\pi}\int_{D}d^{2}x\epsilon^{ij}\left((A_{i}^{a}+2\alpha[Q,D_{i}Q])\dot{A}_{j}^{a}+\alpha[Q,F_{ij}]^{a}\dot{Q}^{a}-A_{0}^{a}G_{ij}^{a}\right).
e_{i}^{(k)}=\frac{e_{i}^{k}}{[k]_{q_{i}}!},~~~f_{i}^{(k)}=\frac{f_{i}^{k}}{[k]_{q_{i}}!}\,,~~~[k]_{q}=\frac{q^{k}-q^{-k}}{q-q^{-1}},~~~q_{i}=q^{(\alpha_{i},\alpha_{i})/2},~~~[k]_{q}!=\prod_{n=1}^{k}[n]_{q}.
\frac{1}{2}\left(\delta_{\mu\lambda}\frac{\partial^{2}}{\partial\tau^{2}}-2ie\bar{F}_{\mu\lambda}\frac{\partial}{\partial\tau}\right){\cal{G}}_{B}^{\lambda\nu}(\tau_{1},\tau_{2})=\delta_{\mu\nu}\left(\delta(\tau_{1}-\tau_{2})-\frac{1}{s}\right)
\left\{-\frac{1}{2\mu}\left[\frac{d^{2}}{dr^{2}}+\frac{1}{r}\frac{d}{dr}-\frac{1}{r^{2}}(l-\alpha)^{2}\right]+\tilde{V}(r)\right\}\psi_{l}(r)=E\psi_{l}(r),
\sum_{r,s}\Bigl(\frac{\partial}{\partialt}b_{(n)}^{(r,s)}\Bigr)\hat{a}^{\daggerr}\hat{a}^{s}=i\sum_{r,s}b_{(n-1)}^{(r,s)}[\hat{a}^{\daggerr}\hat{a}^{s},\hat{H}^{\prime}].
X_{L}(a,b,c;q)=\displaystyle\sum_{\begin{array}{c}{\{\scriptstyle\mu|{\mu_{0}=a,\mu_{L}=b,\mu_{L+1}=c}\}}\\\end{array}}\!\!\!\!\!\!q^{\sum_{k=0}^{L-1}(k+1)H(\mu_{k},\mu_{k+1},\mu_{k+2})},
(v-v^{\prime})^{2}\rightarrow\frac{(v-v^{\prime})^{2}}{\Omega(v)\Omega(v^{\prime})}\,,\quad\vec{x}_{\alpha}\rightarrow\frac{\vec{x}_{\alpha}}{\Omega(v)}\,\quad\vec{x}_{\alpha}^{\prime}\rightarrow\frac{\vec{x}_{\alpha}^{\prime}}{\Omega(v^{\prime})}\,.
v_{\underline{\alpha}}^{~\underline{\beta}}=(v_{\underline{\alpha}}^{~\betar},v_{\underline{\alpha},\betar}),\quad(v^{-1})_{\underline{\beta}}^{~\underline{\alpha}}=(v_{\betar}^{~~\underline{\alpha}},v^{\betar,\underline{\alpha}}).
\hat{S}_{L}=-\intd^{D}x\left[\frac{1}{2m}\eta^{MN}\partial_{M}\hat{\Psi}_{L}^{\dagger}\partial_{N}\hat{\Psi}_{L}-\mu\hat{\Psi}_{L}^{\dagger}\hat{\Psi}_{L}+\frac{1}{2}b\left(\hat{\Psi}_{L}^{\dagger}\hat{\Psi}_{L}\right)^{2}\right]
\Phi_{x}=a_{0}^{-1}\sum_{N\geq0}\frac{\Gamma(4N+1)\;(-y)^{N}}{\Gamma^{3}(N+1)}\;\frac2{2\pii}\int_{C}\frac{\Gamma(2s-N)\Gamma(-s)}{\Gamma(s+1)}(e^{-\pii}x)^{s}\,ds,
{\calH}_{3}={\mathrme}^{-{\mathrmi}\frac{\pi}{2}{\calS}_{2}}{\mathrme}^{-{\mathrmi}\frac{\pi}{2}{\calS}_{3}}{\calH}_{1}{\mathrme}^{{\mathrmi}\frac{\pi}{2}{\calS}_{3}}{\mathrme}^{{\mathrmi}\frac{\pi}{2}{\calS}_{2}}={\calS}_{3}^{2}-{\calS}_{2}^{2}
E^{\mu}(\{q,n\})\;=\;\prod_{a=1}^{N}\Big(\mu\Big)^{\;2n_{a}\sum_{I=2}^{N}(U_{Ia})^{2}}\left\langle\prod_{b=1}^{N}\left[\chi_{+}^{(b)}[Q^{\mu}]\right]^{q_{b}}\left[\chi_{-}^{(b)}[Q^{\mu}]\right]^{2n_{b}-q_{b}}\right\rangle_{Q^{\mu}}\;.