harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g_{y^{\prime}\bar{y}^{\prime}}=g_{y^{\prime}\bar{y}^{\prime}}^{(0)}+{\frac{1}{\theta}}g_{y^{\prime}\bar{y}^{\prime}}^{(1)}+{\frac{1}{\theta^{2}}}g^{\prime}{}_{y^{\prime}\bar{y}^{\prime}}^{(2)}+\cdots
\partial_{\mu}T\left(j_{A\alpha}^{\mu}(x)\bar{j}_{B\dot{\beta}}^{\nu}(y)\right)=2\left(\sigma^{\rho}\right)_{\alpha\dot{\beta}}T_{\rho}^{\nu}\delta^{4}(x-y)\delta_{AB}+2\left(\sigma^{\nu}\right)_{\alpha\dot{\beta}}C_{AB}\delta^{4}(x-y),
0=\sum_{j}[\{z^{i},z^{j}\}_{S}-\{z^{i},z^{j}\}_{P}]\frac{\partialG^{\alpha}}{\partialz^{j}}=\sum_{j}[\{z^{i},z^{j}\}_{S}-\{z^{i},z^{j}\}_{P}]\M_{j}^{~\alpha}\varepsilon_{\alpha}.
\int_{t_{1}}^{t_{2}}\,\left(\frac{\partialL}{\partialx}\;\deltax+\frac{\partialL}{\partial{\calV}}\;\left[\,\frac{1-i}{2}\;\frac{d}{dt_{+}}+\frac{1+i}{2}\;\frac{d}{dt_{-}}\,\right]\,\deltax\right)\,dt=0,
\begin{array}{ccccccc}{}&{H_{A}\psi_{s}(t)}&{=}&{D_{2}D_{1}\psi_{s}(t)}&{=}&{sD_{2}\psi_{s}(t)}&{=}\\{}&{sD_{2}\psi_{1-s}(1/t)}&{=}&{s(1-s)\psi_{1-s}(1/t)}&{=}&{s(1-s)\psi_{s}(t),}&{}\\\end{array}
J^{(N)}(N+1;\{1+\delta_{il}\})=\frac{\pi}{2\;m_{l}\;D^{(N)}}\sum_{j=1}^{N}\frac{(-1)^{j+l}}{m_{j}}\;D_{jl}^{(N-1)}\;\left.J^{(N-1)}(N-1;1,\ldots,1)\right|_{\mathrm{\scriptsize~without}\;j}.
\widetilde{\mathrm{Inh}}\left(\chi_{0},\ldots,\chi_{n-1}\right)=\partial_{\bar{z}}\theta\partial_{z}\bar{\chi}_{n}-\partial_{z}\theta\partial_{\bar{z}}\bar{\chi}_{n}-\frac{1}{2}\theta^{n-1}\sum_{p=2}^{n-2}\left(\!\!\begin{array}{c}{n}\\{p}\\\end{array}\!\!\right)\!\bar{A}_{p+1}A_{n-p+1}-\frac{1}{2}\theta^{n-1}\partial_{\varphi}^{n}V(\varphi_{0})\,.
u\left(x_{\perp}\right)=\int_{0}^{L}dx_{3}\left\{i\Phi_{\mu}^{\dagger}\left(x\right)\stackrel{\leftrightarrow}{\partial}_{3}\Phi^{\mu}\left(x\right)-\bar{\psi}\left(x\right)\frac{\tau_{3}}{2}\gamma_{3}\psi\left(x\right)\right\}\,
\frac{\phi^{\prime}}{\left[1-(\phi^{\prime})^{2}+(y^{\prime})^{2}\right]^{\frac{1}{2}}}=-\frac{e}{r^{p-1}},\;\;\frac{-y^{\prime}}{\left[1-(\phi^{\prime})^{2}+(y^{\prime})^{2}\right]^{\frac{1}{2}}}=\frac{c}{r^{p-1}}
B_{ij}=\frac{\varepsilon}{2\pi\alpha^{\prime}}\left(\begin{array}{ccccc}{0}&{b_{1}}&{}&{}&{}\\{{}-b_{1}}&{0}&{}&{}&{}\\{}&{}&{0}&{b_{2}}&{}\\{}&{}&{-b_{2}}&{0}&{}\\{}&{}&{}&{}&{\ddots}\\\end{array}\right)\quad(i,j=1,\ldots,p^{\prime})~,\quad\mathrm{otherwise~B_{\mu\nu}=0~}~.
=\;2e\sum_{I=1}^{N}\left(\varepsilon_{\mu\nu}\partial_{\mu}A_{\nu}^{(I)},K_{II}\sum_{b=1}^{N}\sum_{j=1}^{n_{b}}U_{Ib}\Big[\delta_{n}(x_{j}^{(b)})-\delta_{n}(y_{j}^{(b)})\Big]\right)\;.
<0|N|0>_{\mathrm{loc}}=\int{\frac{d^{3}\vec{q}}{(2\pi)^{3}}}{\frac{1}{2\omega(\vec{q})}}\left[|J_{\mu}^{\mathrm{cl}}(\vec{q})|^{2}-{\frac{1}{M^{2}}}\left|J_{\mu}^{\mathrm{cl}}(\vec{q})q^{\mu}\right|^{2}\right]
\delta_{n,j}^{(1)}(k)=-\frac{\pi}{2}\int_{0}^{\infty}drV(r)r\left(J_{\frac{n}{2}+j-\frac{3}{2}}(kr)^{2}+J_{\frac{n}{2}+j-\frac{1}{2}}(kr)^{2}\right)
F(D,\beta_{H})\equiv\:\frac{V_{D-1}a(D)}{(2\pi)^{\frac{(D-1)}{2}}}\:\frac{(c^{2})^{\frac{(D-3)}{2}}}{\beta_{H}^{\frac{(D+1)}{2}}\:(\hbarc)^{(D-1)}}\:\equivA(D)\beta_{H}^{-\frac{D+1}{2}}
V_{\calN}^{-}(q)=\frac{C_{1}^{2}}{2}q^{6}+C_{1}C_{2}q^{4}+\frac12\{C_{2}^{2}-(4{\calN}-1-2C_{3})C_{1}\}q^{2}+\frac{C_{3}(C_{3}+1)}{2}\frac{1}{q^{2}}.
\partial_{t}(e^{-a+b+3c}\dot{\lambda})-\partial_{r}(e^{a-b+3c}\lambda^{\prime})=e^{a+b-3c}[2Q_{K}^{2}e^{\frac{8\lambda}{3}}-P^{2}e^{\frac{4\lambda}{3}+4\nu}-Q^{2}e^{\frac{4\lambda}{3}-4\nu}].
{\renewcommand{\arraystretch}{1.4}\begin{array}{ccc}{\partial_{1\dot{1}}\xi^{1}+\partial_{1\dot{2}}\xi^{2}}&{=}&{0,}\\{\partial_{2\dot{1}}\xi^{1}+\partial_{2\dot{2}}\xi^{2}}&{=}&{0,}\\{\partial^{1\dot{1}}\eta_{\dot{1}}+\partial^{2\dot{1}}\eta_{\dot{2}}}&{=}&{0,}\\{\partial^{1\dot{2}}\eta_{\dot{1}}+\partial^{2\dot{2}}\eta_{\dot{2}}}&{=}&{0.}\\\end{array}}
\partial{\bar{\partial}}\hat{{\calA}_{r}^{0}}\sim(-1+a_{r})a_{r}^{3}{\frac{b_{r}^{2}c_{r}e_{r}^{2}f_{r}(\sqrt{3}c_{r}-\sqrt{3}f_{r}){\bar{z}}^{2}+\sqrt{3}(b_{r}e_{r}c_{r})^{2}f_{r}z^{2}}{|z|^{4}}}
\left(\begin{matrix}{Q_{E}}\\{Q_{M}}\\\end{matrix}\right)={\frac{e}{4\pi}}\intdS_{\mu}^{(k)}\,\sqrt{-g}\,Tr\Big\{\left(\begin{matrix}{F^{\mut}}\\{\tilde{F}^{\mut}}\\\end{matrix}\right)\tau_{3}\Big\},
\hat{G}_{a}=f^{\frac{1}{2}}\xi_{a}^{i}\hat{p}_{i}f^{-{\frac{1}{2}}}=\xi_{a}^{i}\hat{p}_{i}-{\frac{i}{2}}\xi_{a,i}^{i}+{\frac{i}{2}}C_{ab}^{b}.
e^{ikX_{<}(z)}=\sum_{m=0}^{\infty}z^{m}\sum_{\sum_{i=1}^{\infty}ir_{i}=m}\frac{(ka_{-1})^{r_{1}}}{1^{r_{1}}r_{1}!}\frac{(ka_{-2})^{r_{2}}}{2^{r_{2}}r_{2}!}...\qquad.
\delta_{k}(\Lambda_{k})\Psi=\sum_{m_{1},\ldots,m_{k}=0}^{\infty}\frac{k^{m_{1}+\ldots+m_{k}}}{m_{1}!\ldotsm_{k}!}[\Psi^{m_{1}},[\ldots,[\Psi^{m_{k}},\Lambda_{k}]\ldots]].
F(r)={\frac{5r^{8}+(l^{2}-28n^{2})r^{6}+5n^{2}(14n^{2}-l^{2})r^{4}+5(3l^{2}-28n^{2})r^{2}-10mrl^{2}+5n^{6}(l^{2}-7n^{2})}{5l^{2}(r^{2}-n^{2})^{3}}}
(Sf)(y,x_{0},\dots,x_{d-1})=\int\frac{\prod_{i=0}^{d-1}dp_{i}}{(2\piy)^{d/2}}\,f(y,p_{0},\dots,p_{d})\,e^{\frac{i}{y}\sum_{i=0}^{d-1}p_{i}x_{i}}
\left\langle\,\prod_{j}T^{\mathrm{\scriptscriptstyleL}}(w_{j})\prod_{k}\widetildeT^{\mathrm{\scriptscriptstyleL}}(\bar{w}_{k})\prod_{r=1}^{N}P(z_{r},\bar{z}_{r})\,\right\rangle^{\!\!\!\mathrm{\scriptscriptstyleL}}\.
a_{l}=-\displaystyle\frac{1}{2}\displaystyle\frac{d}{dx}\left(\displaystyle\frac{K_{l-1}}{K_{l}}\right)+\displaystyle\frac{l}{x^{2}}\left[\left(\displaystyle\frac{K_{l-1}}{K_{l}}\right)^{2}-\left(\displaystyle\frac{K_{l-2}}{K_{l}}\right)^{2}+\ldots+\displaystyle\frac{1}{2}(-1)^{l+1}\displaystyle\frac{K_{0}^{2}}{K_{l}^{2}}\right].
\begin{aligned}{W^{\prime}}&{{}=\{w\inW|(w(\lambda),f(\lambda))=0\}\inGr^{(k+1)}(H),\quad{W^{\prime}}^{\perp}=W^{\perp}+\mathbb{C}f,}\\{{W^{\prime\prime}}}&{{}=W+\mathbb{C}g\inGr^{(k-1)}(H),\qquad{{W^{\prime\prime}}}^{\perp}=\{w\inW^{\perp}|(w(\lambda),g(\lambda))=0\}.}\\\end{aligned}
S_{k_{1},\,k_{2}\ldotsk_{N}\rightarrowp_{1}\,p_{2}\ldotsp_{M}}=\left[\frac{\delta}{\deltaa_{k_{1}}}\frac{\delta}{\deltaa_{k_{2}}}\cdots\frac{\delta}{\deltaa_{k_{N}}}\frac{\delta}{\deltaa_{p_{1}}^{*}}\frac{\delta}{\deltaa_{p_{2}}^{*}}\cdots\frac{\delta}{\deltaa_{p_{M}}^{*}}{\calF}[\varphi]\right]_{\varphi=0}\,.
\chi_{1,i+1}^{(2,2k+2)}(q)=\chi_{2,(i+1)/2}^{(4,k+1)}(q)=\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k}^{2}+n_{i+1}+\cdots+n_{k}}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-1}-n_{k}}(q)_{n_{k}}(-q)_{n_{k}}}.
\delta_{\zeta}z_{A}=\{z_{A},\zeta^{\alpha}\phi_{\alpha}(z_{A})\}\\\,\\\\delta_{\zeta}\lambda^{\alpha}=\dot{\lambda}^{\alpha}+\lambda^{\gamma}\zeta^{\beta}{C_{\beta\gamma}}^{\alpha}-\zeta^{\beta}{C_{\beta}}^{\alpha}\,
\overrightarrow{j_{e}}=\mathrm{grad}E^{0},\quad\overrightarrow{j}_{mag}=-\mathrm{grad}H^{0},\quad\stackrel{\sim}{\rho_{e}}=-\mu\partial_{0}E^{0},\quad\stackrel{\sim}{\rho_{mag}}=-\epsilon\partial_{0}H^{0}.
<\phi_{\omega},\phi_{\sigma}>=(\phi_{\omega},\phi_{\sigma})+\int_{\Sigma_{r}}\sqrt{h}d^{D-1}\left(i\phi_{\omega}^{*}a^{i}(\nabla_{i}+i\sigmaa_{i})\phi_{\sigma}-ia^{i}(\nabla_{i}-i\omegaa_{i})\phi_{\omega}^{*}\phi_{\sigma}\right)
K_{t}^{\infty}(\bar{r},\bar{r}^{\prime}|-\triangle_{c})=\int_{0}^{\infty}\frac{dk}{\pi}e^{ik(\varphi-\varphi^{\prime})}\int_{0}^{\infty}e^{-\nu^{2}t}J_{k}(\nur)J_{k}(\nur^{\prime})\nud\nu
S_{int}=\frac{i}{2\pi\alpha^{\prime}}\int_{z^{2}=0}dz^{1}\left(\frac{1}{2}\partial_{\nu}F_{\mu\lambda}\xi^{\nu}\xi^{\lambda}\partial_{1}\bar{X}^{\mu}+\frac{1}{3}\partial_{\nu}F_{\mu\lambda}\xi^{\nu}\xi^{\lambda}\partial_{1}\xi^{\mu}\right).
\tau_{0}=\frac12\left(\begin{array}{cc}{-i}&{0}\\{0}&{i}\\\end{array}\right)\,,\qquad\tau_{1}=\frac12\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,,\qquad\tau_{2}=\frac12\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\,.
\Phi_{+}^{+Y^{\prime}}(x,\theta^{+},u)=\rho_{+}^{+Y^{\prime}}(x,u)+\theta_{+}^{+A^{\prime}}\phi_{A^{\prime}}^{Y^{\prime}}(x,u)+i(\theta_{+}^{+})^{2}\chi_{-}^{-Y^{\prime}}(x,u)\;,
\tilde{\partial}_{\theta_{W}}-\partial_{\theta_{W}}v\int\Bigl({\frac{\delta}{\deltaH}}+\hat{\zeta}{\frac{\delta}{\delta\hat{H}}}\Bigr)=\tilde{\partial}_{\theta_{W}}-\frac2eM_{Z}cos2\theta_{W}\int\Bigl({\frac{\delta}{\deltaH}}+\hat{\zeta}{\frac{\delta}{\delta\hat{H}}}\Bigr)
(E-\frac{\vec{p}^{~\!2}}{2M})G(\vec{p},\vec{p}^{~\!\prime})\nonumber\,-\frac{iI\mu}{\pi}\intd^{2}p^{\prime\prime}\frac{\sigma_{i}\\varepsilon_{ij}(\vec{p}-\vec{p}^{~\!\prime\prime})_{j}}{|\vec{p}-\vec{p}^{~\!\prime\prime}|^{2}}G(\vec{p}^{~\!\prime\prime},\vec{p}^{~\!\prime})=\delta(\vec{p}-\vec{p}^{~\!\prime})
\phi_{0}^{(-1)}={\frac{1}{{J_{H}^{-1}}^{\prime}}},\qquad\;\phi_{1}^{(-1)}={\frac{J_{H}^{-1}}{{J_{H}^{-1}}^{\prime}}},\qquad\;\phi_{2}^{(-1)}={\frac{\left(J_{H}^{-1}\right)^{2}}{{J_{H}^{-1}}^{\prime}}}.
b(z)\equiv\frac{(z^{2}-{\calM}^{2})(1-y)}{(z^{2}-{\calM}^{2})(1-y)+(2\mu)^{2}},\quad{\calM}^{2}\equiv\frac{(2\mu)^{2}}{1-y}+\frac{\mu^{2}}{y}+\frac{s^{2}}{y(1-y)}\ge(3\mu)^{2}.
\begin{array}{l}{\deltaC=\frac{i}{\xi}g^{\dagger}\partial_{\mu}A^{\mu},\nonumber}\\{\delta\bar{C}=-\frac{i}{\xi}f^{\dagger}\partial_{\mu}A^{\mu},}\\{\delta\psi=-ig_{0}(fC+g\bar{C})\psi,}\\{\delta\bar{\psi}=ig_{0}\bar{\psi}(Cf^{\dagger}+\bar{C}g^{\dagger}),}\\\end{array}
\overline{{\mathcal{D}_{g}\left(\begin{array}{ccc}{p_{1}}&{\ldots}&{p_{N}}\\{r_{1}}&{\ldots}&{r_{N}}\\\end{array}\right)}}=\mathcal{D}_{g}\left(\begin{array}{ccc}{\overline{{p_{1}}}}&{\ldots}&{\overline{{p_{N}}}}\\{1-r_{1}}&{\ldots}&{1-r_{N}}\\\end{array}\right)
\mathrm{Tr}e^{iF_{9}}-\mathrm{tr}e^{iF_{9}}\,\mathrm{tr}e^{iF_{\bar{9}}}+\mathrm{Tr}_{S}e^{iF_{\bar{9}}}={\frac{1}{2}}\,(\mathrm{Str}e^{iF_{9}})^{2}-{\frac{1}{2}}\,\mathrm{Str}e^{2iF_{9}}.
q=\frac{q_{0}}{\sqrt{2\pi}}+\sum_{k>0}\frac{1}{\sqrt{2\omega_{k}}\sqrt{2\pi}}(a_{k}e^{-ikx}+\bar{a}_{k}e^{ikx}+a_{k}^{+}e^{ikx}+\bar{a}_{k}^{+}e^{-ikx}),
L_{(0,1)}^{\kappa\lambda\mu\nu}=\delta^{\kappa\lambda}\delta^{\mu\nu}\,,\,\,L_{(0,2)}^{\kappa\lambda\mu\nu}=\delta^{\kappa\mu}\delta^{\lambda\nu}\,,\,\,L_{(0,3)}^{\kappa\lambda\mu\nu}=\delta^{\kappa\nu}\delta^{\lambda\mu}\,,
\begin{array}{ccll}{[G_{t,i},G_{t,k}]}&{=}&{0,}&{\mathrm{~if~i-k~even}}\\{G_{t,i}G_{\tilde{t},k}}&{=}&{q^{-\frac{m}{2}}G_{\tilde{t},k}G_{t,i},}&{\mathrm{~if~i-k~odd,~i<k,\;~|i-k|<2p~},}\\{[G_{t,i},{M^{(k)}}_{t}]}&{=}&{0}&{\mathrm{~(~i-k~odd}),}\\{G_{t,i}{M^{(k)}}_{\tilde{t}}}&{=}&{q^{-m}M_{t}^{(k)}G_{t,i}}&{\mathrm{~(~i-k~even),}}\\{[{M^{(1)}}_{t},{M^{(2)}}_{\tilde{t}}]}&{=}&{0}\\\end{array}
Z_{t}^{K3}(\tau)=\frac{1}{4}\frac{1}{\eta^{8}(2\tau)}{\tilde{Z}}_{3}(\tau)+4\left(Z_{even}^{S_{0}}(\tau){\tilde{Z}}_{2}(\tau)+Z_{odd}^{S_{0}}(\tau){\tilde{Z}}_{1}(\tau)\right).
{\cal{G}}_{1}^{(\tilde{\eta}_{o})}(x_{1},x_{2},x_{3},x_{4})=\frac{g_{*}^{2}}{C_{\tilde{O}}}\frac{1}{(x_{12}^{2}x_{34}^{2})^{\eta}}\Bigl[u^{\frac{1}{2}\tilde{\eta}_{o}}\,{\cal{H}}_{\tilde{\eta}_{o}}(u,v)+C(d-\tilde{\eta}_{o})\,u^{\mu-\frac{1}{2}\tilde{\eta}_{o}}\,{\cal{H}}_{d-\tilde{\eta}_{o}}(u,v)\Bigl],
{\calB}(T_{1},\ldots,T_{L})=\prod_{n=1}^{L}(\oint\frac{dw_{n}}{2\pii}\frac{dw_{n}}{d\rho})\langle\prod_{r=1}^{N}{\calO}_{r}(z_{r})\prod_{n=1}^{L}b(w_{n})\prod_{i=1}^{V-L_{0}}Z(z_{0i})\prod_{i=1}^{L_{1}}W(z_{0i}^{\prime})\rangle.
A=\sum_{i\neqj\neqk=1}^{3}A_{u_{u}u_{j}{\bar{u}}_{k}}du_{i}\wedgedu_{j}\wedged{\bar{u}}_{k}+\sum_{i\neqj=1}^{3}A_{u_{i}u_{j}{\bar{u}}_{i}}du_{i}\wedgedu_{j}\wedged{\bar{u}}_{i},\nonumber\,
S=-\frac{1}{8\piG}\intd^{3}x\sqrt{-h}\frac{c_{0}}{l_{\mathrm{eff}}},\\\c_{0}=2\sqrt{\frac{3}{a^{2}(1+a^{2})}},\\\\frac{1}{l_{\mathrm{eff}}}=e^{b\phi}\sqrt{\frac{\Lambda}{3}}.
\pi_{q}^{1+}=-\frac{i}{2}e^{+2}\nabla_{+}\lambda_{q}^{1}+e^{+}\lambda_{q}^{1},\\pi_{q}^{2-}=-\frac{i}{2}e^{-2}\nabla_{-}\lambda_{q}^{2}+e^{-}\lambda_{q}^{2},
{\calL}(n){\calL}(n)^{T}=1\quad\Longrightarrow\quad\left(\frac{\partial}{\partialn^{\mu}}{\calL}(n)\right){\calL}(n)^{T}+{\calL}(n)\frac{\partial}{\partialn^{\mu}}{\calL}(n)^{T}=0.
\partial_{z}\partial_{\bar{z}}\tilde{\psi}+\frac{1}{2}\partial_{\varphi}^{2}V(\varphi_{0})\tilde{\psi}=0\,,\quad\,\,\partial_{z}\partial_{\bar{z}}\left(\frac{1}{\tilde{\psi}}\right)-m^{2}\left(\frac{1}{\tilde{\psi}}\right)=0\,.
ds^{2}=r_{+}^{2}\left[\left(\rho^{2}-\frac{1}{\rho^{2}}\right)dt^{2}+\left(\rho^{2}-\frac{1}{\rho^{2}}\right)^{-1}d\rho^{2}+\rho^{2}\sum_{i=1}^{3}dx_{i}^{2}+d\Omega_{5}^{2}\right].
\psi(q^{i})=\left(\frac{1}{2\pi}\right)^{n/2}\int_{Q^{\prime}}\left(\frac{\mu^{\prime}}{\mu}\right)^{1/2}d^{n}q^{\prime}\sqrt{\mathrm{det}\left(\frac{\partial^{2}S}{\partialq^{j}\partialq^{\primek}}\right)}e^{iS}\psi(q^{\primei}).
d\,R_{g}(A|_{h})=a^{M}(h)\frac{\partialx^{N}(hg)}{\partialx^{M}(h)}\frac{\partial}{\partialx^{N}(hg)}=a^{M}(h)R_{M}^{N}(hg,h)\frac{\partial}{\partialx^{N}(hg)}\,,
8\int\frac{d^{4}\tilde{s}}{(2\pi)^{4}}\frac{\tilde{s}_{i}\tilde{s}_{j}\tilde{s}^{2}\tilde{s}^{2}\tilde{s}^{2}}{[\tilde{s}^{2}+\tilde{s}_{0}^{2}(1-x_{1}-x_{3}-x_{5})+\tilde{M}^{2}]^{6}}=315\frac{\delta_{ij}}{(4\pi)^{2}(1-x_{1}-x_{3}-x_{5})^{\frac{1}{2}}}\Gamma(\epsilon).
S\approx\intd^{10}x\sqrt{|\tilde{g}_{10}|}\left[e^{-\tilde{\Phi}}(\tilde{R}_{10}+|\tilde{\nabla}\tilde{\Phi}|^{2})-e^{-\tilde{\Phi}/2}\tilde{F}_{2}^{2}-|d\tilde{B}_{2}|^{2}\right]
\begin{array}{l}{i\hbar\partial_{\tau}\Phi=-\frac{\hbar^{2}}{2M}\Box_{g}\Phi-\frac{i\hbar}{M}g^{\mu\nu}(\partial_{\mu}W_{\tau}+A_{\mu})\partial_{\nu}\Phi_{\tau}-\frac{i\hbar}{2M}\Box_{g}\Phi_{\tau}}\\\end{array}
\left.{\frac{z_{1}dz_{2}-z_{2}dz_{1}}{z_{0}^{*}}}\right|_{S}=\left.{\frac{z_{0}dz_{2}-z_{2}dz_{0}}{z_{1}^{*}}}\right|_{S}=\left.{\frac{z_{1}dz_{0}-z_{0}dz_{1}}{z_{2}^{*}}}\right|_{S}
{\calL}_{em-H}=-\frac{1}{M_{P}e^{kr_{c}\pi}}A^{[\mu}F^{\nu\lambda]}(x)H_{\mu\nu\lambda}^{0}(x)-\frac{1}{\Lambda_{\pi}}\frac{2J_{0}(x_{n})}{\pix_{n}}A^{[\mu}F^{\nu\lambda]}(x)\sum_{n=1}^{\infty}H_{\mu\nu\lambda}^{n}(x)
F^{(-2)}=\frac{1}{j(T)-j(U)}\{-\frac{j_{U}^{2}(U)}{j(U)(j(U)-j(i))}\}\frac{j_{U}^{3}(U)}{j(U)^{2}(j(U)-j(i))}(\frac{j_{U}^{6}}{j^{4}(j(U)-j(i))^{3}})^{-1}.
K_{kl}(t,x)=\delta_{kl}\int\frac{d\omegad^{2}p}{(2\pi)^{3}}\frac{e^{-i\omegat+i{\vec{p}}\cdot{\vec{x}}}}{(\frac{\hbar^{2}p^{2}}{4m\rho_{0}}+\lambda)p^{2}-\frac{m}{\rho_{0}}\omega^{2}}
T_{H}^{\mathrm{BTZ}}=(\rho_{+}^{2}-\rho_{-}^{2})/2\piR^{2}\rho_{-}=T_{H},~~{\frac{1}{T_{L/R}^{\mathrm{BTZ}}}}={\frac{1}{T_{H}^{\mathrm{BTZ}}}}\left(1\pm{\frac{\rho_{+}}{\rho_{-}}}\right)={\frac{1}{T_{L/R}}}.
Q^{1}=-\frac{J}{2}(\xi^{0}\xi^{1*}+\xi^{1}\xi^{0*}),Q^{2}=-i\frac{J}{2}(\xi^{0}\xi^{1*}-\xi^{1}\xi^{0*}),Q^{3}=-\frac{J}{2}(\xi^{0}\xi^{0*}-\xi^{1}\xi^{1*})
\tilde{U}_{N_{s}N_{r}}(\{\sigma_{p}\})=\int_{C_{v_{r}}}\Psi_{\sigma,N_{s}}(t)\frac{d\thetadz}{2\pii}P_{N_{r}}(t)+\int_{\tilde{C}_{r}}\Psi_{\sigma,N_{s}}(t)\frac{d\thetadz}{2\pii}P_{N_{r}}^{(a)}(t)
\intd^{4}x\mathrm{Tr}{\calR}_{\mu\nu}F_{3}(\Box){\calR}_{\mu\nu}=\intd^{4}x\left[\frac{1}{2}RF_{3}(\Box)R-2R_{\mu\nu}F_{3}(\Box)R_{\mu\nu}+O({\calR}^{3})\right]
\frac{a_{1}}{x_{12}^{4}x_{34}^{4}}=\frac{32N_{c}^{2}}{2^{8}\pi^{10}}\,\Bigl[-\frac{1}{2}\frac{1}{x_{34}^{2}}\,D_{2211}+(\frac{t}{s}+\frac{1}{s})x_{12}^{2}\,D_{3322}+\frac{3}{2}\,D_{2222}\Bigr]
{_2}\phi_{1}\left[\begin{array}{c}{\begin{array}{cc}{a,}&{b}\\\end{array}}\\{c}\\\end{array};q,u\right]=\sum_{n=0}^{\infty}\frac{(a;q)_{n}(b;q)_{n}}{(q;q)_{n}(c;q)_{n}}u^{n},
S=\int\left(\frac{R}{16\piG}-\frac{1}{2e^{2}}\mathrm{Tr}(F_{\mu\nu}F^{\mu\nu})-\frac{1}{4}\mathrm{Tr}(D_{\mu}\PhiD^{\mu}\Phi)-\frac{1}{4}\lambda\mathrm{Tr}(\Phi^{2}-\eta^{2})^{2}\right)\sqrt{-g}d^{4}x\,
\tilde{I}_{\mathrm{DE}}=-\frac{1}{16\pi}\intd^{D}x\sqrt{g}\left(R-\frac{1}{2}(\nabla\phi)^{2}-\frac{e^{\alpha\phi}}{2(\tilde{d}+1)!}\tilde{F}_{\tilde{d}+1}^{2}\right)+\frac{1}{8\pi}\intd^{D-1}x\sqrt{h}(K-K_{o}),
{\calL}=\left({\frac{dU}{d\tau}}\pme^{U}|Z|\right)^{2}+\left|{\frac{dz^{i}}{d\tau}}\pme^{U}G^{i\bar{k}}\bar{D}_{\bar{k}}\bar{Z}\right|^{2}\pm2{\frac{d}{d\tau}}\left(e^{U}|Z|\right)\.
\deltaT_{N}^{M}=\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{-\delta\rho}&{-(\rho+p)e^{\alpha_{0}}v_{,i}}\\{0}&{(\rho+p)e^{-\alpha_{0}}v_{,i}}&{\deltap\:\delta_{ij}}\\\end{array}\right)\:\delta(y),
\deltau_{\underline{{m}}}^{(0)}=u_{\underline{{m}}}^{i}i_{\delta}f^{i},\qquad\deltau_{\underline{{m}}}^{i}=-u_{\underline{{m}}}^{j}i_{\delta}A^{ji}+u_{\underline{{m}}}^{(0)}i_{\delta}f^{i},
\Phi^{M}*_{b_{0}}\Psi^{M}[\vec{X}^{3}]=\int\prod_{A=1,2}[d\vec{x}^{A}\d\vec{y}^{A}]\Phi^{M*}[\vec{X}^{1}]\Psi^{M*}[\vec{X}^{2}]K[\vec{X}^{1},\vec{X}^{2},\vec{X}^{3}]
I[A_{i},A_{0},\tau]=\frac{k}{4\pi}\int_{M}\epsilon^{ij}\mathrm{Tr}\,(-A_{i}\dot{A}_{j}+A_{0}F_{ij})-\frac{k\tau}{4\pi}\int_{T_{\infty}^{2}}\mathrm{Tr}A_{\varphi}^{2}\,,
\{f,g\}_{\kappa}=i\left(\frac{\partial^{R}f}{\partial\bar{z}^{A}}g_{\kappa}^{{\bar{A}}B}\frac{\partial^{L}g}{\partialz^{B}}-(-1)^{(p(A)+\kappa)(p(B)+\kappa)}\frac{\partial^{R}f}{\partialz^{A}}g_{\kappa}^{{\bar{A}}B}\frac{\partial^{L}g}{\partial\bar{z}^{B}}\right),
\beta{\Lambda^{\prime}}_{\mu}=\frac{\partialx^{\nu}}{\partial{x^{\prime}}^{\mu}}(\beta\Lambda_{\nu})+\frac{\partial^{2}x^{\nu}}{{\partial{x^{\prime}}^{\mu}}{\partial{x^{\prime}}^{\sigma}}}\frac{\partial{x^{\prime}}^{\sigma}}{\partialx^{\nu}}.
ds^{2}=e^{-\Phi/2}\left(\nud\tau^{2}+N\alpha^{\prime}\left[\frac{1}{\nu}dr^{2}+R^{2}d\Omega_{3}^{2}\right]+\sum_{i=1}^{2}dx^{i}dx^{i}+\frac{N\alpha^{\prime}}{4}\sum_{i=1}^{3}(\hat{\theta}^{i}-A^{i})^{2}\right)\,.
\frac{\Lambda^{6}}{p^{2}(\Lambda^{6}+p^{6})}\epsilon_{\mu\rho\nu}p^{\rho}-\frac{\xi_{0}}{p^{4}f^{2}(p)}p_{\mu}p_{\nu}+\frac{\Lambda^{3}}{(\Lambda^{6}+p^{6})}(p^{2}g_{\mu\nu}-p_{\mu}p_{\nu}).
{\calG}\equiv{\frac{\dot{G}}{G}}=-2{\frac{\dot{\phi}}{\phi}}=-{\frac{48\pi^{2}\rho_{0}a_{0}^{3}}{K\lambda^{2}\phi^{4}a^{3}}}{\frac{\dot{a}}{a}}={\frac{3}{2}}\left({\frac{\left<\phi\right>^{4}}{\phi^{4}}}-1\right)H,
\{Q_{a}\,,\,Q_{b}\}\,=\,2\,i\,P^{\mu}\,(\gamma_{\mu})_{ab}\,+\,2\,i\,\epsilon^{i\,j}\,\int\,d^{2}\vec{x}\,(\partial_{i}\,A)\,(\gamma_{j})_{ab}\,\left(m\,A\,+\,\frac{\lambda}{2}A^{3}\right)\,.
W(\Phi)=\sum_{i=0}^{N}M_{i}W(\imathe_{i})+\frac{1}{2}\alpha\sum_{l=0}^{N}s_{N-l}\sum_{m=0}^{l}\mathrm{Tr}(\Psi\Phi_{0}^{m}\Psi\Phi_{0}^{2l-m})+{\calO}(\Psi^{3})\,.
\left\langle0\right|T\left[\bar{\psi}^{\uparrow}(\vec{t})\psi^{\uparrow}(0)\right]\left|0\right\rangle=\left\langle0\right|T\left[\bar{\psi}^{\downarrow}(\vec{t})\psi^{\downarrow}(0)\right]\left|0\right\rangle=0
D_{z}=\left(\begin{array}{ccc}{(\mathbf{p}_{1})^{\dagger}\mathbf{x}^{1}+(\mathbf{p}_{1}^{\prime})^{\dagger}\mathbf{x}^{2}-(\mathbf{a}_{1})}&{\ldots}&{0}\\{\vdots}&{\ddots}&{\vdots}\\{0}&{\ldots}&{(\mathbf{p}_{k})^{\dagger}\mathbf{x}^{1}+(\mathbf{p}_{k}^{\prime})^{\dagger}\mathbf{x}^{2}-(\mathbf{a}_{k})}\\{-\lambda_{1}A_{1}}&{\ldots}&{-\lambda_{k}A_{k}}\\\end{array}\right)\,,
g=\left(\begin{array}{cc}{1}&{b}\\{0}&{1}\\\end{array}\right)\,\left(\begin{array}{cc}{e^{a/2}}&{0}\\{0}&{e^{-a/2}}\\\end{array}\right)\,(-1)^{\epsilon}\,\left(\begin{array}{cc}{1}&{0}\\{x}&{1}\\\end{array}\right)\,,
\beta={\frac{1}{T_{H}}}={\frac{4\pir_{+}\sqrt{\prod_{i=1}^{4}(r_{+}+q_{i})}}{3r_{+}^{4}+2r_{+}^{3}\sum_{i=1}^{4}q_{i}+r_{+}^{2}(\sum_{i<j}^{4}q_{i}q_{j}+1)-\prod_{i=1}^{4}q_{i}}}.
\sum_{j}\langley_{i},x_{j}\rangle_{\calB}\langley_{j},x_{k}\rangle_{\calB}=\sum_{j}\langley_{i},x_{j}\langley_{j},x_{k}\rangle_{\calB}\rangle_{\calB}=\langley_{i},\sum_{j}\langlex_{j},y_{j}\rangle_{\calA}x_{k}\rangle_{\calB}=\langley_{i},x_{k}\rangle_{\calB}.
D_{2}\eta_{2}^{\alpha_{1}}=\frac12C_{\;\;\beta_{1}\gamma_{1}}^{\alpha_{1}}\eta_{2}^{\beta_{1}}\eta_{2}^{\gamma_{1}}+C_{\;\;\beta_{1}\gamma_{3}}^{\alpha_{1}}\eta_{2}^{\beta_{1}}\eta_{2}^{\gamma_{3}},
A_{x_{||}}^{11}=\frac{1}{\sqrt{\rho}}\int_{\tilde{T}^{2}}d^{2}z\,K_{-}(\omega,z;x)e^{\phi(z)}\left(D_{x}^{\dagger}(\hat{A})\partial_{x_{||}}D_{x}(\hat{A})\right)^{11}\frac{e^{\phi(z)}}{\sqrt{\rho}}K_{-}(z,\omega;x).
i\frac{db_{m}}{dt}=-\sum_{n\neqm}e^{-i\lambda\int_{0}^{t}dt^{\prime}[E_{n}(t^{\prime})-E_{m}(t^{\prime})]}e^{i[\gamma_{n}(t)-\gamma_{m}(t)]}<m;t|i\frac{\partial}{\partialt}|n;t>b_{n}(t).
d\hat{s}^{2}=(-H^{-\frac{1}{2}}+H^{\frac{1}{2}}\dot{r}^{2}+H^{\frac{1}{2}}r^{2}h_{ij}\dot{\varphi}^{i}\dot{\varphi}^{j})dt^{2}+H^{-\frac{1}{2}}(d\vec{x})^{2}
\vert\Xi_{0,p}[\alpha,\beta]\vert^{2}=\frac{\prod_{j=3}^{p-1}(-1)^{2m_{\beta_{j}}}q^{4im_{\beta_{j}}\rho_{\beta_{j}}}}{{\psi}_{2}{}_{(\alpha_{1},\alpha_{2},\beta_{3})}\prod_{j=3}^{p-2}{\psi}_{2}{}_{(\beta_{j},\alpha_{j},\beta_{j+1})}{\psi}_{2}{}_{(\beta_{p-1},\alpha_{p-1},\alpha_{p})}}\Xi_{0,p}[\alpha,\beta]^{2}
g_{ij,kl}=\frac{-q^{-1}}{\left[2\right]}D_{si}\hat{R}_{js,kl}^{\epsilon}=\frac{q^{-1}}{\left[2\right]}\epsilon_{im}^{q}\hat{R}_{jm,kt}\epsilon_{lt}^{q\;-1}\quad,\quad(g=\frac{-q^{-1}}{\left[2\right]}D_{1}{\calP}\hat{R}_{12}^{\epsilon})\quad,
\left[\frac{27}{2}\left(\frac{\partial^{2}W}{\partial\Phi^{2}}\right)^{2}-24W\frac{\partial^{2}W}{\partial\Phi^{2}}-\frac{\partial^{2}V}{\partial\Phi^{2}}\right]_{\left|\Phi=\Phi_{c}^{\pm}\right.}=0.
C_{N}^{2}=\frac1{\left(2\pi\right)^{d-1}}\frac{I_{i\omega}^{\prime}(k\xi_{1})}{I_{i\omega}^{\prime}(k\xi_{2})\frac{\partialN_{i\omega}^{\prime}(k\xi_{1},k\xi_{2})}{\partial\omega}}|_{\omega=\omega_{Nn}}.
ds_{n}^{2}=-\sum_{\mu,\nu=1}^{n}d\chi^{\mu}d\chi^{\nu}\left(\frac{\delta_{\mu\nu}}{1-\displaystyle\sum_{\alpha=1}^{n}\chi^{\alpha}\chi^{\alpha}}+\frac{\chi^{\mu}\chi^{\nu}}{(1-\displaystyle\sum_{\alpha=1}^{n}\chi^{\alpha}\chi^{\alpha})^{2}}\right)~.
(\frac12[\xi^{*},\xi])^{n}=n!\;\sum_{i_{1}<\cdots<i_{n}}\xi_{i_{1}}^{*}\cdots\xi_{i_{n}}^{*}\xi_{i_{1}}\cdots\xi_{i_{n}},\;\cdots,(\frac12[\xi^{*},\xi])^{p}=p!\;\xi_{1}^{*}\cdots\xi_{p}^{*}\xi_{1}\cdots\xi_{p},
\begin{array}{l}{\displaystyle{\frac{A_{n}}{B_{n}}}\rightarrow_{r_{0}\rightarrow0}0\qquad\mathrm{for}\quadn\leqk-1}\\{\displaystyle{\frac{B_{n}}{A_{n}}}\rightarrow_{r_{0}\rightarrow0}0\qquad\mathrm{for}\quadn\geqk}\\\end{array}
_\partial\langlej|\partial_{\mu}\psi(b_{1})\partial^{\mu}\psi(b_{2})|j\rangle_{\partial}=\frac{jC}{2\pi^{4}a}\,\mathrm{Im}\,\left\lbrace\frac{1}{(2(t-a)+i\epsilon)^{3}}-\frac{1}{(2(t+a)+i\epsilon)^{3}}\right\rbrace.
\frac{\partialH}{\partial\rho}+m_{CS}\frac{\partial\phi}{\partial\rho}-\gammaC_{2}\bigg(A_{\varphi}+\frac{1}{\rho^{2}}\int_{0}^{\rho}A_{\varphi}(\rho^{\prime})\rho^{\prime}d\rho^{\prime}\bigg)=0.
G_{N\{(n_{v_{q}})\}}^{\{r_{q},r_{v_{q}}\}}(z_{q})=\sum_{\{p_{v_{q}}\}}^{\{n_{v_{q}}\}}G_{N\{(p_{v_{q}})\}}^{\{\alpha_{q},\alpha_{v_{q}}\}}(z_{q})G_{N\{(n_{v_{q}}-p_{v_{q}})\}}^{\{K_{q},K_{v_{q}}\}}(z_{q})
\begin{array}{cc}{\theta^{i}=\left(\begin{array}{c}{\vartheta_{-}^{a}}\\{(\bar{\vartheta}_{a}^{-})^{t}}\\\end{array}\right)~~~~~}&{~~~~~\tilde{\theta}^{i}=\left(\begin{array}{c}{\vartheta_{+}^{a}}\\{(\bar{\vartheta}_{a}^{+})^{t}}\\\end{array}\right)}\\\end{array}
\Phi_{0}=\alpha_{1}^{2}l^{2}\phi_{0}+\alpha_{1}e^{\alpha_{0}}l\dot{b}_{0}+\left(1+\frac{\ddot{\alpha}_{0}}{\alpha_{1}^{2}}\right)\alpha_{1}e^{2\alpha_{0}}\hat{E}_{1}-2\dot{\alpha}_{0}e^{2\alpha_{0}}\dot{\hat{E}}_{0}-e^{2\alpha_{0}}\ddot{\hat{E}}_{0}.
P^{0}=E=\frac{1}{2e^{2}{\phi_{0}}^{2}}\intd^{2}x\left[\frac{1}{2}F_{ij}^{2}+\vertD_{i}\phi\vert^{2}+\frac{e^{2}}{4}(\vert\phi\vert^{2}-{\phi_{0}}^{2})^{2}\right]
\bar{\cal{H}}\,\sum_{i=1}^{n}d_{\Lambda\Omega^{(i)}}m_{\Omega^{(i)}}=\sum_{i=1}^{n}d_{\Lambda\Omega^{(i)}}\left(\varepsilon_{\Omega^{(i)}}\,m_{\Omega^{(i)}}+\sum_{\Gamma<^{s}\Omega^{(i)}}v_{\Omega^{(i)}\Gamma}m_{\Gamma}\right)=\varepsilon_{\Lambda}\,\sum_{i=1}^{n}d_{\Lambda\Omega^{(i)}}m_{\Omega^{(i)}}\,.
S_{\chi}\,=\,\int_{t^{\prime}}^{t^{\prime\prime}}\,dt\left(p_{i}\dot{q}^{i}+\mu_{a}\dot{\lambda}^{a}+i\zeta_{a}\dot{\eta}^{a}+i\zeta^{\times}{\!}_{a}\dot{\eta}^{\timesa}\,-H\,-i\{\Omega,\phi\}\right).
{\calL}=-\frac{1}{36}\left\{\left(F_{(\mu\nu\sigma)\tau}\right)^{2}-3\left(F_{(\mu\nu\sigma)}{}^{\mu}\right)^{2}-3M^{2}\left[\left(T_{(\mu\nu)\sigma}\right)^{2}-2\left(T_{(\mu\nu)}{}^{\mu}\right)^{2}\right]\right\}
M_{K}=\left(\begin{array}{cccc}{f^{-1}}&{0}&{-\chif^{-1}}&{0}\\{0}&{-1}&{0}&{0}\\{-\chif^{-1}}&{0}&{f+\chi^{2}f^{-1}}&{0}\\{0}&{0}&{0}&{-1}\\\end{array}\right),
D=tH-\frac{1}{2}\int_{-\infty}^{\infty}dx(x-vt){\calP}(x-vt)-\frac{v}{2}t\int_{\infty}^{\infty}dx{\calP}(x-vt)=t(H-\frac{v}{2}P)-\frac{1}{2}D_{0},
{\mathsfL}_{kin}^{45-Higgs}=-\partial^{A}{\mathsfH}\partial_{A}{\mathsfH}^{\dagger}-\frac{1}{2!}\partial^{A}{\mathsfH}_{ij}\partial_{A}{\mathsfH}_{ij}^{\dagger}-\frac{1}{2!}\partial^{A}{\mathsfH}^{ij}\partial_{A}{\mathsfH}^{ij\dagger}-\partial_{A}{\mathsfH}_{j}^{i}\partial^{A}{\mathsfH}_{j}^{i\dagger}.
g^{\zeta}(\xi,\eta)=2\left(\frac{(\xi-\zeta)(\bar{\eta}-\bar{\zeta})}{(1+\bar{\zeta}\xi/R^{2})(1+\zeta\bar{\eta}/R^{2})}+\frac{(\bar{\xi}-\bar{\zeta})(\eta-\zeta)}{(1+\zeta\bar{\xi}/R^{2})(1+\bar{\zeta}\eta/R^{2})}\right).\nonumber
M_{BPS}^{2}=(\vec{\tilde{V}}_{I}.\vec{\tilde{V}}_{J})(\vec{\tilde{A}}_{I}.\vec{\tilde{A}}_{J})+(\vec{\tilde{V}}_{I}\times\vec{\tilde{V}}_{J}).(\vec{\tilde{A}}_{I}\times\vec{\tilde{A}}_{J}).