harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left({\frac{\calZ}{|Z|}}C\right)^{\prime}=\left(\begin{matrix}{U(Y_{0})_{sp}}&{0}\\{0}&{U(Y_{0})_{sp}^{*}}\\\end{matrix}\right)\left({\frac{\calZ}{|Z|}}C\right)=0\,\qquad|Z|^{\prime}=|Z|\.
\begin{array}{lll}{[(sxu\wedge[r])\,m\,(szu\wedge[r])]\,j\,[(sxu\wedge[r])\,m\,(rzd\wedge[s])]}&{=}&{}\\{(\{s\}\wedge[r])\,j\,(sxu\wedgerzd)=sxu\wedge[r],}&{}&{}\\\end{array}
\bar{\chi}^{\mu_{1}\ldots\mu_{p-2k}}=\partial_{\mu}V^{(1)\mu\mu_{1}\ldots\mu_{p-2k}}+\frac1{p-2k}\partial^{\left[\mu_{1}\right.}V^{(1)\left.\mu_{2}\ldots\mu_{p-2k}\right]},\;k=1,\cdots,b,
(\psi_{\omega}^{(\lambda)},\psi_{\sigma}^{(\lambda)})={\frac{1}{(\omega-\sigma)}}\left[(\psi_{\sigma}^{(\lambda)},H(\lambda)\psi_{\omega}^{(\lambda)})^{*}-(\psi_{\omega}^{(\lambda)},H(\lambda)\psi_{\sigma}^{(\lambda)})\right]~~~
S_{\mathrm{kin}}=\intd^{4}x\,\frac{1}{2}\left[\hat{\Phi}^{\dagger}\hat{\Phi}\right]_{D}\longrightarrow\intd^{4}x\,\frac{1}{2}\left[\hat{\Phi}^{\dagger}e^{\pmc\hat{V}}\Phi\right]_{D}.
g(x)=\frac{1}{\pi}\int_{c^{2}}^{b^{2}}dy\,\sqrt{\frac{b^{2}-y}{y-c^{2}}}\frac{f(y)}{y-x}\\\\\\\\f(x)=\frac{1}{\pi}\int_{c^{2}}^{b^{2}}dy\,\sqrt{\frac{y-c^{2}}{b^{2}-y}}\frac{g(y)}{x-y}
\psi=\left(c_{+}(\bar{z})e^{\frac12\omegaz}+c_{-}(\bar{z})e^{-\frac12\omegaz}\right)e^{\int^{\bar{z}}\bar{f}(\bar{\zeta})\,d\bar{\zeta}-\int^{z}W(\zeta,\bar{z})\,d\zeta},
ds^{2}=Md{\bfx}^{2}+{\frac{16\pi^{2}}{Me^{4}}}d\chi^{2}+{\frac{4\pi\kappa}{e^{2}}}\left[{\frac{da^{2}}{a}}+a(\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2})\right].
0=F_{l}^{\prime\prime}+\frac{2}{r}F_{l}^{\prime}\left[1+rL_{-}^{\prime}(kr)\right]-\frac{s^{\prime}}{s-\frac{\omega}{m}}\left[F_{l}^{\prime}+F_{l}L_{-}^{\prime}(kr)\right]+W(r,m)F_{l}-\left[K,F_{l}\right]
\bar{S}=\intd^{4}x\,\Bigl[{\frac{1}{2}}\,A_{\mu}\,\bigl(\Box-m^{2}\bigr)\,A^{\mu}-\,{\frac{1}{2}}\,\partial_{\mu}\,A^{\mu}\,\Bigl((1-{\frac{1}{\alpha}}-{\frac{m^{2}}{\Box}}\Bigr)\,\partial_{\nu}\,A^{\nu}\Bigr]+S_{ghost}\,,
\begin{array}{rcl}{\delta_{\sigma}L_{m}{}^{i}}&{=}&{-\sigmaL_{n}{}^{i}\mathsf{m}^{n}{}_{m}\,,}\\{}&{}&{}\\{\delta_{\sigma}A^{m}{}_{\mu}}&{=}&{\sigma\mathsf{m}^{m}{}_{n}A^{n}{}_{\mu}\,,}\\{}&{}&{}\\{\delta_{\sigma}B_{\mu\nu\,m}}&{=}&{-\sigmaB_{m\,\mu\nu}\mathsf{m}^{n}{}_{m}+2\eta_{mn}\partial_{[\mu}\sigmaA^{n}{}_{\nu]}\,,}\\\end{array}
{\calF}_{t}=\left(\begin{array}{cc}{F_{A}+(\phi^{*}\phi+\phi+\phi^{*})}&{D\phi^{*}+A-A^{\prime}}\\{D\phi+A^{\prime}-A}&{F_{A^{\prime}}+(\phi\phi^{*}+\phi+\phi^{*})}\\\end{array}\right)
{\bfR}=\left(\begin{array}{cc}{\zeta^{-1}\eta}&{-\zeta^{-1}}\\{-\zeta^{-1}}&{-\eta^{*}\zeta^{-1}}\\\end{array}\right),\qquad{\bfL}={\bfR}^{*}\left(\begin{array}{c}{-\zeta^{-1}{\bfd}}\\{{\bfd}^{*}-\eta^{*}\zeta^{-1}{\bfd}}\\\end{array}\right).
\sigma(XA,1)=\\sum_{I}\M_{I}(X)\\big(\\sum_{J\preceqI}\R_{J}(A)\\big)=\\sum_{J}\\big(\\sum_{I\succeqJ}\M_{I}(X)\\big)\R_{J}(A)=\\sum_{J}\F_{J}(X)\,R_{J}(A)\.
{\calL}=\left\{\frac{1}{8\,e^{2}}\int\!\mathrm{d}^{2}\theta\,W^{2}+\mathrm{H.c.}\right\}+\frac{1}{4}\int\!\mathrm{d}^{4}\theta\left(\bar{S}e^{V}S+\bar{T}e^{-V}T\right)+\left\{\frac{m}{2}\int\!\mathrm{d}^{2}\theta\,ST+\mathrm{H.c.}\right\}\,,
H_{L}^{(1M)}=m_{ML}\;\intd^{3}\vec{x}\;(\bar{\psi}_{L}+\bar{\psi}_{L}{}^{C})(\psi_{L}+\psi_{L}{}^{C})=\sum_{(\vec{p})^{+}}H_{\vec{p},L}^{(1M)},\nonumber
\begin{array}{rcl}{\Phi}&{=}&{\phi+\phi_{0},}\\{\phi}&{=}&{\left(\pi_{1},\dots,\pi_{N-1},(\sigma+i\pi_{0})/\sqrt{2}\right)^{T},}\\{\phi_{0}}&{=}&{(0,\dots,0,v/\sqrt{2})^{T},}\\\end{array}
\biggl(-\frac{1}{2}\frac{d^{2}}{dr^{2}}-\frac{N^{2}-1}{2r}\frac{d}{dr}+\frac{1}{2}r^{2}+\sum\frac{c_{K}g^{K}}{N^{4K-2}}(r^{2})^{2K}\biggr)\psi=\epsilon_{0}\psi.
\tilde{S}_{1f}=b_{1}\frac{1}{N}\mathrm{Tr}\,B_{1}^{4}+b_{2}\frac{1}{N}\mathrm{Tr}\,B_{2}^{4}+b_{12}\frac{1}{N^{2}}\mathrm{Tr}\,B_{1}^{2}\mathrm{Tr}\,B_{2}^{2}
\frac{\partial}{\partial\theta}\Gamma\left[\Phi_{\mu}^{\prime}(\bar{x}),\Phi_{5}^{\prime}(\bar{x}),<A_{i,\theta}(\bar{x})>,<A_{i,\lambda}^{\prime}(\bar{x})>,<\zeta(\bar{x})>,<\zeta_{,\theta}>\right]=0
D^{1\;\alpha}\,J_{\alpha\alpha_{2}\ldots\alpha_{2j_{1}};\\dot{\alpha}_{1}\ldots\dot{\alpha}_{2j_{2}}}^{[a_{1},a_{2},a_{3}]}=\bar{D}_{4}^{\dot{\alpha}}\,J_{\alpha_{1}\ldots\alpha_{2j_{1}};\\dot{\alpha}\dot{\alpha}_{2}\ldots\dot{\alpha}_{2j_{2}}}^{[a_{1},a_{2},a_{3}]}=0
V_{E}(\beta,L,\phi)=\mu^{4}\biggl(2\pi^{2}c^{2}\phi^{2}+\frac{1}{3}\pi^{2}g\phi^{4}-\frac{1}{2\mu^{2}}\deltam^{2}\phi^{2}-\frac{1}{4!}\delta\lambda\phi^{4}\biggr)+G_{E}(\beta,L,\phi)
H(t)=\intd\mathrm{x}\left[\frac{1}{2}\frac{1}{R^{3}(t)}\pi^{2}(\mathrm{x},t)+\frac{1}{2}R(t)(\nabla\phi(\mathrm{x},t))^{2}+\frac{1}{2}R^{3}(t)\left(m^{2}+\xi{\bfR}\right)\phi^{2}(\mathrm{x},t)\right],
\overline{{S}}(u_{j}^{i})=\overline{{\Omega}}{}_{m}^{i}\overline{{u}}{}_{n}^{m}\,\Omega_{j}^{n}\,,\\\\overline{{S}}(\overline{{u}}{}_{j}^{i})=u_{j}^{i}\,,\\\,overline{S}(F^{i})=-F^{j}\overline{{S}}(u_{j}^{i})\,.
S=\intd^{D}x\,\sqrt{-g}\left[-\sum_{k=1}^{N}\overline{{\psi}}_{k}\gamma^{\mu}\nabla_{\mu}\psi_{k}+\frac{\lambda_{0}}{2N}(\sum_{k=1}^{N}\overline{{\psi}}_{k}\psi_{k})^{2}\right],
\psi_{x}=\prod_{y<x}[2iS_{3}(y)]S^{-}(x)~~,~\psi_{x}^{\dagger}=\prod_{y<x}[-2iS_{3}(y)]S^{+}(x)~~,~\psi_{x}^{\dagger}\psi_{x}=S^{3}(x)+1/2
\frac{\partial^{|l|}}{(\partialX)^{|l|}}\int_{0}^{\infty}\frac{\mathrm{d}\rho}{\rho^{|l|}}\mathrm{e}^{-\rhob\{aX+Y\}}=(-ab)^{|l|}\int_{0}^{\infty}\mathrm{d}\rho\mathrm{e}^{-\rhob\{aX+Y\}}
\left[q(t,\vec{x}),\,p(t,\vec{x}^{\prime})\right]=\delta(\vec{x}-\vec{x}^{\prime}),\quad\left[\Psi(t,\vec{x}),\,\bar{\Psi}(t,\vec{x}^{\prime})\right]=-i\delta(\vec{x}-\vec{x}^{\prime})
=A_{\infty}(k_{\infty}^{(1)},\cdots,k_{\infty}^{(N)})\prod_{p\in{\bfS}}A_{p}(k_{p}^{(1)},\cdots,k_{p}^{(N)})\prod_{p\not\in{\bfS}}A_{p}(k_{p}^{(1)},\cdots,k_{p}^{(N)}),
\Psi_{G}^{(+)}(\vec{x})=\left(\begin{matrix}{ig_{1}(r){\calY}_{G+\frac{1}{2},G}(\hat{x})}\\{f_{1}(r){\calY}_{G+\frac{1}{2},G+1}(\hat{x})}\\\end{matrix}\right)+\left(\begin{matrix}{ig_{2}(r){\calY}_{G-\frac{1}{2},G}(\hat{x})}\\{-f_{2}(r){\calY}_{G-\frac{1}{2},G-1}(\hat{x})}\\\end{matrix}\right)\.
M^{(-)}(\nu,k)=2\xi(\nu+1/2)+M_{o}^{(-)}(k)+gM_{1}^{(-)}(k)+g\int_{1}^{\infty}d\alpha(\sum_{\ell=1}^{5}\{\frac{1}{(\alpha-ik)^{\ell}}-\frac{1}{\alpha^{\ell}}\}\Phi_{\ell}^{(2)}(\alpha)[\alpha^{\frac{\nu}{2}}+\alpha^{\frac{-\nu}{2}}])
\begin{array}{rcl}{K(z):\Omega_{z}}&{\rightarrow}&{\Omega_{-z}}\\{K(z)v^{(b,a)}}&{=}&{\displaystyle\sum_{a^{\prime}}v^{(b,a^{\prime})}K\left(\left.b\begin{array}{c}{a}\\{a^{\prime}}\\\end{array}\right|z\right),}\\\end{array}
A_{\mu}^{\mu^{\prime}}A_{\nu}^{\nu^{\prime}}(\delta_{\mu^{\prime}}^{\lambda^{\prime}}\delta_{\nu^{\prime}}^{\rho^{\prime}}-{\bar{B^{\prime}}}_{\mu^{\prime}\nu^{\prime}}^{\lambda^{\prime}\rho^{\prime}})\frac{\partial{\bar{g}^{\prime}}_{\lambda^{\prime}\rho^{\prime}}}{\partiall_{i}}
\left(\begin{matrix}{{\bfE}}\\{{\bfB}}\\\end{matrix}\right){}={}\left(\begin{matrix}{\tilde{{\bfE}}}\\{\tilde{{\bfB}}}\\\end{matrix}\right){}+{}\left(\begin{matrix}{\underline{{{\bfE}}}}\\{\underline{{{\bfB}}}}\\\end{matrix}\right)\quad,
\alpha_{\{n_{1},n_{2},\cdots,n_{N}\}}=x^{n_{1}+n_{2}+\cdots+n_{N}}\left[\prod_{i=1}^{N}\frac{(N-i)!}{(N+n_{i}-i)!}\right]d_{\{n_{1},n_{2},\cdots,n_{N}\}},
W[Y]=\frac{1}{2}\intd^{4}x\int\frac{d^{4}p}{(2\pi)^{4}}\int_{0}^{\infty}\frac{dT}{T^{3}}\rho(T,\Lambda^{2})e^{-p^{2}}\mathrm{tr}\left(e^{-T(m^{2}+A)}\right)\cdot1,
\epsilon_{eacd}\,t_{L}^{cd}\,h_{\,\,\bar{a}}^{b}\,A_{b}^{+-a}=0\,,\qquad\epsilon_{\bar{e}\bar{b}\bar{c}\bar{d}}\,t_{R}^{\bar{c}\bar{d}}\,h_{a}^{\,\,\bar{a}}\,A_{\bar{a}}^{-+\bar{b}}=0
\Psi_{s,l}^{\gamma}(z,\bar{z})={\textstyle{\sqrt{\frac{(2|\gamma|-2s-1)\Gamma(s+1)\Gamma(2|\gamma|+l-s)}{4\pi\Gamma(l+s+1)\Gamma(2|\gamma|-s)}}}}z^{l}(1-z\bar{z})^{|\gamma|}\tilde{P}_{s}^{(l,-2|\gamma|-l)}\!\left(\frac{1\!+\!z\bar{z}}{1\!-\!z\bar{z}}\right),
B^{I}(\omega,{\bfk}_{n},{\bfk}_{n^{\prime}})\equiv\int_{0}^{\infty}d\omega^{\prime}\rho(\omega^{\prime}+\omega)\rho(\omega^{\prime})\Omega_{I}^{2}(\omega^{\prime}+\omega,\omega^{\prime},{\bfk}_{n},{\bfk}_{n^{\prime}})e^{-\omega^{\prime}/T},
K_{ij}a_{j}^{(\dagger)}=a_{i}^{(\dagger)}K_{ij},\qquadK_{ij}a_{k}^{(\dagger)}=a_{k}^{(\dagger)}K_{ij}\quad(k\nei,j),\qquadI_{r}a_{i}^{(\dagger)}=-a_{i}^{(\dagger)}I_{r}.
{\calM}_{0}\sim-{\textstyle{\frac{1}{2}}}(m+\bar{m})\hat{Q}_{v}(0)-{\textstyle{\frac{1}{2}}}(2\cdot2^{r/2}\cdotk_{1}k_{2}+2)(n+\bar{n})\hat{Q}_{v}(\zeta\tau)\,
\begin{array}{rcl}{\left[V_{-b}\right]\times\left[V_{\alpha}\right]}&{\rightarrow}&{\left[V_{\alpha-b}\right]+\left[\psi\bar{\psi}V_{\alpha}\right]+\left[V_{\alpha+b}\right],}\\{\left[\psi\bar{\psi}V_{-b}\right]\times\left[V_{\alpha}\right]}&{\rightarrow}&{\left[\psi\bar{\psi}V_{\alpha-b}\right]+\left[V_{\alpha}\right]+\left[\psi\bar{\psi}V_{\alpha+b}\right].}\\\end{array}
V_{-\textstyle{\frac{1}{2}};\chi}=e^{-\textstyle{\frac{\phi}{2}}}\prod_{j=1}^{3}\sigma_{1-\nu}^{j}e^{i(\textstyle{\frac{1}{2}}-\nu_{j})H_{j}}e^{ik_{\mu}X^{\mu}}\,,\\\V_{-\textstyle{\frac{1}{2}};\chi^{*}}=e^{-\textstyle{\frac{\phi}{2}}}\prod_{j=1}^{3}\sigma_{\nu}^{j}e^{-i(\textstyle{\frac{1}{2}}-\nu_{j})H_{j}}e^{ik_{\mu}X^{\mu}}\,.
S_{2}=\sum_{x}\sum_{a\neqb}\left(\frac{1}{(\phi_{a}-\phi_{b})^{2}}-d\sigma_{ab}\right)\delta\phi_{a}(x)\delta\phi_{b}(x)-\frac{1}{2}\eta_{ab}\delta\phi_{a}(x)\sum_{\mu=-d}^{d}\delta\phi_{b}(x+\mu)
F_{g}=-\epsilon_{ij}^{(1)}\epsilon_{ij}^{(2)}k_{\parallel}^{2}{\frac{\Gamma(1+2k_{\parallel}^{2})\Gamma(k\cdotp)}{\Gamma(k\cdotp+2k_{\parallel}^{2}+1)}}\prod_{A=0}^{p}\delta(k^{A}+p^{A})\,
S=\frac{1}{2}\sum_{n=0}^{\infty}\int\\dot{A}_{ai}^{(n)}D_{ai}^{(n)}-\frac{(-1)^{n}}{2}(E_{ai}^{(n)}E_{ai}^{(n)}+B_{ai}^{(n)}B_{ai}^{(n)})-{\Lambda}_{ai}^{(n+1)}{\Psi}_{ai}^{(n+1)}-{\Lambda}_{i}^{(n)}{\calG}_{i}^{(n)}\,
L^{(2)}(\phi,\chi)=\frac{1}{2}\intd^{3}x\left[\sum_{i=1}^{3}\dot{\phi}_{i}^{2}+\sum_{i,j=1}^{3}\dot{\chi}_{i}G_{ij}^{-1}\dot{\chi}_{j}-V(\phi,\chi)\right]-\theta\intd^{3}x\,\,Q^{(2)}(\phi,\chi)\,,
<(S_{i}^{z})^{2}(S_{j}^{z})^{2}>_{\mathrm{VBS}}\stackrel{N\rightarrow\infty}{=}<(S_{i}^{z})^{2}>_{\mathrm{VBS}}<(S_{j}^{z})^{2}>_{\mathrm{VBS}}\stackrel{N\rightarrow\infty}{=}\frac{4}{(q^{2}+1+q^{-2})^{2}}
\oint\frac{d\eta}{2\pii}[\left({\calM}\frac{\delta}{\delta\hat{K}}\right)_{1}(\eta)+\psi_{1}(\eta)]=\int\frac{d\eta}{2\pii}\hat{K}_{1}(\eta)\frac{1}{N^{2}}\frac{1}{\sqrt{5c}}+\hat{\oint}\frac{d\eta}{2\pii}\frac{\delta}{\delta\hat{K}_{1}(\eta)}.
\left[e^{-\bar{H}_{i}^{2}t}\right]_{\mathrm{diag}}\simeq{\frac{1}{(4\pit)^{(D-1)/2}}}e^{-m^{2}\kappa^{2}\rho^{2}t}\left(r_{i}+\bar{b}_{i,1}t+\bar{b}_{i,2}t^{2}+...\right)~~~,
g_{MN}=e_{M}^{m}e_{N}^{n}\eta_{mn}\;\;,\;\;\Gamma_{M}=e_{M}^{m}\hat{\Gamma}_{m}\;\;,\;\;\left\{\Gamma_{M},\Gamma_{N}\right\}=2g_{MN}\;\;,\;\;\left\{\hat{\Gamma}_{m},\hat{\Gamma}_{n}\right\}=2\eta_{mn}
\{Q_{\alpha}^{A},Q_{\beta}^{B}\}\,=\,\frac{\delta^{AB}}{2}(\gamma^{ab}C\gamma^{5})_{\alpha\beta}M_{ab}+\delta^{AB}(\gamma^{5}\gamma^{a}C\gamma^{5})_{\alpha\beta}P_{a}+\mathrm{i}(C\gamma^{5})_{\alpha\beta}T^{AB},
k_{1}=\left[\frac{\Lambda(\zeta_{1},\zeta_{2},t)}{4t}\right]^{\frac{1}{2}}z\+\\frac{\zeta_{1}-\zeta_{2}+t}{2t}K,\\k_{2}=-\left[\frac{\Lambda(\zeta_{1},\zeta_{2},t)}{4t}\right]^{\frac{1}{2}}z\-\\frac{\zeta_{1}-\zeta_{2}-t}{2t}K,\tag{2.16.b}
\tilde{\Phi}_{\mathrm{far}}(x)={\frac{1}{x}}\left(\alphax^{\sqrt{1+\Lambda}}+\betax^{-\sqrt{1+\Lambda}}\right)=\tilde{\Phi}_{\mathrm{far}}^{\mathrm{non-nor}}+\tilde{\Phi}_{\mathrm{far}}^{\mathrm{nor}}
(\Gamma^{\prime})_{a}^{c}\equiv{\frac{1}{(p+1)!}}i^{{{\frac{p(p-1)}{2}}}}\epsilon^{f_{p}\ldotsf_{1}f_{0}}\left(C\Gamma_{\{f_{0}\}\ldots\{f_{p}\}}C^{-1}\right)_{a}^{c}
\intdt_{1}dt_{2}\frac{\lambda^{2}t_{1}t_{2}}{(2\pi)^{2}}\left[\begin{array}{c}{\theta(t-t_{1})K_{0}(\mut)I_{0}(\mut_{1})}\\{+\theta(t_{1}-t)K_{0}(\mut_{1})I_{0}(\mut)}\\\end{array}\right]\left[\begin{array}{c}{\theta(t_{1}-t_{2})K_{1,2,3}(\mut_{1})I_{1,2,3}(\mut_{2})}\\{+\theta(t_{2}-t_{1})K_{1,2,3}(\mut_{2})I_{1,2,3}(\mut_{1})}\\\end{array}\right]K_{0}(\mut_{2}),
{\calA}^{\mu}\Sigma_{\mu}=\left(\begin{matrix}{-\Sigma_{z}}&{\Sigma_{0}}\\{-l\,(1+m\chi^{2})\,\Sigma_{0}+lm\,\psi\chi\,\Sigma_{z}}&{lm\,\psi\chi\,\Sigma_{0}+m\,(1-l\,\psi^{2})\,\Sigma_{z}}\\\end{matrix}\right)\,.
\hat{U}(\Omega,t)={\frac{e^{-{\frac{4t}{A}}}}{\sqrt{At}}}{\frac{1}{2\sqrt{\pi}}}\int_{0}^{+\infty}[e^{-{\frac{(\Omega-\xi)^{2}}{4At}}}-e^{-{\frac{(\Omega+\xi)^{2}}{4At}}}]\phi(\xi)d\xi,
e_{0}^{0}=-\omegat_{c}\,,\quade_{j}^{i}=-\delta_{j}^{i}\,\omegat_{c}\,,\quad\hat{e}_{0}^{0}=-\frac{1}{\omegat_{c}}\,,\quad\hat{e}_{j}^{i}=-\delta_{j}^{i}\,\frac{1}{\omegat_{c}}
g^{2}\,\langle\langleF_{\mu\nu}(z_{1})F_{\rho\sigma}(z_{2})\rangle\rangle=\frac{1}{W_{q\overline{{q}}}}\frac{\delta^{2}W_{q\overline{{q}}}}{\deltaS^{\mu\nu}(z_{1})\deltaS^{\rho\sigma}(z_{2})}
F(D_{x^{2}})f(x^{2})=[-q^{2N+2p}x^{2}D_{x^{2}}^{2}-q^{N}({\frac{N}{2}}+p)_{q^{2}}D_{x^{2}}+{\frac{\omega^{2}}{\mu^{2}}}x^{2}-{\frac{E}{\mu^{2}}}]f(x^{2})=0\.
R(\lambda,t)=\left(\begin{array}{cccc}{a(\lambda)}&{0}&{0}&{0}\\{0}&{c^{\prime}(\lambda)}&{b(\lambda)}&{0}\\{0}&{b(\lambda)}&{c^{\prime\prime}(\lambda)}&{0}\\{0}&{0}&{0}&{a(\lambda)}\\\end{array}\right)\;,
\frac{1}{2{\calR}^{n}}\frac{d}{d{\calR}}\Big(H^{2}{\calR}^{n+1}\Big)=-\frac{\kappa^{4}}{8n^{2}}\varrho\left(2np+(n-1)\varrho\right)-\frac{n+1}{2\ell^{2}}-\frac{n-1}{2}\frac{k}{{\calR}^{2}}+\frac{{\calQ}^{2}}{n{\calR}^{2n}}.
(\vec{Y}^{AB},\rho^{AB},N^{AB},W_{a}^{AB})=\left(\frac{1}{2}(M^{A}\vec{X}^{B}+M^{B}\vec{X}^{A}+\frac{1}{2}\bar{\Omega}^{A}\vec{\tau}\Omega^{B}-4M^{A}M^{B}\vec{t}),\ldots\right).
\begin{array}{rcl}{R^{(1)}K_{1}R^{(2)}K_{2}}&{=}&{R^{(1)}(X_{1}Z_{1}^{\dagger}+Z_{1}X_{1}^{\dagger})R^{(2)}(X_{2}Z_{2}^{\dagger}+Z_{2}X_{2}^{\dagger})}\\{\;}&{=}&{(X_{2}Z_{2}^{\dagger}+Z_{2}X_{2}^{\dagger})R^{(3)}(X_{1}Z_{1}^{\dagger}+Z_{1}X_{1}^{\dagger})R^{(4)}}\\{\;}&{=}&{K_{2}R^{(3)}K_{1}R^{(4)}}\\\end{array}
\begin{array}{cl}{E_{a}{\circ_{1,Z}}E_{b}}&{=(e_{a}Z)(Z^{\dagger}(e_{b}Z))}\\{}&{=e_{a}ZZ^{\dagger}e_{b}Z}\\{}&{=e_{a}e_{b}Z}\\{}&{=-e_{b}e_{a}Z}\\{}&{=-E_{b}{\circ_{1,Z}}E_{a}.}\\\end{array}
V(r,u)=-(8\pi^{2}\alpha^{\prime})^{-1/2}\int_{0}^{\infty}dl~e^{-r^{2}l/2\pi\alpha^{\prime}}{\frac{l^{1/2}\mathrm{tanh}(u)}{\mathrm{Sin}(ul)}}\left[12+4\mathrm{Cos}(2ul)-16\mathrm{Cos}(ul)\right]\quad.
\omega=-\frac{1}{3}\frac{R_{0}^{2}-r_{-}^{2}}{R_{0}^{2}-(1-b)r_{-}^{2}}\left(2+\frac{r_{-}^{2}}{R_{0}^{2}-r_{-}^{2}}+\frac{r_{+}^{2}}{R_{0}^{2}-r_{+}^{2}}\right).
S_{g}=-M_{Pl}^{2}\intR\sqrt{-g}d^{4}x=M_{Pl}^{2}\int\left(\Box\phi^{\xi}\Box\phi^{\xi}-\partial_{\alpha}\partial_{\beta}\phi^{\xi}\partial^{\alpha}\partial^{\beta}\phi^{\xi}\right)\sqrt{-g}d^{4}x~,
\langle{\calW}(1){\calW}(2)\rangle|_{\theta=0}=\langle\phi(1)\phi(2)\rangle=\frac{z_{1i}\delta^{ij}z_{2j}}{x_{12}^{2}}\equiv\frac{(12)}{x_{12}^{2}}=\frac{(21)}{x_{21}^{2}}\;.
L_{L2}={\frac{K\,v^{2}}{2\,R}}~\left[1+\sum_{L=1}^{\infty}\,d_{LL}(0,0)~\left({\frac{N\,v^{2}}{R^{2}M_{P}^{9}V_{T}~b^{7-p}}}\right)^{L}~(1+S_{L2})\right]
{\hat{\calD}}_{+}=\hat{E}_{+}^{~M}\partial_{M}={\frac{\partial}{\partial\theta}}-\theta{\frac{\partial}{\partialz}};~~~~\hat{\calD}_{-}=\hat{E}_{-}^{~M}\partial_{M}={\frac{\partial}{\partial\bar{\theta}}}+\bar{\theta}{\frac{\partial}{\partial\bar{z}}}
{\calL}=-\frac14\kappah\left(\partial_{\mu}A_{\alpha}\partial^{\mu}A^{\alpha}-\partial_{\alpha}\partial^{\alpha}A^{\mu}\right)+\frac12\kappah^{\mu\nu}\left(\partial_{\mu}A_{\alpha}\partial_{\nu}A^{\alpha}+\partial_{\alpha}A_{\mu}\partial^{\alpha}A_{\nu}-\partial_{\alpha}A_{\mu}\partial_{\nu}A^{\alpha}-\partial_{\alpha}A_{\nu}\partial_{\mu}A^{\alpha}\right)
ds^{2}=\frac{4\ell^{4}}{(\ell^{2}-\rho^{2})^{2}}\left[-\left(\frac{\ell^{2}+\rho^{2}}{2\ell^{2}}\right)^{2}dt^{2}+d\rho^{2}+\rho^{2}d\phi^{2}\right]\,
E_{\mathrm{in(D)}}^{\mathrm{(surf)}}=\frac{4\xi-1}{2^{D}a\Gamma(D/2)}\sum_{k=0}^{[(D-1)/2]}\frac{(-1)^{k}(ma)^{2k}}{k!\Gamma(D/2-k)}\sum_{l=0}^{\infty}\frac{\Gamma(l+D-2k)}{l!},
ds^{2}=-\frac{r^{2}}{(r^{2}+r_{-}^{2})(r^{2}-r_{+}^{2})}dr^{2}+r^{2}(d\phi+\frac{r_{+}r_{-}}{r^{2}}dt)^{2}+\frac{(r^{2}+r_{-}^{2})(r^{2}-r_{+}^{2})}{r^{2}}dt^{2}\,.
|Z_{1,2}|^{2}={\frac{1}{(4G_{4})^{2}}}e^{-\varphi_{\infty}}\left[\vec{Q}_{R}^{2}+\vec{P}_{R}^{2}\pm2\{\vec{Q}_{R}^{2}\vec{P}_{R}^{2}-(\vec{Q}_{R}^{T}\vec{P}_{R})^{2}\}^{\frac{1}{2}}\right],
{\calL}=-{\frac{1}{4}}\mathrm{F}_{\mu\nu}\mathrm{F}^{\mu\nu}+\bar{\psi}i{\hbar}{\gamma}^{\mu}{\partial}_{\mu}\psi+e_{+}{\hbar}\bar{\psi}_{+}{\gamma}^{\mu}{\psi_{+}}A_{\mu}+e_{-}{\hbar}\bar{\psi}_{-}{\gamma}^{\mu}{\psi_{-}}A_{\mu},
x\sqrt{\beta-x}(16x-\beta)=\frac{5\beta-6x}{\sqrt{\beta-x}}\left(4x^{2}-\frac{\beta}{2}x+\frac{\beta^{2}}{48}+\frac{8}{3}\frac{x^{3}(5\beta-6x)^{2}}{\beta-x)(16x-\beta)^{2}}\right)
S=\intd^{4}x\sqrt{\midg\mid}\left({\frac{1}{2}}D_{\mu}\Phi^{\ast}D^{\mu}\Phi-{\frac{\lambda}{4}}(\Phi^{\ast}\Phi-v^{2})^{2}-{\frac{1}{4}}{F}_{\mu\nu}{F}^{\mu\nu}+\frac{1}{16\piG}{\calR}\right)
F(V)=\sum_{g=0}^{\infty}\sum_{h=1}^{\infty}\sum_{w_{1},\cdots,w_{h}}{\frac{i^{h}}{h!}}g_{s}^{2g-2+h}F_{g,w}(t)\mathrm{Tr}\,V^{w_{1}}\cdots\mathrm{Tr}\,V^{w_{h}},
(C_{ij}^{n}(q))=(C_{ji}^{n}(q))=\frac{n}{[n]_{q}}\,\frac{[2]_{q}^{2}}{q^{2n}+1+q^{-2n}}\,\left(\begin{array}{cc}{q^{n}+q^{-n}}&{(-1)^{n}}\\{(-1)^{n}}&{q^{n}+q^{-n}}\\\end{array}\right)
U_{k_{1},k_{1z};k_{2},k_{2z};I_{z},I_{z}^{\prime}}=\sum_{m,I_{z}}\langlek,m;I,I_{z}|k_{2},k_{2z}\rangleD_{m,k_{1z}}^{k_{1}}(\alpha,\beta,\gamma)D_{I_{z},I_{z}^{\prime}}^{I}(\alpha_{I},\beta_{I},\gamma_{I})|I,I_{z}^{\prime}\rangle.
\begin{array}{c}{[s_{AB},s_{CD}]=-\eta_{AC}s_{BD}+\eta_{AD}s_{BC}+\eta_{BC}s_{AD}-\eta_{BD}s_{AC}}\\{{}}\\{{[}{t_{i}^{~j}},{}{t_{k}^{~l}}{]}=-\delta_{k}^{~j}t_{i}^{~l}+\delta_{i}^{~l}t_{k}^{~j}-{\calE}^{jl}t_{ki}+\bar{\calE}_{ik}t^{jl}}\\\end{array}
f_{\mathrm{in}}^{\mathrm{TM}}=-{\frac{i}{\pi^{(D+1)/2}2^{D}a^{D+1}\Gamma({\frac{D-1}{2}})}}\int_{0}^{\infty}{\frac{dx}{x}}\sum_{n=0}^{\infty}w(n,D)(x^{2}-n(n+D-2)){\frac{s_{n}(x)}{s_{n}^{\prime}(x)}},
\prod_{\nu\neq\mu}^{M}\frac{\lambda_{\mu}-\lambda_{\nu}+ic}{\lambda_{\mu}-\lambda_{\nu}-ic}\cdot\frac{\lambda_{\mu}+\lambda_{\nu}+ic}{\lambda_{\mu}+\lambda_{\nu}-ic}=\prod_{j=1}^{N}\frac{\lambda_{\mu}-k_{j}-ic/2}{\lambda_{\mu}-k_{j}+ic/2}\cdot\frac{\lambda_{\mu}+k_{j}-ic/2}{\lambda_{\mu}+k_{j}+ic/2}
\begin{array}{lcr}{\sum_{i=1}^{3}(|\varphi_{i}^{1}|^{2}-|\varphi_{i}^{2}|^{2})}&{=\xi^{3}\qquad}&{(a)}\\{\sum_{i=1}^{3}\varphi_{i}^{1}\overline{{\varphi}}_{i2}}&{=\xi^{1}+i{\xi^{2}}\qquad}&{(b)}\\{\sum_{i=1}^{3}\varphi_{i}^{2}\overline{{\varphi}}_{i1}}&{=\xi^{1}-i{\xi^{2}}\qquad}&{(c).}\\\end{array}
k^{2}=\int_{0}^{y_{c}}dy\,\gamma^{\frac{1}{2}}\left|\chi_{(k\ell)}^{\prime}\right|^{2}+\ell^{2}\int_{0}^{y_{c}}dy\,\gamma^{-\frac{1}{2}}\left|\chi_{(k\ell)}\right|^{2}\ge0~.
\sum_{l}\alpha_{il}\Lambda^{*}\mathrm{~\binom{i_1~\dots~i_{m-1}~l}{j_1~j_2~\dots~j_n}~}=(-)^{n+m}\sum_{r=1}^{n}(-)^{r}\beta_{ij_{r}}\Lambda^{*}\mathrm{~\binom{i_1~i_2~\dots~i_{m-1}}{j_1~\dots~\check{j}_r~\dots~j_n}~}\notag
\varepsilon=\{\varepsilon^{ij}\}=\left(\begin{array}{ccccc}{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{\varepsilon^{12}}&{\varepsilon^{13}}&{0}\\{0}&{-\varepsilon^{12}}&{0}&{\varepsilon^{23}}&{0}\\{0}&{-\varepsilon^{13}}&{-\varepsilon^{23}}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\\end{array}\right)
ds^{2}\approx\left[1-\frac{4\piG_{D}-2|\Lambda|}{d(p+d-1)}r^{2}\right]\bar{g}_{\mu\nu}(x)dx^{\mu}dx^{\nu}+\frac{dr^{2}}{1+B_{2}r^{2}}+r^{2}d\Omega_{d-1}^{2},
\begin{array}{ccccccccccl}{D(p-1):}&{1}&{2}&{.}&{p-1}&{\_}&{\_}&{\_}&{\_}&{\_}&{\quad\mathrm{probe}\nonumber}\\{F1:}&{\_}&{\_}&{\_}&{\_}&{\_}&{y}&{\_}&{\_}&{\_}&{\quad\mathrm{soliton.}}\\\end{array}
Z=\intD\phiD\chiD\rho\mathrm{Exp}\left[i\{\frac{1}{2}[\partial_{\mu}\phi_{i}]^{2}-\frac{1}{2}\chi{\vec{\phi}\cdot\vec{\phi}}-NV[{\rho}]+\frac{N}{2}\rho\chi+\vec{j}\cdot\vec{\phi}+NS\chi+NJ\rho\}\right]
S(spin-2)=-\frac{1}{4\kappa^{2}}\intd^{D}X\sqrt{-G}\left[e^{-A}\eta^{\mu\nu}\partial_{\mu}\tilde{h}_{i}^{j}\partial_{\nu}\tilde{h}_{j}^{i}+\partial_{r}\tilde{h}_{i}^{j}\partial_{r}\tilde{h}_{j}^{i}+D_{m}\tilde{h}_{i}^{j}D^{m}\tilde{h}_{j}^{i}\right]~.
\begin{aligned}{n_{1}}&{{}=s_{1}\,,}\\{n_{2}}&{{}=s_{2}\,,}\\{m_{3}}&{{}=s_{3}\,,}\\\end{aligned}\qquad\begin{aligned}{m_{1}}&{{}=s_{1}-Y_{1}+Y_{2}+Y_{3}\pmr,}\\{m_{2}}&{{}=s_{2}+Y_{1}-Y_{2}+Y_{3}\pmr,}\\{n_{3}}&{{}=s_{3}+Y_{1}+Y_{2}-Y_{3}\pmr.}\\\end{aligned}
{\phi^{(-)}}_{j,m}=\left(\begin{matrix}{{({\frac{l+{\frac{1}{2}}-m}{2l+1}})^{\frac{1}{2}}}Y_{l,{m-{\frac{1}{2}}}}}\\{-{({\frac{l+{\frac{1}{2}}+m}{2l+1}})^{\frac{1}{2}}}Y_{l,{m+{\frac{1}{2}}}}}\\\end{matrix}\right),
G_{\mu\nu}^{ab}(p)=\frac{-i\delta^{ab}}{(2\pi)^{2\omega}(p^{2}+i\epsilon)}\Big[g_{\mu\nu}-\frac{p_{\mu}n_{\nu}+p_{\nu}n_{\mu}}{p\cdotn}+\frac{n^{2}p_{\mu}p_{\nu}}{(p\cdotn)^{2}}\Big],\\n^{2}>0,\,epsilon>0,
|b_{n0}|\sim\frac{|b_{n-1,2}|}{n}\sim\cdots\sim\frac{(n-1)!}{2^{n-1}}|b_{1,2n-2}|=\frac{(n-1)!}{2^{n-1}}\frac{|V^{(2n-2)}|}{(2n-1)!}.
\gamma^{0}=\left(\begin{array}{rr}{0}&{-i}\\{i}&{0}\\\end{array}\right)\;\;\;\;\;\gamma^{1}=\left(\begin{array}{rr}{0}&{i}\\{i}&{0}\\\end{array}\right)\;\;\;\;\;\gamma^{2}=\left(\begin{array}{rr}{i}&{0}\\{0}&{-i}\\\end{array}\right)
\delta\omega_{\nu}{}^{mn}=\bar{\epsilon}\Gamma_{\nu}{}^{\phi\rhomn}\hat{\psi}_{\phi\rho}+4\,\bar{\epsilon}\Gamma^{mnp}\hat{\psi}_{\nup}-\frac{4}{9}\,\,\bar{\epsilon}\Gamma^{rs[m}\hat{\psi}_{rs}e_{\nu}{}^{n]}-4\,\bar{\epsilon}{\calS}^{mn}\cdot\hat{H}\psi_{\nu}+{\calO}(\epsilon\psi^{3})+{\calO}(T)\,.
p_{\mu}\;=\;q^{-3/2}\Bigl(\frac{\partialL_{0}^{m}}{\partial\dot{x}^{\mu}}\Bigr)\equiv(\dot{x}_{\mu}+q^{1/2}i\chi\psi_{\mu})[(\dot{x}+q^{1/2}i\chi\psi)^{2}]^{-1/2}\;m,
\Gamma_{\nu}=\varepsilon^{\mu,[\mu\nu]}+\varepsilon^{[\mu\nu],\mu}+\varepsilon^{\nu,0}+\varepsilon^{o,\nu}+\varepsilon^{\widetilde{\nu},\widetilde{0}}+\varepsilon^{\widetilde{0},\widetilde{\nu}}+\frac12e_{\mu\nu\rho\omega}\left(\varepsilon^{\widetilde{\mu},[\rho\omega]}+\varepsilon^{[\rho\omega],\widetilde{\mu}}\right)
R_{\bcd}^{a}=\overline{R}_{\bcd}^{a}+2\delta_{\[d}^{a}\sigma_{;c]b}+2\bar{g}_{b[c}\sigma_{;d]^{a}}+2\bar{g}_{b[d}\sigma_{c]}\sigma^{a}+2\delta_{\[c}^{a}\sigma_{d]}\sigma_{b}+2\delta_{\[d}^{a}\bar{g}_{c]b}\sigma_{e}\sigma^{e}\,,
<\psi_{f}|e^{-i/\hbar(t_{2}-t_{1})\hat{H}_{\mathrm{eff}}}|\psi_{i}>=i\delta_{n,m}(t_{2}-t_{1})\int_{-\infty}^{\infty}d\lambda\,\tilde{\psi}_{f}^{*}(\lambda)\,e^{-i/\hbar\lambda(t_{2}-t_{1})(\hbar\omega(n+1/2)-\Lambda)}\,\tilde{\psi}_{i}(\lambda)\,
(\partial-q_{1}^{(n+\frac{l}{k})})(\partial-q_{1}^{(n+\frac{l}{k})}-p_{n+\frac{l}{k}})^{-1}=(\partial-q_{1}^{(n+1+\frac{l}{k})}-p_{n+\frac{l}{k}})^{-1}(\partial-q_{1}^{(n+1+\frac{l}{k})}).
S_{S}={\frac{1}{2}}\intd^{4}xdy|g|^{\frac{1}{2}}\left\{g^{\mu\nu}\partial_{\mu}\Phi_{I}^{*}\partial_{\nu}\Phi_{I}-m_{S}^{2}\Phi_{I}^{*}\Phi_{I}-\xiR^{2}\Phi_{I}^{*}\Phi_{I}\right\}~,
{\hat{\lambda}}+{\frac{1}{J_{\nu}^{2}(ka)}}\left({\frac{N_{\nu}(ka)}{J_{\nu}(ka)}}-{\frac{N_{\nu}(kb)}{J_{\nu}(kb)}}-{\frac{1}{{\hat{\gamma}}J_{\nu}^{2}(kb)}}\right)^{-1}=-\pi\nu+O(a^{2\nu}).
\begin{array}{c}{s\left(\lambda,\mu\right)=\fract{\lambda-\mu}\left(\phi\left(\lambda\right)^{-1}-\phi\left(\mu\right)^{-1}\right),}\\{a\left(\lambda,\mu\right)=\fract{\lambda-\mu}\left(\phi\left(\lambda\right)^{-1}-\phi\left(\mu\right)^{-1}\right).}\\\end{array}
e^{2\phi}=\sqrt{\frac{C^{2}}{8\kappa\phi_{0}}}\left(a^{2\sqrt{\phi_{0}/K}}+a^{-2\sqrt{\phi_{0}/K}}\right)\approx\sqrt{\frac{C^{2}}{8\kappa\phi_{0}}}a^{-2\sqrt{\phi_{0}/K}}.
{\frac{1}{a+bz}}\left(\begin{array}{l}{1}\\{z^{\prime}}\\{{z^{\prime}}^{2}}\\{\vdots}\\\end{array}\right)={\frac{1}{a+bz}}\left(\begin{array}{l}{1}\\{\left({\frac{c+dz}{a+bz}}\right)}\\{\left({\frac{c+dz}{a+bz}}\right)^{2}}\\{\vdots}\\\end{array}\right).
\begin{array}{c}{\displaystyle\displaystyle\frac{(-1)^{\delta}e^{iZ_{N}(x-i\eta)}iZ_{N}^{\prime}(x-i\eta)}{1+(-1)^{\delta}e^{iZ_{N}(x-i\eta)}}=}\\{\displaystyle\frac{-(-1)^{\delta}e^{-iZ_{N}(x-i\eta)}iZ_{N}^{\prime}(x-i\eta)}{1+(-1)^{\delta}e^{-iZ_{N}(x-i\eta)}}+iZ_{N}^{\prime}(x-i\eta)}\\\end{array}
\int_{0}^{2\piR}dx_{4}i\bar{\Psi}\gamma^{\mu}{\calD}_{\mu}^{(q)}\Psi=\sum_{j=1}^{3}\left(i\bar{\psi}_{L}^{j}\gamma^{\mu}D_{\mu}^{(q)L}\psi_{L}^{j}+i\bar{\psi}_{R}^{j}\gamma^{\mu}D_{\mu}^{(q)R}\psi_{R}^{j}\right),
a_{m}^{\dagger}a_{m}+\psi_{\dot{\alpha}}^{\dagger}\psi_{\dot{\alpha}}=\tilde{a}_{m}^{\dagger}\tilde{a}_{m}+\tilde{\psi}_{\dot{\alpha}}^{\dagger}\tilde{\psi}_{\dot{\alpha}}=1-\mid0\rangle\langle0\mid
P_{JJ}^{-}=\frac{g^{2}}{2}\int_{-\infty}^{\infty}\frac{dk}{k^{2}}\tilde{J}_{ij}^{+}(k)\tilde{J}_{ji}^{+}(-k)=\frac{g^{2}}{2}\int_{0}^{\infty}\frac{dk}{k^{2}}\left\{\tilde{J}_{ij}^{+}(k),\tilde{J}_{ji}^{+}(-k)\right\}.
{A^{\prime}}^{a}T^{a}=U\,A^{a}T^{a}\,U^{-1}\\\,\\\{\phi^{\prime}}^{a}T^{a}=U\,\phi^{a}T^{a}\,U^{-1}\,+\,\frac{i}{g}\,U\,\frac{d}{dt}U^{-1}\\\,