harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A_{k}(t,r,\theta,\varphi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\Bigr[c_{l}(r)\partial_{k}Y_{lm}(\theta,\varphi)+T_{l}(r)\varepsilon_{kp}\partial^{p}Y_{lm}(\theta,\varphi)\Bigr]e^{i\omegat}.
\left(D_{i}\right)_{\;\;b}^{a}=\delta_{\;\;b}^{a}\partial_{i}+gf_{\;\;bc}^{a}A_{i}^{c},\;\left(D_{i}\right)_{b}^{\;\;a}=\delta_{b}^{\;\;a}\partial_{i}-gf_{\;\;bc}^{a}A_{i}^{c}.
E_{i}(x)\mathrm{~}\Psi(y)\midE\rangle=\left(\varepsilon_{i}(x)+e\int_{0}^{1}d\alpha\mathrm{~}y_{i}\mathrm{~}\delta^{(3)}\left(\alpha\mathbf{y}-\mathbf{x}\right)\right)\Psi(y)\midE\rangle.
\mathrm{either}\;\;\tilde{{\calS}}^{ab}=-\frac{i}{4}[dx^{a}\;\wedge+\;e^{a},\;dx^{b}\;\wedge+\;e^{b}]=-\frac{i}{4}[\tilde{\gamma}^{a},\tilde{\gamma}^{b}],\quad\mathrm{or}\quad\tilde{\tilde{\calS}}{}^{ab}=\frac{i}{4}[dx^{a}\;\wedge-\;e^{a},\;dx^{b}\;\wedge-\;e^{b}].
\int_{-\varepsilon_{\alpha}}^{Q-\sum_{\beta=1}^{\alpha-1}p^{\beta}+\varepsilon_{\alpha}}dp^{\alpha}\to\int_{\tilde{\varepsilon}_{\alpha}}^{2\lambda-\sum_{\beta=1}^{\alpha-1}\tilde{p}^{\beta}-\tilde{\varepsilon}_{\alpha}}d\tilde{p}^{\alpha}\.
\begin{array}{l}{\overline{{R}}(\zeta)_{++}^{++}=\overline{{R}}(\zeta)_{--}^{--}=1,}\\{\displaystyle\overline{{R}}(\zeta)_{+-}^{+-}=\overline{{R}}(\zeta)_{-+}^{-+}=b(\zeta)={\frac{x(\zeta^{2}-1)}{1-x^{2}\zeta^{2}}},}\\{\displaystyle\overline{{R}}(\zeta)_{-+}^{+-}=\overline{{R}}(\zeta)_{+-}^{-+}=c(\zeta)={\frac{(1-x^{2})\zeta}{1-x^{2}\zeta^{2}}}.}\\\end{array}
\Big\langle\overline{{\psi}}_{\alpha_{0}}^{(a_{0})}(x_{0})\psi_{\beta_{0}}^{(b_{0})}(x_{0})\overline{{\psi}}_{\alpha_{1}}^{(a_{1})}(x_{1})\psi_{\beta_{1}}^{(b_{1})}(x_{1}).....\overline{{\psi}}_{\alpha_{n}}^{(a_{n})}(x_{n})\psi_{\beta_{n}}^{(b_{n})}(x_{n})\Big\rangle_{c}\;.
X^{i}=\left(\begin{array}{cccc}{x_{1}^{i}+v_{1}^{i}t}&{0}&{0}&{\ddots}\\{0}&{x_{2}^{i}+v_{2}^{i}t}&{\ddots}&{0}\\{0}&{\ddots}&{\ddots}&{0}\\{\ddots}&{0}&{0}&{x_{N}^{i}+v_{N}^{i}t}\\\end{array}\right)\,.
ds^{2}=\alpha^{\prime}\,h(r)^{-\frac{1}{4}}\left(\eta_{ij}+\rho_{ij}\right)\,dx^{i}\,dx^{j}+\alpha^{\prime}\,h(r)^{\frac{3}{4}}\left(dr^{2}+r^{2}d\Omega_{3}^{2}\right),
\rho={\bf\nabla}.{\bfE}=-\left({\frac{\partial^{2}\phi}{\partialx^{2}}}+{\frac{\partial^{2}\phi}{\partialy^{2}}}\right)\qquad;\qquad{\bfj}={\bf\nabla}\times{\bfB}=-\left({\frac{\partial^{2}\phi}{\partialx^{2}}}+{\frac{\partial^{2}\phi}{\partialy^{2}}}\right)\,{\hat{\bfz}},
d\tilde{s}^{2}=-{\frac{T_{p}^{2}V_{p}^{2}}{E^{2}}}(h_{0}r_{0}^{2}\rho^{2})^{p}e^{-2\Phi}g_{tt}^{2}dt^{2}+h_{0}r_{0}^{2}\rho^{2}d\Omega_{p}^{2},
c(l_{0},l_{1}+1;l_{0},l_{1})=c(l_{0},l_{1};l_{0},l_{1}+1)=c_{0}\sqrt{\frac{(k_{1}-l_{1}-1)(k_{1}+l_{1})}{(l_{1}-l_{0})(l_{1}-l_{0}+1)(l_{1}+l_{0})(l_{1}+l_{0}+1)}},
{\calG}=M_{ab}d{\bfx}_{a}\cdotd{\bfx}_{b}+\frac{g^{4}}{16\pi^{2}}(M^{-1})_{ab}(d\xi_{a}+{\bfW}_{ac}\cdotd{\bfx}_{c})(d\xi_{b}+{\bfW}_{bd}\cdotd{\bfx}_{d}).
\frac{\partial\tilde{U}}{\partialR}=(1-\frac{N^{2}}{2R^{2}})(1-\frac{2m}{\sqrt{R^{2}+Z^{2}}})^{1/2}+\frac{mR}{(R^{2}+Z^{2})^{3/2}}(R+\frac{N^{2}}{2R})(1-\frac{2m}{\sqrt{R^{2}+Z^{2}}})^{-1/2}=0,
\left(\begin{array}{cccc}{\hat{p}_{0}-m}&{0}&{\hat{p}_{3}}&{\hat{p}_{1}-i\hat{p}_{2}}\\{0}&{\hat{p}_{0}-m}&{\hat{p}_{1}+i\hat{p}_{2}}&{-\hat{p}_{3}}\\{-\hat{p}_{3}}&{-\hat{p}_{1}+i\hat{p}_{2}}&{-\hat{p}_{0}-m}&{0}\\{-\hat{p}_{1}-i\hat{p}_{2}}&{\hat{p}_{3}}&{0}&{-\hat{p}_{0}-m}\\\end{array}\right)\left(\begin{array}{cccc}{\Psi_{1}}\\{\Psi_{2}}\\{\Psi_{3}}\\{\Psi_{4}}\\\end{array}\right)_{D}=0\quad,
+\frac12\Biggl[\partial_{\lambda}\Biggl(x_{\mu}\delta_{\nu\rho}-x_{\rho}\delta_{\nu\mu}\Biggr)+\partial_{\nu}\Biggl(x_{\rho}\delta_{\lambda\mu}-x_{\mu}\delta_{\lambda\rho}\Biggr)\Biggr]{\calD}_{1}^{\mathrm{full}}\left(x^{2}\right),
\theta^{\alpha}\to\theta^{\alpha}+br^{-1}(\theta\zeta+\bar{\zeta}\bar{\theta})(\bar{\zeta}\tilde{p})^{\alpha}\,,\quad\bar{\theta}^{\dot{\alpha}}\to\bar{\theta}^{\dot{\alpha}}+br^{-1}(\theta\zeta+\bar{\zeta}\bar{\theta})(\tilde{p}\zeta)^{\dot{\alpha}}
\mathrm{k}^{\mu}=\frac{(\mathrm{p}_{A}\mathrm{k})}{(\mathrm{p}_{A}\mathrm{p}_{B})}\mathrm{p}_{B}^{\mu}+\frac{(\mathrm{p}_{B}\mathrm{k})}{(\mathrm{p}_{A}\mathrm{p}_{B})}\mathrm{p}_{A}^{\mu}+k_{\bot}^{\mu}\nonumber
4\oint_{\gamma_{1}}dx^{\mu}\oint_{\gamma_{2}}dy^{\nu}\left[\varepsilon_{\beta\rho\lambda}\partial^{\lambda}\delta^{3}(x-y)\right]\left[\varepsilon_{\mu}{}^{\rho\tau}\partial_{\tau}\delta^{3}(x-y)\right]\left[\varepsilon^{\beta\sigma}{}_{\nu}\partial_{\sigma}\delta^{3}(x-y)\right]=0\,,
I_{d+3}=\frac{1}{16\pi}\intd^{d+3}x\sqrt{-g}\left(R-\frac{1}{2}(\nabla\rho)^{2}-\frac{1}{2}(\nabla\sigma)^{2}-\frac{1}{2(d+1)!}e^{\beta\rho}F_{d+1}^{2}\right),
S_{tot}=S_{H}={\frac{(p-1)V_{p}}{4G}}\left[{\frac{(p-3)^{2}+16}{(p-3)^{2}}}\right]^{\frac{p-3}{4}}\bar{\calE}^{-{\frac{p-3}{2}}}\bar{\calP}_{\phi}^{\frac{p-1}{2}}.
T_{\hat{r}}{}^{\hat{s}\hat{t}\hat{u}\hat{v}}\equiv\frac{1}{288}(\Gamma_{\hat{r}}{}^{\hat{s}\hat{t}\hat{u}\hat{v}}-8\delta_{\hat{r}}^{[\hat{s}}\Gamma^{\hat{t}\hat{u}\hat{v}]})~,~~~S_{\hat{r}\hat{s}}{}^{\hat{t}\hat{u}\hat{v}\hat{w}}\equiv\frac{1}{72}(\Gamma_{\hat{r}\hat{s}}{}^{\hat{t}\hat{u}\hat{v}\hat{w}}+24\delta_{\hat{r}}^{[\hat{t}}\delta_{\hat{s}}^{\hat{u}}\Gamma^{\hat{v}\hat{w}]})~.
S=\frac{1}{g^{2}}\intd^{2}x\left\{\partial_{a}\sigma^{i}\partial^{a}\sigma^{i}-\alpha(\sigma^{i}\sigma^{i}-1)+g^{2}\theta\epsilon^{ijk}\epsilon^{ab}\sigma^{i}\partial_{a}\sigma^{j}\partial_{b}\sigma^{k}\right\}
K_{\bfa}^{(1)}(x,x^{\prime},t)={\frac{\bfa}{4}}e^{{\frac{{\bfa}^{2}}{4}}t}e^{-{\frac{\bfa}{2}}(|x|+|x^{\prime}|)}\left(1-\Phi({\frac{|x|+|x^{\prime}|}{2\sqrt{t}}}-{\frac{\bfa}{2\sqrt{t}}})\right)~~,
\frac{dc_{x}}{\sqrt{2\pi}}=\kappa\sqrt{c}\frac{dx_{0}}{\sqrt{2\pi}},\:\:\frac{dc_{y}}{\sqrt{2\pi}}=\kappa\sqrt{c}\frac{dy_{0}}{\sqrt{2\pi}},\:\:\frac{dc_{\theta}}{\sqrt{2\pi}}=\sqrt{\frac{4\pi}{\beta}}\frac{d\theta_{0}}{\sqrt{2\pi}}.
\theta_{1}\theta_{2}=-\theta_{2}\theta_{1}-h(r_{\pm}^{12}\theta_{1}\theta_{2}+\theta_{1}\theta_{2}r_{\pm}^{12}-\theta_{1}r_{\mp}^{12}\theta_{2}+\theta_{2}r_{\pm}^{12}\theta_{1})+O(h^{2}).
\frac{\partial^{2}\tilde{A}}{\partialp_{i}\partial\tilde{p}_{j}}\,dp_{i}\,d\tilde{p}_{j}=\frac{\partial^{2}\tilde{A}}{\partialp_{k}^{\prime}\partial\tilde{p}_{l}^{\prime}}\left(\frac{\partialp_{k}^{\prime}}{\partialp_{i}}\right)\left(\frac{\partial\tilde{p}_{l}^{\prime}}{\partial\tilde{p}_{j}}\right)\,dp_{i}\,d\tilde{p}_{j}
\zeta_{0}(\nu)=-\nu{\frac{\varrho^{-2\nu}}{2\pi}}\sum_{\omega}\int_{C_{+}}dz{\frac{z\breve{\chi}^{\prime}(\breve{\omega},z)}{(z^{2}+\breve{\omega}^{2}(z))^{\nu+1}}}e^{i\epsilonz}.
\begin{array}{c}{\lbrackA^{-},A^{+}\rbrack_{q^{-1}}=q^{N_{A}}}\\{\lbrackA^{-},A^{+}\rbrack_{q}=q^{-N_{A}}}\\{\lbrackN_{A},A^{\pm}\rbrack=\pmA^{\pm}}\\\end{array}
I_{0}=\int_{{\calM}_{0}}{\calL}_{p+1}^{0}\equiv\int_{{\calM}_{0}}({\frac{1}{(p+1)!}}E^{a_{0}}\wedgeE^{a_{1}}\wedge...\wedgeE^{a_{p}}\epsilon_{a_{0}a_{1}...a_{p}}e^{-{\frac{p-3}{2}}\phi}\sqrt{-det(\eta_{ab}+F_{ab})}
\phi_{(a)}=\sqrt{{\frac{N\,V}{g_{0}^{2}}}\,\mathbf{F^{(0)}}{}_{cd\,\mu\nu}\mathbf{F}^{(0)}{}^{cd\,\mu\nu}}\equiv\sqrt{\,{\frac{N\,V}{g_{0}^{2}}}\,\mathrm{Tr}\,\mathbf{F^{(0)}}{}_{\mu\nu}\mathbf{F}^{(0)}{}^{\mu\nu}}
\rho+p_{2}=-\frac{\ddot{a}_{1}}{a_{1}}-\frac{\ddot{a}_{3}}{a_{3}}+\frac{\dot{a}_{1}}{a_{1}}\frac{\dot{a}_{2}}{a_{2}}+\frac{\dot{a}_{2}}{a_{2}}\frac{\dot{a}_{3}}{a_{3}}\ge0
\begin{array}{l}{E_{4}{\circ_{1,Z}}E_{1}=E_{3},\;\;\;\;E_{4}{\circ_{1,Z}}E_{5}=E_{2},}\\{E_{6}{\circ_{1,Z}}E_{1}=E_{2},\;\;\;\;E_{6}{\circ_{1,Z}}E_{3}=E_{5}.}\\\end{array}
T_{s}=\frac{1}{4}\left(\!\!\!\begin{array}{ccccccccc}{2}&{0}&{-2}&{-4}&{2}&{2}&{3}&{-1}&{-2}\\{-1}&{3}&{-2}&{2}&{-1}&{-1}&{0}&{-1}&{1}\\{0}&{6}&{-6}&{0}&{0}&{0}&{3}&{-3}&{0}\\\end{array}\right)^{\!\!\sim}
S_{\mathrm{ct}}=\frac{1}{8\piG}\int_{\partial\calM^{+}}d^{n+1}\sqrt{h}{\calL}_{\mathrm{ct}}+\frac{1}{8\piG}\int_{\partial\calM^{-}}d^{n+1}x\sqrt{h}{\calL}_{\mathrm{ct}},
\zeta^{(0)}(z,\beta,\sigma)=\sum_{n=0}^{\infty}\sum_{l=1}^{\infty}D_{n}^{(0)}(\Lambda_{n,l}^{T}(\sigma))^{-z}+\sum_{n=1}^{\infty}\bar{D}_{n}^{(0)}(\Lambda_{n,0}^{T}(\sigma))^{-z}~~~,
\frac{\deltaS}{\deltae_{i}^{a}}=2\pi^{ij}e_{j}^{a}+\frac{1}{2}\chi^{b}\epsilon_{ijk}M^{-1}[\theta]_{c}^{b}((e-{\bf1}\,tr\,e)^{-1})_{kl}\,O^{ja}O^{lc}.
{\calS}=-{\frac{1}{2\pi\beta^{\prime}}}\intd^{6}\sigma{\frac{1}{6!}}\epsilon^{abcdef}\partial_{a}X^{M}\partial_{b}X^{N}\partial_{c}X^{P}\partial_{d}X^{Q}\partial_{e}X^{R}\partial_{f}X^{S}A_{MNPQRS}.
\overline{{D}}\gamma_{p}{\bfJ}+\overline{{\bfJ}}\gamma_{p}{\bfJ}=i~\overline{{\theta}}\theta~\tau^{j}\varepsilon^{\mu\nu}(\partial_{\mu}J_{\nu}^{j}+\varepsilon^{jkl}J_{\mu}^{k}J_{\nu}^{l})=0~.
\langle\,\partial_{-}f(\sigma_{0})\,\partial_{-}f(\sigma)\,\rangle\,_{\mathrm{{in}}}\:\rightarrow\:-\,\frac{1}{4\pi}\,\frac{2\lambdaA^{-2}/M}{(B-x_{H})^{2}}\,\frac{e^{-\,2\lambda\sigma^{0}}}{(\sigma+B)^{2}}\,.
\{\overline{{W_{n}^{m}}},\overline{{W_{p}^{q}}}\}\,=\,(mp-nq)\{\overline{{W_{n+p}^{m+q-1}}}+\overline{{W_{n+p-1}^{m+q}}}\}+N^{-1}\:{\calP}(\overline{{W_{b}^{a}}})
\tilde{\phi}=\phi-{\frac{R_{y}}{\ell^{2}}}\,a_{2}\varphi-{\frac{R_{y}}{\ell^{3}}}\,a_{1}\tau,\quad\tilde{\psi}=\psi-{\frac{R_{y}}{\ell^{2}}}\,a_{1}\varphi-{\frac{R_{y}}{\ell^{3}}}\,a_{2}\tau,
F(m,\mu)=\frac{2}{3}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{4\mu(p^{2}/m^{2})(1-2\mu^{3}p^{2}/m^{2})+3}{f(\mu,p^{2})[p^{2}-m^{2}f^{2}(\mu,p^{2})]^{2}}.
\frac{1}{2sL}\int_{-sL}^{sL}dx^{-}\varphi_{i}^{n}(x,sL)\varphi_{j}^{m}(x,sL)=\frac{1}{2L}\int_{-L}^{L}dx^{-}\varphi_{i}^{n}(x,L)\varphi_{j}^{m}(x,L)\quad,
(\partial-{\bar{\partial}})(\hat{{\cal\phi}^{0}},\hat{{\calA}_{z}^{0}})\sim{\frac{\sqrt{3}(-1+a_{i})a_{i}^{3}\biggl[(b_{i}e_{i})^{2}c_{i}f_{i}(c_{i}-f_{i}){\bar{z}}+(-b_{i}c_{i}^{2}e_{i}^{2}f_{i}(f_{i}-b_{i})+e_{i}b_{i}^{2}c_{i}f_{i}^{2}(c_{i}-e_{i}))z\biggr](z-{\bar{z}})}{|z|^{4}}},
W_{n_{1}n_{2}m}^{l}(\delta_{1},\delta_{2})=\frac{(-1)^{\frac{m-|m|}{2}}}{\Gamma(|m|+\delta_{1}+1)\Gamma(2l+\delta_{1}+\delta_{2}+2)}E_{n_{1}n_{2}}^{lm}K_{nn_{1}}^{lm},
\frac{1}{2}y_{sB}\frac{\partial}{\partialy_{sB}}\psi_{0}=-\frac{1}{2}y_{sB}y_{sB}\psi_{0}=-\frac{1}{2}\sum_{sB}(y_{sB}^{2}-\frac{1}{2})\psi_{0}-\frac{1}{4}\cdot2(d-1)\psi_{0}\;,
A_{4}=\frac{g^{2}\sqrt{2}}{2\pi}\intd^{2}z\\langle\mathrm{left~part}\rangle\\langle\bar{G}_{1}\bar{G}_{2}\bar{G}_{3}\bar{G}_{4}\rangle\\langle\prod_{i}e^{i/2\K_{i}X}e^{i/2\K_{i}\bar{X}}\rangle
\zeta+a_{0}\frac{1}{\zeta}x_{1}^{4-n}z^{8-n}\to\zeta+a_{0}\frac{1}{\zeta}\prod_{i=1}^{N_{s}}(x_{1}-m_{i}^{4})\prod_{j=1}^{N_{f}}(z-m_{j}^{2}).
K^{(1)}\left(T,U,\overline{{T}},\overline{{U}}\right)=-{\frac{1}{16\,\pi^{2}}}\,Y\left(T,U,\overline{{T}},\overline{{U}}\right)+\kappa\Big(T,U\Big)+{\bar{\kappa}}\left(\overline{{T}},\overline{{U}}\right)\,;
S=\intdx^{D}\sqrt{g_{D}}\left[-\frac{M_{*}^{D-2}}{16\pi}(R+2\Lambda)-\frac12g^{AB}\nabla_{A}{\bf\Phi}\cdot\nabla_{B}{\bf\Phi}-\frac{\lambda}{4}({\bf\Phi}^{2}-v^{2})^{2}\right],
\left(\begin{array}{cc}{0}&{\partial_{y}+xy}\\{\partial_{y}-xy}&{0}\\\end{array}\right)\Psi_{n}+x^{\kappa}\left(\begin{array}{cc}{\partial_{x}}&{0}\\{0}&{-\partial_{x}}\\\end{array}\right)x^{-\kappa}\Psi_{n-1}=0\;,\qquadn=1,2,...\;.
[x_{r},p_{r}]=i\hbar+i\hbar(q^{2}-1)\sum_{s\ler}\left(\frac{q^{2}+1}{2}\right)^{s-1}\left(\frac{x_{s}^{2}}{4L_{s}^{2}}+\frac{p_{s}^{2}}{4K_{s}^{2}}\right)
\psi_{1}^{-}=\frac{i}{\sqrt{2R(R+r_{5})}}\left[\begin{array}{c}{-r_{3}-ir_{4}}\\{r_{1}-ir_{2}}\\{R+r_{5}}\\{0}\\\end{array}\right],\psi_{2}^{-}=\frac{i}{\sqrt{2R(R+r_{5})}}\left[\begin{array}{c}{-r_{1}-ir_{2}}\\{-r_{3}+ir_{4}}\\{0}\\{R+r_{5}}\\\end{array}\right].
=-\bigg[\frac{(d-1)\beta+4\gamma}{4(d-2)}\bigg]{\calR}_{GB}^{2}+\bigg(\frac{d-1}{d-2}\bigg)(\beta/4+\gamma)C^{2}+\bigg[\frac{4d\alpha+(d+1)\beta+4\gamma}{4d}\bigg]R^{2}
a_{n}(q^{\prime},q)=\int_{0}^{1}\eta^{n-1}d\eta\left\{\frac{1}{2}\frac{\partial^{2}}{\partialx^{2}}-V(x)\right\}a_{n-1}(x,q)\Biggl.\Biggr|_{x=q+(q^{\prime}-q)\eta}.
\frac{\beta\alpha^{6}}{m}H_{ij}^{\dagger}H_{kl}^{\dagger}\psi^{ij}\psi^{kl}\,\qquad\frac{\beta\alpha^{6}}{m}H_{ik}^{\dagger}H_{jl}^{\dagger}\psi^{ij}\psi^{kl}\.
x_{-}=p^{\frac{1}{2}}+\frac{p^{\frac{n-1}{2}}}{1!\,2}\,+\frac{\left(2n-1\right)p^{\frac{2n-3}{2}}}{2!\,2^{2}}\,+\frac{\left(3n-1\right)\left(3n-3\right)p^{\frac{3n-5}{2}}}{3!\,2^{3}}\,+\frac{\left(4n-1\right)\left(4n-3\right)\left(4n-5\right)p^{\frac{4n-7}{2}}}{4!\,2^{4}}\,+\dots
\epsilon_{(n)}^{\vec{\alpha}\beta\gamma}=(-1)^{\frac{n}{2}-1}\delta_{\alpha[1]\alpha[2]}\,\delta_{\alpha[3]\alpha[4]}\,\cdots\,\delta_{\alpha[n-3]\alpha[n-2]}\,\epsilon^{\alpha[n-1]\betab}\,\epsilon^{b\alpha[n]\gamma}\;
x_{+}=1-{\frac{p^{\frac{1}{2}}}{1!\,(n-2)}}-{\frac{\left(n+1\right){p}^{2}}{2!\,\left(n-2\right)^{2}}}-{\frac{\left(n+3\right)\left(2\,n+1\right){p}^{3}}{3!\,\left(n-2\right)^{3}}}-{\frac{\left(n+5\right)\left(2\,n+3\right)\left(3\,n+1\right){p}^{4}}{4!\,\left(n-2\right)^{4}}}-\dots.
\beta(P_{0};\kappa)=-P_{0}^{2}\,\frac{e^{P_{0}/\kappa}}{2f(P_{0})}\left(\frac{n\Phi(P_{0})}{\kappa}+\Phi^{\prime}(P_{0})-\frac{e^{-P_{0}/\kappa}\Phi(P_{0})}{f(P_{0})}\right).
{\textstyle\frac{1}{2}}|{\calZ}_{1,2}|^{2}=\sum_{n}|\Gamma^{(n)}|^{2}\pm\left[\left(\sum_{n}|\Gamma^{(n)}|^{2}\right)^{2}-|\sum_{n}\Gamma^{(n)2}|^{2}\right]^{\frac{1}{2}}\,.
\overline{{\vec{n}}}+\overline{{\vec{m}}}=\frac{1}{2}\left({\calI}_{\mathrm{\scriptsize~E}_{7}}\overline{{\vec{m}}}+(2\overline{{n}}_{1}-1)\vec{e}_{2}+\vec{e}_{i}\right)\qquadi=2,\ldots,8,
J^{(N)}(n;1,\ldots,1)=2\mathrm{i}^{1-2N}\pi^{n/2}\;\Gamma\left(N-{\textstyle{\frac{n}{2}}}\right)\;\frac{m_{0}^{n-N-1}}{\sqrt{\Lambda^{(N)}}}\;\begin{array}{c}{{}}\\{\int\ldots\int}\\{{}^{\Omega^{(N)}}}\\\end{array}\frac{\prod\mathrm{d}\gamma_{i}}{\gamma_{N}^{n-N}}\;\delta\left(\sum\gamma_{i}^{2}-1\right),
\int_{t_{0}}^{t_{0}^{*}}dt\sqrt{\mu^{2}+\phi^{2}(t)}=\int_{t_{0}}^{t_{0}^{*}}dt{\frac{dS_{0}}{dt}}\sqrt{\Omega+S_{0}^{2}}=\int_{i\sqrt{\Omega}}^{-i\sqrt{\Omega}}dS_{0}\sqrt{\Omega+S_{0}^{2}}=-{\frac{i\Omega\pi}{2}}.
H_{gonimetric}^{2d}=-\sum_{\vec{r},\vec{\alpha}}\sigma_{\vec{r}}\sigma_{\vec{r}+\vec{\alpha}}+\frac{1}{2}\sum_{\vec{r},\vec{\alpha},\vec{\beta}}\sigma_{\vec{r}}\sigma_{\vec{r}+\vec{\alpha}+\vec{\beta}}
\sum_{m=0}^{2r}D[f_{2l+ml}]-\sum_{m=0}^{2r}D[f_{ml}]=4A\sum_{m=0}^{2r}\frac{d}{d\tau}\left\{K_{1}\sqrt{\dot{x}_{+ml}^{2}(0)}+K_{2}\sqrt{x_{l+ml}^{2}(l)}\right\}.
H_{orb}=\frac{1}{2}p_{i}g^{ij}p_{j}=\frac{1}{2}\left(p_{\mu}-\frac{i\hbar}{2}\Gamma_{\mu}\right)g^{\mu\nu}\left(p_{\nu}+\frac{i\hbar}{2}\Gamma_{\nu}\right)=\frac{1}{2}p_{\mu}g^{\mu\nu}p_{\nu}+V_{orb}
I=\frac{1}{2\kappa^{2}}\intd^{D}X\sqrt{-g}\left(R-\frac{1}{2}(\nabla\vec{\phi})^{2}-\sum_{I=1}^{k}\frac{e^{-\vec{\alpha}^{(I)}\vec{\phi}}}{2(d_{I}+1)!}F_{d_{I}+1}^{(I)2}\right).
(A\otimesB)=\left(\begin{array}{cccc}{A_{11}B}&{A_{12}B}&{\cdots}&{A_{1N}B}\\{.}&{.}&{.}&{.}\\{.}&{.}&{.}&{.}\\{A_{M1}B}&{A_{M2}B}&{\cdots}&{A_{MN}B}\\\end{array}\right)
{\calI}_{\mathrm{1-loop}}^{I}=-\frac{iV^{(8)}B^{4}}{2\pi}\sum_{w}\frac{T_{2}}{w^{I}G_{IJ}w^{J}}\bigg[\sum_{ij}e^{2\pii(a_{i}+a_{j})_{I}w^{I}}(q_{i}+q_{j})^{4}-\sum_{i}e^{4\piia_{iI}w^{I}}(2q_{i})^{4}\bigg]
S_{D3}=\intd^{4}x\lambda\sqrt{-g}\left[1+{\frac{1}{2}}{\calF}\cdot\bar{\calF}-{\frac{1}{16}}({\calF}\cdot^{\,*}{\calF})(\bar{\calF}\cdot^{\,*}\bar{\calF})+{\frac{1}{4}}H\cdotH\right],
A(\Lambda_{1},K_{1};\ldots;\Lambda_{N},K_{N})=g^{(N-2)}\int\prod_{i=1}^{N}d^{2}z_{i}\biggl\langle\prod_{j=1}^{N}e^{i/2\K_{j}\cdotX}W_{\Lambda}(z_{j},{\bar{z}}_{j})e^{i/2\K_{j}\cdot\bar{X}}\biggl\rangle\.
\delta_{F}(\left[v\theta^{6}\right])+\delta_{B}(\left[v^{2}\theta^{4}\right])+v^{i}\epsilon\left[\dot{N}^{i(4)}\theta^{4}\right]\theta+\frac{i}{2}\theta\left[v\dot{M}^{(4)}\theta^{4}\right]\epsilon\simeq0~,
{\calJ}=-i\int{d{\bfx}\,\psi^{\dagger}\epsilon_{ln}x_{l}(\partial_{n}-ieA_{n})\psi}-\int{d{\bfx}\,\psi^{\dagger}\frac{\gamma_{0}}{2}\psi}-\frac{1}{2}\int{d{\bfx}\,x_{n}F_{n0}\epsilon_{ij}F_{ij}},
{\frac{1}{f_{k}(\xi,\tau)}}={\frac{k-3/2}{\xi^{k-1/2}}}+\sum_{n=1}^{k-1}a_{n}\tau^{n}{\frac{k+2n-3/2}{\xi^{k+2n-1/2}}}+b\tau^{k-1/2}{\frac{3k-5/2}{\xi^{3k-3/2}}}+O(\tau^{k})\,
\begin{array}{lccclcc}{{\mathrmRe}\Gamma_{+-}^{T}(p^{2})\big|_{p^{2}=M_{W}^{2}}}&{=}&{0}&{\qquad}&{{\mathrmRe}\Gamma_{ZZ}^{T}(p^{2})\big|_{p^{2}=M_{Z}^{2}}}&{=}&{0}\\{\phantom{\mathrmRe}\Gamma_{AA}^{T}(p^{2})\big|_{p^{2}=0}}&{=}&{0}&{\qquad}&{{\mathrmRe}\Gamma_{HH}(p^{2})\big|_{p^{2}=m_{H}^{2}}}&{=}&{0}\\\end{array}
V(x_{1},x_{2})=\left(\begin{matrix}{\alpha(x_{1},x_{2})}&{\bar{\beta}(x_{1},x_{2})}\\{-\beta(x_{1},x_{2})}&{\bar{\alpha}(x_{1},x_{2})}\\\end{matrix}\right),\;\;\;|\alpha|^{2}+|\beta|^{2}=1\;.
{[}L_{\overline{{0}}},L_{\overline{{0}}}{]}\subseteqL_{\overline{{0}}},\quad{[}L_{\overline{{0}}},L_{\overline{{1}}}{]}\subseteqL_{\overline{{1}}},\quad{\{}L_{\overline{{1}}},L_{\overline{{1}}}{\}}\subseteqU(L_{\overline{{0}}}).
\phi_{1}^{2}=p_{2}q_{2},~~~~\phi_{2}^{2}=p_{2}q_{3}-\alpha\frac{\sqrt{g}}{{q_{2}}^{2}},~~~~\phi_{3}^{2}=\varepsilon(p_{2}q_{2}q_{3})-\beta\frac{q_{2}q_{3}}{\sqrt{q_{2}^{2}}}.
g\simeq\tilde{\Lambda}^{2}\,(-A^{2}\eta^{2}ds^{2}+d\eta^{2}+d\rho^{2})+\tilde{\Lambda}^{-2}\rho^{2}d\phi^{2},\qquad\tilde{\Lambda}\equiv1+\frac{B^{2}\rho^{2}}{4},
\frac{x^{i}}{\rho}F_{\hat{i}\hat{0}}=\partial_{\rho}\psi=-\frac{Q_{elec}}{\rho^{2}}+{\calO}(\rho^{-3})\,\qquad\frac{x^{i}}{\rho}\tilde{G}_{\hat{i}\hat{0}}=\partial_{\rho}\chi=-\frac{P_{elec}}{\rho^{2}}+{\calO}(\rho^{-3})\{}.
\Pi_{\mu_{1}\mu_{2}}^{\mathrm{ht}}(k,u)=-\frac{1}{\left(2\pi\right)^{3}}\int\frac{\mathrm{d}^{3}Q}{2|{\bfQ}|}N(|{\bfQ}|)\left\{P_{\alpha\beta}(Q)A_{\mu_{1}\mu_{2}}^{\alpha\beta}\left(Q,k,-k\right)+(Q\rightarrow-Q)\right\}_{Q_{0}=|{\bfQ}|}.
\Phi^{-1}(p)~\sim~\frac{1}{A}\left[1~-~\frac{A^{\prime}}{(p^{2})^{\omega}}\right]~~~,~~~\sigma^{-1}(p)~\sim~\frac{1}{B}\left[1~-~\frac{B^{\prime}}{(p^{2})^{\omega}}\right]
\sum_{A=1}^{r}a_{I}^{A}q_{A}=\sum_{A=1}^{r-1}\tilde{a}_{I}^{A}\tilde{q}_{A}+\tilde{a}_{I}^{0}\frac{\sum_{A=1}^{r}m_{A}q_{A}}{\sum_{A=1}^{r}m_{A}}
\Pi_{\mathrm{T\,(HTL)}}^{(1)(\beta)}(0)\simeq-C(D)e^{2}\int\frac{d^{D-1}k}{(2\pi)^{D-1}}\,\frac{n_{F}(\omega_{k})}{\omega_{k}}\left((D-3)+\frac{\vec{k}^{2}\vec{p}^{\,2}}{(\vec{k}\cdot\vec{p}\,)^{2}}\right)=0
\Psi_{E}\left(r,\theta\right)=\sum_{n=-\infty}^{\infty}A_{n}\left(\begin{array}{l}{J_{n-\kappa}\left(|E|r\right)e^{in\theta}}\\{i\frac{|E|}{E}J_{n+1-\kappa}\left(|E|r\right)e^{i\left(n+1\right)\theta}}\\\end{array}\right)
\begin{array}{rcl}{\mathrm{Tr}\Bigl(E(l_{4})F(l_{4})\Bigr)}&{=}&{\mathrm{Tr}\Bigl(V^{-1}E(l_{4})VV^{-1}F(l_{4})V\Bigr)}\\{}&{\rightarrow}&{\mathrm{Tr}\Bigl(E^{\prime}(l_{4})U_{2}F(l_{4})U_{2}^{-1}\Bigr)=\mathrm{Tr}\Bigl(E^{\prime}(l_{4})F^{\prime}(l_{4})\Bigr),}\\\end{array}
\|\partial^{p}A_{2m}^{(s)}(\Lambda)\|\leq{\frac{1}{\Lambda^{p}}}\int_{\Lambda}^{\Lambda_{0}}{\frac{d\Lambda^{\prime}}{\Lambda^{\prime}}}P\left(ln{\frac{\Lambda^{\prime}}{\Lambda_{R}}}\right)\left({\frac{\Lambda}{\Lambda^{\prime}}}\right)^{p+2m-4}\leq{\frac{1}{\Lambda^{p}}}\tilde{P}\left(ln{\frac{\Lambda}{\Lambda_{R}}}\right)
S_{\lambda}^{\mathrm{R}}(\tau,\rho,\sigma,\nu)~:~\qquad\boxed{A_{\lambda}^{\mathrm{R}}(\tau,\rho,\sigma,\nu)=B_{\lambda}^{\mathrm{R}}(\tau,\rho,\sigma,\nu)}\qquad\qquad\lambda=0,\dots,2u-1,
F_{\mu\nu}^{+ij}+\frac{1}{2}\big(\overline{M}_{(\mu}^{j}M_{\nu)}^{i}-\frac{\delta^{ij}}{N}\overline{M}_{(\mu}^{k}M_{\nu)}^{k}\big)=0,\;\;\;\;\;\;\gamma^{\mu}D_{\mu}M=0,
\{A_{1}^{\pm\pm}(n_{1},n_{2}),A_{2}^{\pm\pm}(m_{1},m_{2})\}=\frac{2}{N_{k}}\,f^{\pm\pm}(n_{1},n_{2})\,\delta_{n_{1},m_{1}}\,\delta_{n_{2},m_{2}}\\\,
\Biggr\langleP_{a_{1}b_{1}}(x_{1})\cdotsP_{a_{n}b_{n}}(x_{n})e^{i\oint_{C}\omega}\Biggr\rangle_{TQFT_{4}}=\Biggr\langleP_{a_{1}b_{1}}(x_{1})\cdotsP_{a_{n}b_{n}}(x_{n})e^{i\oint_{C}\omega}\Biggr\rangle_{NLSM_{2}},
\begin{array}{c}{A_{\mu}=\left(\frac12\left(l_{\mu}^{1}+l_{\mu}^{2}\right),\frac12\left(l_{\mu}^{1}+l_{\mu}^{2}\right)\right),}\\{B_{\mu}=\left(\frac12\left(l_{\mu}^{1}-l_{\mu}^{2}\right),-\frac12\left(l_{\mu}^{1}-l_{\mu}^{2}\right)\right),}\\\end{array}
ds_{P}^{2}={\frac{\left|{\partial^{3}{\calF}/\partiala^{3}}\right|^{2}}{\left(\mathrm{Im}\,{\partial^{2}{\calF}/\partiala^{2}}\right)^{2}}}|da|^{2}={\frac{\left|{\partial^{3}{\calF}/\partialu\partiala\partiala}\right|^{2}}{\left(\mathrm{Im}\,{\partial^{2}{\calF}/\partiala^{2}}\right)^{2}}}|du|^{2}=e^{\varphi}|du|^{2}.
b_{1}(x,y)=\sum_{n=1}^{\infty}\Bigl[b_{1}^{(c,2n-1)}(x)Ec_{1}^{2n-1}(\omegay,k)+b_{1}^{(s,2n-1)}(x)Es_{1}^{2n-1}(\omegay,k)\Bigr].
\tilde{I}_{\mu_{1}\ldots\mu_{n}}\sim\int_{\frac{-\pi}{a}}^{\frac{\pi}{a}}\frac{d^{4}l}{(2\pi)^{4}}\sum_{r=-\infty}^{+\infty}\sum_{j=0}^{n-1}\frac{A_{j}^{\mu_{1}\ldots\mu_{n}}(l,Q,M_{r})}{P^{2}(l+Q_{j})+M^{2}r^{2}},\quadQ_{j}=p_{1}+\ldots+p_{j}.
\overline{{\omega}}_{r}(\varepsilon^{s})={\frac{\partial{\calM}(\tau)}{\partial\overline{{\calS}}^{r}}}=F_{rs}(\overline{{\calS}}^{u},q^{\mu},p_{\mu})\,\overline{{\calS}}^{s}+G_{r}(\overline{{\calS}}^{u},q^{\mu},p_{\mu}),
N=OO^{*}=c^{2}t^{2}+\frac{\hbar^{2}}{P_{x}^{2}}+\frac{\hbar^{2}}{P_{y}^{2}}+\frac{\hbar^{2}}{P_{z}^{2}}-x^{2}-y^{2}-z^{2}-\frac{c^{2}\hbar^{2}}{E^{2}}~,
\Gamma_{3}=\frac{\lambda^{3}}{\left(2\pi\right)^{5}}\int\frac{\mathrm{d}^{5}Q}{2|{\bfQ}|}\left\{\left[\frac{1}{2Q\cdotk_{1}+k_{1}^{2}}\;\frac{1}{-2Q\cdotk_{2}+k_{2}^{2}}+permutations\right]+Q\rightarrow-Q\right\}N(|{\bfQ}|),
P_{\epsilon}^{(i)}(z;r)=\oint_{C_{\epsilon}^{(i)}}\frac{\epsilondw}{2\pi\sqrt{-1}}\frac{1-wzq^{-2}}{1-wrq^{-2}}\frac{\Theta_{q^{4}}(q^{2}w^{2})}{\Theta_{q^{2}}(wz)\Theta_{q^{2}}(wz^{-1})}
h_{\mathrm{open}}^{\prime}\equiv{\frac{1}{m}}\left[\sum_{r=2}^{M}{\bfq}_{r}^{2}-\sum_{r=2}^{M-1}{\bfq}_{r}\cdot{\bfq}_{r+1}+{\frac{1}{2}}\sum_{r=2}^{M}V({\bfy}_{r})\right].
v_{m}(z)=\frac{\sqrt{m/2k}\left[J_{1}(m/k)Y_{2}(me^{k|z|}/k)-Y_{1}(m/k)J_{2}(me^{k|z|}/k)\right]}{\sqrt{J_{1}(m/k)^{2}+Y_{1}(m/k)^{2}}}.
ds^{2}=\left(\frac{l}{z}\right)^{2}\left(dz^{2}-(1+2\phi)d\tau^{2}+2b_{,i}dx^{i}d\tau+\left((1-2\hat{\Psi})\delta_{ij}+2\hat{E}_{,ij}\right)dx^{i}dx^{j}\right),
\xi_{1}=\Phi_{1}^{l_{1}},{}~~~~\xi_{i}=\frac{\Phi_{i}}{\Phi_{1}^{l_{i}/l_{1}}},~i=2,\cdots,5,{}~~~~\xi_{6}=\Phi_{1}\Psi_{1}^{2},{}~~~~\xi_{7}=\frac{\Psi_{2}}{\Psi_{1}}.
\langle{\bfk}^{\prime}\vertV\vert{\bfk}\rangle={\frac{1}{(2\pi)^{3}}}{\frac{m_{1}m_{2}}{\sqrt{w_{1}({\bfk})w_{2}({\bfk})w_{1}({\bfk}^{\prime})w_{2}({\bfk}^{\prime})}}}\tilde{I}_{\mathrm{inst}}({\bfk}^{\prime},{\bfk})
\begin{array}{lll}{\left[u,v\right]=q\lambdavz\;,\qquad}&{vz=q^{2}zv\;,\qquad}&{[u,z]=0\;,}\\{\left[w,u\right]=q\lambdazw\;,\qquad}&{[v,w]=q\lambda(u-z)z\;,\qquad}&{wz=q^{-2}zw\;,}\\\end{array}
{\calL}=\partial_{\mu}\psi_{1}^{\dagger}\widetilde\gamma_{\mu\nu}\partial_{\nu}\psi_{2}+\partial_{\mu}\psi_{2}^{\dagger}\gamma_{\mu\nu}\partial_{\nu}\psi_{1}+m^{2}\psi_{2}^{\dagger}\psi_{1}+m^{2}\psi_{1}^{\dagger}\psi_{2}\quad.
\left(L_{n}^{1/2}(x)\right)_{ik}{\calL}_{n}(y,l)K_{kj}(y-x+\frac{i}{2})=K_{ik}(y-x+\frac{i}{2}){\calL}_{n}(y,l)\left(L_{n}^{1/2}(x)\right)_{kj},
\int_{\Lambda}\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{[(k+k_{1})^{2}-m_{t}^{2}][(k+k_{2})^{2}-m_{t}^{2}]}=I_{log}(m_{t}^{2})-\left(\frac{i}{(4\pi)^{2}}\right)\left[Z_{0}(m_{t}^{2},m_{t}^{2};(k_{1}-k_{2})^{2};m_{t}^{2})\right]
\pi_{i}^{(1)a}=-\,\frac{1}{m^{2}}\,\eta^{1b}\,G_{1i}^{(0)ba}+\frac{g}{2m^{2}}\,f^{bcd}\,\eta^{2b}\,(D_{i}\pi^{i})^{c}\,G_{1i}^{(0)da}-\eta^{2b}\,G_{2i}^{(0)ba}\,,
\bigl\{T_{2}^{(2)},4\,\eta+\eta^{3\prime\prime}+2\,\lambda\,\eta^{3}\bigr\}_{(\eta)}=-\,8\,\phi^{A}\pi^{A}\,\eta^{1}\delta+2\,\phi^{A}\pi^{A}\,\eta^{4}\delta^{\prime\prime}+4\,\lambda\,\phi^{A}\pi^{A}\,\eta^{4}\delta+\cdots
H=\int~d^{2}\sigma~\biggl[\frac{1}{2}P_{I}^{2}+\frac{1}{2}h^{ab}\lambda^{\dagger}{\gamma_{a}\gamma\partial_{b}}\lambda+\frac{1}{4}~\biggl\{X^{I},X^{J}\biggr\}~\biggl\{X^{I},X^{J}\biggr\}\biggr]
\frac{W_{\mu}}{m}=\left(\frac{(\vec{{\calP}^{D}}\vec{S})}{m},\,\,{S}_{i}+\frac{{{\calP}^{D}}_{i}(\vec{{\calP}^{D}}\vec{S})}{m({\calE}+m)}\right)\nonumber\,=\left(\gamma(\vec{v}\vec{S}),\,\,{S}_{i}+\gamma^{2}\frac{{v}_{i}(\vec{v}\vec{S})}{\gamma+1}\right),
\begin{array}{rl}{0}&{=\nabla_{\rho}h^{\rhoI_{2}\ldotsI_{s}}=}\\{}&{=z^{d+2(s-1)}\partial_{\rho}\left(z^{-d-2(s-1)}h^{\rhoI_{2}\ldotsI_{s}}\right)=\partial_{z}h^{zI_{2}\ldotsI_{s}}+\partial_{I}h^{II_{2}\ldotsI_{s}}-{\frac{d+2(s-1)}{z}}h^{zI_{2}\ldotsI_{s}}}\\\end{array}
\sum_{p,s\in{\calI}}T^{*}(i,p)T^{*}(j,s){\hat{\calR}}\left(l,k;s,p\right)=\sum_{p,s\in{\calI}}{\hat{\calR}}\left(s,p;j,i\right)T^{*}(p,k)T^{*}(s,l)\;,
(\vec{\sigma}\wedge\vec{\bigtriangledown}_{p})(1+n_{0})^{\frac{1}{2}}(1+n_{0}^{\prime})^{\frac{1}{2}}\hat{U}(n)D\\hat{U}^{-1}(n^{\prime})=-ip_{0}\left(\frac{1+n_{0}}{p_{0}}\right)^{2}\delta(\Omega-\Omega^{\prime}).