harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
J_{y}=J_{(3)}={\frac{-i}{\sqrt{2}}}({\bar{\alpha}}_{A}{\bar{\alpha}}^{B}+{\bar{\beta}}_{A}{\bar{\beta}}^{B}){\bar{\pi}}_{B}\omega^{A}+{\frac{i}{\sqrt{2}}}(\alpha_{A^{\prime}}\alpha^{B^{\prime}}+\beta_{A^{\prime}}\beta^{B^{\prime}})\pi_{B^{\prime}}{\bar{\omega}}^{A^{\prime}},
[K_{m}^{ij},K_{n}^{lp}]=i(K_{m+n}^{ip}\,\delta^{jl}+K_{m+n}^{jl}\,\delta^{ip}-K_{m+n}^{il}\,\delta^{jp}-K_{m+n}^{jp}\,\delta^{il})+2mk\delta_{pl}^{ij}\delta_{m+n,o}
G_{10}^{(p+1)}\to\left(\begin{array}{cc}{G_{9\,ij}^{(p)}-2\bar{\theta}_{2}\partial_{i}\theta_{2}\bar{\theta}_{1}\partial_{j}\theta_{1}}&{\partial_{i}S-2\bar{\theta}_{2}\partial_{i}\theta_{2}}\\{-\partial_{j}S-2\bar{\theta}_{1}\partial_{j}\theta_{1}}&{-1}\\\end{array}\right)\,.
S_{E}=-\frac{1}{16\piG_{d-1}}\int_{\Sigma}d^{d-1}x\sqrt{g}[R-\frac{4}{d-3}(\partial\phi)^{2}-me^{\frac{4\phi}{\sqrt{d-2}(d-3)}}-\frac{1}{4}e^{\frac{-4\sqrt{d-2}}{d-3}\phi}F^{2}].
\begin{array}{rclrclrclrcl}{A}&{=}&{T_{a}A^{a},\quad\hat{\psi}}&{=}&{T_{a}\hat{\psi}^{a},\quad\psi}&{=}&{\Sigma_{\alpha}\psi^{\alpha},\quad\hat{A}}&{=}&{\Sigma_{\alpha}\hat{A}^{\alpha},}\\{\hat{a}}&{=}&{T_{a}\hat{a}^{a},\quad\alpha}&{=}&{T_{a}\alpha^{a},\quad\hat{\alpha}}&{=}&{\Sigma_{\alpha}\hat{\alpha}^{\alpha},\quada}&{=}&{\Sigma_{\alpha}a^{\alpha}.}\\\end{array}
S_{nl}[J]=-\frac{1}{2}\intd^{3}x\int_{0}^{L}ds_{1}\intd^{3}y\int_{0}^{L}ds_{2}\,J_{\alpha}(x,s_{1})K_{\alpha\beta}(x-y,s_{1}-s_{2})J_{\beta}(y,s_{2})\,.
W=\sum_{i=1}^{k}Q_{i}(X-a_{i})\tilde{Q}^{i}+\sum_{a=1}^{l}T_{a}(Y-b_{a})\tilde{T}^{a}+\sum_{ia}Q_{i}h_{a}^{i}\tilde{T}^{a}+\tilde{Q}^{i}\tilde{h}_{i}^{a}T_{a},
|P\rangle\equiv\sum_{N=0}^{\infty}\int\left[\prod_{i=1}^{N}dp_{i}\right]\delta\left(\sum_{i=1}^{N}p_{i}-P\right)\psi_{N}(p_{1},p_{2},\dots,p_{N})\left[\prod_{i=1}^{N}a^{\dagger}(p_{i})\right]|0\rangle,
e^{iA}=\frac{\Gamma[\frac{ib^{2}}{4v\pi\alpha^{\prime}}+\frac{1}{2}-\frac{i\pi\epsilon}{2v}]\Gamma[\frac{ib^{2}}{4v\pi\alpha^{\prime}}+\frac{1}{2}+\frac{i\pi\epsilon}{2v}]}{\Gamma[\frac{ib^{2}}{4v\pi\alpha^{\prime}}+1]\Gamma[\frac{ib^{2}}{4v\pi\alpha^{\prime}}]}.
Z=-\frac{1}{e^{2}{\phi_{0}}^{2}}\intd^{2}x\left[\frac{e}{2}\epsilon^{ij}F_{ij}(\vert\Phi\vert^{2}-{\Phi_{0}}^{2})+i\epsilon^{ij}(D_{i}\Phi)(D_{j}\Phi)^{*}\right]
(i\hbarD_{i}^{(1)}\psi^{(1)})^{*}\equiv\psi^{(1)^{+}}(i\hbarD_{i})^{+}\equiv-i\hbar\partial_{i}\psi^{(1)+}-{\frac{e}{c}}\psi^{(1)+}a_{i}^{[1]eff}
\frac{\partialL_{0}^{m}}{\partial\dot{x}^{\mu}}=\frac{\partial(\dot{x}+q^{1/2}i\chi\psi)^{2}}{\partial\dot{x}^{\mu}}\;\frac{\partial[(\dot{x}+q^{1/2}i\chi\psi)^{2}]^{1/2}}{\partial(\dot{x}+q^{1/2}i\chi\psi)^{2}}\;\frac{\partialL_{0}^{m}}{\partial[(\dot{x}+q^{1/2}i\chi\psi)^{2}]^{1/2}}.
\rho={\frac{1}{|x-X_{1}|^{2}}}+{\frac{\lambda_{2}}{|x-X_{2}|^{2}}}+{\frac{\lambda_{3}}{|x-X_{3}|^{2}}}\approx{\frac{1}{|x-X_{1}|^{2}}}+{\frac{\lambda_{2}}{|x-X_{2}|^{2}}}
{\frac{1}{\beta_{c}}}=b(d-2)(1-\kappa){\int}_{-1}^{1}\;d\xi\;H_{c}(\xi)\;{\frac{\sqrt{1-\xi^{2}}}{\sqrt{(\xi+\frac{3+\kappa}{1-\kappa})(\xi+\frac{1+3\kappa}{1-\kappa})}}}\;.
\int\frac{d{\bfp}}{p_{0}}\,\sum_{\mu=-s}^{s}|{\Psi}_{\mu}^{(s)}({\bfp})|=\int\frac{d{\bfp}}{p_{0}}\,\sum_{\mu=-s}^{s}|{\Psi}_{\mu}^{(s)}({\bfp},t,{\bfx})|<\infty,
\beta_{2}=\beta_{2}^{\mu}+\beta_{2}^{\eta}+\beta_{2}^{(4)}+\beta_{2}^{(6)}=\left[-\frac{3}{4}-\frac{1}{12}+\frac{9}{4}\right]K_{4}^{2}=\frac{17}{12}K_{4}^{2}\;,
D^{\mu\nu}(x,y)=i\int\frac{d^{4}k}{(2\pi)^{4}}\,frac{e^{-ik_{\lambda}(x-y)^{\lambda}}}{k_{\lambda}k^{\lambda}+i\epsilon}\left[g^{\mu\nu}-\frac{\gammak^{\mu}k^{\nu}}{k_{\lambda}k^{\lambda}+i\epsilon}\right];
S_{1}=\intd\tau\left(\begin{array}{c}{p_{1}\cdot\dot{x}_{1}-p_{1}\cdot\lambda_{2}\,A_{1}}\\{+\bar{\theta}_{1}\gamma^{\mu\nu}\dot{\theta}_{1}\,p_{1\mu}\lambda_{2\nu}-\frac12e_{1}p_{1}^{2}}\\{+\bar{\theta}_{1}\gamma^{11,\nu}\dot{\theta}_{1}\,m_{1}\lambda_{2\nu}-\frac12e_{1}m_{1}^{2}}\\\end{array}\right).
{\calL}_{1/m^{2}}^{(3+1)spin}=\frac{\alpha}{720\pim^{2}}\left[18F^{\nu\lambda}F_{\nu\lambda,\mu}^{~~~~\mu}+7F^{\nu\lambda,\mu}F_{\nu\lambda,\mu}-2F_{~~~\lambda}^{\nu\lambda,}F_{\nu\mu,}^{~~~\mu}\right],
f*g=f\cdotg+\sum_{k=1}^{\infty}\left(-\frac{{i\hbar}}{2}\right)^{k}\frac{1}{k!}r^{i_{1}j_{1}}\ldotsr^{i_{k}j_{k}}(X_{i_{1}}\ldotsX_{i_{k}}f)(X_{j_{1}}\ldotsX_{j_{k}}g)\,,
S^{I}(\tau)\;=\;\int_{-\infty}^{\infty}\Bigl[\;-{\frac{1}{2}}\;\varepsilon\;{\frac{x^{2}}{\tau^{2}}}\,\left({\partial_{x}\phi}\right)^{2}\;+\;{\frac{\lambda}{4}}{\frac{1}{1+{\frac{x^{2}}{\tau^{2}}}\;(1-\varepsilon)}}\;\phi^{4}\;\Bigr]\;dx.
M_{L}=-{\nabla}^{2}-\frac{24\lambda^{2}}{[1+\lambda^{2}(x-x_{0})^{2}]^{2}}\,,\,\,M_{T}=-{\nabla}^{2}-\frac{8(1-\delta)\lambda^{2}}{[1+\lambda^{2}(x-x_{0})^{2}]^{2}}
\frac{\partialF}{\partialw_{n}^{j}}=-\frac{1}{2\pii}\oint_{c}\frac{\partialH^{\prime}}{\partial\alpha^{j}}\zeta^{n}\frac{d\zeta}{\zeta^{2}}-\frac{1}{2\pii}\oint_{0}\frac{\partialH}{\partial\alpha^{j}}\zeta^{n}\frac{d\zeta}{\zeta^{2}}
\begin{aligned}{S_{e}}&{{}=\frac{1}{2\kappa^{2}}\int{d^{4}x\sqrt{-g}R},}\\{S_{v}}&{{}=\frac{1}{2\kappa^{2}}\int{d^{4}x\sqrt{-g}G_{\mu\nu}\pi^{\mu\nu}},}\\\end{aligned}
\begin{array}{ccccc}{{\bfs\}{\calL\}{\bft}}&{\Longleftrightarrow}&{\exists{\bfu,v\inS,}}&{{\bfu*s=t,}}&{{\bfv*t=s,}}\\{{\bfs\}{\calR\}{\bft}}&{\Longleftrightarrow}&{\exists{\bfu,v\inS,}}&{{\bfs*u=t,}}&{{\bft*v=s.}}\\\end{array}
(\hat{Y}_{lm})_{ss^{\prime}}=<\frac{N}{2},s|\hat{Y}_{lm}|\frac{N}{2},s^{\prime}>=(-1)^{\frac{N}{2}-s}\left(\begin{array}{ccc}{\frac{N}{2}}&{l}&{\frac{N}{2}}\\{-s}&{m}&{s^{\prime}}\\\end{array}\right)\sqrt{(2l+1)(N+1)}
E_{i\mathrm{~}\mathrm{quasilocal}}=l_{p}^{2}\left(E_{i+}-E_{i-}\right)=l_{p}^{2}\left(r\left[1-\left|r,_{y}\right|\right]\right)_{y=y_{i+}}-l_{p}^{2}\left(r\left[1-\left|r,_{y}\right|\right]\right)_{y=y_{i-}}.
\Sigma_{a}=\left\{\left(\begin{matrix}{0}&{I}\\{-I}&{0}\\\end{matrix}\right),\quad\left(\begin{matrix}{0}&{-\sigma_{1}}\\{-\sigma_{1}}&{0}\\\end{matrix}\right),\quad\left(\begin{matrix}{\sigma_{1}}&{0}\\{0}&{-\sigma_{1}}\\\end{matrix}\right)\right\}
{\binom{x}{t}}\sim{\binom{x}{t}}+{\binom{\sqrt{{\frac{R^{2}}{2}}+R_{s}^{2}}}{-{\frac{R}{\sqrt{2}}}}}\approx{\binom{x}{t}}+{\binom{{\frac{R}{\sqrt{2}}}+{\frac{R_{s}^{2}}{\sqrt{2}R}}}{-{\frac{R}{\sqrt{2}}}}}
\left[x_{e}^{\mu},p_{e^{\prime}}^{\nu}\right]_{*}=i\theta\eta^{\mu\nu}\delta_{ee^{\prime}},\;\;\left[x_{e}^{\mu},x_{e^{\prime}}^{\nu}\right]_{*}=\left[p_{e}^{\mu},p_{e^{\prime}}^{\nu}\right]_{*}=0.
=-2\pib_{0}\int_{\Sigma}+\beta^{-1}\int_{\Sigma}\left[\gamma_{1}+\gamma_{2}R_{\Sigma}+(\gamma_{3}+\gamma_{4}\beta^{-2})R_{\mu\nu}n_{i}^{\mu}n_{i}^{\nu}+(\gamma_{5}+\gamma_{6}\beta^{-2})R_{\mu\nu\lambda\rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}\right]~~~
\begin{aligned}{\theta_{n}^{1}}&{{}=\left[n-\omega_{n}+i{\mu}\OmegaI\right]\theta_{n}\,,}\\{\tilde{\theta}_{n}^{1}}&{{}=\left[n+\omega_{n}-i{\mu}\OmegaI\right]\theta_{n}\,,}\\{\theta_{n}^{2}}&{{}=\left[n+\omega_{n}+i{\mu}\OmegaI\right]\Omega\theta_{n}\,,}\\{\tilde{\theta}_{n}^{2}}&{{}=\left[n-\omega_{n}-i{\mu}\OmegaI\right]\Omega\theta_{n}\,,}\\\end{aligned}
\left({\frac{\dot{R}}{R}}\right)^{2}={\frac{-1}{R^{2}}}+\left({\frac{8\piG}{3}}\right)\rho+\ell^{2}\left({\frac{4\piG}{3}}\rho\right)^{2}+{\frac{\alpha^{2}}{R^{4}}},
\frac{\dot{a}^{2}}{a^{2}}+\frac{k}{a^{2}}=\frac{\kappa^{2}}{3}(\rho_{\Phi}+e^{3\Phi}\rho_{T})=\frac{1}{3}\dot{\Phi}^{2}+\frac{\kappa^{2}}{3}T_{3}\;e^{3\Phi}V(T)\frac{1}{\sqrt{1-e^{-2\Phi}\dot{T}^{2}}},
\left(\Box_{x}+\frac{e^{2}}{\pi}\right)\varepsilon^{\mu\nu}\partial_{\mu}^{x}\frac{\deltaW}{\deltaJ^{\nu}(x)}-\varepsilon^{\mu\nu}\partial_{\nu}^{x}J_{\mu}(x)-ie\overline{{\eta}}_{a}(x)\gamma_{5}^{ab}\frac{\deltaW}{\delta\overline{{\eta}}_{b}(x)}-ie\eta_{a}(x)\gamma_{5}^{ba}\frac{\deltaW}{\delta\eta_{b}(x)}=0\;.
U_{(0,\,r-1)}\equiv\frac{1}{2}(Q_{0})_{\alpha_{0}\beta_{0}}(Q_{1})_{\alpha_{1}\beta_{1}}(Q_{2})_{\alpha_{2}\beta_{2}}\cdots(Q_{r-1})_{\alpha_{r-1}\beta_{r-1}}\epsilon^{\beta_{0}\alpha_{1}}\epsilon^{\beta_{1}\alpha_{2}}\cdots\epsilon^{\beta_{r-1}\alpha_{0}},
\phi_{left}(y)=\frac{1}{2}\left(\frac{\phi_{0}(y)}{\phi_{0}(0)}+\frac{\phi_{light}(y)}{\phi_{light}(0)}\right)\,\quad\phi_{right}(y)=\frac{1}{2}\left(\frac{\phi_{0}(y)}{\phi_{0}(0)}-\frac{\phi_{light}(y)}{\phi_{light}(0)}\right)\.
\left(L_{n}^{(1)}-\tilde{L}_{-n}^{(2)}\right)\vert{\calB}\gg_{\mathrm{transmit}}\=\\left(L_{n}^{(2)}-\tilde{L}_{-n}^{(1)}\right)\vert{\calB}\gg_{\mathrm{transmit}}\=\0\.
\begin{array}{rcl}{({\bf1})_{abcd}}&{=}&{S_{ac}S_{bd}}\\{(Q-1)(P_{0})_{abcd}}&{=}&{S_{ab}S_{cd}}\\{\mathrm{and\;\;}(Q-2)(P_{1})_{abcd}}&{=}&{Q\sum_{\alpha}S_{a\alpha}S_{b\alpha}S_{\alphac}S_{\alphad}-S_{ab}S_{cd}\;.}\\\end{array}
V_{nm}^{>}(B)=\frac{-1}{2\pi^{2}l^{3}}\int_{0}^{1}d\eta\int_{0}^{1}du\,\frac{u^{4(n-1)}\left[F(u;r)\right]^{n-2m}\left[G(u;r)\right]^{m}}{u^{2m}\left[(r+\frac{\pi}{2}u)^{2}+\bar{c}^{2}u^{2}\right]^{n}}.
e^{iS_{\mathrm{IF}}[\phi,\phi^{\prime};t_{f}]}=\intd\varphi_{f}d\varphi_{i}d\varphi_{i}^{\prime}\int_{\varphi_{i}}^{\varphi_{f}}{\calD}\varphi\int_{\varphi_{i}^{\prime}}^{\varphi_{f}}{\calD}\varphi^{\prime}\;e^{i(S[\varphi]+S_{\mathrm{int}}[\phi,\varphi]-S[\varphi^{\prime}]-S_{\mathrm{int}}[\phi^{\prime},\varphi^{\prime}])}\rho_{\mathrm{h}}(\varphi_{i},\varphi_{i}^{\prime};t_{i})
\Psi^{\prime}=\Psi-\frac{\delta\Psi}{\delta\Phi^{m}}\left(\frac{\deltaF}{\delta\Phi_{m}^{*}}\right)_{\Phi^{*}=\delta\Psi/\delta\Phi+K}+(F)_{\Phi^{*}=\delta\Psi/\delta\Phi+K}\;.
\hat{h}_{\mathrm{red}}=\sum_{\vec{b}}\left\{-\frac{g^{2}LM}{4\sqrt2}\frac{1}{J}\frac{\delta}{\delta\varphi_{\vec{b}}}J\frac{\delta}{\delta\varphi_{\vec{b}}}+\left(\frac{4\sqrt2}{g^{2}LM}\right)\sum_{\vec{\varepsilon}}(\varphi_{\vec{b}}-\varphi_{\vec{b}+\vec{\varepsilon}})^{2}\right\}\,.
\deltax(\tau)=\{q_{1}(\tau),G\}\left(q_{1}=x,q_{2}=\dot{x},q_{3}=\ddot{x},p_{3}=-\frac{\partialL}{\partial{\stackrel{\ldots}{x}}}\right),~~~\delta{\dot{x}}(\tau)=\frac{d}{d\tau}\deltax(\tau).
\Big(\partial_{\sigma}\partial^{\sigma}H_{\mu\nu}+\partial_{\mu}\partial_{\nu}H-\partial_{\mu}\partial^{\sigma}H_{\sigma\nu}-\partial_{\nu}\partial^{\sigma}H_{\sigma\mu}-\eta_{\mu\nu}\left[\partial_{\sigma}\partial^{\sigma}H-\partial^{\sigma}\partial^{\rho}H_{\sigma\rho}\right]+{\widehatM}_{P}^{3-D}T_{\mu\nu}\Big)\delta(z)=0~.
<G(p,\tau)>=\frac{-i}{(m^{2}-p^{2})}\sum_{n=0}^{\infty}\frac{(-e^{2}A)^{n}}{n!\epsilon^{n}}\Gamma(1+n\epsilon)(\frac{2i\mu^{2}}{m^{2}-p^{2}})^{n\epsilon}
\left\{F,G\right\}_{PB}=\int_{S}d^{p}\sigma\,\sum_{i}\left[\left(-1\right)^{\phi^{i}}F\frac{\overleftarrow{\delta}}{\delta\phi^{i}\left(\sigma\right)}\frac{\overrightarrow{\delta}}{\delta\Lambda_{i}\left(\sigma\right)}G-F\frac{\overleftarrow{\delta}}{\delta\Lambda_{i}\left(\sigma\right)}\frac{\overrightarrow{\delta}}{\delta\phi^{i}\left(\sigma\right)}G\right]\quad;
\begin{array}{l}{\rho^{\mu}\partial_{\mu}x^{AA^{\prime}}o_{A}=0,}\\{\partial_{\mu}x^{AA^{\prime}}o_{A}\bar{o}_{A^{\prime}}=0,}\\{\partial_{\mu}(\rho^{\mu}o_{A}\bar{o}_{A^{\prime}})+3\kappa\varepsilon^{\mu_{1}\mu_{2}}\partial_{\mu_{1}}x^{A_{1}{A^{\prime}}_{1}}\partial_{\mu_{2}}x^{A_{2}{A^{\prime}}_{2}}\partial_{[AA^{\prime}}B_{A_{1}A_{2}{A^{\prime}}_{1}{A^{\prime}}_{2}]}=0,}\\\end{array}
F_{B}={\frac{1}{3}}H_{ijl}^{(1)}H_{ijl}^{(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})\,
{F_{(0)\mu\nu}^{a-}(x^{\mu};x_{0}^{\mu},\rho_{0})=-\frac{4}{g_{_{YM}}}{\frac{\eta_{\mu\nu}^{a}\rho_{0}^{2}}{\left(\rho_{0}^{2}+(x-x_{0})^{2}\right)^{2}}},}
\begin{array}{ll}{(A):}&{n>0,}\\{(B):}&{(i)\k>0,\\mathrm{or}\(ii)\k=0,\l>0,}\\{(C):}&{(i)\k>0,\\mathrm{or}\(ii)\k=0,\l>0,\\mathrm{or}\(iii)\k=l=0,\b<0\,,}\\\end{array}
\begin{array}{rcl}{C}&{=}&{C^{(0)}+C^{(1)}+C^{(2)}+\ldots}\\{}&{}&{}\\{G}&{=}&{G^{(0)}+G^{(1)}+G^{(2)}+\ldots}\\{}&{}&{}\\{\Lambda^{(\cdot)}}&{=}&{\Lambda^{(0)}+\Lambda^{(1)}+\ldots\,,}\\\end{array}
\{\varepsilon^{\mu\nu}\}\rightarrow\{\varepsilon^{\mu\nu}\}=W(p,q)\{\varepsilon^{\mu\nu}\}W^{T}(p,q)=\{\varepsilon^{\mu\nu}\}+\varepsilon^{12}\left(\begin{array}{cccc}{0}&{-q}&{p}&{0}\\{q}&{0}&{0}&{q}\\{-p}&{0}&{0}&{-p}\\{0}&{-q}&{p}&{0}\\\end{array}\right)
\hat{N}=\frac{1}{2}(\hat{\eta}^{a}\hat{\zeta}_{a}-\hat{\zeta}_{a}\hat{\eta}^{a})+\frac{1}{2}(\hat{\eta}^{\timesa}\hat{\zeta}^{\times}{\!}_{a}-\hat{\zeta}^{\times}{\!}_{a}\hat{\eta}^{\timesa})
S_{cm}=-{\frac{M_{0}}{2}}\int_{0}^{T}d\tau\,\left[\,-{\frac{\dot{x}^{\mu}\,\dot{x}_{\mu}}{e(\tau)}}+e(\tau)\right]\equiv\int_{0}^{T}d\tau\,L_{cm}\left(\,\dot{x}^{\mu}\;e(\tau)\,\right)\,
\begin{array}{l}{\ell_{3}^{+}=\ell^{+},\ell^{-}=0}\\{\hat{h}=\frac{k^{-}}{k}\ell^{+}=\frac{\hat{q}}{2}}\\{\hat{h}=\hat{h}^{\prime},\hat{q}=\hat{q}^{\prime}}\\\end{array}
S=-\frac{1}{\kappa_{D}}\intd^{D}y\,\sqrt{|G|}e^{-2\Phi}\,\left[\hat{R}-4(D\Phi)^{2}+4D^{2}\Phi-\frac{2\Lambda}{\alpha^{\prime}}+\frac{\rho}{\alpha^{\prime}}e^{2\Phi}\right].\,
d\tilde{\tau}^{2}\simeq\left(\eta_{\mu\nu}-\frac{q^{2}}{A^{2}m^{2}}F_{\mu\lambda}F_{\,\,\,\nu}^{\lambda}\right)dx^{\mu}dx^{\nu}\equiv\tilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}\,{.}
z(1-z)\frac{\partial^{2}}{\partialz^{2}}U_{\epsilon}(z,\bar{z})-\left[c_{\epsilon}-(a_{\epsilon}+b_{\epsilon}+1)z\right]\frac{\partial}{\partialz}U_{\epsilon}(z,\bar{z})-a_{\epsilon}b_{\epsilon}U_{\epsilon}(z,\bar{z})=0,
{\calI}_{\mathrm{1-loop}}^{I}=\frac{V^{(8)}T_{2}}{2^{8}}t_{8}F_{+}^{4-l}F_{-}^{l}\int\frac{dt}{t^{6}}\sum_{j,p\neq0}e^{\frac{-\piT_{2}}{tU_{2}}|j-pU|^{2}}\frac{T_{2}^{2}}{U_{2}^{2}}(j-pU)^{4-l}(j-p\bar{U})^{l}\frac{1}{2}\Big(N^{2}-N+2^{4}\Big)
\left(\varphi_{\alpha},\varphi_{\alpha^{\prime}}\right)=-i\intd{\mathbfx}\int_{\xi_{1}}^{\xi_{2}}\frac{d\xi}{\xi}\varphi_{\alpha}\stackrel{\leftrightarrow}{\partial}_{\tau}\varphi_{\alpha^{\prime}}^{*}=\delta_{\alpha\alpha^{\prime}}.
\frac{\partial}{\partial{t}}{U}_{k}(\rho)=\frac{1}{2}\int\frac{{d}^{d}{q}}{(2\pi)^{d}}\frac{\partial}{\partial{t}}{R}_{k}({q})\left(\frac{{N}-1}{{M}_{0}}+\frac{1}{{M}_{1}}\right)
\Big(\tau^{2}\Big)^{\frac{1}{N}\Big[\big(\sum_{a}n_{a}\big)\big(\sum_{a}m_{a}^{\prime}\big)+\big(\sum_{a}m_{a}\big)\big(\sum_{a}m_{a}^{\prime}\big)-\big(\sum_{a}n_{a}\big)\big(\sum_{a}n_{a}^{\prime}\big)-\big(\sum_{a}n_{a}\big)\big(\sum_{a}m_{a}^{\prime}\big)\Big]}\;.
Q(\Psi)_{ij}=Q^{inst}\Psi_{ij}+c_{0}\frac{1}{4\pi^{2}\alpha^{\prime}}g_{IJ}[X^{I},[X^{J},\Psi]]_{ij}+\frac{\sqrt{2}}{2\pi\sqrt{\alpha^{\prime}}}\sum_{n=-\infty,n\neq0}^{\infty}c_{n}g_{IJ}\alpha_{-n}^{I}[X^{J},\Psi]_{ij}\.
{\calS}(x,0)=\int\!{\displaystyle\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}}\,\,{\displaystyle\frac{\mathrm{e}^{\mathrm{i}kx}}{k^{2}+m^{2}}}\quad=\,\frac{m}{(2\pi)^{2}|{\bfx}|}\\mathrm{K}_{1}(m|{\bfx}|)\,.
\cdots\stackrel{a(l-1)^{\dagger}}{\longrightarrow}{\calE}(l-1)\stackrel{a(l)^{\dagger}}{\longrightarrow}{\calE}(l)\stackrel{a(l+1)^{\dagger}}{\longrightarrow}{\calE}(l+1)\stackrel{a(l+2)^{\dagger}}{\longrightarrow}\cdots
d(e^{-2\phi_{0}}*dA_{3})+\frac{\lambda^{2}}{4}e^{2\phi_{0}}g{\vertv\vert}^{4}*(gA_{3}-g^{\prime}A)+\frac{g^{2}}{4}{\vertv\vert}^{2}*A_{3}-\frac{gg^{\prime}}{4}{\vertv\vert}^{2}*A=0
\langleH|S_{\mu}^{\\nu}|H\rangle\to-\frac{k_{2}\alpha}{96M^{4}}\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{2}&{0}\\{0}&{0}&{0}&{2}\\\end{matrix}\right)
\vertp,\sigma,q,\lambda\colonf(p)\rangle=exp\biggl[-e\int\sum_{\lambda=1,2}{\frac{dk^{+}}{\sqrt{2k^{+}}}}\int{\frac{d^{2}k_{\perp}}{(2\pi)^{\frac{3}{2}}}}\bigl[f(k,\lambda\colonp)a^{\dagger}(k,\lambda)-f^{*}(k,\lambda\colonp)a(k,\lambda)\bigr]\biggr]\vertp,\sigma,q,\lambda\rangle\;.
g_{kl}g_{pq}\left(\frac{1}{g+F}\right)^{ij}\partial_{i}\left(\frac{1}{g+F}\right)^{kp}\partial_{i}\left(\frac{1}{g+F}\right)^{lq}=-(h_{S})^{ij}\mathrm{Tr}\left(h_{S}(\partial_{i}F)h_{S}(\partial_{j}F)\right),
I_{1}=(\Pi_{23})^{2}\intd^{4}x_{4,5}d^{4}u_{4,5}d^{4}\theta_{4,5}^{+}\;\Pi_{14}\Pi_{42}\Pi_{35}\Pi_{51}\;(D_{4}^{+})^{4}\left({\frac{\delta^{8}(\theta_{4}-\theta_{5})}{x_{45}^{2}}}\right)\delta^{(-2,2)}(u_{4},u_{5})\.
F^{II}=-{\frac{1}{6}}C_{abc}t^{a}t^{b}t^{c}-{\frac{\chi\zeta(3)}{2(2\pi)^{3}}}+{\frac{1}{(2\pi)^{3}}}\sum_{d_{1},...,d_{h}}n_{d_{1},...,d_{h}}^{r}Li_{3}({\bfe}[i\sum_{a}d_{a}t^{a}]),
{\mathrm{reg}}\,q(a,r)=-\frac{1}{8\pi^{2}a^{4}}\sum_{l=1}^{\infty}\int_{0}^{\infty}{\chi_{\mu}(z/a)F_{l}^{(q)}(z,x)dz},\quadr<a,\quadq=\varepsilon,p,p_{\perp}.
({\mathcalA}_{2}^{R})_{m}(x_{\perp})=\int\,dx_{-}(\partial^{i}R_{i}^{a}(x_{-},x_{\perp})R_{+}^{b}(x_{-},x_{\perp})-R_{+}^{a}(x_{-},x_{\perp})\partial^{i}R_{i}^{b}(x_{-},x_{\perp}))C_{mba},
\Theta_{\(j_{1}\cdotsj_{k-1})}^{(i_{1}\cdotsi_{k-1}klm)\J=k}=Tr\Big[\underbrace{\Theta_{\j_{1}}^{i_{1}}\otimes_{h.w.}\cdots\otimes_{h.w.}\Theta_{\j_{k-1}}^{i_{k-1}}}_{k-1\\mathrm{objects}}\otimes_{h.w.}\Theta^{(klm)}\Big]\,.
E=i\mu\lambda^{\frac{1}{2}}q^{Q}(q-q^{-1})\left[\kappa\sum_{i=1}^{Q-1}z_{i}-i\frac{\lambda^{\frac{1}{2}}-\tau\lambda^{-\frac{1}{2}}}{q^{1/2}-q^{-1/2}}\right].
\overline{{\tilde{F}}}_{\sigma}^{(n)}(\zeta)^{kk_{1}\cdotsk_{2n}}=\prod_{a\inA^{\prime\prime}}\oint_{C_{a}}\frac{dw_{a}}{2\pi\sqrt{-1}w_{a}}\tilde{\Psi}_{\sigma}^{(n)}(w|\zeta)Q^{(n)}(w|\zeta)^{kk_{1}\cdotsk_{2n}}.
i\Pi_{\mu\nu(non-planar)}^{(abc)}=-e^{2}\mu^{4-d}\int_{0}^{1}dz\int\frac{d^{d}k}{(2\pi)^{d}}(cosp\wedgek)\frac{\textit{N}_{\mu\nu}^{(abc)}}{[k^{2}-M^{2}]^{2}}
F_{-\gamma}(p)\,F_{\gamma}(p)=\left(\frac{\Gamma(1-\alpha-i\frac{\gammap}{2\pi})\,\Gamma(1-\alpha+i\frac{\gammap}{2\pi})}{\Gamma(1+\alpha-i\frac{\gammap}{2\pi})\,\Gamma(1+\alpha+i\frac{\gammap}{2\pi})}\right)^{\frac{1}{2}}\,.
\left[{\frac{d^{2}}{dr^{2}}}+\left({\frac{d^{\prime}}{d}}+{\frac{3}{r}}\right){\frac{d}{dr}}\right]\delta\tilde{\phi}+{\frac{\omega^{2}f}{d^{2}}}\delta\tilde{\phi}-{\frac{8R^{4}}{r^{2}(r^{2}+R^{2})^{2}d}}\delta\tilde{\phi}=0.
\begin{array}{ll}{\tilde{T}_{(0)tt}=U\delta(x)\delta(y)+\frac{(J^{2}+\nuS^{2})}{4\pi}\nabla^{2}\left(ln\frac{r}{r_{0}}\right)^{2},}\\{\tilde{T}_{(0)zz}=-T\delta(x)\delta(y)+\frac{(J^{2}-\nuS^{2})}{4\pi}\nabla^{2}\left(ln\frac{r}{r_{0}}\right)^{2},}\\{\tilde{T}_{(0)ij}=(J^{2}+\nuS^{2})\delta_{ij}\delta(x)\delta(y)-\frac{(J^{2}+\nuS^{2})}{2\pi}\partial_{i}\partial_{j}ln(r/r_{0}),}\\\end{array}
\zeta_{0}^{(p)}(\nu)={\frac{1}{\pi\Gamma(\nu)}}\int_{0}^{\infty}dx\int_{0}^{1}dt~t^{\nu-1}e^{-tx^{2}}{\frac{1}{(4\pit)^{(D-1)/2}}}\sum_{n=0}^{\infty}\breve{a}_{n}(x)t^{n}.
{\calL}:=\frac{(2\pi)^{4}}{N^{3}}Tr\\{\frac{1}{3}\theta[\partial_{x}\theta,\partial_{y}\theta]-\frac{1}{2}\\big((\partial_{x}\theta)(\partial_{\tilde{x}}\theta)+(\partial_{y}\theta)(\partial_{\tilde{y}}\theta)\big)\}
\begin{array}{ccccccc}{g_{ij^{\star}}}&{=}&{-\mathrm{i}\langle\,{\hat{U}}_{i}\,\vert\,{\hat{\bar{U}}}_{j^{\star}}\,\rangle}&{;}&{C_{ijk}}&{=}&{\langle\,\partial_{i}{\hat{U}}_{j}\,\vert\,{\hat{U}}_{k}\,\rangle}\\\end{array}
A(1,2,3,4)={\frac{1}{256}}e^{E_{3}^{(4)}}\theta_{43}^{\alpha}\bar{\theta}_{43}^{2}\left(\theta_{23,\alpha}\bar{\theta}_{23,\dot{\alpha}}+{\frac{1}{2}}(\sigma^{\mu}\bar{\theta}_{23})_{\alpha}p_{2,\mu}\bar{\theta}_{23,\dot{\alpha}}\theta_{23}^{2}\right)\bar{\theta}_{13}^{\dot{\alpha}}\theta_{13}^{2}.
\sum_{b=1}^{N}\Big(a_{\mu}^{(b)},j_{\mu}^{(b)}\Big)=\sum_{I=1}^{N}\sum_{b=1}^{N}\Big(U_{Ib}A_{\mu}^{(I)},j_{\mu}^{(b)}\Big):=\sum_{I=1}^{N}\Big(A_{\mu}^{(I)},J_{\mu}^{(I)}\Big)\;,
S_{I}\equivS_{I}[\phi^{(0)}],\qquadS_{I;p}^{\prime}\equiv\frac{\deltaS_{I}}{\delta\phi_{p}}[\phi^{(0)}],\qquadS_{I;-p,p}^{\prime\prime}\equiv\frac{\deltaS_{I}}{\delta\phi_{-p}\delta\phi_{p}}[\phi^{(0)}].
\begin{array}{lcr}{{q^{\prime}}_{i}^{n-1}=q_{i}^{n-1}+q_{i}^{n}}\\{{q^{\prime}}_{n-1}^{n-1}{V^{\prime}}_{n-1}=(q_{n-1}^{n-1}V_{n-1}+q_{n}^{n}V_{n})+(q_{n-1}^{n}V_{n-1}+q_{n}^{n-1}V_{n})}\\{{q^{\prime}}_{n-1}^{n-2}{V}_{n-1}^{\prime}=q_{n-1}^{n-2}V_{n-1}+q_{n}^{n-2}V_{n}.}\\\end{array}
\sigma_{E}\,\longleftrightarrow\,(-1)^{\sumJ+j}\:\:\left\{\begin{array}{ccc}{J_{1}}&{j}&{J_{2}}\\{J_{3}}&{J_{4}}&{J_{5}}\\\end{array}\right\}\,\doteq\,(-1)^{\sumJ+j}\:\{6j\}(\sigma_{E}),
\left(\begin{array}{c}{\theta_{0}}\\{\theta_{1}}\\\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{\pm1}&{\pm1}\\{1}&{-1}\\\end{array}\right)\left(\begin{array}{c}{{\theta_{0}}^{\prime}}\\{{\theta_{1}}^{\prime}}\\\end{array}\right)
D_{\frac{1}{2}~\frac{1}{2}}^{\frac{1}{2}}=\frac{\partial}{\partial\xi_{1}^{*}}~(\xi_{1}^{*}\psi_{1}+\xi_{2}^{*}\psi_{2})=\psi_{1}=e^{i(\alpha+\gamma)/2}~cos(\frac{\beta}{2})
\delta\!\langle\Upsilon_{0}(z_{0})\cdots\Upsilon_{n}(z_{n})\rangle_{\Sigma}=\frac{1}{\pi}\int_{\Sigma}\langle\Psi(z)\Upsilon_{0}(z_{0})\cdots\Upsilon_{n}(z_{n})\rangle_{\Sigma}\,d^{2}\!z.
I\left(\begin{array}{c|c}{a_{1}...a_{A}}&{b_{1}...b_{B}}\\{}&{}\\\hline{}&{}\\{c_{1}...c_{C}}&{d_{1}...d_{D}}\\\end{array}\right)(z)={\frac{1}{2\pii}}\int_{\gamma}ds{\frac{\prod_{i=1}^{A}\Gamma(a_{i}-s)\prod_{j=1}^{B}\Gamma(b_{j}+s)}{\prod_{k=1}^{C}\Gamma(c_{k}-s)\prod_{l=1}^{D}\Gamma(d_{l}+s)}}z^{s},
W_{b}=\int_{r_{-}(\tau=-\frac{\beta}{2})}^{r_{-}(\tau=\frac{\beta}{2})}p_{b}dr=\frac{2\sigmaS_{1}r_{+}}{3r_{0}}\left[(r_{+}^{2}+r_{-}^{2}){\calE}(k)-2r_{-}^{2}{\calK}(k)\right]
L=\{\left(\begin{matrix}{-i}&{0}\\{0}&{-i}\\\end{matrix}\right),\left(\begin{matrix}{0}&{i}\\{i}&{0}\\\end{matrix}\right),\left(\begin{matrix}{0}&{1}\\{-1}&{0}\\\end{matrix}\right),\left(\begin{matrix}{i}&{0}\\{0}&{-i}\\\end{matrix}\right)\}
\int_{0}^{1}dt\delta_{\vec{u}}(t)<\mathrm{Tr}(H_{12}(\gamma)H_{34}(\gamma))>={\frac{i\pi}{k}}\mathrm{Tr}(H_{12}(\gamma))\mathrm{Tr}(H_{34}(\gamma))+{\frac{i\pi}{k}}\mathrm{Tr}(H_{12}(\gamma)H_{34}^{-1}(\gamma)).
\zeta_{A}(s/2)=\frac{1}{2\sqrt{\pi}}\,\frac{\Gamma(s-1/2)}{\Gamma(s)}\,\sum_{n=-\infty}^{\infty}\frac{y_{-}}{(y_{+}y_{-})^{s}}\,F(s/2,1/2;s;1-\eta),\\\\\eta\equiv\frac{y_{-}^{2}}{y_{+}^{2}}.
=\;2^{N}\prod_{a=1}^{N}\sum_{n_{a}=0}^{\infty}\frac{(\beta^{(a)})^{2n_{a}}}{(2n_{a})!}\prod_{b=1}^{N}\sum_{q_{b}=0}^{2n_{b}}{\binom{2n_{b}}{q_{b}}}E^{\mu}(\{q,n\})\;,
Z_{3}^{A}(q^{2})=iC_{A}\frac{g^{2}\nu^{4-n}}{2^{n+1}\pi^{n/2}}\;\frac{3n-2}{n-1}\;\frac{\Gamma^{2}(n/2-1)\Gamma(2-n/2)}{\Gamma(n-2)}\;(q^{2})^{(n-4)/2},
F_{k}(\tau,q,\dot{q},\ddot{q})=0\qquad\dot{q}^{j}=\frac{dq^{j}}{d\tau}\qquad\ddot{q}^{j}=\frac{d^{2}q^{j}}{d\tau^{2}}\qquadj,k=1,\cdots,n\.
\Phi^{\prime}=\frac{(2\varepsilon(\hat{\bfp}))^{-1/2}}{(\varepsilon(\hat{\bfp})+mc^{2})^{\frac{1}{2}}}\left((\varepsilon(\hat{\bfp})+mc^{2})\Phi(\varepsilon(\hat{\bfp})+mc^{2})+c^{2}\vec{\sigma}{\bf\hat{p}}\Phi\vec{\sigma}{\bf\hat{p}}\right)\frac{(2\varepsilon(\hat{\bfp}))^{-1/2}}{(\varepsilon(\hat{\bfp})+mc^{2})^{\frac{1}{2}}}\,;
\vec{S}=\left[\begin{array}{c}{\mathrm{Re}\left({\mathcal{W}}\right)}\\{\mathrm{Im}\left({\mathcal{W}}\right)}\\\end{array}\right],\qquad\vec{a}=-\left[\begin{array}{c}{\mathrm{Im}(\phi\partial_{x}\bar{\phi})}\\{\mathrm{Im}(\phi\partial_{t}\bar{\phi})}\\\end{array}\right],
\begin{array}{rcl}{\widetilde\phi_{\mathrm{fixed}}(u)}&{=}&{\widehat\phi_{\mathrm{free}}(u/\sqrt{2})\,}\\{\widetilde\phi_{\mathrm{free}}(x)}&{=}&{\widehat\phi_{\mathrm{fixed}}(\sqrt{2}x)\.}\\\end{array}
{\calS}_{\pm}=\intd^{4}k\left[\pm\omega({\bfk})\dot{\varphi}_{a}^{(\pm)}\epsilon^{ab}\varphi_{b}^{(\pm)}\,-\,\omega^{2}({\bfk})\varphi_{a}^{(\pm)}\varphi_{a}^{(\pm)}\,-\,2g{\calA}_{\pm}^{a}\varphi_{a}^{(\pm)}-\frac{g^{2}}{2\,\omega^{2}({\bfk})}{\calA}_{\pm}^{a}\right]+\frac{a\,g^{2}}2{\calS}_{reg},
\frac{1}{\Omega^{2}m_{*}^{2}}V_{\pm}=x^{2}\left(\frac{\mu}{m_{*}^{2}}-\frac{v_{\pm}}{2}(1-v_{\pm})^{-1}+\frac{1}{x^{2}}(1-v_{\pm})^{2}\right)^{2}.
\frac{Q_{t-1}(u_{j}^{t}+2)Q_{t}(u_{j}^{t}-2)Q_{t+1}(u_{j}^{t})}{Q_{t-1}(u_{j}^{t})Q_{t}(u_{j}^{t}+2)Q_{t+1}(u_{j}^{t}-2)}=-1
\overline{{K}}_{de}^{abc}=P_{d}^{(2)i}P_{e}^{(2)j}\left(3C_{ijk}^{a}A_{0}^{b}A^{ck}-C_{ij}^{a}\eta^{(2)b}A_{0}^{c}\right)-P_{d}^{(2)[i}P_{e}^{jk]}C_{ijk}^{a}\eta^{(2)b}A_{0}^{c},
\left[\frac{d^{2}}{dt^{2}}+\vec{k}^{2}+m_{R}^{2}(t)+\frac{\lambda_{R}}{2}\left(\langle\Phi^{2}(t)\rangle-\langle\Phi^{2}(0)\rangle\right)\right]{\cal{U}}_{k}^{\pm}(t)=0
\frac{\int_{S}\sum_{n_{1}=1}^{\infty}\left|\sum_{\gamma\in\Gamma_{\infty}^{2}\backslash\Gamma_{1}^{2}}\nabla_{\tau}\phi_{n_{1}}^{0}(\gamma\circ\tau)\right|^{2}dV}{\int_{S}\sum_{n_{1}=1}^{\infty}\left|\sum_{\gamma\in\Gamma_{\infty}^{2}\backslash\Gamma_{1}^{2}}\phi_{n_{1}}^{0}(\gamma\circ\tau)\right|^{2}dV}<10\lambda.
\left(\begin{array}{ccc}{p^{2}g_{\mu\lambda}}&{-(\overline{\psi}\gamma_{\mu})(p\!\!\!/+m)}&{-(\gamma_{\mu}\psi)^{T}(p\!\!\!/+m)^{T}}\\{(\gamma_{\lambda}\psi)}&{p^{2}+m^{2}}&{0}\\{-(\overline{\psi}\gamma_{\lambda})^{T}}&{0}&{p^{2}+m^{2}}\\\end{array}\right)_{(\mu,\lambda)}\,.
\hat{\Gamma}_{\hat{\mu}}=\hat{\Gamma}_{\hat{\mu}\hat{\nu}\hat{\lambda}}S^{\hat{\nu}\hat{\lambda}}=\frac{i}{4}(C_{\hat{\mu}\hat{\nu}\hat{\lambda}}-C_{\hat{\mu}\hat{\lambda}\hat{\nu}}-C_{\hat{\nu}\hat{\lambda}\hat{\mu}})S^{\hat{\nu}\hat{\lambda}}