harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left(\begin{matrix}x=0 x=1 x=\infty \, \left( \frac12 \right) \begin{matrix}0 \theta^0_1 \theta^0_2 \end{matrix}\begin{matrix}0 0 \theta^1 \end{matrix}\overbrace{\begin{matrix}\sqrt{-t}\theta^\infty_1/2 -\sqrt{-t}\theta^\infty_1/2 0 \theta^\infty_2 \end{matrix}}\end{matrix}\right) ,
\tilde A(y,s):=A(R_0^{k_0}y,s) &\tilde f(y,s):=R_0^{(2k_0(1-\alpha)+k_0\alpha)}f(R_0^{k_0}y,s) &\tilde Q (y):=R_0^{2k_0(1-\alpha)}Q (r_0^{k_0}y).
{R^{\rm{N}}}= \min \left\{{R_{b \to s}^{\rm{N}},R_{b \to b}^{\rm{N}}}\right\}\triangleq W{\log_2}\left({1 + \frac{{{p_b}{{\left|{{{\widetilde h}_b}}\right|}^2}}}{{{p_s}{{\left|{{{\widetilde h}_b}}\right|}^2}+ W{N_0}}}}\right),
\int_{\mathbb{S}^{d-1}}e^{ir \omega \cdot x}d\omega = \sum_{k=0}^{\frac{d-3}{2}}c_k \frac{e^{i r |x|}}{( r|x|)^{\frac{d-1}{2}+k}}+ \sum_{k=0}^{\frac{d-3}{2}}c_k \frac{e^{- i r |x|}}{(- r|x|)^{\frac{d-1}{2}+k}}
Y(T) = \left\{\phi \in C\left([0,T], L^2(\mathbb R^N) \right): \|\phi \|_{L^{\infty}([0, T], L^2)}\leq 2 \|u_0\|_{L^2}, \|\phi\|_{L^{\frac{8}{\gamma_i}}([0,T], L^{\frac{4N}{2N-\gamma_i}})}\lesssim \|u_0\|_{L^2}, i=1,2\right\}
\mathbb{E}[f_{\mathbf X}(y)] = \mathbb{E}\left[\sum_{i=1}^N \mathbb I_{\{y\leq X_i\}}\right] & =\lfloor f(y) \rfloor + \mathbb{E}\left[\mathbb I_{\{y\leq X_{\lfloor f(y) \rfloor}\}}\right] =\lfloor f(y) \rfloor + (f(y) -\lfloor f(y) \rfloor )=f(y).
{1\over n}\log \theta_L(\bar H(n, \delta n)) = h(\delta/2) + o(1)\,,{1\over n}\log \theta_S(\bar H(n, \delta n)) = h(\delta/2) + o(1)\,,{1\over n}\log \theta_L(H(n,\delta n)) = 1-h(\delta/2) + o(1)\qquad\mbox{By Lovasz identity}
i \left( \begin{matrix}{{\langle U_{z}, \bar{U}_{\bar{z}}\rangle}}&{{\langle U_{z}, V \rangle}}\\{{\langle \bar{V}, \bar{U}_{\bar{z}}\rangle}}&{{\langle \bar{V}, V \rangle}}\\ \end{matrix}\right) = \left( \begin{matrix}{{( z + \bar{z})^{- 2}}}&{{\frac{a - 1}{a + 1}( z + \bar{z})^{- 1}}}\\{{\frac{a - 1}{a + 1}( z + \bar{z})^{- 1}}}&{{1}}\\ \end{matrix}\right)
u_n(\theta) = \sup \left\{\sum_{i=1}^n \frac{1}{x_i}: (x_i)_{i=1}^n \in U_n(\theta) \right\}& = \sup \left\{\sum_{i=1}^n \frac{1}{x_i}: (x_i)_{i=1}^n \in U_n(\theta) x_i < x_i^* i = 1,\ldots, n \right\}= \sup \left\{\sum_{i=1}^n \frac{1}{x_i}: (x_i)_{i=1}^n \in U_n^{(n)}(\theta) \right\}
Z(x,t):=\mathcal{F}_{\xi\rightarrow x}^{-1}(e^{-t|f(\xi)|_{p}^{\beta}})=\int\nolimits_{\mathbf{\mathbb{Q}}_{p}^{n}}\chi_{p}(-x,\xi )e^{-t|f(\xi )|_{p}^{\beta}}d^{n}\xi ,t>0,x\in \mathbf{\mathbb{Q}}_{p}^{n}.
\sum_{k=1}^{n/m}\binom{n}{km}^2\binom{n}{m}\cdot 8\cdot 2^k\beta^{pkm^2/2000}\le 8 \sum_{k=1}^{n/m}\big(\tfrac{n}{km}\big)^{3km}2^k\beta^{pkm^2/2000}\le 8\sum_{|X|=\log n}^{n}\exp\big(3|X|\log(en/|X|)+|X|/m+(\log\beta)\cdot C'|X|\log(en/|X|)/2000\big) \le \sum_{|X|=\log n}^{n}\exp\big(-|X|\log(en/|X|)\big)
\sum_{n=1}^{\infty}\mathbb{P}(\eta_1+\ldots+\eta_n\leqslant x)=\sum_{n=1}^{\infty}\mathbb{P}\left(e^{-t(\eta_1+\ldots+\eta_n)}\geqslant e^{-tx}\right)\leqslant e^{tx}\sum_{n=1}^{\infty}\left(\mathbb{E}e^{-t\eta}\right)^n<\infty,
K = *{\frac{( \kappa^{2})}{2}}={\frac{i}{2}}\left[ f_{\mu_{1}\bar{\nu}_{1}}\, f_{\mu_{2}\bar{\nu}_{2}}\right] \epsilon_{\hspace{0 . 2 5 i n}\alpha}^{\mu_{1}\mu_{2}}\, \epsilon_{\hspace{0 . 2 5 i n}\bar{\beta}}^{\bar{\nu}_{1}\bar{\nu}_{2}}\, \, d z^{\alpha}\wedge d \bar{z}^{\beta}
{\mathcal L}(x,\partial_x)=a_{ij}\frac{\partial^2}{\partial x_i\partial x_j}+b_k\frac{\partial}{\partial x_k}=\widetilde a_{ij}\frac{\partial^2}{\partial \tilde x_i\partial \tilde x_j}+\widetilde b_k\frac{\partial}{\partial \tilde x_k}=\widetilde{\mathcal L}(\tilde x,\partial_{\tilde x}).
\chi_{a , \ell}( z , \tau ) = \frac{\displaystyle \sum_{w \in \overline{W}}\mathrm{s g n}( w ) \Theta_{w ( \overline{{a}}+ \overline{{\rho}}) , g + \ell}( z , \tau )}{\displaystyle \sum_{w \in \overline{W}}\mathrm{s g n}( w ) \Theta_{w ( \overline{{\rho}}) , g}( z , \tau )}\: ,
\left( 1 -{\frac{r_{0}}{r}}\right){\frac{r}{r_{0}}}{\frac{d}{d r}}\left( r^{2}{\frac{d f_{\ell}}{d r}}\right) + \left( 2 \ell + \left( 2 \ell + 1 + i \;{\frac{k}{2 \pi T_{H}}}\right){\frac{r_{0}}{r}}\right){\frac{r^{2}}{r_{0}}}\;{\frac{d f_{\ell}}{d r}}+ \ell \left( - \ell + i \;{\frac{k}{2 \pi T_{H}}}\right) \; f_{\ell}= 0
\alpha^m c_A^{\frac{m}{2}}\|S_m\|_{L^1}&\le e^{|W_0|}&\quad\cdot\Bigg(1_{m=0}+\sum_{n=1}^N\frac{1}{n!}\prod_{l=1}^n\Bigg(\sum_{m_l=1}^Nc_A^{\frac{m_l}{2}}\|W_{m_l}\|_{L^1}\sum_{k_l=0}^{m_l}\left(\begin{array}{c}m_l k_l\end{array}\right)\alpha^{k_l}\Bigg)1_{\sum_{l=1}^nk_l=m}\Bigg).
S_{i\tau\varphi}(C) = \{a \in M(C) \mid \varphi_C(a) = \top_C\}&= \{a \in M(C) \mid (\chi_{S_\varphi})_C(a) = \top_C\}&= \{a \in M(C) \mid \{p : D \rightarrow C \mid p^\ast a \in S_\varphi(D)\}= \top_C\}& = \{a \in M(C) \mid p^\ast a \in S_{\varphi}(D), \ \p : D \rightarrow C\}.
\lim_{\varepsilon \rightarrow 0^{+}}| \langle \pi_s, \overleftarrow{i_{\varepsilon}^{u}}\rangle - \rho(s,u) | = \lim_{\varepsilon \rightarrow 0^{+}}| \langle \pi_s, \overrightarrow{i_{\varepsilon}^{u}}\rangle - \rho(s,u) | = 0,
\widetilde{S}= \left( \begin{array}{c c c}{{2 S_{a , b}}}&{{S_{a , f}}}&{{S_{a , f}}}\\{{S_{f , b}}}&{{\frac{1}{2}}}&{{- \frac{1}{2}}}\\{{S_{f , b}}}&{{- \frac{1}{2}}}&{{\frac{1}{2}}}\\ \end{array}\right) ,
\frac{1}{2\pi\sigma_{x}\sigma_{y}\sqrt{1-\rho^{2}}}\iint_{xy}exp[-\frac{1}{2(1-\rho^{2})}(\frac{x^{2}}{\sigma_{x}^{2}}+\frac{y^{2}}{\sigma_{y}^{2}}-\frac{2\rho xy}{\sigma_{x}\sigma_{y}})]\delta(z-(x+y))dxdy.
L_{0}+ \tilde{L}_{0}= \frac{1}{2}k^{2}+ N + \tilde{N}+ \frac{1}{2}\Pi_{i}G^{i j}\Pi_{j}+ \frac{1}{2}L^{i}( G - B G^{- 1}B )_{i j}L^{j}+ L^{i}B_{i j}G^{j k}\Pi_{k}.
\frac{1}{N!}\sum_{h_1,\dots,h_N}\,\prod_{i<j}|\texttt{t}^{h_i}-\texttt{t}^{h_j}| \, \prod_{i=1}^N e^{V({\bf\texttt{p}}^{*},\texttt{t}^{h_i})}\mu(h_1;\texttt{q},\texttt{t})
\int_{\mathbb{R}^{d}}|V(\mathbf{x})\mathrm{e}^{-H_{0}t}f(\mathbf{x})|^{2}\,\textnormal{d}\mathbf{x}=\int_{\mathbb{R}^{d}}|V(\mathbf{x})|^{2}\bigg|\int_{\mathbb{R}^{d}}K_{t}(\mathbf{x},\mathbf{y})f(\mathbf{y})\,\textnormal{d}\mathbf{y}\bigg|^{2}\,\textnormal{d}\mathbf{x}.
\Sigma(\delta) = \left\{\gamma \in \mathcal{P}(\mathcal{S}): \boldsymbol{W}_{p_1}(\gamma_{1}, \mu_1 ) \leq \delta_1^{1/p_1}, \ \boldsymbol{W}_{p_2}(\gamma_{2}, \mu_2 )\leq \delta_2^{1/p_2}\right\}.
\widehat{D}\theta= d \theta + \frac{1}{4}\hat{\omega}^{\hat{r}\hat{s}}\Gamma_{\hat{r}\hat{s}}+ \hat{e}^{\hat{r}}T_{\hat{r}}{}^{\hat{s}\hat{t}\hat{u}\hat{v}}\widehat{F}_{\hat{s}\hat{t}\hat{u}\hat{v}}\theta
{\cal Z}_{open,BPS}^{p-p}\int_0^\infty\frac{ds}{s}\left[{\rm Tr}_{\rm NS}\left(\frac{1+(-1)^F}{2}\right)q^{2L_0-1}-{\rm Tr}_{\rm R}\left(\frac{1+(-1)^F}{2}\right)q^{2L_0}\right]~~,
\kappa^{4}\int E R \overline{{{R}}}\left( \overline{{{\nabla}}}^{2}R \right) \left( \nabla^{2}\overline{{{R}}}\right) d^{4}\theta = - \frac{3}{2}\kappa^{4}\int \epsilon \left( \overline{{{\nabla}}}^{2}+ \frac{1}{3}\overline{{{R}}}\right) \left[ R \overline{{{R}}}\left( \overline{{{\nabla}}}^{2}R \right) \left( \nabla^{2}\overline{{{R}}}\right) \right] d^{2}\theta + \mathrm{h . c .}
S = -\tau_0 \int d\xi \left\{(1-v^2)^{1/2}(1- \phi) -{1 \over 2}(1-v^2)^{-1/2}(h^{IIA}_{00}+ 2h_{0i}v^i + h_{ij}v^iv^j) - C_0 - C_iv^i \right\}
\frac{d}{dt}\|x_\nu^{(\frac{N+1}{2})}(t)-x_\nu^{(\frac{N+1}{2}+2)}(t)\|^2 \leq-\alpha\nu\|x_\nu^{(\frac{N+1}{2})}(t)-x_\nu^{(\frac{N+1}{2}+2)}(t)\|^2+\nu\|x_\nu^{(\frac{N-1}{2})}(t)-x_\nu^{(\frac{N+1}{2}+3)}(t)\|^2 &+\frac{1}{\nu}M_{T_1,T_2}^{\frac{N+1}{2},\frac{N+1}{2}+2,5-\alpha}(\omega)
\int_{0}^\infty |D_\lambda(x,y)|\,\mathrm dx = \frac{1}{2\sqrt{|\lambda|}}\int_{0}^\infty e^{-\operatorname{Im}\sqrt{\lambda}|x-y|}\,\mathrm dx=\frac{2-e^{-\operatorname{Im}\sqrt{\lambda}y}}{2\sqrt{|\lambda|}\operatorname{Im}\sqrt{\lambda}}\leq \frac{1}{\sqrt{|\lambda|}\operatorname{Im}\sqrt{\lambda}},
W_i = - \frac{dz_i}{w_i}\Bigg( \frac{1-\frac{w_i\hbar}{2z_i}}{1+\frac{w_i\hbar}{2z_i}}\, e^{\frac{w_i\hbar}{z_i}}\Bigg)^{-\frac{r\lambda^{r-1}}\hbar}\!\!\exp\Big( \frac{(1+\frac{w_i\hbar}{2z_i})^{1-r}-(1-\frac{w_i\hbar}{2z_i})^{1-r}+(r-1)w_i\hbar z_i^{-1}}{\hbar (r-1)z_i^{r-1}}\Big); D_x = - d \; \frac{z^{r+1}}{r(1-(\lambda z)^{r-1})dz}.
\tan{\delta_r}=\dfrac{\tan{\phi_r}[2(1-\tilde{b}\beta_i)({\beta^2_i}{\tan^2{\phi_i}}-\tan^2{\phi_r})-(\gamma+1)(\beta_i-1)\sec^2{\phi_r}]}{\beta_i(\gamma+1)(1+{\beta_i}{\tan^2{\phi_i}})\sec^2{\phi_r}-2(1-\tilde{b}\beta_i)({\beta^2_i}{\tan^2{\phi_i}}-\tan^2{\phi_r})}.
\|D_\varkappa^\nu\varphi\|^2 -(1 -b)\|D_\varkappa^0\varphi\|^2 =\int_{-\infty}^\infty\big\langle (R^{-1}\mathcal M\varphi)(s), A_\varkappa^\nu(b, s)(R^{-1}\mathcal M\varphi)(s)\big\rangle\,\mathrm ds
2\log\Gamma^{\ast}\left( a,\chi\right) =-\overline{E}_{0,\overline{\chi}}\left( 0\right) \log a-2\ell^{\prime}\left( 0,\chi\right) -\chi\left(-1\right) \frac{\overline{E}_{1,\overline{\chi}}\left( 0\right)}{a}+\chi\left( -1\right) \int\limits_{0}^{\infty}\frac{\overline{E}_{1,\overline{\chi}}\left( x\right)}{\left( x+a\right)^{2}}dx,
\sup_{z \in S (\beta, \, \rho)}\|\Delta (z)\| = \sup_{z \in S (\beta, \, \rho)}\, \max \{\, |\Delta^x(z)|, |\Delta^y(z)| \}\leq \, \max \, \{\tfrac{d}{2}\, \rho^{n}, \, \tfrac{d}{2}\, \rho^{m}\}\, < \frac{d}{2}.
f^{(n_0+1)}(\sigma^2)=-\sum_{k>0}kC_k(f^\prime(\sigma^2))^{-k}\frac{f^{\prime\prime}(\sigma^2)}{f^\prime(\sigma^2)}+\sum_{l>0}C_l l(f^\prime(\sigma^2))^l\frac{f^{\prime\prime}(\sigma^2)}{f^\prime(\sigma^2)}
PU = U\Bigl(\lambda^{3/2}R^{3}e^{i3\theta}-\lambda^{5/2}Re^{i\theta}x_{1}^{2}- b_{0}\lambda^{3}x_{1}^{3}+\lambda^{1/2}Re^{i\theta}A^{2}\frac{w''}{w}(Ax_{1}+B)\Bigr).
H_{F}= 1 + \sum_{i}{\frac{Q_{i}}{[ | \vec{x}- \vec{x}_{0 \, i}|^{2}+ 4 Q | \vec{z}- \vec{z}_{0}| ]^{3}}}, \ \ \H_{p}={\frac{Q}{| \vec{z}- \vec{z}_{0}|}},
\check{R}^{\prime}= \sum_{I \neq J}e_{J I}\otimes e_{I J}+ \sum_{I}( - 1 )^{p_{I}}q^{1 - 2 p_{I}}e_{I I}\otimes e_{I I}+ ( q - q^{- 1}) \sum_{I < J}e_{I I}\otimes e_{J J}
||\beta(F^*_i (\alpha_i) ) - F^*_{i+1}(\alpha_{i+1}) || =||(F^0_{i+1})^* ( \beta(\alpha_i) ) - F^*_{i+1}(\alpha_{i+1}) || & ||(F^0_{i+1})^* ( \beta(\alpha_i) )- F_{i+1}^* ( \beta(\alpha_i) ) +|| F_{i+1}^* ( \beta(\alpha_i) )- F^*_{i+1}(\alpha_{i+1}) || & \leq \delta_i || \beta(\alpha_i) || + || \beta(\alpha_i) - \alpha_{i+1}||
X =\frac{\phi(-q^3)^4}{\phi(-q)^4}=\phi(-q^3)^4\left(\frac{\phi(-q^9)^3}{\phi(-q^3)^4}\left(1+2qw(q^3)+4q^2 w(q^3)^2\right)\right)^4 &=\frac{\phi(-q^9)^{12}}{\phi(-q^3)^{12}}\big(1 + 8 q w(q^3) + 40 q^2 w(q^3)^2 + 128 q^3 w(q^3)^3 + 304 q^4 w(q^3)^4 &\quad+ 512 q^5 w(q^3)^5 + 640 q^6 w(q^3)^6 + 512 q^7 w(q^3)^7 + 256 q^8 w(q^3)^8\big).
(y - x^*)^\alpha \varphi_\lambda^{(t^*)}(y-x^*) = U_0(t^*)\left[(y- x^*- it^* \nabla_y )^\alpha (\varphi_0)_\lambda\right] = \sum_{\beta + \gamma \le \alpha}C_{\beta,\gamma}{t^*}^{|\beta |}\lambda^{b(|\beta |-|\gamma |)}\varphi^{(\beta,\gamma)}_\lambda (t^*,y-x^*),
\omega_\eta(y(t)) - \omega_\eta(y(t-t')) = \int_{t-t'}^t (\mathcal{L}\omega_\eta)(y(s))\, ds + \sqrt{2\beta^{-1}}\int_{t-t'}^t \big[(\nabla \omega_\eta)^T\sigma\big](y(s))\, d\bar{w}(s)\,.
1-h\cdot\Biggl(\frac{\sum_{n=0}^{N-1}v_n}{\sum_{n=0}^{N-1}u_n}\Biggr)^e\le \frac{\sum_{n=0}^{N-1}\bigl|\sum_{i=0}^{\tau-1}\delta_{\tau n+i}\bigr|^q}{\sum_{n=0}^{N-1}\sum_{i=0}^{\tau-1}|\delta_{\tau n+i}|^q}\le 1+h\cdot\Biggl(\frac{\sum_{n=0}^{N-1}v_n}{\sum_{n=0}^{N-1}u_n}\Biggr)^e,~~~
\sum_{k\not=0}\exp(-Td^3(k,0))d^2(k,0)=T^{-\frac23}\sum_{k\not=0}\exp(-Td^3(k,0))\big(T^{\frac13}d(k,0)\big)^2\lesssim(T^\frac{1}{3})^{-\frac{9},{2}}
\int_{S^{3}}u_{N}P_{g,h}^{\left( 1\right)}u_{p}d\mu +4u_{N}^{\left(1\right)}\left( p\right) +2u_{N}\left( p\right) trh\left( p\right) =\alpha_{N}^{\left( 1\right)}u_{p}\left( N\right) .
\nabla_{i}(e^{0}\otimes e^{0})=\coth(r)(e^{0}\otimes e^{i}+e^{i}\otimes e^{0}) &\nabla_{i}\nabla_{i}e^{0}\otimes e^{0}=2\coth^{2}(r)e^{i}\otimes e^{i}-2\coth^{2}(r)(e^{0}\otimes e^{0}),
_i\widetilde{\tau}_j =\sum_{k=1}^i\left(\frac{1}{b_k}-\frac{1}{a_k}\right) \left(\frac{1}{b_k}+\frac{1}{a_k}\right)+\sum_{k=i+1}^j\frac{1}{b_k^2}\le\frac{2}{b_1}\sum_{k=1}^i\left(\frac{1}{b_k}-\frac{1}{a_k}\right)+\frac{2}{b_1}\sum_{k=i+1}^j\frac{1}{b_k}=\frac{2}{b_1}\mathbb{E}_i\widetilde{\tau}_j.
\left( \frac{2 \pi \omega^{+ j}R^{+ j}+ Y^{+ j}}{2 \pi \alpha^{\prime}}\right) \left( \frac{2 \pi \omega_{- j}R_{- j}+ Y_{- j}}{2 \pi \alpha^{\prime}}\right) - \frac{1}{8}+ + N_{B}( D D ) + N_{F}^{N S}( D D ) ,
\mathsf{R}_{k}-\mathsf{R}_{k}\circ T =\mu(E_{k})\left( a_{T}(\varphi_{E_{k}}\circ T+1)-a_{T}(\varphi_{E_{k}}\circ T)\right) & \leq\mu(E_{k})\,a_{T}(1)\{\varphi_{E_{k}}>1\}=T^{-1}E_{k}^{c}
\langle \alpha^\vee, 2\rho\rangle = \frac 12\left(\langle \alpha^\vee, 2\rho\rangle + \langle s_\alpha(\alpha^\vee), s_\alpha(2\rho)\rangle\right) = \frac 12\langle \alpha^\vee, 2\rho - s_\alpha(2\rho)\rangle.
\lambda_{\ell,j}^k=\begin{cases}1, \ell=j,\, k=0; 0, k+\ell<j\, \mathrm{or}\, \ell>j, s_{\ell}\lambda_{\ell,j-1}^{k-1}+v_{\ell}\lambda_{\ell,j}^{k-1}; \mathrm{otherwise}.\end{cases}
\psi_{n}(x) =-AD_{\alpha}\frac{\sin(n\pi/2)}{E_{n}\pi}\ \left( \frac{n\pi\hbar}{2a}\right)^{\alpha}PV(I) & =\frac{AD_{\alpha}\sin^{2}(\frac{n\pi}{2})}{E_{n}}\ \left( \frac{n\pi\hbar}{2a}\right)^{\alpha}\cos\frac{n\pi x}{2a}.
\max_{i=1,\ldots, d}\ \max_{\xi \in \mathbb{T}^d}|g^i_{\boldsymbol{\ell}}(\xi) | \ \to \ 0 \;\; \mbox{as}\;{\boldsymbol{\ell}}\to 0, \mbox{and consequently}\; \| g_{\boldsymbol{\ell}}\|_{(L^2(\mathbb{T}^d))^d}\ \to \ 0 \;\; \mbox{as}\;{\boldsymbol{\ell}}\to 0.
|\cot2\phi_{k}+ \cot2\psi_{k}| = \frac{|\sin 2(\phi_k +\psi_k)|}{|\sin 2\phi_k \sin 2\psi_k|}\geq |\sin 2(\phi_k +\psi_k)| & = 2|\cos (\alpha_k +\beta_k)|\cos(\phi_k +\psi_k) > \left(2-4.937284\delta_k^2\right)\cos(\phi_k +\psi_k) & > 1.9999\cos(\phi_k +\psi_k), 1\leq k\leq5,
\Rightarrow|\psi\rangle=\frac{1}{2}[|0\rangle(|f_{k}\rangle|f_{k}^{\prime}\rangle+|f_{k}^{\prime}\rangle|f_{k}\rangle)+|1\rangle(|f_{k}\rangle|f_{k}^{\prime}\rangle-|f_{k}^{\prime}\rangle|f_{k}\rangle)]
s^{I}( z , \vec{x}) = \frac{\Gamma ( 2 k )}{A \pi^{3}\Gamma ( 2 k - 3 )}\int d \vec{x}^{\prime}\, \left( \frac{z}{z^{2}+ | \vec{x}- \vec{x}^{\prime}|}\right)^{2 k}s_{0}^{I}( \vec{x}^{\prime}) .
u(t,x)= \lim_{r\to 0}I_r[u](t,x) = \lim_{r\to 0}\frac{1}{\beta^{(k)}_0 r}\Omega_r[u](t,x) = \frac{1}{(n-2)!!}\lim_{r\to 0}\frac{\widetilde{v}(r,t;x)}{r}.
w_j(X_j,Y_j,t_j,\tilde X_j, \tilde Y_j, \tilde t_j) \geq u(X_j, Y_j, t_j) -\phi(\tilde X_j, \tilde Y_j,\tilde t_j) = u(X_j, Y_j, t_j) - \phi(X_j, Y_j, t_j) + \phi(X_j, Y_j, t_j) - \phi(\tilde X_j, \tilde Y_j,\tilde t_j) > 2\varepsilon - \varepsilon = \varepsilon.
\;{}_{E}\langle T_{O}| = \mathrm{T r}\left[{}_{E}\langle W | \; \frac{b_{0}}{L_{0}}\right] = \int_{0}^{1}\mathrm{T r}\left[{}_{E}\langle W | \; b_{0}\, k^{L_{0}}\right] \frac{d k}{k}~ ,
\tilde{z}_{s}+\left( \frac{\tilde{z}^{2}}{2}+H^{-1}\tilde{z}\right)_{x}=0,\tilde{z}_{y}+\left( \frac{\tilde{z}^{3}}{3}+H^{-1}\tilde{z}^{2}+(H^{0}+(H^{-1})^{2})\tilde{z}\right)_{x}=0.
\Vert \Psi_t^{G,\hbar}a - a \Vert_{s-\sigma}\leq \sum_{j=1}^\infty \frac{j^{2j}\vert t \vert^j \Vert G \Vert_s^j}{(j!)^2(e \sigma)^{2j}}\Vert a \Vert_s \leq C_{\sigma}|t| \|G\|_s \Vert a \Vert_s,
S_{F}[ \psi , a_{\mu}, b_{\mu}] = \int d^{2}x ( - \overline{{\psi}}\gamma^{\mu}\partial_{\mu}\psi + i \overline{{\psi}}\gamma^{\mu}\psi a_{\mu}+ i \overline{{\psi}}\gamma^{\mu}\gamma_{3}\psi b_{\mu}) .
E_{n}(\tilde{g};[0,1]) \ge \frac{1}{m}\sum_{l=n}^{m}E_{l}(\tilde{g};[0,1]) &\ge \frac{1}{M_1}\omega_{\varphi}^1(\tilde{g},\frac{1}{m})_\infty - \frac{1}{m}\sum_{l=0}^{n-1}E_{l}(\tilde{g};[0,1]) &\ge \frac{1}{M_1}\omega_{\varphi}^1(\tilde{g},\frac{1}{m})_\infty - \frac{n}{m}E_{0}(\tilde{g};[0,1]).
\nabla^{\mathbb{S}}_{u}( \omega \otimes \tau \otimes s) =( \nabla^{F}_{X}\omega ) \otimes \tau \otimes s +\omega\otimes ({\mathcal L}_{X}\tau ) \otimes s & +\frac{1}{3}(\iota_{X}{\mathcal H})\wedge \omega \otimes \tau \otimes s + \omega \otimes \tau \otimes ( \nabla^{0, \mathcal S_{\mathcal G}}_{X}s -\frac{1}{3}(\mathrm{ad}_{r}) (s)) .
(G^T_{{\cal L}}G_{{\cal R}})^{ij}= H^{ik}_{L}(- l^{k\underline{m}}r^l_{\underline{m}}+{2(r^{(++)\underline{n}}l^k_{\underline{n}})(l^{(--)\underline{n'}}r^l_{\underline{n'}})\over(l^{(--)\underline{n''}}r^{(++)}_{\underline{n''}})}) H^{jl}_{R}~,
2x^4(D+12) (D+8)(D+4)T - \frac{1}{2}(D+6)T + 6x^2((D+4)T)^2 = \frac{3}{4}\left( \left. Y_{N+2}\right|_{t_i = x\delta_{i,3}}+ \left. Y_{N-2}\right|_{t_i = x\delta_{i,3}}- \left. 2Y_{N}\right|_{t_i = x\delta_{i,3}}\right).
\beta_0=\frac{g_0(\bar{P}|\bar{P})}{g_0(\bar{P}|\bar{P})+\sum_{\{k|e_{ik}\in\mathcal{U}_e\}}g_j(\bar{P}|\bar{P})}, &\beta_j=\frac{g_j(\bar{P}|\bar{P})}{g_0(\bar{P}|\bar{P})+\sum_{\{k|e_{ik}\in\mathcal{U}_e\}}g_k(\bar{P}|\bar{P})},~{\forall}j\in\{k|e_{ik}\in\mathcal{U}_e\},
\tau_{R S}( M ) = \det ( \phi_{0}^{*}\phi_{0})^{- \frac{1}{2}}\det ( \phi_{3}^{*}\phi_{3})^{\frac{1}{2}}{\det}^{\prime}( d_{0}^{*}d_{0})^{\frac{1}{2}}{\det}^{\prime}( d_{1}^{*}d_{1})^{- \frac{1}{2}}{\det}^{\prime}( d_{2}^{*}d_{2}) .
\begin{array}{ll}\omega= 4 \sum_{i=1}^3 A_i B_i \eta_i^- \wedge \eta_i^+, \Omega_1 = 8 B_1 B_2 B_3 \eta_{123}^- -4 \sum_{i=1}^3 \epsilon_{ijk}A_i A_j B_k \eta_i^+ \wedge \eta_j^+ \wedge \eta_k^-, \Omega_2 = -8 A_1 A_2 A_3 \eta_{123}^+ +4 \sum_{i=1}^3 \epsilon_{ijk}B_i B_j A_k \eta_i^- \wedge \eta_j^- \wedge \eta_k^+,\end{array}
\Lambda_n(H_\theta) \ge \min\Big\{\Lambda_n(Q^{1+\varepsilon}_\theta), -\Big(1+\frac{1}{\varepsilon}\Big)^2\Big\}=\Lambda_n(Q^{1+\varepsilon}_\theta)=-\dfrac{1}{(2n-1)^2\theta^2}+O\Big(\dfrac{1}{\theta}\Big).
\Theta : \mathcal{R}\to \mathcal{R}, \sum_{i = 0}^{n-1}a_ix^i \mapsto \sum_{i = 0}^{n-1}\sigma^{-i}(a_i)x^{-i}= a_0 + \sum_{j=1}^{n-1}\sigma^j(a_{n-j}) x^j,
\int_Mtd\mu =2\pi\int_{-a}^atA(t)dt &=\frac{\omega(F)^3}{96\pi}\left[6\frac{\omega(c_+)-\omega(c_-)}{\omega(F)}-\sum_j\frac{1}{r_js_j}\left(\frac{\omega(r_jE_j')-\omega(-s_jE_j)}{\omega(F)}\right)^3+c_1(Y)[\Sigma_{-a}]+c_1(Y)[\Sigma_a]\right],
\mp \frac{\pi i}{2}\zeta_{{D\!\!\!\!/}^{~2}+m^2}(0)+\frac{1}{2}\lim_{s\to 0}\frac{1}{\Gamma(s)}\int^1_0~dt~t^{s-1}\sum_{n=0}^{\infty}H_nt^{{(n-d)}/{2}}\left[\frac{d K^{(1)}}{ds}\pm 2 m\sqrt{t}\frac{d K^{(2)}}{ds}\right].
[ \tilde{\cal R}_{\mu}, \tilde{\cal R}_{\nu}] = - i \epsilon_{\mu \nu \lambda}\tilde{\cal R}{}^{\lambda}, \quad [ \tilde{\cal R}_{\mu}, \tilde{\cal R}_{\nu}]_{{}_{+}}= - \frac{1}{2}\Pi_{-}\eta_{\mu \nu}, \quad [ \tilde{\cal R}_{\mu}, \Pi_{-}] = 0 ,
S^{c}={\cal C}{\cal K}= e^{i \vartheta_{[ 1 ]}^{c}}\left( \begin{matrix}{{0}}&{{\Theta_{[ 1 ]}}}\\{{- \Theta_{[ 1 ]}}}&{{0}}\\ \end{matrix}\right){\cal K}\, \, , \quad \Theta_{[ 1 ]}= \left( \begin{matrix}{{0}}&{{0}}&{{1}}\\{{0}}&{{- 1}}&{{0}}\\{{1}}&{{0}}&{{0}}\\ \end{matrix}\right)
t\left( 2n,2l\right) =2^{2l}\sum_{m=1}^{j+1-n}\left( V^{-1}\left[j\right] \right)_{2l+1,m}\frac{\left( j+1-m\right) ~\left( j+n-m\right)!}{\left( j+1-n-m\right) !}& +2^{2l}\sum_{m=j+1+n}^{2j+1}\left( V^{-1}\left[ j\right] \right)_{2l+1,m}\frac{\left( m-1-j\right) \left( m+n-j-2\right) !}{\left(m-n-j-1\right) !}\ .
C\left({\mathbf{c}, \mathbf{d}}\right)(\tau)=\begin{cases}\sum_{i=0}^{L-1-\tau}\omega^{c_{1i}-c_{2i+\tau}}, 0 \leq \tau \leq L-1, \sum_{i=0}^{L-1+\tau}\omega^{c_{1i-\tau}-c_{2i}}, -L+1 \leq \tau \leq-1, 0, |\tau| \geq L,\end{cases}
R^\pm(\lambda) = \left(\frac{1}{4}\int_0^{\pm \infty}e^{-2i\lambda y}(q^\sharp)'(1+\eta_{11}^\pm)\right) &+\left(\frac{1}{4}\int_0^{\pm \infty}e^{-2i\lambda y}q^\sharp \left[ q \eta_{21}^\pm + \frac{1}{2i}|q|^2 \eta_{11}^\pm \right]\right).
\geq_n \phi^{(j)}(1-\epsilon) \times \left[ \frac{n}{2}\log \left( \sum_{j'=1}^J k_n^{(j')}\right) - \sum_{j'=1}^J \beta^{(j')}\ell_n H_2 \left( \alpha_n^{(j')}\right) \right]
(x\epsilon_{1,0}^* \circ x\epsilon_{1,0}^*)(\epsilon_{1,0}) = x , (x\epsilon_{1,0}^* \circ y\epsilon_{0,1}^*)(\epsilon_{0,1}) = 0 , (y\epsilon_{0,1}^* \circ x\epsilon_{1,0}^*)(\epsilon_{1,0}) = 0 , (y\epsilon_{0,1}^* \circ y\epsilon_{0,1}^*)(\epsilon_{0,1}) = y.
\frac{\delta{\cal W}}{\delta \psi^{\alpha}}= \frac{\delta \Gamma}{\delta \psi^{\alpha}}, \frac{\delta{\cal W}}{\delta \phi_{A a}^{*}}= \frac{\delta \Gamma}{\delta \phi_{A a}^{*}}, \frac{\delta{\cal W}}{\delta \bar{\phi}_{A}}= \frac{\delta \Gamma}{\delta \bar{\phi}_{A}},
|f(x') - f(y')| = \left\lvert\sum_{j=0}^\infty 2^{- \gamma m j}g_j (x') - \sum_{j=0}^\infty 2^{- \gamma m j}g_j(y')\right\rvert \leq \sum_{j=0}^{n'}2^{- \gamma m j}|g_j(x') - g_j(y')| + \sum_{j=n'+1}^\infty 2^{- \gamma m j}|g_j(x') - g_j(y')|
x_{n}\sum_{a=1}^{\mathsf{N}}g_{a}(\zeta_{n}^{(1)})x_{a}+x_{n}f(\zeta_{n}^{(1)})=q_{n},q_{n}=\frac{\mathrm{det}_{q}K_{+}(\xi_{n})\mathrm{det}_{q}\,\mathcal{U}_{-}(\xi_{n})}{\sinh (\eta +2\xi_{n})\sinh (\eta-2\xi_{n})},\forall n\in \{1,...,\mathsf{N}\},
=\int_{F}\frac{d^{2}\tau}{\tau_{2}^{2}}+ 2 \int_{0}^{\infty}\frac{d\tau_{2}}{\tau_{2}^{2}}\sum_{n>0}e^{\frac{-\pi x}{2 \tau_{2}}n^{2}}=\frac{\pi}{3}+\frac{2\pi}{3x}.
\begin{array}{ll}{\cal L}_\chi =(2\pi R){\cal L}_5,_\chi =\tilde{g}^{MN}(\partial_M\chi)^{*}(\partial_N\chi )-V =e^{\sqrt{\frac 23}\kappa \varphi}g^{\mu \nu}(\partial_\mu \chi)^{*}(\partial_\nu \chi )-V\end{array}
\textrm{Pr}^{\textrm{OFDMA}}\left\lbrace \textrm{FDF=1,FDN=0}\right\rbrace\to \frac{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^{2}}{\lambda_{\textrm{BF}}}\left(\frac{(1-\theta)}{(1-\rho)}\left(2^{\frac{R}{1-\theta}}-1\right)-\frac{\theta}{\rho}\left(2^{\frac{R}{\theta}}-1\right)\right) \frac{1}{P_{\textrm{B}}}.
r^{\pi}_t(x_t|y_{t-M}^{t-1},y_t)=\Big(\frac{q_t(y_t|y_{t-M}^{t-1},x_t)}{\nu^{\pi}_{t}(y_t|y_{t-J}^{t-1})}\Big){\pi}_{t}(x_t|y_{t-J}^{t-1}),~t\in\mathbb{N}_0^n.
\mathcal{E}(\varphi,\varphi)=\sum_{x\in V}\kappa_{x}\varphi_{x}^{2}+\sum_{x\in V}W_{x}\varphi_{x}^{2}-2\sum_{e\in E}J_e(\vert\varphi\vert) \operatorname{sign}(\varphi_{e_{+}})\operatorname{sign}(\varphi_{e_{-}}),
\limsup_{N\to \infty}\frac{1}{N}\#\{1\le n \le N \mid \widetilde{T}^n (x;a) \in C_s \times \{a'\}\}\qquad\le \limsup_{N\to \infty}\frac{1}{N}\#\{1\le n \le N \mid \widetilde{T}^n (x;a) \in C_s \times A\}&\qquad= \mu(C_s) &\qquad\le \frac{1}{|A|}\tilde{\mu}\left( C_s \times \{a \}\right).
\Phi = ( m + \frac{1}{2}) \pi , \, \, \, \, \mathrm{a n d}\, \, \, \, \, \, \, \, \gamma = m^{\prime}+ \frac{1}{2}, \, \, \, \, \mathrm{w i t h}\, \, \, \, \, \, m , m^{\prime}\in \, \, \, \, \mathrm{{\bf ~ Z}},
\left\vert \Phi ( P^{+}) \right\rangle_{f}= \int_{0}^{P^{+}}d k_{1}d k_{2}\delta ( k_{1}+ k_{2}- P^{+}) \phi_{b f}( k_{1}, k_{2}){\frac{1}{N}}\mathrm{t r}[ a^{\dagger}( k_{1}) , b^{\dagger}( k_{2}) ] \left\vert 0 \right\rangle ,
m^{2}t r \ \gamma^{5}G_{L}( x , y ) = - m^{2}\int \frac{d^{4}k}{( 2 \pi )^{4}}e^{i k \cdot ( x - y )}\epsilon^{\lambda \mu \rho \sigma}\frac{1}{( m^{2}- k^{2})^{3}}G_{\lambda \mu}( y ) G_{\rho \sigma}( y ) \ .
f \equiv f(r,t):=-\Delta w +\lambda w -\alpha u -\beta v, g_{1}\equiv g_{1}(r,t):= \frac{u_{r}}{\sqrt{\chi(u+1)}}-\sqrt{\chi(u+1)}w_{r}, g_{2}\equiv g_{2}(r,t):=\frac{v_{r}}{\sqrt{\xi(v+1)}}-\sqrt{\xi(v+1)}w_{r}.
d\tau=\sqrt{[1-(\frac{r\omega}{c})^{2}]dt^{2}-\frac{dr^{2}}{c^{2}}-\frac{r^{2}d\theta^{2}}{c^{2}}-\frac{dz^{2}}{c^{2}}-2\frac{r^{2}\omega dtd\theta}{c^{2}}}.
\sigma = \bar{\sigma}+ \tilde{\sigma}\in \widetilde{L}^{\infty}_{T}(\widetilde{B}^{\frac{N}{2}+1 -\alpha,\frac{N}{2}}) \cap L^1_T(\widetilde{B}^{\frac{N}{2}+1,\frac{N}{2}+2-\alpha}) ,\u = \bar{u}+ u_L \in \widetilde{L}^{\infty}_{T}(\dot{B}^{\frac{N}{2}+1-\alpha}_{2,1}) \cap L^1_{T}(\dot{B}^{\frac{N}{2}+1}_{2,1}).
Q = \frac{1}{2 \pi}\left( \lim_{r{\rightarrow}{\infty}}\overline{{{\cal{N}}}}( r ) - \overline{{{\cal{N}}}}( 0 ) \right) = \frac{1}{2 \pi}\, \lim_{r{\rightarrow}{\infty}}\overline{{{\cal{N}}}}( r ) = \frac{m}{2}\, ,
\begin{aligned}F^1_1(\phi^1_1)=\frac{A(A-3)+B^2}{\cos^2 3\phi^1_1}-\frac{B(2A-3)\sin 3\phi^1_1}{\cos^2 3\phi^1_1}+9\bigg[\frac{2(2A-3)}{2A-3-2B\sin 3\phi^1_1}-\frac{2[(2A-3)^2-4B^2]}{(2A-3-2B\sin 3\phi^1_1)^2}\bigg] G^1_2(\phi^1_2)=G^1_3(\phi^1_3)=\cdots=G^1_{d_i-1}(\phi^1_{d_i-1})=0\end{aligned}
\lim_{|\xi|_{p,q,r}+\theta^{\ell}\to\infty}\frac{|\hat{\eta}(\xi,\theta|\bar{\xi},\bar{\theta})|}{|\xi|_{p,q,r}+\theta^{\ell}}=\lim_{|\xi|_{p,q,r}+\theta^{\ell}\to\infty}\frac{|\hat{\eta}(\xi,\theta)|}{|\xi|_{p,q,r}+\theta^{\ell}}=0.
\therefore S = \frac{2 a^{2}k ( k^{2}+ 1 )}{a k^{2}+ 2}= \frac{8 k ( k^{2}+ 1 )}{( 2 k^{2}+ 1 ) ( k^{2}+ 2 )}= \frac{8 k ( k^{2}+ 1 )}{2 ( k^{2}+ 1 )^{2}+ k^{2}}
\Phi_{\widetilde{F}}(z^\iota_n, x^\iota_n) =\Phi_{\widetilde{F}}\circ \widetilde{\boldsymbol{F}}_{n^2+k_n, n^2+\kappa_0}(z_0,x_0) & = \Phi_{\widetilde{F}}(z_0,x_0) +k_n-\kappa_0 + o(1) &= \Phi_F(z,x)+k_n +o(1),
\langle \vec{k}, 0 | \vec{k}',0 \rangle = i \int d^3 x \,{\varphi}^0_{\vec{k}}(x)^*\stackrel{\leftrightarrow}{\partial_0}{\varphi'}^0_{\vec{k}'}(x) = (2 \pi)^3 2k^0 \delta^{(3)}(\vec{k}-\vec{k}') \, .