harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
J_{k}\leq C \sum_{i,j=1}^d \sum_{1 \leq a,b \leq d}\|c^{ab}_{ij}(\partial v) \|_{L^\infty(\partial \Omega_\lambda)}\int_{\partial \Omega_{\lambda}}|h\cdot \sigma | |\sigma \cdot \nabla \partial_t^\mu v|^2 dS \leq C_\lambda \delta \int_{\partial \Omega_{\lambda}}|h\cdot \sigma | |\sigma \cdot \nabla \partial_t^\mu v|^2 dS (k=12,13).
[ a_{m}, \tilde{a}_{n}] = [ a^{\dagger}_{m}, \tilde{a}^{\dagger}_{n}] = [ \psi_{\dot{\alpha}}, \tilde{\psi}_{\dot{\beta}}]_{+}=[ \psi^{\dagger}_{\dot{\alpha}}, \tilde{\psi}^{\dagger}_{\dot{\beta}}]_{+}= 0
\dfrac{d}{dt}\dfrac{1}{2}\left(\displaystyle\int_\Omega|u_t|^2dx+\eta\displaystyle\int_\Omega| \Delta u|^2dx+\displaystyle\int_\Omega|\theta|^2dx-2\displaystyle\int_\Omega\int_0^u f(t,s)dsdx\right) &=-\kappa\displaystyle\int_\Omega|\nabla \theta|^2dx,
{}_pF_q \left( c_1, \cdots, c_p;\, d_1, \cdots, d_q;\, z \right):= \sum_{n=0}^{\infty}\frac{(c_1)_n \cdots (c_p)_n}{(d_1)_n\cdots (d_q)_n}\frac{z^n}{n!},
\left\|\vec f\right\|_{\dot A^{s,\tau}_{p,q}(W,\mathbb{R}^n)}\lesssim\left\|\vec u\right\|_{\dot a^{s,\tau}_{p,q}(W,\mathbb{R}^n)}\sim\left\|\vec u\right\|_{\dot a^{s,\tau}_{p,q}(\mathbb{A}(W),\mathbb{R}^n)}&=[\ell(Q(I,k_0))]^{-n\tau-s+n(\frac{1}{p}-\frac{1}{2})}\left|A_{Q(I,k_0),W}\vec u_{Q(I,k_0)}\right|=\left|A_{Q(I,k_0),W}\vec z\right|,
\left( \begin{matrix}{{a^{\prime}}}&{{b^{\prime}}}\\{{c^{\prime}}}&{{d^{\prime}}}\\ \end{matrix}\right) x = \left( \begin{matrix}{{x a^{\prime}+ ( 1 - q^{2}) y c^{\prime}}}&{{q^{- 1}x b^{\prime}+ ( q - q^{- 1}) y ( a^{\prime}- d^{\prime})}}\\{{q x c^{\prime}}}&{{x d^{\prime}+ ( 1 - q^{2}) y c^{\prime}}}\\ \end{matrix}\right)
L\left[ \partial_{x}\left( xP_{n}^{2}\right) \right] =2zP_{n}^{2}\left(z;z\right) e^{-z^{2}}+2\left[ h_{n+1}\left( z\right) +\gamma_{n}^{2}\left( z\right) h_{n-1}\left( z\right) \right] .
E(w;f) = E(y;f) &+ \int\limits_0^1 \int\limits_\Omega (1-t)D^2 W_{\rm CB}( (1-t)\nabla y(x) + t \nabla w(x) )[\nabla w(x) - \nabla y(x)]^2 \,dx \,dt &\geq E(y;f) + \frac{\lambda}{2}\int\limits_\Omega \lvert \nabla w(x) - \nabla y(x)\rvert^2 \,dx.
\left| \frac{{\partial \varphi}_{m+j}}{{\partial z}_{k}}\right|\le \frac{l_{j}^{y}}{l_{1}^{y}}\frac{\left( M_{1}^{y}M_{j}^{y}-\mu_{1}^{y}\mu_{j}^{y}\right)}{{(\mu_{j}^{y})}^{2}}\frac{l_{1}^{z}}{\mu_{1}^{z}\sum\limits_{k=1}^p \frac{l_{k}^{z}}{M_{k}^{z}}},\thinspace j=1,\thinspace \mathellipsis ,\thinspace n;k=1,\mathellipsis ,p
a_{n+1}\leq \|\varphi_n (u_1+u_3)\|_{L^{p_0}(\Omega_{n,n+2})}\leq \frac{C}{1-\frac{|d_1 - d_3|}{d_1+d_3}C_{\rm{mr}, p_0'}}\|u_1+u_3\|_{L^{p_0}(\Omega_{n,n+1})}^{1-\alpha}=\mathcal{C}a_n^{1-\alpha}.
\langle T_{\mu}^{\mu}\rangle_{R}- ( T_{\mu}^{\mu})_{0}= 2 \pi m^{2}\int_{- \infty}^{+ \infty}\frac{d \theta}{2 \pi}\left[ \frac{e^{- m R \cosh \theta}}{1 + e^{- m R \cosh \theta}}+ 2 \frac{e^{- 2 m R \cosh \theta}}{( 1 + e^{- m R \cosh \theta})^{2}}+ \frac{9}{2}\frac{e^{- 3 m R \cosh \theta}}{( 1 + e^{- m R \cosh \theta})^{3}}+ \cdots \right] \, \, \, ,
\begin{array}{l l l}{{J ( z )}}&{{=}}&{{\frac{1}{k + \check{g}}\{S_{a}^{a}- \frac{1}{( \check{g}- \check{s})}\Omega_{c a}^{c b}S_{b}^{a}+ k \psi^{a}\psi_{a}}}\\{{}}&{{}}&{{+ \Omega_{a b}^{b c}\psi^{a}\psi_{c}+ \frac{( k - \check{g}+ \check{s})}{2 ( \check{g}- \check{s})}\psi^{a b}\psi_{a b}\}( z )}}\\ \end{array}
|y_0-y_{k+1}|>|y_i-y_{k+1}|>b-b_0=b_0-a>|y_0-y_i|>\left|y_0^2-y_i^2\right|/4 = \left|(1+|u_i|/\ln n)^2-1\right|S(I)^2/4\geq|u_i|/\ln n\cdot S(I)^2/2 \geq \varepsilon_1S(I)^2/(2\ln n)\geq \varepsilon_0 (\ln n)^{-1}.
\mathbb{E}\left[f|\mathcal{A}^{(k)}\right]=\left( \mathbb{E(}f|A_{1}^{(k)}),\dots,\mathbb{E}(f|A_{m_{k}}^{(k)})\right)^{t},\quad\mathbb{G}\left[\mathcal{A}^{(k)}\right]=\left( \mathbb{G}(A_{1}^{(k)}),\dots,\mathbb{G}(A_{m_{k}}^{(k)})\right)^{t},
\begin{aligned}\eta(U|U_l) \le a\eta(U|U_r) &~ \Longleftrightarrow v>c_*~\mbox{and}~ |q_2|+|q_3|+ |u_1|+ |u_2|+|u_3|\le c^*\quad\mbox{for some constants}~c_*,~c^*>0, \end{aligned}
\int_{\sum_{i = 1}^{3}\nu_{i}\alpha_{i}+ 2 K l > 0}\prod_{i = 1}^{3}\frac{d \alpha_{i}}{\sqrt{2 K | q_{i}|}}\exp \frac{i \pi}{2 K q_{i}}\left[ p_{i}\alpha_{i}^{2}- 2 \alpha_{i}( 2 K ( n_{i}+ q_{i}\lambda ) + \mu_{i}) + s_{i}( 2 K n_{i}+ \mu_{i})^{2}\right]
\delta \omega^{\alpha}[ \eta ] = \int_{- \infty}^{+ \infty}d x^{1}\left( A^{\alpha}( x^{1}) \, \delta \phi^{\alpha}( x^{1}) \, \, + \, \, B^{\alpha}( x^{1}) \, \delta \pi^{\alpha}( x^{1}) \right) ,
\sum_{\mathfrak{a}\in \mathcal{E}_k}\overline{\langle W, E_{\mathfrak{a}}(.,1/2+it)\rangle}\theta(z,.)*E_{\mathfrak{a}}(.,1/2+it) = \sum_{\mathfrak{a}^{\prime}\in \mathcal{E}^{\prime}_{4k}}\langle \theta*W, E_{\mathfrak{a}^{\prime}}(.,1/2+it/2)\rangle E_{\mathfrak{a}^{\prime}}(z,1/2+it/2)
\big[\Lambda_k(x) \Phi\big](z_1,\ldots,z_k) = r_k(x) \prod_{i=1}^{k-1}[z_i-z_{i+1}]^{1-2s}&\times\prod_{i=1}^{k-1}\int d^2 w_i\,[w_i-z_i]^{s+ix-1}[w_i-z_{i+1}]^{s-ix-1}\,\,\Phi(w_1\,,\ldots\,,w_{k-1})\,.
\widetilde V_r^{(n)}(z):= \begin{pmatrix}\widetilde \alpha_r^{(n)}(z) \widetilde\beta_r^{(n)}(z) \widetilde\gamma_r^{(n)}(z) \widetilde\delta_r^{(n)}(z) \end{pmatrix}, z\in{\mathbb C},
\lambda_{1}(t) E_{0,1}(\lambda_{1}(t),\lambda_{1}'(t),\lambda_{1}''(t)) - \lambda_{2}(t) E_{0,1}(\lambda_{2}(t),\lambda_{2}'(t),\lambda_{2}''(t)) &\leq \frac{C ||e_{1}-e_{2}||_{X}}{t^{2}\log^{b+1}(t) \sqrt{\log(\log(t))}}
\rho_{0}^{(2)}= 1 \mbox{and}\rho_{n}^{(2)}= \frac{S_n(1)}{S_n^{\ast}(1)}= \frac{\rho_{n-1}^{(2)}- \overline{\alpha}_{n-1}}{1 - \rho_{n-1}^{(2)}\alpha_{n-1}}, \ \n \geq 1,
X|_{\hat t\times\hat s}= \begin{pmatrix}X_{11}X_{12}X_{21}X_{22}\end{pmatrix}, Y|_{\hat s\times\hat r}= \begin{pmatrix}Y_{11}Y_{12}Y_{21}Y_{22}\end{pmatrix}, Z|_{\hat t\times\hat r}= \begin{pmatrix}Z_{11}Z_{12}Z_{21}Z_{22}\end{pmatrix}
[\Pi\eta_1,\Pi\eta_2]_{A}= [\Pi\eta_1, \iota_{\eta_2}(X_1)\wedge X_2- X_1\wedge \iota_{\eta_2}(X_2)]_{A}= [\Pi\eta_1, \iota_{\eta_2}(X_1)\wedge X_2]_{A}- [\Pi\eta_1,X_1\wedge \iota_{\eta_2}(X_2)]_{A}
Au^{\prime\prime2}+(B_1+B_2u^\prime+B_3u^{\prime2}+B_4u^{\prime3})u^{\prime\prime}&+(C_0+C_1u^\prime+C_2u^{\prime2}+C_3u^{\prime3}+C_4u^{\prime4}+C_5u^{\prime5}+C_6u^{\prime6})=0
\mathbb E [ \overline{b}(s, x, \overline{X}_{s, j})] \, =\, \int_{\mathbb R}\widetilde{b}(s, x, z) \mathrm m_{s}({\mathrm d}z) - \int_{\mathbb R}\widetilde{b}(s, x, y) \mathrm m_{s}({\mathrm d}y) \, =\, 0 \, ; s \ge 0, \, \, x \in \mathbb R \, .
{\left[{\bf{B}}\right]_{11}}={\left[{{{\bf{F}}_2}}\right]_{11}}\left({\left[{{\bf{G}}}\right]_{{\rm{11}}}- \sum\limits_{k = 2}^K{\left[{{\bf{G}}}\right]_{{{k1}}}\frac{{\left[{{\bf{G}}}\right]_{{{1k}}}}}{\omega}}}\right) + \sum\limits_{k = 2}^K{{{\left[{{{\bf{F}}_2}}\right]}_{1k}}\left[{{\bf{G}}}\right]_{{{k1}}}},
{\hat{R}}= -{\frac{2}{k}}{\frac{[ 2 ( R^{4}- 1 ) \operatorname{s i n}^{2}x - 5 ( R^{2}- 1 ) - 3 ]}{( 1 + ( R^{2}- 1 ) \operatorname{s i n}^{2}x )^{2}}}\; .
\mathrm{i}{\cal P}_{\Lambda}={\o{1}{2}}\left( k_{\Lambda}^{i}\partial_{i}{\cal K}- k_{\Lambda}^{i^{\star}}\partial_{i^{\star}}{\cal K}\right) = k_{\Lambda}^{i}\partial_{i}{\cal K}= - k_{\Lambda}^{i^{\star}}\partial_{i^{\star}}{\cal K}
\|f_{3,j,> 0}\|_{L^2(Q)}^2 \lesssim \ \| \chi_Qf_{3,j,> 0}\|_{L^2}^2 \lesssim\int_{-\infty}^\infty \| \chi_Q 2^{\sigma k}u_k 2^{\sigma j}P_j F'(u_{<k})\|_{L^2}^2 \, dk \lesssim \ 2^{(\sigma-N)j}\| \chi_Q S_{>0}(|D|^\sigma u)\|_{L^2}^2 \lesssim 2^{(\sigma-N)j}\|u\|_{BMO^\sigma}^2.
\frac{2}{\sqrt{2\pi \epsilon}}e^{-\frac{2}{3}((\epsilon+t)^3-t^3)}e^{-\frac{x^2}{2\epsilon}}\mathbb{E}\left[\left(-2\sqrt{\epsilon^3}\int_{0}^1 B(z)dz \right) \vert B(1)=-\frac{x}{\sqrt{\epsilon}}\right]+F^{\epsilon}(x)
\sum_{j=1}^\infty \|b_j f\|_{D(\mu_{l+1})}^2 &\leq \frac{2}{|\Phi(\zeta_{l+1}))|^2}\Big[\|f\|_{D(\mu_{l+1})}^2 + \sum_{j=1}^\infty \sum_{i =1}^\infty \|[\varphi_i - \varphi_i(\zeta_{l+1}))] e_j f\|_{D(\mu_{l+1})}^2\Big].
\begin{array}{l l}{z_{1}= \operatorname{c o t}\omega \operatorname{c o s}\frac{\theta}{2}e^{i \frac{\psi + \phi}{2}}}{z_{2}= \operatorname{c o t}\omega \operatorname{s i n}\frac{\theta}{2}e^{i \frac{\psi - \phi}{2}}}\end{array}
\langle w,\nabla^2h_j(y)w\rangle =\lim_{\tau\downarrow 0}\frac{2[h_j(y+\tau w)-h_j(y)-\langle\nabla h_j(y),w\rangle]}{\tau^2}=\lim_{\tau\downarrow 0}\frac{2[h(y+\tau w)-h(y)-dh(y)(w)]}{\tau^2}.
O_{\omega\omega'}^0 = \frac{1}{4\pi}\{\int_0^{\theta}\!(\tilde{f}_\omega^E(\sigma)\:\tilde{f}_{\omega'}^E(\sigma) + \tilde{f}_{-\omega}^E(\sigma)\:\tilde{f}_{-\omega'}^E(\sigma)) d\sigma + \int_{\theta}^{2\pi}\!(\tilde{f}_\omega^L(\sigma)\:\tilde{f}_{\omega'}^L(\sigma) + \tilde{f}_{-\omega}^L(\sigma)\:\tilde{f}_{-\omega'}^L(\sigma) ) d\sigma\}
\bold{G}=\left[\begin{array}{c|c|c|c}\bold{F}_{1,1}\bold{F}_{1,2}\dots \bold{F}_{1,p_0}\hline\bold{F}_{2,1}\bold{F}_{2,2}\dots \bold{F}_{2,p_0}\hline\vdots \vdots \ddots \vdots \hline\bold{F}_{p_0,1}\bold{F}_{p_0,2}\dots \bold{F}_{p_0,p_0}\end{array}\right],
[z^n u^{r-1}] Q_d(z,u) = [v^n] (1-v^2) (1+v^2)^{n-1}\frac{2^{r-1}v^{2(r-1)+d}}{(1+v^2)^{r-1}}=2^{r-1}\,[v^{n-d-2(r-1)}] (1-v^2) (1+v^2)^{n-r}.
G' =\vec{e}_{R}\cdot \vec{g}=\left(\begin{array}{cccc}0 -g_{3}g_{2}-g_{1}g_{3}0 -g_{1}-g_{2}-g_{2}g_{1}0 -g_{3}g_{1}g_{2}g_{3}0 \end{array}\right) = \vec{e}^{~'}\cdot \vec{g}
c^{-1}=-m \det\Big[ \left[h_{i_1,j_1}(v_1)\right]_{\substack{i_1=1,...,p j_1=1,...,p_1}}~~~\cdots~~~ \left[h_{i_K,j_K}(v_K)\right]_{\substack{i_K=1,...,p j_K=1,...,p_K}}\Big].
\beta(r)_{Nut}^2=\frac{\lambda\,c\,\,2^{n}\,r\,(r-L)\,L^{n}}{(n+1)(r+L)^{n}}\,{\rm{Appell}}F_{1}[n+1,2,-n,n+2,1-\frac{r}{L},\frac{L-r}{2L}].
\forall \mathcal{O}\in \mathcal{B}(\mathbb{R}^d), \forall z \in \mathbb{R}^d,\: \forall t>0, \: P_{n,t}(z,\mathcal{O}) \triangleq \mathbb{P}(Z_t^n \in \mathcal{O}\vert Z_0^n = z)
\Big\Vert \partial_{\nu}\Tilde{\mathrm{H}}\Big\Vert_{\mathbb{H}^{\frac{1}{2}}(\partial \mathrm{B})}= \mathcal{O}\Big(1\Big) + \mathcal{O}\Big(\delta^2\Big) + \mathcal{O}\Big(|\log{\delta}|\delta^{3-h}\Big) + \mathcal{O}\Big(\delta^{3-h}\Big) + \mathcal{O}\Big(\delta^{4-h}\Big) = \mathcal{O}\Big(1\Big).
\tilde{\zeta}_{T_n}(t) =G_{T_n}(x_0)+\int_{0}^{t}a_0 \bigl(\tilde{\zeta}_{T_n}(s) \bigr)\,ds+\tilde{\alpha}^{(0)}_{T_n}(t)+\tilde{\alpha}^{(1)}_{T_n}(t)+\tilde{\eta}_{T_n}(t), \langle\tilde{\eta}_{T_n}\rangle(t) =\int_{0}^{t}\sigma_0^2 \bigl(\tilde{\zeta}_{T_n}(s) \bigr)\,ds+\tilde{\alpha}^{(2)}_{T_n}(t),
A = \sum_{e_{1}, e_{2}\in Q}\lim_{\delta \rightarrow 0}V T{\frac{| \beta_{1}+ \beta_{2}|}{( 2 \pi )^{2}}}\int_{\delta}^{\infty}d t t^{- 1}\int{\frac{d^{2 4}p}{( 2 \pi )^{2 4}}}e^{- \pi t p^{2}/ 2}T^{\perp}T^{\parallel}.
\frac{1}{2}\frac{d}{dt}Q(t) =\int_0^1 [X_1(t,S_1)-X_2(t,S_2)]\,[V_1(t,S_1)-V_2(t,S_2)]\,ds &+\int_0^1 [V_1(t,S_1)-V_2(t,S_2)]\,\bigl[\nabla\Psi_1\bigl(t,X_1(t, S_1)\bigr)-\nabla\Psi_2\bigl(t,X_2(t, S_2)\bigr)\bigr]\,ds.
\left(n-3\right)+\left(n-2\right)=2n-5=d+2n-3, \left(n-3\right)+\left(n-1\right)=2n-4=d+2n-3, \left(n-2\right)+n=2n-2=d+2n-3, \left(n-1\right)+n=2n-1=d+2n-3.
\begin{array}{c c c}{g_{A B}}&{=}&{\tilde{g}_{A B}( y^{C}) ,}\\{}&{}&{}\\{g_{i j}}&{=}&{a_{k}( y^{C}) a_{l}( y^{C}) \eta_{i j}^{k l},}\\{}&{}&{}\\{g_{A i}}&{=}&{0 .}\\ \end{array}
\frac{d X(s)}{ds}= \beta_\varepsilon\big(V(s)\big) + \big[V(s) - \beta_\varepsilon\big(V(s)\big)\big] \eta_\varepsilon\big(X(s)\big), X(t) = x begin{align*}5pt] \frac{d V(s)}{ds}= -\mathbf{B}\big(s,X(s),V(s)\big), V(t) = v
H_n(z) := (T_n\psi_n)(z)\frac{m(z)}{m_n(z)}\frac{B_n^2(z)S_{\dot\mu}^2(z)}{G_{\lambda_n}D_{\lambda_n}^2(z)}\left( \frac{\varphi(z)}\rho\right)^{2(n-d_n)}.
\sigma_{\infty,w_X}=\lim_{\epsilon \to 0}\frac{\int_{m<z^2-4d_1d_2xy<m+\varepsilon}w_{X/d_1}(x)w_{X/d_2}(y) dx dy dz}{\epsilon}, &\sigma_p(V_{m,d_1d_2}):=\lim_{k\to \infty}\frac{|V_{m,d_1d_2}(\mathbb{Z}/p^k\mathbb{Z})|}{p^{2k}}.
\partial_{tt}u-c^{2}\Delta u = f(\boldsymbol{x},t), \boldsymbol{x}\in\Omega,0<t<T, u(\boldsymbol{x},0) = u_{0}(\boldsymbol{x}),\; u_{t}(\boldsymbol{x},0) = v_{0}(\boldsymbol{x}), \boldsymbol{x}\in\Omega, u(\boldsymbol{x},t) = g(\boldsymbol{x},t), \boldsymbol{x}\in\partial\Omega,0<t<T,
\mathbb E_x^{(t)}(Z_t(Z_t-1)...(Z_t-n) s^{Z_t-n-1}, T_1\in dt_1, ..., T_n\in dt_n)/ dt_1...dt_n= & x\psi_t^{'}(s) \psi_t(s)^{x-1}\prod_{i=1}^n \psi_{t_i}^{'}(s) \big[ \sum_{k\geq 2}k(k-1)\mu (k) \psi_{t_i}(s)^{k-2}\big], s\in [0,1).
w_{n}:=w_n(p_1,\ldots,p_n) ={\frac{1}{n+1}}\sum_{p_1,\ldots, p_n \ge 0 \atop \sum j p_j=n}\binom{n+\sum p_j}{n,p_1, \ldots,p_n}y_1^{p_{1}}y_2^{p_{2}}\cdots y_n^{p_{n}},
A =\dfrac{1}{2}\left| \begin{matrix}F_{n+2k}-F_n F_{n+3k}-F_{n+k}F_{n+4k}-F_n F_{n+5k}-F_{n+k}\end{matrix}\right| =\dfrac{1}{2}\left| \begin{matrix}F_{n+2k}-F_n F_{n+3k}-F_{n+k}F_{n+4k}-F_{n+2k}F_{n+5k}-F_{n+3k}\end{matrix}\right|.
\hat{\mathbf{c}}^{\mbox{\tiny (\itshape k\upshape)}}\in \mathcal{S}_k := \Big\{\mathbf{c}' \in \mathbb{R}^{n}: \mathbf{B}^{\mbox{\tiny (\itshape k\upshape)}}\mathbf{c}' \leq \mathbf{b}^{\mbox{\tiny (\itshape k\upshape)}}\Big\}\subseteq \mathcal{S}_0
E(y) = 5 w_{\mu_i^{\prime},\zeta_i^{\prime}}(y)^4\Bigl( \varepsilon^{\frac{1}{2}}\pi_i(\varepsilon\, y)+ \sum_{j\not=i}w_{\mu_j^{\prime},\zeta_j^{\prime}}(y)+\varepsilon^{\frac{1}{2}}\pi_j(\varepsilon\, y)\Bigr) & + O( w_{\mu_i^{\prime},\zeta_i^{\prime}}(y)^3 \varepsilon^2) + O(\varepsilon^5), y \in B_{\delta/\varepsilon}(\zeta_i').
\left[1+\tau\beta_{k,\max}(2-\beta_{k,\min})\right]\left\Vert z^k-x^k\right\Vert^2-\frac{\beta_{k,\min}(2-\beta_{k,\max})}{C_g^2}\left(1-\frac{1}{\tau}\right)\sum_{j=1}^m\left(g^+_{\omega_{k,j}}(j|x^k_j)\right)^2.
\mathbb{P}_{\rm FA}+ \mathbb{P}_{\rm MD}= E_U[\mathbb{P}_{\rm FA}({\mathbf U}) + \mathbb{P}_{\rm MD}({\mathbf U})] & \geq E_U[\mathbb{P}_{\rm FA}({\mathbf U}) + \mathbb{P}_{\rm MD}({\mathbf U}) |{\mathcal A}^c] P({\mathcal A}^c) & > 1 - \epsilon.
\begin{array}{c}\ [h_\lambda, h_\mu]=0, \ \ \ [h_\lambda,k_\mu]=0, \ \ \ [k_\lambda,k_\mu]=0, \ [h_\lambda, x_\alpha]=-(\Phi_-\lambda|\alpha)x_\alpha,\ \ \ [k_\lambda,x_\alpha]=(\Phi_+\lambda|\alpha)x_\alpha, \ [x_\alpha,x_\beta]=2^{-1}([x_\alpha,x_\beta]_{\frak g}+[x_\alpha,x_\beta]_{\frak g'})\end{array}
\dim(\hat{\sigma}(X))^2=3^a(u_1^{-(b+c+d)+2}u_2^{b+2})^2\beta^d\qquad\qquad\dim(\hat{\sigma^2}(X))^2=3^a(u_1^{c-2}u_2^{-(b+c+d)+4})^2\beta^d.
a_1:= \frac{4(p+1)}{3(p-1)}, m_1:= \frac{2(p-1)(p+1)}{5-p}, n_1:= \frac{2(p-1)(p+1)}{3p^2-5p-2}, a_2:= \frac{4(q+1)}{3(q-1)}, m_2:= \frac{2(q-1)(q+1)}{5-q}, n_2:= \frac{2(q-1)(q+1)}{3q^2-5q-2},
PL_b(s_{v_0})((P)) =R_{y_1}(s_{v_0})\cdots R_{y_r}(s_{v_0}) ((P)) &=p_b^*(P) +\sum_{i=1}^r\sum_{j=1}^{e_i}\left(e_iw_0^{y_i}+\sum_{j=1}^{e_i-1}(k-j)(w_{-j}^{y_i}+w_j^{y_i})\right) &= p_b^*((P)+v_0)=p_b^*(s_{v_0}((P)).
\int_\Omega (-\psi ) d \mu =\sum_{j\geq 1}\int_\Omega (1+ \gamma_\Omega ) dd^c u_{j,k_j}=\sum_{j\geq 1}\int_{\Omega \cap \{\gamma_\Omega < -1+ \frac{1}{2^{j}}\}}(1+ \gamma_\Omega ) dd^c u_{j,k_j}& \leq \sum_{j\geq 1}\frac{1}{2^{j}}\int_{\Omega'}dd^c u_{j,k_j}\leq 1.
\begin{aligned}G^{[3]}= W_\alpha\P\left[Q_\alpha\bar W_\alpha - \bar Q_\alpha W_\alpha\right] - \P\left[(Q_\alpha \bar W_\alpha - \bar Q_\alpha W_\alpha)(W_\alpha + \bar W_\alpha)\right], &K^{[3]}= Q_\alpha^2 W_\alpha + Q_\alpha\P\left[Q_\alpha\bar W_\alpha - \bar Q_\alpha W_\alpha\right] +\P\left[Q_\alpha\bar Q_\alpha(W_\alpha+\bar W_\alpha)\right],\end{aligned}
\mathcal{B}(x,y)=\frac{1}{2}\left(\frac{x+y}{|x+y|}-\frac{x-y}{|x-y|}\right)=\frac{1}{2}\left((x+y)-(x-y)\right)={\begin{cases}(y){}|x| < |y| (y)/2{}|x|=|y|, y\ne 0 0{}\end{cases}}
\Vert z-y\Vert \leq 2\lambda \frac{\left( \beta -\alpha +\frac{r}{2}\right)\left( 1-2\delta \right)}{\left( 1-2\left( \delta +r\lambda \right) \right)}:=M_{\lambda}\ \forall\y\in P_{\lambda}(f)(z)\ \forall\z\in B_{\delta}(x).
g_3(C^{\rm R}_n,C^{\rm I}_n) = 1- \left(\frac{1}{2}+\frac{1}{2}{\rm{erf}}\big(\sqrt{C^{\rm R}_n}-\sqrt{P/2}\big)\right)\left(\frac{1}{2}+\frac{1}{2}{\rm{erf}}\big(\sqrt{C^{\rm I}_n}+\sqrt{P/2}\big)\right).
[ \hat{x}_{1}^{\prime}, \hat{x}_{2}^{\prime}] = i \alpha^{2}, \hspace{0 . 4 c m}[ \hat{p}_{1}^{\prime}, \hat{p}_{2}^{\prime}] = i \alpha^{- 2}, \hspace{0 . 4 c m}[ \hat{x}_{i}^{\prime}, \hat{p}_{j}^{\prime}] = i \delta_{i j}
\langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}})-1}{r^{2}}\right)v_{1}\vert_{r=R\lambda(t)},\phi_{0}\rangle = \frac{-16}{\lambda(t)^{3}}\int_{t}^{\infty}ds \lambda''(s) K(s-t,\lambda(t)) + \frac{-16}{\lambda(t)^{3}}\int_{t}^{\infty}ds \lambda''(s) K_{1}(s-t,\lambda(t))
- \Bigl[ \left[ Q(\xi) , Q(\eta) \right] , b (k, \sigma) \Bigr] \stackrel{!}{=}\Bigl[ \left[ Q(\eta) , b (k, \sigma) \right] , Q(\xi) \Bigr] + \Bigl[ \left[ b (k, \sigma) , Q(\xi) \right] , Q(\eta) \Bigr]
\frac{d}{ds}w(e^{-\theta_*s},s) =\frac{d}{2}w(e^{-\theta_*s},s)+\frac{U_d'(r(s))}{U_d(r(s))}r'(s)w(e^{-\theta_*s},s) +e^{\frac{d}{2}s}U_d(r(s))(\partial_t u_*)(0,t(s))t'(s) +e^{\frac{d}{2}s}U_d(r(s))[(\partial_r F_N^0)(r(s),t(s))r'(s)+(\partial_tF_N^0)(r(s),t(s))t'(s)]
I_2 = \int_{s/4<|y-x|\le r_0/4}\left(2^{\eta}\left(\frac{4|y-x|}{s}\right)^{\eta}-1\right) \frac{f(|y-x|^{-1})}{|y-x|^d}dy & = \omega_d \int_s^{r_0}\left(2^{\eta}(t/s)^{\eta}-1\right)\frac{f(4/t)}{t}dt
\left\langle \int\, d^4x\, \sqrt g\, s\, \Delta_4\,f_{\epsilon}(x,x_0)\right\rangle_{_V}= \left\langle \int\, d^4x\, \sqrt{\overline g}\, s\, \overline\Delta_4 e^{\beta_8\sigma}\overline f_{\epsilon}(x,x_0)\right\rangle_{_V}\sim s V^{\beta_8\over \beta_0}
\mathbb{E}\left[e^{-u\eta}\right] =\left(1-e^{-\alpha}\right)^{-1}\left(\widehat{m}_{\alpha,\beta}(-u)-e^{-\alpha}\right) =\left(1-e^{-\alpha}\right)^{-1}\left(\exp\left\lbrace \frac{-\alpha u}{\beta+u}\right\rbrace -\exp\left\lbrace -\alpha\right\rbrace \right).
\Delta S=\int_{i}^{f}dS=\int_{V_{i}}^{V_{f}}\frac{PdV}{T}=\int_{V_{i}}^{V_{f}}\frac{nRdV}{V}=nRln\frac{V_{f}}{V_{i}}=Nk_{B}ln\frac{V_{f}}{V_{i}}.
\lim_{u\to D}\frac{g_{\kappa,u}(g^{-1}(u)x)}{q(u)}=\lim_{u\to D}\frac{\sum_{i=0}^\kappa(g(M^i g^{-1}(u)x)-u)}{q(u)}=\sum_{i=0}^\kappa(1-(M^{1/\alpha})^i x^{1/\alpha}) &=\kappa+1-\frac{1-M^{(\kappa+1)/\alpha}}{1-M^{1/\alpha}}x^{1/\alpha}
\frac{2n+2}{3}f(1,4)+\frac{n-5}{3}f(4,4)= \frac{2}{3}(n+1) e^3+\frac{1}{3}(n-5) e^{3 \sqrt{2}}, begin{align*}2mm]2f(1,3)+f(3,4)+\frac{2n-5}{3}f(1,4)+\frac{n-7}{3}f(4,4)= 2 e^2+e^{\sqrt{13}}+\frac{1}{3}(2 n-5) e^3+\frac{1}{3}(n-7)e^{3 \sqrt{2}},
{\overline{\zeta_n}(z)=\int_M\zeta_n\mathrm{d}\mu+O(1)\Big(\sum_{{l}=1}^m (|A_{n,{l}}|+e^{-\frac{\lambda_{n,j}^{(1)}}{2}})(\frac{e^{\frac{\lambda_{n,j}^{(1)}}{2}}}{|z|}+1)\Big),}\ \ \textrm{and thus}
T = \tilde{O}\left( \left[ \frac{2^\frac{r^2 + 1}{r}\sigma_q}{\mu_r^{1/r}}\cdot \frac{1}{\varepsilon^{\frac{(r-1)}{r}}}\right]^\frac{1+\kappa}{\kappa}\right), T_k = \tilde{O}\left( \left[ \frac{\sigma_q 2^{(1+r)}}{\mu_r R_0^{r-1}}2^{k(r-1)}\right]^\frac{1+\kappa}{\kappa}\right).
\mathbb F^0_{p,q}:=\mathbb F_{p,q}\cap\left(\mathbb C^n\times\{0\}^{\tau(1)-1}\times\mathbb C\times\{0\}^{m-\tau(1)}\right), \mathbb F^0_{\tilde p,q_{\tau}/t}:=\mathbb F_{\tilde p,q_{\tau}/t}\cap\left(\mathbb C^{n+1}\times\{0\}^{m-1}\right).
\hat{z}_j={\bf g}_{jB}^H ({\bf x}_B+{\bf e}_B)+\sum_{\substack{i=1 i\neq j}}^{J}{\bf g}_{ji}^H ({\bf x}_i+{\bf e}_i)+{\bf g}_{jj}^H{\bf e}_j + w^{'}_j + n^{'}_j, \forall j\in\mathcal{D},
\lim_{N \rightarrow \infty}\left. N \, P_{2 n + 1}( \frac{x}{N}) \right|_{n = N t / 2}= \frac{\sin u x}{u}= x \, \Gamma ( \frac 3 2 ) \frac{J_{\frac 1 2}\left( u x \right)}{\left( u x / 2 \right)^{\frac 1 2}}.
N_{(1,1)}(n) = 16\,\sigma(n) - 64\,\sigma(\frac{n}{4}) + 64\, W_{(1,1)}(n) + 1024\, W_{(1,1)}(\frac{n}{4}) - 512\,W_{(1,4)}(n) = 16\,\sigma_{3}(n) - 32\,\sigma_{3}(\frac{n}{2}) + 256\,\sigma_{3}(\frac{n}{4}),
\varphi_{1}\left( n_{0}\right) :=2\,B_{1}C_{3}\,c\,P_{n}^{\beta-\kappa(\beta-\mu)}+C_{1}\left( P_{n}^{\sigma-\delta-\kappa(\beta-\mu)}+g^{(\sigma-\delta)_{+}}P_{n}^{\kappa(\sigma-\delta)_{+}-(\delta-\sigma)_{+}-\kappa(\beta-\mu)}\right)
\Phi (\Phi^{-1}(\kappa)) \leq \mathbb{P}(X_i < 0 \mbox{for all}i=1, \dots, k) &= \mathbb{P}( X_i + \Phi^{-1}(\kappa) < \Phi^{-1}(\kappa) \mbox{for all}i=1, \dots, k ) &= F_{Y^*}(\Phi^{-1}(\kappa)),
\lim_{\varepsilon \searrow 0}\epsilon(q^2) \:\int_{-\infty}^\infty d\tau\:\frac{1}{((\tau^2 -\frac{1}{4})\: q^2 \pm i \varepsilon)^{\alpha+1}}\:H^{(\beta)}_z \:{\mathcal{P}}(\tau) \:e^{-iq \tau(y-x)}\;.
\mathrm{E}_{1}\left( \eta_{1}, \cdots , \eta_{p}\right) = \eta_{1}^{a_{1}}\cdots \eta_{p}^{a_{p}}\, \, h^{a}( \phi ) \partial_{a}\, \, \partial_{a_{1}}^{( 1 )}\cdots \partial_{a_{p}}^{( p )}F_{p}( \phi , \cdots , \phi )
\begin{array}{l}{{{I_{+}}^{T}={I_{+}}^{*}={( I_{+})}^{+}, \quad{( I_{+})}^{2}= 1 , \quad \mid I_{+}\mid = 1 ,}}\\{{({I_{+}})_{( 1 , 1 , \alpha )}^{( 2 , 1 , \alpha )}= ({I_{+}})_{( 2 , 2 , \alpha )}^{( 1 , 2 , \alpha )}= 0 , \quad ({I_{+}})_{( 1 , 1 , \alpha )}^{( 1 , 1 , \alpha )}= ({I_{+}})_{( 2 , 2 , \alpha )}^{( 2 , 2 , \alpha )}= 1 .}}\\ \end{array}
{\lim_{\left| k_{2 \mu}\right| \rightarrow \infty}}T_{\mu \nu}^{( 3 ) a b c d}T_{\mu^{\prime}\nu^{\prime}}^{( 3 ) a^{\prime}b^{\prime}c d *}Q^{\mu \mu^{\prime}}( k_{1}) Q^{\nu \nu^{\prime}}( k_{2}) = \widetilde{S}
\begin{aligned}\partial_t\textbf{u}+(\textbf{u}\cdot\nabla)\textbf{u}+\nabla P=\varepsilon^2\Delta\textbf{u}-\varepsilon^4\nabla\cdot(\nabla Q \odot\nabla Q+(\Delta Q)\cdot Q-Q\cdot\Delta Q) \end{aligned}
\frac{1}{- \frac{\pi}{4}+ 2 m \pi}e_i =(-1) \times \frac{1}{- \frac{7}{4}\pi + 2 (- m+1) \pi}e_i, \frac{1}{- \frac{5}{4}\pi + 2 m \pi}e_i =(-1) \times \frac{1}{- \frac{3}{4}\pi + 2 (- m+1) \pi}e_i,
\left(j_{w\lambda}^{\vee}\left(u\right),y\right)_{L}=\left(u,\pi_{w\lambda}^{\vee}\left(y\right)\right)_{\lambda}^{\varphi}(=\left(u,y^{\vee}.u_{w\lambda}\right)_{\lambda}^{\varphi}=\left(\varphi\left(y^{\vee}\right).u,u_{w\lambda}\right)_{\lambda}^{\varphi}).
\mathbf{Z}= \bigcap_{n \in \mathbb{N}}\mathop{\bigcup_{s \in \mathbb{N}}}_{\sigma \in \mathbb{N}^{< \mathbb{N}}}\bigcap_{t \geq s}[I_{\sigma,n,t}] = \bigcap_{n \in \mathbb{N}}\bigcup_{\sigma \in \mathbf{C_n^{\infty}}}[\mathbf{I_{\sigma,n}^{\infty}}]
(B^2 \cdot E) + \frac{a_{i_1}}{a_{\xi}^3}E^3 = \frac{a_{\xi}+ a_{i_1}}{a_{\xi}^3}E^3 = \frac{a_{\xi}+ a_{i_1}}{a_{i_2}a_{i_3}a_{\xi}\bar{a}_{\zeta}},
\mathcal{L}= e^{5 f}G^{-1}\left( \frac{\dot{G}^2}{2}+ 25 \dot{f}^2 G^2+10 G \dot{G}\dot{f}- \dot{l}^2 G^2-2 \dot{k}^2 G^2-\frac{G}{2}(\dot{a}^2+\dot{\tilde{a}}) \cosh2k - G \dot{a}\dot{\tilde{a}}\sinh2k\right.
x_{1,2}= \pm\alpha_2, x_{5,6}= \pm\sqrt{|s|^2 - |ac| + \sqrt{(|s|^2 - |ac|)^2 - 1}}, x_{3,4}= \pm |s - \sqrt{ac}|, x_{7,8}= \pm\sqrt{|s|^2 - |ac| - \sqrt{(|s|^2 - |ac|)^2 - 1}},
6 g^{6}{\frac{p^{4}}{M^{4}}}l n{\frac{m_{0}^{2}+ p^{2}}{m_{0}^{2}}}- 2 g^{6}l n{\frac{\Lambda^{2}}{m_{0}^{2}}}+ 2 g^{6}l n{\frac{m_{0}^{2}+ p^{2}}{m_{0}^{2}}}+ . . .
D^S_{nq}:= \sum_{p = 1}^{q-1}\bar{U}^{(pq)}_h, D^T_{nq}:= \sum_{p = 1}^{q-1}W_h^{(pq)}, D^Z_{nq}:= \sum_{p = 1}^{q-1}U_h^{(pq)}
\sum_{\substack{{d'|d}{d'\geq p^{\frac{C}{\log\log p}}}}}\left|\mu\left(\frac{d}{d'}\right)\right| d' \leq \sum_{d'|d}\left|\mu\left(\frac{d}{d'}\right)\right| d' \leq d \prod_{p|d}\left(1+\frac1p\right) =d \prod_{p|d}\frac{1+\frac1p}{1-\frac1p}\left(1-\frac1p\right) \leq \phi(d) 3^{\omega(d)}
(a;q)_n = \prod_{1\leq t < n}(1-aq^t), (a;q)_\infty = \prod_{1\leq t}(1-aq^t), (a_1,a_2,\dots,a_j;q)_m =(a_1;q)_m(a_2;q)_m\cdots(a_j;q)_m.
\sum_{n=u_{\ell}}^{u_{\ell+1}-1}h(n) =\sum_{n=u_{\ell}}^{v_{\ell+1}-1}h(n)+ \sum_{n=v_{\ell+1}}^{u_{\ell+1}-1}h(n) \leq v_{\ell+1}\psi(u_\ell) + u_{\ell+1}\|v_{\ell+1}\theta \| &\leq \frac{v_{\ell+1}}{2(\ell+1)^2v_{\ell+1}}+\frac{u_{\ell+1}}{2(\ell+1)^{2\tau+2}v_{\ell+1}^{\tau}}\leq\frac{1}{(\ell+1)^2},
Z_{m}( \kappa ) ={\frac{\Gamma ( 1 / ( 2 m ) ) e^{- 1 / ( 4 \kappa )}}{2^{1 - 1 / m + ( - 1 )^{m}/ 2}\sqrt{2 \pi}\kappa^{1 / 2 - 1 / ( 2 m )}}}{}_{1}F_{1}( 1 / ( 2 m ) , 1 / 2 ; 1 / ( 4 \kappa ) )
\alpha(\nu) := \left( \begin{array}{cc|cc}\nu 0 \sqrt{\nu^2-\frac{1}{4}}0 0 \nu 0 -\sqrt{\nu^2-\frac{1}{4}}\hline \sqrt{\nu^2-\frac{1}{4}}0 \nu 0 0 -\sqrt{\nu^2-\frac{1}{4}}0 \nu \end{array}\right)\;,
f^{\ast}(u) =f^{\ast}(f^{\prime}(x))=\sup_{x\in\mathbb{R}}[xu-f(x)]=xf^{\prime}(x)-f(x)=\sum_{n\geq1}e^{\sigma_{n}x}\left( \sigma_{n}x-1\right) & =\sum_{n\geq1}f_{n}^{\ast}(f_{n}^{\prime}(x))=\min\bigg\{\sum_{n\geq1}f_{n}^{\ast}(u_{n})\mid(u_{n})_{n\geq1}\subset\mathbb{R}_{+},\u=\sum_{n\geq1}u_{n}\bigg\},
\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right]n(-1)^{n-1}d^nq^{n(n+1)/2}+(dq)_N\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right]\frac{d^nq^{n(n+1)}}{(dq)_n(1-q^n)}=\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right]\frac{(dq)^n(dq)_{N-n}}{1-q^n}.