harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{\{\alpha,\beta;\epsilon\}\in\mathcal{S}_{(d\varphi,\iota)}\times\{\pm\}}\left(e^{\frac{1}{2}(\ell_{\alpha}(\phi)+\ell_{\beta}(\phi))}+(-1)^{\alpha\cdot\beta}\right)^{-1}=t\;\mathrm{mod}_q(d\varphi)-t\;\mathrm{mod}_q(d\varphi)=0.
\int_{\mathbb{R}^3_v}fdv = 1 + (a+ 3c), &\int_{\mathbb{R}^3_v}vfdv = b, &\int_{\mathbb{R}^3_v}\vert v\vert^2fdv = 3 + 3(a+5c), &\int_{\mathbb{R}^3_v}v_i v_j fdv = \delta_{ij}+ (a+5c)\delta_{ij}+ (v_i v_j \mu^{1/2}, g_2), &\int_{\mathbb{R}^3_v}\vert v\vert^2 vfdv = 5b + (\vert v \vert^2 v \mu^{1/2}, g_2).
\hat\theta_n^{(4)}(m)=\frac{X_{n^{m-1}}^2}{2\int_0^{n^{m-1}}X^2_t\,dt+2\vartheta_n}=\left(\frac{1}{\hat\theta_{n^{m-1}}^{(2)}}+2\cdot\frac{e^{2\theta n^{m-1}}}{X_{n^{m-1}}^2}\cdot\frac{\vartheta_n}{e^{2\theta n^{m-1}}}\right)^{-1}\to\theta
m_{h^{0}}^{2}\le m_{Z^{0}}^{2}cos^{2}2\beta+\frac{3}{\pi^{2}}\frac{m_{t}^{4}sin^{4}\beta}{v^{2}}(log\frac{m_{\tilde{t}}}{m_{t}}+a^{2}(1-a^{2}/12))
S^{-1}(\tilde{C}) = \sum_{i = 1}^{N}q^{-3i}S^{-1}(E_{\xi_{i}}) S^{-1}(E_{\xi_{i}}^{*}) = \sum_{i = 1}^{N}q^{-3i}S^{-1}(E_{\xi_{i}}) S(E_{\xi_{i}})^{*}.
\phi_1 := \phi_{12B}(\tau, z) = 2 \zeta^{\pm 1}-4 + (-4 \zeta^{\pm 2}+ 8 \zeta^{\pm 1}- 8) q + (2 \zeta^{\pm 3}- 8 \zeta^{\pm 2}+ 20 \zeta^{\pm 1}- 28)q^2 + O(q^3)
\begin{array}{cc}{\rm y}_{\pm}=\left(\begin{array}{cc}0 -(y_{\pm}{\cdot\sigma})^{t}y_{\mp}{\cdot\sigma}0\end{array}\right)~~~~ ~~~~y^{\mu}_{\pm}=y^{\mu}\pm i\bar{\phi}{}^{a}\tilde{\sigma}^{\mu}\phi_{a}=x^{\mu}_{\pm}\end{array}
A_{(b)}=g^4\int d^{26}k \int_{0}^{1}dx_1 dx_3 dz_2 dz_4~x_{1}^{-\alpha (-(k-p_1)^2) -1}\,x_{3}^{-\alpha (-(k-p_3)^2) -1}\,(z_{2}z_{4})^{-\alpha (-(p_1-p_3)^2) -1}.
\boldsymbol{\Phi}_{\boldsymbol{\phi}}^{H}\mathbf{r}= \sqrt{N_c}\mathbf{F}\mathrm{Diag}\left(e^{j\boldsymbol{\theta}}\right)^H\mathbf{F}^H\mathbf{r}= \sqrt{N_c}\mathbf{F}\mathrm{Diag}\left(e^{j\boldsymbol{\theta}}\right)^H\mathbf{y},
\frac{\mathbf{P}\Big( a_n S_n \geq x \sqrt{v_n + na_n^2 \sigma^2}\, \Big)}{1-\Phi \left( x\right)}=1+o(1) \ \ \and \ \ \ \frac{\mathbf{P}\Big( a_n S_n \leq- x \sqrt{v_n + na_n^2 \sigma^2}\, \Big)}{\Phi \left( -x\right)}=1+o(1)
\left( \begin{array}{rr}x y - \bar{y}\bar{x}\end{array}\right) \left( \begin{array}{cc}\exp [(r i-1) f(x,y)] g(x,y) 0 \exp [(1- r i) f(x,y)] \end{array}\right) = \left( \begin{array}{rr}u v - \bar{v}\bar{u}\end{array}\right).
\sum_{\nu=0}^N\left[\omega_\nu^2\eta_{\nu\rho}+\sum_{k=1}^N\lambda_k\left(\eta_{k\rho}\delta_{0\nu}+\eta_{0\rho}\delta_{k\nu}\right)\right]\eta_{\nu\sigma}=\delta_{\rho\sigma}\Omega_\sigma^2\;.
\varphi_{Y}^{2}(\alpha_{n}^{(N)})={\displaystyle\sum\limits_{i=1}^{d}}{\displaystyle\sum\limits_{j=1}^{d}}\left( \alpha_{n}^{(N)}(f_{i}f_{j})\right)^{2}, \varphi_{Y}^{2}(\mathbb{G}^{(N)})={\displaystyle\sum\limits_{i=1}^{d}}{\displaystyle\sum\limits_{j=1}^{d}}\left( \mathbb{G}^{(N)}(f_{i}f_{j})\right)^{2}.
\int_{-\infty}^{\infty}p_{_1}(y) \log \frac{1}{p_{_1}(y)}dy + \eta_0 \left[\mathcal{D}(p_{_0}||p_{_1})-2\epsilon^2\right] &+ \eta_1 \left[\int_{-\infty}^{\infty}p_{_1}(y) dy -1\right] + \eta_2 \left[\int_{-\infty}^{\infty}y^2 p_{_1}(y) dy - P_y\right] & = \int_{-\infty}^{\infty}\overline{\mathcal{L}}(y, p_{_1}(y)) dy - c,
u_x^{m,n}= \frac{\delta y_{m,n}\, \Delta u_{m,n}- \Delta y_{m,n}\, \delta u_{m,n}}{\Delta x_{m,n}\, \delta y_{m,n}- \Delta y_{m,n}\, \delta x_{m,n}},u_y^{m,n}= \frac{\Delta x_{m,n}\, \delta u_{m,n}- \delta x_{m,n}\, \Delta u_{m,n}}{\Delta x_{m,n}\, \delta y_{m,n}- \Delta y_{m,n}\delta x_{m,n}},
\langle \Psi |\textit{\textbf{G}}_2|\Psi \rangle =\frac{i}{2\pi}\sum_{n\neq 0}\frac{q_0+q_\pi}{2n}\langle \Psi |\mbox{\boldmath $\alpha_n\times \alpha_{-n}$}|\Psi \rangle =\frac{q}{2\pi}\langle \Psi |\textit{\textbf{S}}|\Psi \rangle .
h(k+1,l+1)-h(k,l+1) &=\left(\frac{\ell}{\ell+1}-\frac{f(k+1)}{\ell+1}\right)h(k+1,\ell)+\frac{f(k+1)}{\ell+1}h(k,\ell) -\left(\frac{\ell}{\ell+1}-\frac{f(k)}{\ell+1}\right)h(k,\ell)-\frac{f(k)}{\ell+1}h(k-1,\ell) &=\left(\frac{\ell}{\ell+1}-\frac{f(k+1)}{\ell+1}\right)\left(h(k+1,\ell)-h(k,\ell)\right)+\frac{f(k)}{\ell+1}\left(h(k,\ell)-h(k-1,\ell)\right),
( Z_{\bf m}^{N})_{\bf k l}\equiv \int d^{2}\sigma f_{\bf k}^{*}( \sigma ) \, Z_{\bf m}^{N}( \sigma ) \, f_{\bf l}( \sigma ) = - 2 i \sin \left( \frac{2 \pi}{N}{\bf m \times l}\right) \delta_{{\bf m}\, ({\bf k - l})}\, .
0= \langle \tau_{H}, \alpha_{1}\rangle \{(2m_{2}+m_{1}+m_{3}) (1-(\cot \langle \alpha_{1}, H \rangle)^{2}) & +m_{3}(1-(\tan \langle \alpha_{1}, H \rangle)^{2})+4m_{3}\}(\alpha_{1}+\alpha_{2}).
H_{0}= \int d p f ( p ) A_{f}^{\dagger}( p ) A_{f}( p ) = \int_{x}[{\frac{1}{2}}\Pi_{x}^{2}+{\frac{1}{2}}( \partial_{x}\phi_{x})^{2}+{\frac{1}{2}}\mu^{2}( \phi_{x}- \varphi )^{2}-{\frac{1}{2}}I_{0}( \mu ) ] \; .
A_{k}^{i}\left. \left| 0 \right\rangle \! \right\rangle = B_{m}^{j}\left. \left| 0 \right\rangle \! \right\rangle = \tilde{A}_{k}^{i}\left. \left| 0 \right\rangle \! \right\rangle = \tilde{B}_{m}^{j}\left. \left| 0 \right\rangle \! \right\rangle = 0 .
\frac{d}{d\tau}(\rho^2(m_\tau)) = d(\rho^2)(-I_1T(m_\tau)) = 4x_1(m_\tau), \frac{d}{d\tau}(x_1(m_\tau)) = dx_1(-I_1T(m_\tau)) = -g_0(I_1T,-I_1T) = |T|^2_{g_0}(m_\tau).
\begin{aligned}R_{c,k}^{Oip}=\left(1-\frac{K}{2T}\right)\eta\log_2\left(1+\frac{\lambda_{k,g}^O\beta_g^k\gamma_{k,g}^OM}{p_d\beta_g^k+1}\right), & k=1,\ldots, K/2\end{aligned}
A \simeq \frac{1}{\sqrt{2}}\begin{pmatrix}0 g \tfrac{y}{2}\dot{g}0 -g 0 -\tfrac{x}{2}\dot{g}0 -\tfrac{y}{2}\dot{g}\tfrac{x}{2}\dot{g}0 \dot{g}0 0 -\dot{g}0\end{pmatrix}, \det A = \frac{g^2 \dot{g}^2}{4}> 0.
\phi_R \rightarrow B^{\prime}e^{-\gamma a x}e^{\gamma a L}\frac{\Gamma(2\gamma) \Gamma(1+2\sigma)}{\Gamma (1+\gamma +\sigma +\lambda) \Gamma (\gamma + \sigma - \lambda)}+B^{\prime}e^{\gamma a x}e^{-\gamma a L}\frac{\Gamma(-2\gamma) \Gamma(1+2\sigma)}{\Gamma (1-\gamma +\sigma -\lambda) \Gamma (-\gamma + \sigma - \lambda)}
S(H)(t):= \left \{\mathbf{y_3}\in \{0, \ldots, t-1\}^{n_1}\left\vert \begin{array}{ll}\left(\forall u \in H^{-1}(\mathbb{Z}), y_u=I(H(u))(t)\right) \, \wedge \left(\forall u \not \in H^{-1}(\mathbb{Z}), H(u)_1 \le y_u \le t+H(u)_2\right) \end{array}\right. \right\}.
(s_{\mu\gamma}s_{(\mu\gamma)^*}s_{\mu}s_{\nu^*})^*(r s_{\mu\gamma}s_{(\mu\gamma)^*}s_{\mu}s_{\nu^*}) = (r s_\nu s_{\mu^*})s_{\mu\gamma}s_{(\mu\gamma)^*}s_{\mu\gamma}s_{(\mu\gamma)^*}s_{\mu}s_{\nu^*}& = s_{\mu\gamma}s_{(\mu\gamma)^*}(r s_{\nu}s_{\mu^*}) s_\mu s_{\nu^*}&= r s_{\mu\gamma}s_{(\mu\gamma)^*}s_{\nu}s_{\nu^*}.
\{(\imath, -\imath +2-2k)\mid k=0,\dots, 2n-1-\imath\ \ \imath=n+1,\dots, 2n-1\}&\cup\{(n, -n+\frac{3}{2}-k)\mid k=0, \dots, 2n-2\}&\cup\{(\imath, -\imath+1-2k)\mid k=0,\dots, 2n-2-\imath\ \ \imath=1,\dots, n-1\}.
\Psi'_N(t) ~ =2t H(t)\left(1+t^2+\varphi_N(t^2) \right)+2t(1+g(t))\left(\frac{t^{2N}}{N!}-t^2 \right) &=2t H(t)\varphi_N(t^2)+2t\left(1+\frac{t^{2N}}{N!}\right)(1+g(t))+g'(t)(1+t^2)\,.
\lim_{N\rightarrow \infty}\frac{1}{N}\sum_{n=0}^{N-1}1_B(T_{1}^{3n^2}T_{2}^{8n^2}x)(1_{A}-\mathbb{E}(1_A|\mathcal{A}))(T_{1}^{n^2}T_{2}^{-n^2}x)=0.
S(x)= S^{k+2, k+2}= \left(\begin{array}{cccc}0_{k-1, k-1}0_{k-1, 3}begin{align*}6pt]0_{3, k - 1}S^{3,3}\end{array}\right) \mbox{where}S^{3,3}= \left(\begin{array}{cccc}0 0 \lambda_{k+2}\alpha begin{align*}6pt] 0 0 \lambda_{k+2}\beta begin{align*}6pt] 0 0 0\end{array}\right).
\xi_{s_{\alpha_\tau}\cdot z,v}= \psi_{\sigma_v(3),v}z^{h_{\sigma_v(3),v}-h_{\tau_v(3),v}}\varepsilon^2 \otimes \psi_{\sigma_v(2),v}z^{h_{\sigma_v(2),v}-h_{\tau_v(2),v}}\varepsilon \otimes \psi_{\sigma_v(1)}z^{h_{\sigma_v(1),v}-h_{\tau_v(1),v}}.
\begin{array}{l}{{\Theta_{1}\equiv \pi_{0}\approx 0 \hspace{1 . 5 c m}\Theta_{2}\equiv E^{\prime}+ \phi^{\prime}+ \pi + A_{1}\approx 0}}\\{{\Theta_{3}\equiv E \approx 0 \hspace{1 . 5 c m}\Theta_{4}\equiv - \pi - \phi^{\prime}- 2 A_{1}+ A_{0}\approx 0}}\\ \end{array}
d\rho_i(t)+\sum_{j\in N(i)}\theta(\rho_i,\rho_j)(S_j-S_i)dt+\sum_{j\in N(i)}\theta(\rho_i,\rho_j)(\Sigma_j-\Sigma_i) \circ dW_t=0, &dS_i(t)+\sum_{j\in N(i)}\frac 12(S_j-S_i)^2 \frac{\partial \theta}{\partial \rho_i}(\rho_i,\rho_j)dt+\sum_{j\in N(i)}(S_i-S_j)(\Sigma_i-\Sigma_j)\frac{\partial \theta}{\partial \rho_i}(\rho_i,\rho_j) \circ dW_t=0.
{\textrm{SINR}}_d^i = \frac{{|h_{si}|^2 |h_{id}|^2 \gamma_s}}{{|h_{id}|^2 (|h_{pi}|^2 \alpha \gamma_p + 1) + |h_{si}|^2 (|h_{pd}|^2 \alpha \gamma_p + 1)}},
Kir_k \Phi(x) = \frac{1}{P_k(x)}\int_{y\in\mathbb{R}}\Phi(x,y)\mathcal{X}_{[-\tfrac{1}{k},\tfrac{1}{k}]}(x-y) dy &= \frac{\pi}{k}(1+k^2x^2)\int_{x-\tfrac{1}{k}}^{x+\tfrac{1}{k}}\Phi(x,y) dy &= 2\pi\left(\frac{1}{k^2}+x^2\right)\frac{k}{2}\int_{x-\tfrac{1}{k}}^{x+\tfrac{1}{k}}\Phi(x,y) dy.
\hat{p}(t,x,\xi)=\mathbb{E}\left[e^{2\pi i\langle \xi, X_t^x\rangle}\right],\quad\hat{p}_l^\eta(t,x,\xi)=\mathbb{E}\left[\eta(U_t)e^{2\pi i\langle \xi, X_l(t,x)\rangle}\right].
\Gamma_{\nu}= \varepsilon^{\mu , [ \mu \nu ]}+ \varepsilon^{[ \mu \nu ] , \mu}+ \varepsilon^{\nu , 0}+ \varepsilon^{o , \nu}+ \varepsilon^{\widetilde{\nu}, \widetilde{0}}+ \varepsilon^{\widetilde{0}, \widetilde{\nu}}+ \frac 1 2 e_{\mu \nu \rho \omega}\left( \varepsilon^{\widetilde{\mu}, [ \rho \omega ]}+ \varepsilon^{[ \rho \omega ] , \widetilde{\mu}}\right)
\sum_{k=0}^\infty\sum_{j=0}^\infty \ell^{2k+1}E_{2k}\frac{(-1)^j (\pi^2/\alpha^2 - \omega)^j}{j!}\left[ \int_{\gamma}\xi^{2k}\frac{\mathrm{d}^j}{\mathrm{d}\omega^j}m_\omega (\xi) \ \mathrm{d}\xi \right]_{\omega = \pi^2/\alpha^2}
A_{77,\ell}= 8^2 \int_{C/\ell}^{\pi -C/\ell}\frac{\gamma_4^2}{\ell^8}\left( 1+2 \frac{\alpha_2^2}{\ell^4}\right) \sin \phi d \phi- \frac{8^3}{\ell^{12}}\int_{C/\ell}^{\pi -C/\ell}\gamma_4 \alpha_2 \beta_3^2 \sin \phi d \phi +O(\ell^{-2}).
{\rho}/{\rho_0}=1+{\epsilon}\rho_i^{(1)}+{\frac{\epsilon}{\pi}}\left({\frac{\sqrt{2}\mu\sin{2\mu\pi}}{\cos^2\mu\beta-\sin^2\mu\pi}}\right){\sqrt{1-\frac{r}{c_0\kappa_0t}}}+O\left(\frac{\epsilon^2}{\sqrt{1-{r}/{c_0\kappa_0t}}}\right).
\left|\langle h,h_{f}^{\alpha}(g)\rangle_{A_{\alpha}^{2}}\right|=|T_{f}^{\alpha}(g,h)|\le\|T_{f}^{\alpha}\|_{A_{\omega}^{p_{1}}\times A_{\nu}^{p_{2}}\to \mathbb{C}}\|g\|_{A_{\omega}^{p_{1}}}\|h\|_{A_{\nu}^{p_{2}}},g,h\in H^{\infty}.
T_{BR}^{\mu\nu}=\frac{i}{4}(\overline{\psi}\gamma^{\mu}\nabla^{\nu}\psi-(\nabla^{\nu}\overline{\psi})\gamma^{\mu}\psi+\overline{\psi}\gamma^{\nu}\nabla^{\mu}\psi-(\nabla^{\mu}\overline{\psi})\gamma^{\nu}\psi).
\int_0^1 \frac{\sqrt{1-x^2}\ln (x)}{\left(2-x^2\right)^{3/2}}\, dx = \frac{1}{4}\sum_{n = 0}^{\infty}\left(-\frac{1}{2}\right)^n \binom{n}{\left\lfloor \frac{n}{2}\right\rfloor}\frac{(n+1) \left(4 \ln (2)-4 O_{n+1}-\frac{4}{2 n+3}\right)}{2 n+3}.
V(G) = V(G_1) \sqcup V(G_2) - \{v_1\}E(G) = (E(G_1) \sqcup E(G_2) - \{(u, v_1): u \in V(G_1) \}) \cup \{(u,v): u \in V(G_1)-\{v_1\}, (v_1, v) \in E(G_2) \}
\textbf{(ii):}\displaystyle\left( 1 - \alpha^{i-j}\right) \lambda_{i,j,k}= \alpha^{i-3j-3}\left( \sum_{\nu=0}^{j}\alpha^{3\nu}\right) \lambda_{i-2,j+1,k+1}, \displaystyle\left( 1 - \alpha^{i-k}\right) \lambda_{i,j,k}= \alpha^{i-2j-k-3}\left( \sum_{\nu=0}^{k}\alpha^{3\nu}\right) \lambda_{i-2,j+1,k+1}
\lVert u(\tau) - u_{p}\rVert \leq \lVert u_{p-1}- u_{p}\rVert = t_{p-1}\lVert y_{p-1}- r_{p-1}+ w_{p-1}\rVert, \lVert u_{q}- u(t)\rVert \leq \lVert u_{q}- u_{q+1}\rVert = t_{q}\lVert y_{q}- r_{q}+ w_{q}\rVert.
\int_{\mu_{1}}^{\infty}d x \left( x^{2}- \mu_{1}^{2}\right)^{\nu - 1}e^{- \mu_{2}x}= \frac{1}{\sqrt{\pi}}\left( \frac{2 \mu_{1}}{\mu_{2}}\right)^{\nu - 1 / 2}\Gamma \left( \nu \right) K_{\nu - \frac{1}{2}}\left( \mu_{1}\mu_{2}\right) ,
\|e^{k}\|_H^2\le 2\|e^2\|_H^2+2\|\widetilde{M}\|_2^2\|r^2\|_H^2-2\langle e^2, M_2r^2\rangle-(\frac{1}{2}-2\|\widetilde M\|_2^2)\sum_{i=2}^{k-1}\|r^{i+1}\|_H^2,
D_{\mathrm{H}}^{2}[ \Phi , \Phi^{\prime}] = \sum_{x}\sum_{\alpha = 0}^{3}\left[ ( ( \phi_{\alpha})_{x}- ( \phi_{\alpha}^{\prime})_{x})^{2}+ ( ( \dot{\phi}_{\alpha})_{x}- ( \dot{\phi}_{\alpha}^{\prime})_{x})^{2}\right] ,
\partial_{t}(v-\langle v \rangle)+v\partial_{x}v - \frac{{\Delta u}^2+{\Delta v}^2}{{\Delta q}^2}(x-\langle q \rangle) +\frac{1}{2}\partial_{x}\bigg( u^2+\frac{\hbar}{m}\partial_{x}u\bigg)\,=\,0 \:.
Pr(x|x_{0},\alpha_{0},\alpha)=\frac{\Gamma(x_{\bullet})}{\Gamma(x_{0})\prod_{i=1}^{m}\Gamma(x_{i}+1)}\times\frac{\Gamma(\alpha_{\bullet})}{\prod_{i=0}^{m}\Gamma(\alpha_{i})}\times\frac{\prod_{i=0}^{m}\Gamma(x_{i}+\alpha_{i})}{\Gamma(x_{\bullet}+\alpha_{\bullet})}
(\lambda+\nu-n)A-(\lambda+\nu-n)B+(\lambda+\nu-n)_2C =(\lambda+\nu-n)(\nu-\lambda+1), -2(\nu+\frac 12) A+4(\nu+\frac 12)B-2(\nu+\frac 12)(2\lambda-n+1)C =0, A-2(\nu-\frac{n-3}{2})B+2(\lambda+\nu-n+1)C =0,
\begin{aligned}F^{ij}(b_{n-1}(A_1 v - A_2 w) \pm (u - \varphi))_{ij}\geq \, 0 \mbox{in}\omega b_{n-1}(A_1 v - A_2 w) \pm (u - \varphi) \leq \, 0 \mbox{on}\partial \omega\end{aligned}
= 2 \pi \hbar \sum_{j}< \psi_{j}| \bigg \{\frac{- 1}{3 i \hbar}\hat{\theta}[ \partial_{x}\hat{\theta}, \partial_{y}\hat{\theta}] + \frac{1}{2}\big ( ( \partial_{x}\hat{\theta}) ( \partial_{\tilde{x}}\hat{\theta}) + ( \partial_{y}\hat{\theta}) ( \partial_{\tilde{y}}\hat{\theta}) \big ) \bigg \}| \psi_{j}> ,
\frac{d}{dt}\left| D \varphi_t (\omega, x) v \right|^2 = 2 \left\langle Db (\varphi_t (\omega,x)) D\varphi_t (\omega,x) v, D \varphi_t (\omega, x) v \right\rangle &\leq 2 \left| D \varphi_t (\omega, x) v \right|^2.
\prod_{k=\kappa(j_1)+1}^{\kappa(j_2)}(s_{\iota(k)})_{|\, i(g)-\sigma(k-1)}= \prod_{j = j_1+1}^{j_2}\; \prod_{k=\kappa(j-1)+1}^{\kappa(j)}(s_{\iota(k)})_{|\, i(g)-\sigma(k-1)}= \prod_{j = j_1+1}^{j_2}(s_{\iota(\kappa(j))})_{|\, i(g)-\sigma(\kappa(j)-1)}\ne 1,
Q(z|v)=\frac{1}{2^{r_1}}\sum_{\Phi_{\mathcal{S}_1}}P(\Phi_{\mathcal{S}_1})\sum_{e \in \{0,1\}^{r_1}}W_1^N(z|(v,e,\Phi_{\mathcal{S}_1}(v,e))G_N),
\textbf{A}_{\textbf{ST}}=\begin{pmatrix}I_{d\times d}-I_{d\times d}0_{d\times d}0_{d\times d}0_{d\times d}0_{d\times d}I_{d\times d}I_{d\times d}0_{d\times d}0_{d\times d}0_{d\times d}-I_{d\times d}-I_{d\times d}0_{d\times d}0_{d\times d}0_{d\times d}\end{pmatrix}
M_f = \begin{pmatrix}f_0 f_1 \ldots f_m 0 \ldots 0 0 \sigma(f_0) \ldots \sigma(f_{m-1}) \sigma(f_{m}) \ldots 0 & \ddots \ddots 0 \ldots 0 \sigma^{n-m-1}(f_0) \ldots \ldots \sigma^{n-m-1}(f_m)\end{pmatrix}_{(n-m)\times n}.
\begin{cases}\phi_\varepsilon\left(x,y\right)+\psi_\varepsilon\left(L\left(x,y\right)\right)\geq-\varepsilon \forall x,y\in\mathbb{R}\phi_\varepsilon\left(x_{\varepsilon},y_{\varepsilon}\right)+\psi_\varepsilon \left(L\left(x_{\varepsilon},y_{\varepsilon}\right)\right)\leq\varepsilon.\end{cases}
\begin{aligned}t_{m}=\frac{2\eta_0}{2m+1}, T_j =t_{k_j},\,\,1\leq j\leq 3; k_1 =(1-\sigma)k_0, k_2 =\varepsilon_0^{-1/40}\sqrt{\eta_0}, k_3 =\varepsilon_0^{-1/40}k_0.\end{aligned}
E\big[|I^{\prime}_3(t)|^2\big] = (1-\lambda)E\big[|I_3(t)|^2\big] \approx \frac{(1-\lambda)^2\alpha}{\lambda}\cdot \frac{M^2}{\tau}\cdot \beta_{j,k}^2 + (1-\lambda) M \beta_{j,k}\beta_{1,k}.
\begin{array}{c c c c c}{{D \, \Phi ( 0 ) = D_{0}\, \Phi ( 0 )}}&{{;}}&{{Y \, \Phi ( 0 ) = y_{0}\, \Phi ( 0 )}}&{{;}}&{{J_{2}\, \Phi ( 0 ) = j \, \Phi ( 0 ) \, .}}\\ \end{array}
\pi \left( \bigcup_{x\in A}\{x\}\times \left\{\Gamma_x\cap\{(-\infty,x)\cup(x,+\infty)\}\right\}\right) = \int_{x\in A}\pi_x\left(\Gamma_x\cap\{(-\infty,x)\cup(x,+\infty)\}\right) \mu(dx) =0,
V = - w^{2}e^{2 B ( \rho ) - 2 A ( \rho )}+ \frac{1}{\sqrt U}\frac{d^{2}\sqrt U}{d \rho^{2}}+ \frac{l ( l + 7 - p )}{\rho^{2}}e^{2 B ( \rho ) - 2 C ( \rho )}
\frac{{\delta}{\cal H}[ N ]}{{\delta}D_{a i}^{( n )}}= \frac{1}{2}\frac{N}{{\cal H}+ 1}( ( - 1 )^{n}D_{a i}^{( n )}+{\epsilon}_{a b c}B_{b i}^{( n )}{\cal H}_{c}^{M}) \ .
{{\partial^R{\cal H}}\over{\partial \Pi_c^a}}-\partial_{t\mu}{{\partial^R{\cal H}}\over{\partial \left[{\partial_{t\mu}\Pi_c^a}\right]}}+\partial_nc^a=0,\qquad{{\partial^R{\cal H}}\over{\partial c^a}}-\partial_{t\mu}{{\partial^R{\cal H}}\over{\partial \left[{\partial_{t\mu}c^a}\right]}}+\partial_n\Pi_c^a=0,
\hbox{either}t_{j}^{n}=0, \forall n, \mbox{or}t^{n}_{j}\overset{n\rightarrow \pm \infty}\longrightarrow \pm\infty, &\hbox{either}x_{j}^{n}=0, \forall n, \mbox{or}x^{n}_{j}\overset{n\rightarrow \infty}\longrightarrow \pm\infty;
B_{\rho_u\mu_-}(Df_{\mid E^u}(v_u)) \subseteq Df_{\mid E^u}(B_{\rho_u}(v_u))\subseteq B_{\rho_u\mu_+}(Df_{\mid E^u}(v_u)), B_{\rho_s\lambda_-}(Df_{\mid E^s}(v_s)) \subseteq Df_{\mid E^s}(B_{\rho_s}(v_s))\subseteq B_{\rho_s\lambda_+}(Df_{\mid E^s}(v_s)).
\sum_{i}D_{i}E_{i}= O\Big( \sum_i D_i^2 +{n^{-\beta_2}}\sum_{i}{l_i}^{\alpha_2}{D_{i}}^{1-\alpha_2}\Big) = O\Big( \sum_i D_i^2 +{n^{-\beta_2}}\Big(\sum_{i}{l_i}\Big)^{\alpha_2}\Big(\sum_{i}{D_{i}}\Big)^{1-\alpha_2}\Big) =O\big( D^{2}+{n^{-\beta_2}}m^{\alpha_2}D^{1-\alpha_2}\big) .
\left( a;q\right)_{0}=1,\ \ \ \left( a;q\right)_{n}={\displaystyle\prod\limits_{j=0}^{n-1}}\left( 1-q^{j}a\right) ,\ \ \n\in\mathbb{N}, \left( a;q\right)_{\infty}={\displaystyle\prod\limits_{j=0}^{\infty}}\left( 1-q^{j}a\right) ,\ \ \ \ \left\vert q\right\vert <1,\ \a\in\mathbb{C}.
\sum_{i=0}^{3}\sum\limits_{k=0}^{+\infty}\varepsilon^{\alpha_i +k -1}\, \left(w_{\alpha_i +k -1}^{(i)}(x_i) + u_{\alpha_i +k -1}^{(i)}\big( x_i, \tfrac{\overline{x}_i}{\varepsilon}, t \big) \right)+ \sum\limits_{k=0}^{+\infty}\varepsilon^{k}\, \left(w_{k}^{(i)}(x_i) + u_{k}^{(i)}\big( x_i, \tfrac{\overline{x}_i}{\varepsilon}, t \big) \right).
\| \prod_{i=1}^k \mathcal E_\lambda^\gamma f_i \|_{L^2(d\mu)}\le \| \prod_{i=1}^k \mathcal E_\lambda^\gamma f_i \|_{L^2(\mathbb R^d)}\| | \phi_\lambda | \ast \mu\|^{\frac12}_{\infty}\le C \langle \mu \rangle_\alpha^{\,\,\frac12}L^{-\frac{k^2 - k}{4}}\lambda^{-\frac{\alpha - d +k}{2}}\prod_{i=1}^k \| f_i \|_{L^2}
b^i_2 > b^j_3 \ \b^i_1 > b^j_3 \Longrightarrow Q(A_i,A_j) \geq 2 \cdot 6 - 9 = 3, b^i_1 \ > \b^j_2 \Longrightarrow Q(A_i,A_j) \geq 2 \cdot 6 - 9 = 3.
\frac{1}{(2 \pi i)^{g-1}}\int_{\Gamma_g}\cdots \int_{\Gamma_2}\frac{(k_1 p_1) \cdots (k_g p_g)}{(\lambda_1 + k_1 p_1) (\lambda_2 + k_2 p_2) \cdots (\lambda_g + k_g p_g)}\cdot \frac{d\lambda_2 \cdots d\lambda_g}{\lambda_2 \cdots \lambda_g}= (-1)^{g-1}\frac{k_1 p_1}{k_1 p_1 + k_2 p_2 + \cdots + k_g p_g}
[ \hat{\pi}_{\alpha}\; , \; \hat{\pi}_{\beta}]_{-}= \frac{m}{8 s}\epsilon_{\alpha \beta}( 1 + \nu \hat{\cal K}) \qquad [ \hat{\cal K}\; , \; \hat{\pi}_{\alpha}]_{+}= 0 \qquad \hat{\cal K}^{2}= 1 \, ,
3 . 2 B_{\mu \nu \alpha \beta}= R_{\ \mu \ \alpha}^{\rho \ \sigma \}R_{\rho \nu \sigma \beta}+ R_{\ \mu \ \beta}^{\rho \ \sigma \}R_{\rho \nu \sigma \alpha}- \textstyle{\frac{1}{2}}g_{\mu \nu}R_{\alpha}^{\ \rho \sigma \tau}R_{\beta \rho \sigma \tau}.
\begin{array}{l l}{{\kappa_{+}= ( f_{x}+ i g_{x}) + i ( f_{y}+ i g_{y}) ,}}&{{\qquad \kappa_{-}= ( f_{x}+ i g_{x}) - i ( f_{y}+ i g_{y}) ,}}\\{{\lambda_{+}= ( f_{x}- i g_{x}) + i ( f_{y}- i g_{y}) ,}}&{{\qquad \lambda_{-}= ( f_{x}- i g_{x}) - i ( f_{y}- i g_{y}) .}}\\ \end{array}
\big[r_\epsilon(a,b)\big](X) \, =\,-\frac{4i}{\pi^{4}}\int_{\Xi\times\Xi}\,dYdZ\, e^{-2i\sigma(Y,Z)}\left(\int_0^1e^{-4itF_{\epsilon}(x,y,z)}dt\right) \times \quad\quad \times F^\circ_{\epsilon}(x,y,z)\, \Big( \partial_\xi a (X-Y)\wedge\partial_\xi b (X-Z) \Big)\,.
\P_{N,r_N}(U_\varepsilon^x) = \P_{N,h_N}(U_\varepsilon^x)\,\frac{1}{\P_{N,h_N}(U_\varepsilon^x)}\int_{U_\varepsilon^x}\exp\left(-\log\frac{\d \P_{N,h_N}}{\d\P_{N,r_N}}\right)\,\d\P_{N,h_N}.
\frac{\tilde{L}_{i}^{k,k}(e^{x(t)})}{\tilde{L}_{1}^{k,k}(e^{x(t)})}=\tilde{L}_{i}^{k,k}(e^{x(t)}){\frac{dx}{dt}}=L_{i}^{k,k}(e^{t}).
{\underbrace{h_{\beta \alpha}h_{\delta \gamma}}}= \left[ \frac{1}{2}\left( \eta_{\beta \delta}\eta_{\alpha \gamma}+ \eta_{\alpha \delta}\eta_{\beta \gamma}\right) - \frac{1}{8}\eta_{\alpha \beta}\eta_{\gamma \delta}\right] \frac{1}{k_{\bot}^{2}}
\tilde{M}_k:=\sup_{F\in \mathcal{S}_k}\frac{\sum_{m=1}^{k}(\tilde{I}_{2k}^{(m)}(F)+\tilde{I}_{3k}^{(m)}(F))}{\tilde{I}_{1k}(F)}\tilde{r}_k:= \bigg\lceil \frac{m_K\tilde{M}_k}{B}\bigg\rceil
G_{5 5}= 3 \left[ \frac{a^{\prime}}{a}\left( \frac{a^{\prime}}{a}+ \frac{n^{\prime}}{n}\right) - \frac{b^{2}}{n^{2}}\left( \frac{\dot{a}}{a}\left( \frac{\dot{a}}{a}- \frac{\dot{n}}{n}\right) - \frac{\ddot{a}}{a}\right) \right] .
\alpha = | \mathrm{\bf \i t s h a p e ~ A}| , \hspace{3 m m}\beta = | \mathrm{\bf \i t s h a p e ~ B}| , \hspace{3 m m}\cos \varphi = \frac{\mathrm{\bf \i t s h a p e ~ A}\cdot \mathrm{\bf \i t s h a p e ~ B}}{\alpha \beta}, \hspace{3 m m}K = | \mathrm{\bf \i t s h a p e ~ K}| .
Z =\{x^{-1}R^{\bullet}\cap (\widehat{R}^{\bullet}:B^{-1})\mid x\in B^{-1}\}=\{x^{-1}R^{\bullet}\cap (\widehat{R}^{\bullet}:C^{\bullet})\mid x\in C^{\bullet}\}=\{(x^{-1}R)^{\bullet}\cap (\widehat{R}:C)^{\bullet}\mid x\in C^{\bullet}\}=\{(x^{-1}R\cap (\widehat{R}:C))^{\bullet}\mid x\in C^{\bullet}\}=\{I^{\bullet}\mid I\in Z_0\}.
\geq \frac{1}{2}P( \{\mathcal{T}_{\bar D}\cap [t, t +1] \neq\emptyset \}\cap \{\mathcal{T}_{A}\cap [t, t +1] = \emptyset :\forall \, A \in \mathcal{A}_k A\neq \bar D, A \cap(R' \cup\partial^{ext}R ')\neq \emptyset \}),
\widehat{S_{k}}^{\# ,0}( \Gamma_{0}( N) ) :=\dfrac{S_{k}^{\# ,0}( \Gamma_{0}( N) )}{D^{k-1}( M_{2-k}^{\#}( \Gamma_{0}( N) )) \oplus S_{k}( \Gamma_{0}( N) )}.
q^{\alpha^{2}/2}J_{\nu-\alpha}^{(2)}\left(-2iq^{n/2};q\right)\left(iq^{-n/2}\right)^{\nu-\alpha}=\frac{1}{\sqrt{2\pi\log q^{-1}}}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{x^{2}}{\log q^{2}}+\alpha xi\right)}{\left(q;q\right)_{\infty}\left(-q^{\nu+1/2}e^{-ix};q\right)_{n}}dx,
-\lim_{g(y,y)\to 0}\frac{1}{4}\,Tr(\mathcal{M}\cdot \mathcal{F}_A)=\,\lim_{g(y,y)\to 0}\frac{1}{4}\,Tr(\mathcal{M}\cdot \tilde{\mathcal{F}}_A)=\,A_j\,y^j.
\pi = (2^{f_2},4^{f_4},6^{f_6},\dots, (4j-2)^{f_{4j-2}}, (4j-1)^{1},\dots), \intertext{where $f_2,\f_4,\ \dots,\f_{4j-2}$ are all positive, or as}\pi = (2^{f_2},4^{f_4},6^{f_6},\dots, (4j)^{f_{4j}}, (4j+1)^{0},(4j+2)^{0},\dots),
\bigl ( S_{\mu \nu}\bigr )^{- 1}= \left( \begin{array}{c c}{0}&{- 1}\\{1}&{0}\\ \end{array}\right) \Bigl ( \eta_{\mu \nu}+ ( \alpha - 1 ) \eta_{0 \mu}\eta_{0 \nu}\Bigr ) \delta ( \vec{x}- \vec{y}) .
\begin{array}{l l}{{\hat{q}}}&{{= \frac{2}{( k^{+}+ 1 )}\ell_{3}^{+}}}\\{{}}&{{}}\\{{\hat{h}}}&{{= \frac{1}{( k^{+}+ 1 )}\{\ell^{+}( \ell^{+}+ 1 ) - ( \ell_{3}^{+})^{2}\}}}\\ \end{array}
u_{t}+F\left(t,x,u,Du,D^{2}u\right) =0\,\Omega^{\circ}, \frac{\partial u}{\partial \widetilde{\gamma}}+f\left(t,x,u\left(t,x\right)\right) =0\,\partial \Omega , u\left(0,x\right) =g\left(x\right)x\in \overline{\Omega}_{0},
P_{l}^{| m |}( x ) = \frac{( - 2 )^{| m |}}{\sqrt{\pi}}\Gamma \left( | m | + \frac{1}{2}\right) ( 1 - x^{2})^{\frac{| m |}{2}}C_{l - | m |}^{| m | + \frac{1}{2}}( x )
q^{+}( \zeta , u ) = f^{i}( x ) u_{i}^{+}+ \theta^{+ \alpha}\psi_{\alpha}( x ) + \bar{\theta}_{\dot{\alpha}}^{+}\bar{\kappa}^{\dot{\alpha}}( x ) + 2 i \theta^{+}/ \! \! \! \partial \bar{\theta}^{+}f^{i}( x ) u_{i}^{-},
\mathcal{J}_{D,II}|Z_{3}=\Pr\{X_{2}\geq \frac{Y_{2}Z_{3}\gamma_{th}}{P_{\mathcal{I}}},Y_{2}\geq\frac{P_{\mathcal{I}}}{\rho Z_{2}}\}&=\int_{\frac{P_{\mathcal{I}}}{\rho Z_{2}}}^{\infty}e^{-\frac{y_{2}Z_{3}\gamma_{th}}{P_{\mathcal{I}}\lambda_{2}}}\frac{M}{\omega_{2}}\sum_{k}^{M-1}{M-1 \choose k}(-1)^{k}e^{-(\frac{k+1}{\omega_{2}})y_{2}}dy_{2}
\begin{aligned}|\bar u(x,t)-\bar{H}(x,t)|\leq C_k |(x,t)|^{k+1+\alpha}, ~~\forall ~(x,t)\in Q_{1}^+, \|\bar{H}\|\leq C_k, |\bar{H}_t-\bar F(D^2\bar{H},x,t)|\leq C_k|(x,t)|^{k}, ~~\forall ~(x,t)\in Q_{1}^+, \end{aligned}
\int_{0}^{\infty}x^{\nu-1}(x+\beta)^{\nu-1}e^{-\mu x}\,dx=\frac{1}{\sqrt{\pi}}\biggl(\frac{\beta}{\mu}\biggr)^{\nu-\frac{1}{2}}e^{\frac{\beta\mu}{2}}\varGamma(\nu)K_{\frac{1}{2}-\nu}\biggl(\frac{\beta\mu}{2}\biggr)
O(at)^T [(O(at)x-y)\otimes (O(at)x-y)] &=(x-O(at)^Ty)\otimes (O(at)x-y) &=A_0+(\cos at)\,A_c+(\sin at)\,A_s+\frac{\cos 2at}{2}\,\widetilde A_c+\frac{\sin 2at}{2}\,\widetilde A_s
\begin{cases}\partial_t w -\partial_x^2 w = 0 , t>0,\x \in (0, \, \infty) w(t, 0)=u_0(t) , t>0 w(0, x)=f, x\in (0, \infty).\end{cases}&\begin{cases}\partial_t w -\partial_x^2 w +x^2 w = 0 , t>0,\x \in (0, \, \infty) w(t, 0)=u_0(t) , t>0 w(0, x)=f, x\in (0, \infty).\end{cases}
\begin{cases}\mathbb{L}'_e(\hat{{\mathbf U}},\hat{\Psi})\dot{{\mathbf U}^\natural}=\mathbf{F}\Omega_T, \mathbb{B}'_e(\hat{{\mathbf U}},\hat{\varphi})(\dot{{\mathbf U}^\natural},\varphi)=0 \Gamma_T, (\dot{{\mathbf U}^\natural},\varphi)=0 t<0,\end{cases}
\hat{H}_{II}=\frac{\mu_{0}\mu_{N}^{2}}{4\pi}\sum_{\alpha\ne\alpha^{\prime}}\frac{g_{\alpha}g_{\alpha^{\prime}}}{R_{\alpha\alpha^{\prime}}^{3}}\{I_{\alpha}\cdot I_{\alpha^{\prime}}-3(I_{\alpha}\cdot\hat{R}_{\alpha\alpha^{\prime}})(I_{\alpha^{\prime}}\cdot\hat{R}_{\alpha\alpha^{\prime}})\}.
|\{u\leq M\}\cap G(y,3r/2)| =|T\bigr(\{\tilde{u}\leq M\}\cap G((y_1/r,0),3/2)\bigl)| &=r^3|\{\tilde{u}\leq M\}\cap G((y_1/r,0),3/2) | &\geq r^3 \frac{\nu}{2}|G((y_1/r,0),3/2)| &= \frac{\nu}{2}|G(y,3r/2)|.