type: math

math: \underline{\varepsilon} \cdot \underline{x} = \sum_{i=1}^{5} \varepsilon_i x_i \leq 1, \quad \underline{x}\in \mathbb{H}_5


math: |L| = |C|\cdot t \le \frac{|U|}{10r}\cdot t


math: p^*i_0^*(i_1^*b_J + \delta \phi) = \delta v


math: \mathbb{E}[h(\hat{x}_\eta) + \eta f(\hat{x}_\eta)] -f^*_\eta \leq \texttt{Err}_{\eta}


math: E\in \big| \omega_X^{\otimes k} \otimes \alpha \big|


math: \{i_k\}_{k \in \mathbb{N}} \subset I


math: S_k=0, \ \text{if}\ k \in \partial\mathcal{L}


math: \tilde{\mathbb{F}}^r := \{\tilde{\mathscr F}_t^r\}_{t\geq 0}


checked: verified

math: \underline{u}(0, x) \leq u_0(x) \quad \text{for } x \in [-h_0, h_0]


math: \sum_{n=1}^N \sum_{m=1}^N \int_0^T a_n a_m w(nf,t) w(mf,t)\,dt


math: \Delta{t}_{\text{Filter}} = T/ Nt_{\text{Filter}}


math: \hat S_{1,K}(\textbf{k}) = \cos K\ \tau^z + \sin K\ \tau^x


math: |\langle L_I,h\rangle| \leq \alpha \int_0^T \int_{\mathbb{R}^N} |\xi(t,x)|dx dt + 2\beta \|I\|_{L^\infty}\|h\|_{L^\infty}T + \gamma \int_{\mathbb{R}^N} |\xi(T,x)|dx \leq C_1 \|h\|_{L^\infty(0,T)}


math: K = \big\{ u \in V \mid u \cdot \tilde{n} \leq \phi_1 \text{ on } \Gamma_\mathrm{T} \big\} \,


math: BS_n\backslash (BS_{n-1}^1\oplus BS_{n-1}^2\oplus BS_{n-1}^3)


math: \langle \tau , d \rangle \to -\infty


math: \argmin_{L\in C_2} D(L,M) = \left(\frac{m_{ij}}{\sum_{k=1}^n m_{kj}}\right)_{1\le i, j\le n}


math: \prod_{j = 1}^{P_{\Delta_n}} F_{i, j} \leq b_{i, \Delta_n}^{P_{\Delta_n}}


math: \widetilde{M}=\widehat{\Omega}\times Y


math: Conv\bigg\{\displaystyle \bigcup_{j\in \Lambda}\partial H_j(x^*)\bigg\}


math: 0\neq(a_1 \wedge a_2) \wedge n_i = (a_1 \wedge a_2) \wedge (n_i^1 \wedge n_i^2) =(a_1 \wedge n_i^1) \wedge (a_2 \wedge n_i^2)


math: \epsilon : H^1(\Omega,\mathbb{R}^3) \to L^2(\Omega,\mathbb{R}^{3\times 3}_{\mathrm{sym}})


math: \mathbf{G}, N_r, N_{\text{init}}, \Delta n


math: A_F f = \sum_{\alpha\in \mathbb N^n} \left( \sum_{\beta\in\mathbb N^n} {A_F}_{\alpha,\beta} f_\beta \right)e_\alpha\\


math: \{\delta^i_\eta : \eta < \lambda^{+k}\}


math: \varphi(x, \xi') - \varphi(y, \xi') = \Sigma(x, y', \xi')(x' - y')


math: G = \bigsqcup_{A, B \in \mathcal{C}} \mathrm{Hom} (A, B)


math: p(\tau, \tau \sqcup \{ij\}) = \left\{ \begin{array}{cl} \displaystyle \frac{\alpha_{ij} w_{i, ij}}{\sum_{k=1}^d \alpha_{ik} w_{i, ik}} \alpha_i , & \text{if } i \in \partial \tau, \\[15pt] \alpha_{ij}, & \text{if } i \notin \partial \tau \text{ and } ij \notin \tau, \end{array} \right.


math: \mathcal{F}: \mathbb{R}^d \to \mathbb{R}^{n}


math: e\in \operatorname{int}\left(\mathcal{B}_{G} \right )


math: E_j(\theta) = (1+\mu_j) E_{\Gamma_0}(\theta)+\mu_j


math: \mathbb{P}(\underline{\hat{V}}_{\mathcal{L}}(\underline{Y}_{\mathcal{L}}) \neq \underline{V}_{\mathcal{L}}) \leq \epsilon_e


math: f^2=(h + \overline{g})^2 = h^2 +\overline{g}^2+ 2h\overline{g}


math: {\bar {\mathcal{A}}}_u \in \mathbb{R}^{nm \times nm}


math: x \in \complement \overline{V_i} \cap U_i


math: \argmin_{L\in C_1} D(L,M) = \left(\frac{m_{ij}}{\sum_{k=1}^n m_{ik}}\right)_{1\le i, j\le n}


math: \tilde{\varepsilon}^i \sim \mathcal{N}(0, \pmb{I}_l)


math: C^\bullet(E) \simeq R\text{Hom}(C^\bullet(E), \mathbb{Q}/\mathbb{Z}(1))[1]


math: (\bm{u}\otimes\bm{v})_{ij}=u_i v_j, ~(\bm{C}\otimes\bm{u})_{ijk}= C_{ij}u_k


math: 0 < \overline{\epsilon} \leq \epsilon^*


math: \frac{d}{d\epsilon} J_1(u_1 + \epsilon v_1, u_2) \Big|_{\epsilon=0}


math: \mathbb{R}\times S^1\times\mathbb{C}


math: i: D ^{2} \hookrightarrow \mathcal{D}


math: E_2 = \operatorname{Span}\{b_1, b_2\}


math: \langle L_1 \phi , \chi_b(r) r^{\frac12}\rho_1\rangle = \lambda_0 \langle \phi, \chi_b(r) r^{\frac12}\rho_1\rangle


math: C_T \left(u^T_n X^{\text{Test}}_n,\overline{ \text{x}}^{\text{Test}}\right)


math: (\tau,z)=(\sum_{j=1}^{b^{2}(X)} t_j \phi_j, z) \in H^2(X) \times \mathbb{C}^\times


math: \Theta: {\Sigma_\vartheta} \times \mathbb{T} \to {\Sigma_\vartheta} \times \mathbb{T}


math: \begin{aligned} H(\sigma) &= \sum_{i \neq j} Q_{ij} \cdot \frac{(\sigma_i + 1)(\sigma_j + 1)}{4} + \sum_i q_i \cdot \frac{\sigma_i + 1}{2} \\ &= \sum_{i \neq j} \frac{Q_{ij}}{4} \sigma_i \sigma_j + \sum_i \left( \frac{q_i}{2} + \frac{\sum_{i \neq j} (Q_{ij} + Q_{ji})}{4} \right) \sigma_i \\ &\quad + \left( \frac{\sum_i q_i}{2} + \frac{\sum_{i \neq j} Q_{ij}}{4} \right) \end{aligned}


math: a_1 \in \widetilde{S}_s^{m-1, k}